Chezy s Resistance Coefficient in a Circular Conduit

Transcription

1 Send Orders for eprints to Te Open Civil Engineering Journal Cezy s esistance Coefficient in a Circular Conduit Open Access Bacir Acour esearc Laboratory in Subterranean and Surface Hydraulics University of Biskra PO Box 15 P Biskra Algeria Abstract: In te literature tere is no explicit metod for calculating te resistance coefficient of Cezy especially for a circular conduit. Existing relationsips are eiter implicit or do not take into account all parameters influencing tlow suc as kinematic viscosity or te slope of te conduit. In many practical cases one affects arbitrarily a constant valuor Cezy s coefficient. It is a pysically unjustified approac because Cezy s coefficient varies wit flow parameters especially tilling rate of te conduit and te absolute rougness. In tis paper simple and explicit relationsips are presented for te calculation of Cezy s resistance coefficient in a circular conduit. Tese relationsips ave been establised based on te roug model metod. Te Cezy s resistance coefficient is expressed in terms of known ydraulic parameters of tlow in a referential roug model. For fast calculation of Cezy s coefficient te simplified metod is te most appropriate since it requires only four parameters wic are te discarge te absolute rougness te slope and te kinematic viscosity. Te study also sows tat te Cezy s resistance coefficient reaces a maximum wose expression is well defined. Some examples are presented sowing ow to calculate Cezys coefficient in a circular conduit wit a minimum practical data. Keywords: Cezy s coefficient circular conduit discarge energy slope ydraulic radius roug model metod. 1. INTODUCTION eferring to te literature we can see tat few formulae exist for expressing Cezy s resistance coefficient C. Te most frequently cited are te old formulae of Guanguillet- Kutter [1] Bazin [2] and Powell []. Tese relationsips are well summarized and discussed by Cow []. Te Guanguillet-Kutter formula expresses C in terms of te ydraulic radius te coefficient of rougness n known as Kutter s n and te slope S. In Englis units tis formula is: C = S n n 1 S " Tis relationsip does not contain a term relating to te kinematic viscosity. Tus it can not be applied to te entire domain of turbulent flow. Its application seems to be restricted to te roug domain for wic te kinematic viscosity as no effect. Bazin formula expresses te coefficient C as a function of ydraulic radius but not of te slope S. Tis formula is: Address correspondence to tis autor at te esearc Laboratory in Subterranean and Surface Hydraulics University of Biskra PO Box 15 P Biskra Algeria; Tel: ; Fax: ; bacir.acour@laryss.net C = 87 1 m 2 Were m is a coefficient of rougness wose values are given by a table as a function of te type of te material forming te cannel or te conduit. As for te Guanguillet- Kutter formula Bazin formula contains no terms of kinematic viscosity. It does not terefore apply to te wole domain of turbulent flow. Te Powell formula is more complete as it contains te ydraulic radius te absolute rougness ε and te eynolds number. However tis formula is implicit expressing C as: C C = 2 log " According to tis relationsip C depends especially on te eynolds number and terefore on te kinematic viscosity ν. In tis relation tere is no term tat expresses te influence of te slope S on te coefficient C. Its application seems to be suitablor te entire domain of turbulent flow. It is interesting to note tat Powell formula contains te absolute rougness ε wic is a measurable parameter in practice. To determine te coefficient C by te Powell formula it is necessary to use a trial-and-error procedure. More recently Swamee and atie [5] ave attempted to propose a general relationsip for Cezy s coefficient C / Bentam Open

