Open Access The Rough Model Method (RMM) Application to The Computation of Normal Depth in Circular Conduit

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1 Sed Orders for Reprits to Te Ope Civil Egieerig Joural, 04, 8, Ope Access Te Roug Model Metod (RMM) Applicatio to Te Computatio of Normal ept i Circular Coduit Bacir Acour ad Saba Setal Researc Laboratory i Subterraea ad Surface Hydraulics (LARHYSS) Uiversity of Biskra, PO Box 4 RP Biskra, Algeria Abstract: A ew metod is preseted to compute te ormal dept i circular coduit. Tis is te roug model metod (RMM). It states tat te liear dimesio of a coduit or cael is equal to te liear dimesio of a referetial roug model corrected by te effect of a o-dimesioal correctio factor. Te metod is based o te Colebrook-Wite ad arcy-weisbac relatiosips, applicable to te etire domai of turbulet flow. From te relatiosip goverig te flow i te roug model, te ormal dept i a circular coduit is explicitly deduced. Keywords: Circular coduit, discarge, eergy slope, ormal dept, roug model metod.. INTROUCTION Te metods of calculatig te uiform flow i coduits ad caels are ot umerous. Some of tem are grapics ad oter iterative [-]. For circular ad o-circular coduits, explicit calculatio metods ave bee proposed i te past ad oe may quote as a example te metod of Swamee ad Swamee [4]. I tis metod, te desig of ocircular coduits is based o a similar relatio to tat establised for te circular pipe by Swamee ad Jai []: k L () Wit L / ( gi) 8 / ( gi) () were is te vertical dimesio of te coduit, k is a costat depedig o te sape of te coduit ad L is a liear dimesio wose pysical meaig as ot bee specified. Note tat relatio () is oly applicable to te fillig rate of 7%. I Eq. (), is te discarge, is te absolute rougess caracterizig te state of te ier wall of te coduit, g is te acceleratio due to gravity, i is te logitudial slope of te coduit ad is te Kiematic viscosity. I te study of Swamee ad Swamee [4], a explicit approximate relatio is proposed for estimatig te fillig rate i circular ad o-circular coduits: / k max k k k Address correspodece to tis autor at te Researc Laboratory i Subterraea ad Surface Hydraulics, Biskra Uiversity, PO Box 4 RP Biskra Algeria; Tel: ; Fax:007448; bacir.acour@laryss.et () were y /, y is te ormal dept, is te diameter of te coduit, k, k ad k are costat values wic deped o te sape of te geometric profile of te cosidered coduit. Note tat te applicatio of Eq. () requires te determiatio of te maximum discarge wose expressio is establised o te basis of statistical aalysis of several curves. Accordig to te autors, Eq. () is applicable for fillig rate iger ta 0% ad it causes a maximum relative error of about %. Te last kow metod for determiig ormal dept i circular coduit was publised te years ago [6]. Usig Lagrage s teorem [7], te autors gave expressio of te fillig rate i te form of a ifiite series, depedig owever o te coefficiets of Cezy ad Maig tat are determied wit great difficulty usig iterative procedure, because tese coefficiets deped particularly o te fillig rate. Te fial result is obtaied by trucatig te series, leadig to a approximate fillig rate. As we ca see below, te relatiosip determiig te fillig rate is ot easy to adle for te use of te egieer. We Cezy s equatio is used, fillig rate is give as: 64 / cos M / / M 8 M /6 / si... 4 were M / C i ad C is Cezy s coefficiet. We Maig s equatio is used, te fillig rate is expressed as: (4) /4 04 Betam Ope