2 188 Te Open Civil Engineering Journal 2015 Volume 9 Bacir Acour applicable in te entire domain of turbulent flow and for all sapes of cannels and conduits. However tis relationsip is implicit requiring also a trial-and-error procedure especially wen te linear dimension of te cannel or conduit is not given or wen it comes to compute te normal dept of tlow. Swamee and atie suggested for C a logaritmic formula as: C = 2.57 " g ln gs Apart from its implicit form tis relationsip as te advantage of being very complete. All tlow parameters are included in tis relationsip. According to te literature several tests were performed on corrugated pipes or large scale rougness in cannels of non circular cross section tat ave not led to a convincing formula for Cezy s coefficient. Among tese studies we can mention tose of Streeter [6] Ead and al. [7] Pyle and Novak [8] Marone [9] Perry and al. [10] Naot and al. [11]. More recently Giustolisi [12] used a genetic programming to determine Cezy s resistance coefficient for full circular corrugated cannels. For commercial pipes or artificial cannels te literature does not indicate specific studies. Tat is wy tis article is proposed wic aims to establis simple relationsips for calculating Cezy s coefficient based on practical data. Te calculation approac is based on te roug model metod MM tat as been proven in te recent past by contributing successfully to te design of conduits and cannels and to te calculation of normal dept [1-25]. Two explicit metods of calculating Cezy s coefficient are proposed. Tirst metod considers tilling rate of te conduit wile te second one is more simplified. It takes no account of tilling rate of te conduit or its diameter. Bot metods give similar results. In tis article examples are provided to better appreciate te ease of te metod and calculation. 2. HYDAULIC POPETIES Te caracteristics of tlow in a circular conduit partially occupied Fig. 1 are in particular: y n Fig. 1. Flow in a circular conduit. 1. Te wetted area: D A = D 2 1 2" 2 1 2" cos1 " 1 " 5 It tus appears tat te wetted area is depending on te diameter D of te conduit and tilling rate = y n / D were y n is te normal dept. Eq. 5 can be written as: A = D 2 were: " " 6 " = cos 1 1 2" 7 " 1 " 2 1 2" " = 1 cos 1 1 2" For a circular conduit completely filled corresponding to = 1 one can deducrom Eq. 7 and Eq. 8 respectively tat "=1= and " = 1=1. 2. Te Wetted perimeter P = D cos 1 1 2" 9 Tis can be written simply as: P = D " 10. Te ydraulic radius = A / P is tus: = D " 11. GENEAL ELATIONSHIP OF CHEZY S ESIS- TANCE COEFFICIENT Cezy s relation gives te discarge Q as: Q = CA S 12 To igligt te variation of Cezy s coefficient based on all parameters governing tlow Acour and Bedjaoui formula [2] is very useful. Tis relationsip applicable to all geometric profiles was establised in te wole domain of turbulent flow encompassing smoot transition and roug regimes. According to Acour and Bedjaoui [2] te discarge Q is given by tollowing formula: " Q g A S log were S is te slope of te conduit is a eynolds number ε is te absolute rougness and g is te acceleration due to gravity. Te eynolds number is governed by tollowing equation: = 2 2 g S were ν is te kinematic viscosity. 8 1

3 Cezy s esistance Coefficient in a Circular Conduit Te Open Civil Engineering Journal 2015 Volume Inserting Eq. 11 into Eq. 1 leads to: g S D " / 2 15 For a circular conduit completely filled te ydraulic radius is = D /. Tus Eq. 1 becomes: g S D 16 Te subscript f refers to tull state of te conduit. Taking into account Eq. 16 Eq. 15 can be rewritten as: = " / 2 17 Comparing Eq. 12 and Eq. 1 it is obvious tat Cezy s coefficient is suc tat: C = C C g g " 2g log or in dimensionless form : " log Inserting Eq. 11 and Eq. 17 into Eq. 19 leads to: log. / It tus appears tat C depends on te relative rougness / D tilling rate and te eynolds number. Wen tese parameters are given relation 20 allows te explicit determination of te coefficient C. However wen it comes to design te conduit D is not a given data and only Q η S ε and ν are te known parameters. In tis case Eq. 20 does not allow determining explicitly te coefficient C. However tis problem can be solved using te roug model metod MM.. COMPUTATION OF CHEZY S ESISTANCE CO- EFFICIENT.1. Te oug Model Metod Te roug model is a circular conduit of diameter D greater tan D in wic tlow is caracterized by a friction factor f = 1 / 16 arbitrarily cosen. Tis ig friction factor implies tat tlow in te model is roug. In te roug model te discarge is Q te slope is S te kinematic viscosity is ν and tilling rate is. Taking into account tat Cezy s resistance coefficient in te roug model is C = 8g / f one may write: C = 8 2g = constant 21 According to te MM [1 2 25] D and D are related by tollowing equation: D = D 22 were is a non-dimensional correction factor of linear dimension less tan unity. It was demonstrated tat can be written as: = 16 f 1/5 2 = 1.5 "log / /./ "2 / 5 2 were and are respectively te ydraulic radius and te eynolds number in te roug model. Te Cezy s resistance coefficient C and triction factor f are as C = C = 8 8g / f. As a result Eq. 2 leads to: 2g 5 / 2 25 wic can be rewritten as: C = C 26 5 / 2 Inserting Eq. 11 and Eq. 17 into Eq. 2 leads to: - / D = 1.50"log / 1 / 2 0 / 1. "2 / 5 Combining Eq. 25 and Eq. 27 one can write: C = 5. g log. / Te eynolds number is given by Eq. 16 as: g S D Eq. 28 will be used wen te diameter D of te conduit is not a given data of te problem. Te coefficient C is explicitly calculated provided te discarge Q te slope S te absolute rougness ε and tilling rate η are given. To express te diameter D apply Cezy s relation to te roug model. Hence:

4 190 Te Open Civil Engineering Journal 2015 Volume 9 Bacir Acour Q = C A S 0 Taking into account Eq. 6 Eq. 11 and Eq. 21 Eq. 0 leads to: D = 2 " 0.6 Q ". g S - 1 Eq. 1 permits a direct determination of te diameter D since Q S and g are te known parameters of te problem. Tus all relationsips are establised for te explicit determination of te Cezy s coefficient C troug te following steps provided Q S g and are given: 1. For te given value of tilling rate compute " and " according to Eq. 7 and Eq. 8 respectively. 2. For te given values of Q S and g compute te diameter D of te roug model by applying Eq. 1.. For te given values of D S g and Eq. 29 gives te eynolds number.. Finally using Eq. 28 Cezy s resistance coefficient C is worked out for te known values of ε D and g..2. Example 1 For tollowing data compute Cezy s resistance coefficient: Q m / s = " 6 2 = 10 m / s. S = " 10 " = 10 m 0. 6 = 1. According to Eq. 7 and Eq. 8 " and " are respectively: " = cos 1 1 2" = = cos cos " = 1 " 1 cos 1 2 " " 1 2 " " = 1 = In accordance wit te relationsip 1 te diameter D of te roug model is: D = 2 " ["] 0.6 Q g S. - 0 " = " = m. Applying Eq. 29 te eynolds number is ten : = 2 = g S D " "6. Finally according to Eq. 28 te Cezy s resistance coefficient C is: C = 5. g log = 5. " 9.81 " [ ] / 2 log 10 / " " / 2 = m 0.5 / s 79.8 m 0.5 / s.. Simplified Metod In wat follows a simplified metod is proposed for fast calculation of Cezy s coefficient C wit a reduced number of data. Neiter te diameter D of te conduit nor tilling rate is required. Only four parameters are needed to evaluate Cezy s coefficient namely te discarge Q te slope S te absolute rougness ε and te kinematic viscosityν. All tese parameters are easily measurable in practice. Tis simplified metod also based on te teory of te roug model causes a maximum relative deviation of about 1.25 compared to te metod described in section.1. Tis relative deviation is less tan te relative error wit wic te absolute rougness is measured in practice. Assuming " and applying Eq. 1 for te roug model leads to: Q = 2 " " Were Q = Q / / 2 Q is te relative conductivity expressed as: 5 gs D 2