2 8 Te Ope Civil Egieerig Joural, 04, Volume 8 Acour ad Setal N / / cos /6 / / N 64 / si N / were N / i ad is Maig s coefficiet. I tis study, a simple metod is proposed for te desig of coduits ad caals as well as te determiatio of ormal dept. It is based o fittig a sigle curve, tat of a referetial roug model avig te same sape as of te cosidered coduit. All ydraulic flow caracteristics i te cosidered coduit are directly derived from tose of te referetial roug model wic are kow caracteristics. Te calculatio metod is preseted troug practical examples applied to te circular coduit.. LOCATION OF THE PROBLEM Normal dept is a importat parameter i te desig of te pipes ad caels, as well as i te varied flows aalysis. I te literature, te ormal dept is referred to as y. Fig. () sows scematically te ormal dept i a circular coduit of diameter. y Fig. (). efiitio sketc of ormal dept i circular coduit. Te calculatio of ormal dept is based primarily o te ope cael resistace equatios. Te most commoly used equatios i practice are amely te arcy-weisbac relatiosip, Cezy ad Maig equatios. Te first oe uses te frictio factor as defied by Colebrook-Wite wic is implicit. Te solutio ivolves may trials ad tedious computatios or laborious grapical procedure. Te secod ad te tird resistace equatios use Cezy ad Maig rougess coefficiets tat are ot costat but vary depedig especially o te fillig rate ad terefore o te ormal dept. Te calculatio of tese coefficiets is doe wit a great difficulty usig a icoveiet iterative process. Te problem is to compute ormal dept usig measurable data i practice suc as te discarge, te slope i of te coduit, te diameter of te coduit, te kiematic viscosity ad maily te absolute rougess wic reflects te state of te iteral wall of te coduit. To () solve te problem accordig to tese data oly, te RMM is te most appropriate metod. Tis is wat attempt to demostrate te result of tis study.. FUNAMENTAL RELATION OF THE RMM Cosider o oe ad a circular coduit of diameter, flowig te discarge of a fluid wit as te kiematic viscosity, uder a logitudial slope i. Te ier wall of te coduit is caracterized by te absolute rougess. O te oter ad, cosider a referetial roug model of te same sape defied by te diameter, te discarge, te kiematic viscosity ad a logitudial slopei i. Te ier wall of te model is caracterized by te relative rougess / 0.07, were is te ydraulic diameter. ue to te ig relative rougess, te flow i te model is roug ivolvig a frictio factor f / 6 accordig to Colebrook-Wite relatiosip [9] for te Reyolds umber R. Te diameters ad are ot oly differet but are govered by te iequality. Betwee te diameters ad, oe ca write te followig equatio: (6) were is te o-dimesioal correctio factor of liear dimesios suc as 0. Eq. (6) is te fudametal relatio of te RMM. It ca be geeralized to all coduits ad caels, writig tat: L L (7) were L is ay liear dimesio, suc tat te widt of a rectagular cael, te ydraulic radius, te diameter of a circular pipe or te vertical dimesio of a ocircular closed coduit etc Eq. (7) ca be writte as ad sice L ad L L L are respectively proportioal to te water areas A ad A, te we ca write : A A (8) 4. NON-IMENSIONAL CORRECTION FACTOR OF LINEAR IMENSIONS Te Colebrook-Wite relatiosip [8] is applicable to ay geometric profile of coduits or caels, sice te form factor R / R as a secod-order effect [], were R, e e, is te effective ydraulic radius. Colebrook-Wite relatiosip [4] ca te be writte as: /. log f.7 R f were f is te frictio factor ad R is te Reyolds umber. Accordig to Eq. (7), oe ca write te followig equatios: (9)

3 Te Roug Model Metod (RMM) Te Ope Civil Egieerig Joural, 04, Volume 8 9 R (0) 4 4 R () P P Takig ito accout te arcy-weisbac relatiosip [8], te logitudial slope of te coduit ca be writte as: f f i i () f A ga ga Hece: () 6A Isertig equatios (8) ad (0) ito Eq. (), oe ca deduce: 6 f / (4) Combiig equatios (8), (9), () ad (4), leads to: / / 0.04 log /.7 R () Eq. () is implicit towards te correctio factor wic must te be grapically estimated or computed wit te aid of a iterative procedure. Oe way to avoid tis is to use te followig derived explicit relatiosip, obtaied by a statistical aalysis [0, ]: / 8.. log 4.7 R / (6) A compariso was made betwee equatios () ad (6), varyig / from 0 to 0.0. It revealed tat maximum deviatio is less ta % for R 00 correspodig to R 00. Eq. (6) is applicable i all te field of turbulet flow, correspodig to R 00 ad to te wide rage 0 / 0.0. Eq. (6) is applicable to ay form of coduits ad caels. It is valid for ay fillig rate ad sape parameter caels. It is more geeral ta is Eq. ().. NORMAL EPTH COMPUTATION IN CIRCULAR CONUIT USING RMM For te roug model, Cezy s equatio ca be writte as: C A R i (7) Sice f /6, oe ca write C 8 g / f 8 g. Te wetted area A is govered by te followig relatio: A cos ad 4 (8) Were y / is te fillig rate i te roug model y is te ormal dept. Eq. (8) ca be writte as: A ( ) ( ) (9) 4 were : ( ) cos (0) cos ( ) Te wetted perimeter P is give by: P cos or : () () P ( ) () R Te ydraulic radius R A / P is te: ( ) (4) 4 Takig ito accout Eqs. (9) ad (4) ad te fact tat C 8 g, Eq. (7) is writte as: gi ( ) ( ) / Te relative coductivity / gi ( ) ( ) / Te relative coductivity is te: () (6) i te roug model is tus fuctio of te fillig rate. Eq. (6) was plotted i Fig. (). It sows tat te relative coductivity i te roug 0, ,8 0, ,6 0, 0,4 0, 0. 0, 0, 00 y ,.,,.,., 4. Fig. (). Plot of Eq. (6). ( ) Maximum relative coductivity correspodig to 0.9 gi y C 8 g