5 Cezy s esistance Coefficient in a Circular Conduit Te Open Civil Engineering Journal 2015 Volume Consider a referential roug model aving a diameter D equal to tat of tull-model state corresponding to = 1; Eq. 7 and Eq. 8 give respectively " = and " = 1. As a result Eq. 2 leads to Q = 2. For tis value of te relative conductivity Eq. 2 indicates a second value of tilling rate equal to " We tus obtain a roug model wit a diameter equal to tat of tull-model state caracterized by tilling rate " Consequently te wetted perimeter P and te ydraulic radius are given by Eq. 9 and Eq. 11 respectively as: P = 2.52D = 0.01D 5 Te diameter D of tull roug model is obtained for te relative conductivity Q = " Q D = 2 gs 2 implying wat follows: 6 Te calculation of Cezy s coefficient is readily carried out using tollowing steps: 1. Compute te diameter D of tull model using Eq Compute ten te wetted perimeter P and te ydraulic radius by te use of Eq. and Eq. 5 respectively.. Te eynolds number = Q / P in te roug model is ten worked out.. Wit te computed values of and e te nondimensional correction factor ψ is explicitly determined using Eq Finally Cezy s coefficient C is directly deduced from Eq Example 2 Let us consider te data of example 1 to compute Cezy s resistance coefficient using te simplified metod. Q = m " " 6 2 / s S = " 10 = 10 m = 10 m / s. 1. According to Eq. 6 te diameter D of tull roug model is: " Q D = 2 gs = " " "10 = m 2. Using Eq. and Eq. 5 te wetted perimeter P and te ydraulic radius are respectively: P = 2.52D = =.17116m = 0.01D = = m. Te eynolds number in te roug model is: = Q / P = " / " 10 6 = According to Eq. 2 te non-dimensional correction factor ψ is ten: = 1.5 "log / /./ "2 / 5 = 1.5 "log 10" / = "2 / 5 5. Using Eq. 25 te required value of Cezy s coefficient is: C = 8 2g = 8 " 2 " 9.81 = / / 2 m0.5 / s Tus comparing tis result wit tat obtained in example 1 we can observe tat te relative deviation is less tan 0.9 only. 5. MAXIMUM OF CHEZY S COEFFICIENT 5.1. General elationsip According to Eq. 20 te Cezy s resistance coefficient C depends on tree dimensionless variables namely te relative rougness / D tilling rate of te conduit and te eynolds number. Its grapical representation is not easy but it can be sown as an indication its variation for a fixed value of te relative rougness / D. Tis as been performed for different values of / D and for eynolds number varying between 10 and Among all te obtained graps tose of Figs. 1 and 2 are representative. Fig. 2 translates te variation of C / g versus tilling rate and te eynolds number for te value / D = 0 corresponding to a smoot

6 192 Te Open Civil Engineering Journal 2015 Volume 9 Bacir Acour f C/ g Fig. 2. Variation of C / g versus and according to Eq. 20 for / D = 0. Maximum value / g obtained for " f C/ g Fig.. Variation of C / g versus and according to Eq. 20 for / D = Maximum value / g obtained for " inner wall of te conduit. Fig. sows te variation of C / g versus tilling rate and te eynolds number for te value / D = corresponding to a state of te roug inner wall of te conduit. Te cosen values of te relative rougness / D correspond in fact to te extreme values of te Moody diagram. Fig. 2 clearly sows tat for a given value of te eynolds number C / g increases wit te increase of tilling rate up to a maximum value represented by te full sign on tigure. Beyond tis maximum value C / g decreases wit te increase of tilling rate and te decrease continues until tull state of te conduit corresponding to = 1. It sould also be noted tat watever te value of te cange in C / g according to is carried out rapidly at first and undergoes a sligt variation in a second time. Te rapid variation of C / g is observed for a narrow range of tat can be defined as 0 " 0.2. Beyond te value = 0. 2 C / g undergoes a very slow cange in a wide range of independently of te value of te eynolds number. Tis state of cange can also be seen in Fig. 2. It also indicates tat for te ig cosen rougness value / D = te variation curves of C / g versus are very close to eac oter and mergor te values Tis igligts te roug state of tlow were C / almost independent of te eynolds number 5 > 10. g is and depends solely on te value of tilling rate of te conduit. Te calculation reveals tat Cezy s coefficient is te samor te particular values = 0.5 and = 1. In te range 0.5 " " 1 tere are two normal depts for te same value of C wic is owever very close to te maximum value due to te low variation of te curve. Te most significant result obtained wen plotting te variation of C / number g as a function of and te eynolds lies in tact tat te maximum value is acieved for tilling rate " watever te value of te relative rougness / D and tat of te eynolds number. In oter words te maximum value of

7 Cezy s esistance Coefficient in a Circular Conduit Te Open Civil Engineering Journal 2015 Volume 9 19 C / g and tus C max. is acieved at normal dept y n 081D. For = tunction " defined by Eq. 8 takes tollowing value: " = 1 cos = Inserting tis value in Eq. 20 leads to: g or: = log g log wen / D and are te known parameters of te problem Eq. 8 permits a direct determination of te maximum Cezy s resistance coefficient. Wen te diameter D of te conduit is not given te determination of te maximum of Cezy s resistance coefficient is possible by te use of Eq. 28 in wic ϕη = Hence: = 5. g log According to Eq. 9 te maximum of Cezy s resistance coefficient is related to te known parameters of tlow in te roug model wic can be ten calculated in a simple manner even if te conduit diameter D is not given. Tollowing examples sow te steps for calculating te maximum of Cezy s resistance coefficient Example Compute te maximum value of Cezy s coefficient for tollowing data: " D = 1.5m ; = 10 m ; " 6 2 S = " 10 ; = 10 m / s 1. According to Eq. 16 te eynolds number e f tull state of te conduit is ten: = g S D for = " 2 " 9.81" " 10 " Finally applying Eq. 8 one may obtain: 10 6 = 2g log " 2 " 9.81 " log 10 / = m / s 5.. Example Compute te maximum value of Cezy s resistance coefficient in a circular conduit for tollowing data: Q m / s = " 6 2 = 10 m / s. S = " = " = 10 m 1. According to Eq. 7 and Eq. 8 " and " are respectively : " " 1 1 = cos 1 2 = cos = " 1 " 2 1 2" " = 1 cos 1 1 2" " 0.65 " " 1 2 " 0.65 = 1 cos " 0.65 = Eq. 1 gives te diameter D of te roug model as: D = 2 " " = Q ". g S " = m. According to Eq. 29 te eynolds number full state of te roug model is ten : = 2 = g S D " "6 for te

8 19 Te Open Civil Engineering Journal 2015 Volume 9 Bacir Acour. As a result Eq. 9 leads to: = 5. g log = 5. " 9.81 " log 10 / = 80 m 0. 5 / s 6. CONCLUSION Using te general discarge relationsip te expression of Cezy s coefficient C was establised for a circular conduit. Te obtained expression clearly sowed tat C depends on te relative rougness ε/d tilling rate η of te conduit and te eynolds number f caracterizing tull state of tlow. Tis in turn depends on te slope S te diameter D of te conduit and te kinematic viscosityν. All parameters influencing tlow are represented in te expression of C unlike current relationsips. Wen all tese parameters are given te resulting expression is used to calculate explicitly te required value of C. Wen te diameter D of te conduit is not a given data of te problem te explicit calculation of C is still possible troug te use of te roug model metod. C is ten expressed as a function of te known parameters of tlow in te roug model. In tis case te calculation of C requires te discarge Q te slope S te absolute rougness ε tilling rate η and te kinematic viscosity ν. Wen te user does not ave all te data of te problem te explicit calculation of C is still possible tanks to te simplified metod tat was clearly described. Tis metod uses te minimum measurable data in practice wic are te discarge Q te slope S te absolute rougness ε and kinematic viscosityν. Tis simplified metod gives very satisfactory results. Te paper was completed by te particular study of te coefficient C. Te grapical representation sowed a rapid increase in te range 0 < " 0.2. It also sowed a sligt increase in C beyond = 0. 