4 60 Te Ope Civil Egieerig Joural, 04, Volume 8 Acour ad Setal model begis wit a ascedig pase, te reaces a maximum ad fially udergoes a decreasig pase beyod tis maximum. Te calculatio sowed tat te maximum relative coductivity is acieved for te fillig rate 0.9. I te wide rage , correspodig to , a special study of Eq. (6) as sow tat te fillig rate could be expressed, wit a maximum relative deviatio less ta 0.% oly, by te followig approximate relatio: 0. 9 si 0. 4 (7) Cosider a referetial roug model avig a diameter equal to tat of te full-model state correspodig to ; Eqs. (0) ad () give respectively ( ) ad ( ). As a result, Eq. (6) leads to. For tis value, Fig. () idicates a fillig rate We tus obtai a roug model wit a diameter equal to tat of te full-model state, caracterized by te fillig rate For tis fillig rate, Eqs. () ad (4) allow to write respectively tat: P. (8).6 (9) Te roug full-model diameter is obtaied for te relative coductivity, applyig wat follows: gi (0) For te data of te problem, amely,, i, ad, Eqs. (8), (9) ad (0) are used to calculate respectively te caracteristics, P ad of te referetial roug model ad ece te value of te Reyolds umber R 4 / ( P ). For te calculated values of ad R, Eq. (6) allows to compute explicitly te o-dimesioal correctio factor of liear dimesios. If we affect to te referetial roug model te ew liear dimesio /, accordig to Eq. (6), te fillig rate i te roug model would be equal to te fillig rate i te cosidered coduit. Tis fillig rate is give by Eq. (7) for te relative coductivity / gi( / ). Te required ormal dept is te y. 6. EXAMPLE Compute te relative ormal dept i circular coduit for te followig data, usig te RMM: 4 6, m, i 0, 0, 0 m / s m / s Note tat te resistace coefficiet C of Cezy or of Maig is ot required. Calculate te diameter of te full roug model usig Eq. (0). Hece: gi.47m Accordig to Eq. (8), te wetted perimeter P is : P m Te Reyolds umber R is : 6 R 4 / ( P ) 4 / (.8 0 ) Accordig to Eq. (6), te o-dimesioal correctio factor of liear dimesios is te : 8.. log R / / 8.. log Compute te relative coductivity dimesio : / / m. Hece: / gi( / ) 4 / Accordig to Eq. (7), te fillig rate is tus : si ( = si ( ) ) for te ew liear Te required value of te ormal dept is te : y 0.60.m

5 Te Roug Model Metod (RMM) Te Ope Civil Egieerig Joural, 04, Volume 8 6. Ceck te calculatios by determiig te discarge by te use of Eq. (7). For te calculated fillig rate 0.60, Eqs. (0) ad () give respectively:. ( ) cos cos cos ( ) 0.60 cos As a result, Eq. (9) gives te wetted area A as:. 90 A ( ) ( )= R 4. 96m Accordig to Eq. (4), te ydraulic radius.90 ( ). 0.8m Fially, Eq. (7) gives te discarge as: C A R i m / s m s.99 / 4 R is te: As oe ca see, te calculated discarge correspods to te discarge give i te problem statemet, cofirmig te validity of te calculatios. Te calculatios could be also cecked by te geeral relatiosip of te discarge []. 7. EXAMPLE Let us propose aoter example of te ormal dept calculatio i circular coduit for a positive absolute rougess, i order to cofirm te reliability of te RMM. Cosider te followig data: m / s,.7m, 6 0 m / s i 4 4 0, 0.00m,. Usig Eq. (0), te diameter of full roug model is : gi m. Accordig to Eq. (8), te wetted perimeter P is : P m. Usig Eq. (9), te ydraulic diameter is te: m 4. Wit te calculated value of P, Reyolds umber R i te roug model is: 6 R 4 / ( P ) 4/ (.9 0 ) Usig Eq. (6), te o-dimesioal correctio factor of liear dimesio is as: / 8.. log 4.7 R / 0.00 / log / 6. Assig to te roug model te followig ew liear dimesio, accordig to Eq. (6): /.7 / m Te correspodig value of te relative coductivity is te: / gi( / ) 4 / Te required value of te fillig rate is give by Eq. (7) as: si ( = si ( ) 8. Fially, te ormal dept is: y m ) Verify te validity of te calculatios by determiig te discarge usig Eq. (7). Compute first ( ) ad ( ) usig Eqs. (0) ad () respectively. Hece:. ( ) cos cos cos ( )

6 6 Te Ope Civil Egieerig Joural, 04, Volume 8 Acour ad Setal cos Accordig to Eq. (9), te water area A is:. 8 A ( ) ( )= R. 897m Accordig to Eq. (4), te ydraulic radius.8 ( ) m Usig Eq. (7), te discarge is te: C A R i m / s m s / 4 R is as: Tis secod example sows oce agai tat te calculated discarge is equal to te discarge give i te problem statemet, cofirmig te validity of te calculatios. 8. CONCLUSION A ew computatio metod of ormal dept i circular coduit is preseted. Tis is te roug model metod (RMM) wic states tat ay liear dimesio of a coduit or cael is equal to te liear dimesio of a referetial roug model, corrected by te effect of a dimesioless factor. Te metod is based o Colebrook-Wite ad arcy- Weisbac relatiosips ad is, terefore, valid trougout te field of turbulet flow. Te applicatio of te metod for te determiatio of ormal dept i a circular coduit is preseted. It leads to te establismet of a excellet approximate equatio expressig te fillig rate, depedig o te relative coductivity i te referetial roug model. Tis is caracterized by a diameter equal to tat of te full-model state ad by a fillig rate of 0.8. Te obtaied relatiosip allows us to deduce explicitly te ormal dept i te cosidered circular coduit, usig te fudametal relatiosip of te RMM. Practical examples are take to explai te process of calculatio. Te proposed metod does ot require te coefficiets of Cezy ad Maig, ulike curret metods of calculatio. Te teoretical developmet as well as te calculatio examples we proposed sow o restrictio i te applicatio of te roug model metod. NOTATION A = C = = = Water area Cezy s coefficiet iameter of te coduit Hydraulic diameter f = Frictio factor g = Acceleratio due to gravity i = Slope of te coduit L = P = = = R = R = y = = Liear dimesio Wetted perimeter iscarge Relative coductivity Reyolds umber Hydraulic radius Normal dept Absolute rougess = Fillig rate equal to y / = No-dimesioal factor = kiematic viscosity CONFLICT OF INTEREST Te autors cofirm tat tis article cotet as o coflicts of iterest. ACKNOWLEGEMENTS Noe declared. REFERENCES [] V.T. Cow, Ed., Ope-Cael Hydraulics. McGraw Hill: New York, 97. [] R.H. Frec, Ed., Ope Cael Hydraulics. McGraw Hill: New York, 986. [] R.O. Siiger ad W.H. Hager, Ed., Costructios ydrauliques. Presses Polyteciques Romades: Suisse, 989. [4] P.K. Swamee ad N. Swamee, esig of ocircular sewer sectios, J. Hydraulic Res., vol. 46, o., pp. 77-8, 008. [] P.K. Swamee ad A.K. Jai, Explicit equatios for pipe-flow problems, J. Hydraulic Eg., ASCE, vol. 0 (HY), o , (HY), pp , 976. [6] P.K. Swamee ad P.N. Ratie, Exact solutios for ormal dept problem, J. Hydraulic Res., vol. 4, o., pp. 4-47, 004. [7] E.T. Wittaker ad G.N. Watso, A Course of Moder Aalysis. Cambridge Uiversity Press, Cambridge: UK, 96. [8] H. arcy, Sur les recerces expérimetales relatives au mouvemet des eaux das les tuyaux, Comptes redus des séaces de l Académie des Scieces, o. 8, pp. 09-, 84. [9] C.F. Colebrook, Turbulet flow i pipes, wit particular referece to te trasactio regio betwee te smoot ad roug pipe laws, J. Ist. Civil Eg., vol., pp. -6, 99. [0] B. Acour ad A. Bedjaoui, Turbulet pipe-flow computatio usig te roug model metod (RMM), J. Civil Eg. Sci., vol., o., pp. 6-4, 0.

7 Te Roug Model Metod (RMM) Te Ope Civil Egieerig Joural, 04, Volume 8 6 [] B. Acour, esig of pressurized vaulted rectagular coduits usig te roug model metod, Adv. Mat. Res., vol , 44-49, pp., 0. [] B. Acour ad A. Bedjaoui, iscussio. Exact solutios for ormal dept problem, J. Hydraulic Res., vol. 44, o., pp. 7-77, 006. Received: Jauary 09, 04 Revised: Marc 4, 04 Accepted: Marc, 04 Acour ad Setal; Licesee Betam Ope. Tis is a ope access article licesed uder te terms of te Creative Commos Attributio No-Commercial Licese (ttp://creativecommos.org/liceses/ by-c/.0/) wic permits urestricted, o-commercial use, distributio ad reproductio i ay medium, provided te work is properly cited.