2 and ten reaced a maximum at tilling rate η = C ten undergoes a sligt diminution to tull state of te conduit corresponding to = 1. ABBEVIATIONS A = Water area C = Cezy s coefficient D = Diameter of te conduit D = Diameter of te roug model D = Hydraulic diameter f = Friction factor g = Acceleration due to gravity S = Slope of te conduit P = Wetted perimeter Q = Discarge Q = elative conductivity = eynolds number f = eynolds number at tull state of te conduit = Hydraulic radius y n = Normal dept ε = Absolute rougness η = Filling rate equal to y / n ψ = Non-dimensional correction factor ν = kinematic viscosity CONFLICT OF INTEEST Te autor confirms tat tis article content as no conflict of interest. ACKNOWLEDGEMENTS Declared none. EFEENCES [1] E. Ganguillet and W. Kutter An investigation to establis a new general formula for uniform flow of water in canals and rivers Zeitscrift des Oesterreiciscen Ingenieur und Arcitekten Vereines vol. 21 no. 1 pp no.2- pp [2] H. Bazin Etude d une nouvellormule pour calculer le débit des canaux découverts Annales des ponts et causses vol. 1 ser.7 ème trimestre pp [].W. Powell esistance to flow in roug cannels Trans. Am. Geopys. Union vol. 1 no. pp [] V.T. Cow Ed. Open-Cannel Hydraulics. McGraw Hill: New York 197. [5] P. K. Swamee and P.N. atie Exact solutions for normal dept problem J. Hydraul. es. vol. 2 no. 5 pp [6] V.L. Streeter Frictional resistance in artificially rougened pipes Trans. ASCE vol. 101 pp [7] S.A. Ead N. ajaratnam C. Katopodis and F. Ade Turbulent open-cannel flow in circular corrugated culverts J. Hydraul. Eng. vol. 126 no. 10 pp [8]. Pyle and P. Novak Coefficient of friction in conduits wit large rougness J. Hydraul. es. vol. 19 no. 2 pp [9] V. Marone Le resistenze al movimento uniforme in un alveo ciuso o aperto di sezione rettangolare e scabrezza definita L Energia Elettrica vol. 1 pp [10] A.E. Perry W.H. Scofield and P.N. Joubert oug wall turbulent boundary layers J. Fluid Mec. vol. 7 no. 2 pp [11] D. Naot I. Nezu and H. Nakagawa Hydrodynamic Beaviour of Partly Vegetated Open Cannels J. Hydraul. Eng. vol. 122 no. 11 pp [12] O. Giustolisi Using genetic programming to determine Cezy resistance coefficient in corrugated cannels J. Hydroinf. vol. 6 no. pp [1] B. Acour and S. Setal Te roug model metod MM. application to te computation of normal dept in circular conduit Open Civil Eng. J. vol. 8 pp [1] B. Acour and M. iabi Design of a pressurized trapezoidal saped conduit using te roug model metod Part 1 Adv. Mater. es. vols pp [15] B. Acour Computation of normal dept in trapezoidal open cannel using te roug model metod Adv. Mater. es. vol pp D

9 Cezy s esistance Coefficient in a Circular Conduit Te Open Civil Engineering Journal 2015 Volume [16] M. iabi and B. Acour Design of a pressurized circular pipe wit bences using te roug model metod MM Adv. Mater. es. vol pp [17] B. Acour and A. Bedjaoui Design of a pressurized trapezoidal saped conduit using te roug model metod part 2 Appl. Mec. Mater. vol pp [18] B. Acour Computation of normal dept in orsesoe saped tunnel using te roug model metod Adv. Mater. es. vol pp [19] B. Acour Design of a pressurized rectangular conduit wit triangular bottom using te roug model metod Open Civil Eng. J. vol. 8 pp [20] B. Acour Computation of normal dept in parabolic cross sections using te roug model metod Open Civil Eng. J. vol. 8 pp [21] B. Acour Design of a pressurized rectangular-saped conduit using te roug model metod Appl. Mec. Mater. vol pp [22] B. Acour and M. Kattaoui Design of pressurized vaulted rectangular conduits using te roug model metod part 2 Adv. Mater. es. vol pp [2] B. Acour and A. Bedjaoui Discussion. exact solutions for normal dept problem J. Hydraul. es. vol. no. 5 pp [2] B. Acour and A. Bedjaoui Turbulent pipe-flow computation using te roug model metod MM J. Civil Eng. Sci. vol. 1 no. 1 pp [25] B. Acour Design of pressurized vaulted rectangular conduits using te roug model metod Adv. Mater. es. vol pp eceived: December evised: Marc Accepted: Marc Bacir Acour; Licensee Bentam Open. Tis is an open access article licensed under te terms of te Creative Commons Attribution Non-Commercial License ttp://creativecommons.org/licenses/ by-nc/.0/ wic permits unrestricted non-commercial use distribution and reproduction in any medium provided te work is properly cited.