ISO 9001:008 Certfed Internatonal Journal of Engneerng Scence and Innovatve Technolog (IJESIT) olume 3 Issue 5 September 014 Mathematcal Model of the flud flow n Crcular Cross-sectonal Open Channels.N. Oambo M. N. Knanu D. M. Theur P. R. Kogora K. Gterere bstract: Open channel flow nvolves the flow of a flud n a channel or condut that s not completel flled. Ths stud eamnes a non-unform and unstead open channel flow wth crcular cross-sectonal area that runs partl flled. The flow varables are veloct and depth whle the flow parameters are cross-sectonal area of the channel channel radus slope of the channel and lateral nflow rate per unt length of the channel. Effects of cross sectonal area of the channel on the flow veloct the effects of channel radus slope of the channel and lateral nflow rate per unt length of the channel on the flow veloct are nvestgated. Ths s done b varng one parameter whle holdng the other parameters constant. The stud further nvestgates the mamum flow depth. The momentum and the contnut equatons are the governng equatons of ths flow. The momentum equaton s epanded usng the unstead state of the contnut equaton. Fnte dfference appromaton method has been used to solve the governng equatons due to ts accurac stablt and convergence characterstcs. The results have been graphcall presented. It s establshed that ncrease n the cross-sectonal area leads to decrease n the flow veloct an ncrease n the radus of the channel leads to decrease n the flow veloct an ncrease n the roughness coeffcent leads to decrease n the flow veloct an ncrease n the flow depth leads to decrease n the flow veloct and a decrease n the lateral nflow rate per unt length of the channel leads to an ncrease n the flow veloct. Inde terms: Open channel veloct depth. I. INTRODUCTION Man researchers have ventured to stud open channel flows but few concentrated on the crcular channel flows because of ther dffcult n constructon. Crcular channel flows ncrease the flow veloct due to the mnmum wetted permeter that the have. Chow [1] studed Open Channel flows and developed man relatonshps such as veloct formula for open channel flow. He s also one of the poneers n the defnton of the geometrc elements of an open channel flow. The predcton of the one-dmensonal flow characterstcs (e.g. free surface profle and local dstrbuton of dscharge) s based upon the dnamcal flow equaton. pplcaton of the longtudnal momentum theorem elds a generalzed backwater relaton as etended b Yen and Wenzel []. The dnamcal flow equaton must be completed b the lateral outflow law and epressons relatng to the momentum correcton coeffcent the lateral outflow veloct component n the longtudnal drecton and the frcton slope. The usual approach assumes a unform veloct dstrbuton and s based on the energ equaton n whch losses are accounted for b Mannng's formula. Strelkoff [3] and Yen [4] derved the energ equaton from ntegraton of Naver stokes equatons. The clearl demonstrated that the losses of the mechancal energ are due to the work done b the nternal stresses to overcome veloct gradent. Snha et al. [5] nvestgated the development of the lamnar flow of a vscous ncompressble flud from the entr of a straght crcular ppe. It was observed that the veloct ncreases more rapdl durng the ntal development of the flow n comparson to the downstream flow. It was observed that durng the ntal stages of the development of the flow the rate of ncrease n stream wse veloct s larger and consequentl the pressure drop s larger n comparson wth ther values further downstream. Ths s because the retarded flud partcles n the boundar laer are pushed towards the core more rapdl near the entr where the boundar laer s thnner as compared to the downstream regon. The nfluence of lateral mass flu on free convecton flow past a vertcal flat plate embedded n a saturated porous medum was studed b Mansour et al. [6].Hager [7] studed the lateral outflow mechansm of sde wers usng a one-dmensonal approach. In partcular the effects of flow depth approachng veloct lateral outflow drecton and sde wer channel shape were ncluded resultng n epressons for the lateral outflow angle and the lateral dscharge ntenst. For vanshng channel veloct the conventonal wer formula for plane flow condtons was reproduced. However for ncreasng local Froude number the lateral outflow ntenst decreases under otherwse fed flow parameters. comparson of the theoretcall determned soluton wth observatons ndcated a far agreement. The results were then appled to open channel bfurcatons b consderng a sde-wer of zero wer heght. Dstncton between sub- and supercrtcal flow condtons s made. gan the computed 335
ISO 9001:008 Certfed Internatonal Journal of Engneerng Scence and Innovatve Technolog (IJESIT) olume 3 Issue 5 September 014 results compare well wth the observatons and allow a smple predcton of all pertnent flow characterstcs. Crossle [8] nvestgated the strateges developed for the Euler s equatons for applcaton to the Sant enant equatons of open channel flow n order to reduce run tmes and mprove the qualt of solutons n the regons dscontnutes. Nallur et al. [9] analzed epermental date on resstance to flow n smooth channels of crcular cross secton. The results of the tests showed that the measured frcton factors are larger than those for a ppe of equvalent dameter (D- 4R). Through the analss of the data for the range of flow depths 0 < < 0.85d the geometrc parameter /P was found more appropratel representng the shape effects on resstance to flow n crcular channels than usng a sngle parameter lke the hdraulc mean radus R. Carlos and Santos [10] consdered an adont formulaton for the non-lnear potental flow equaton. Tutoek and Hcks [11] modeled unstead flow n compound channels wth an am of controllng floods. Kwanza et al. [1] analzed the effects of the channel wdth slope of the channel and lateral dscharge on flud veloct and channel dscharge for both rectangular and trapezodal channels. The noted that the dscharge ncreases as the specfed parameters are vared upwards. han [13] set up Neutralzed Based Fuzz Reference Sstem (NFIS) for predcton of frcton coeffcent n open channel flows n whch Renold s number veloct and dscharge are used as nputs to determne frcton coeffcent of an open channel flow. It was shown that when provded wth correct and suffcent samples NFIS model can be used to predct the non-lnear relatonshp between frcton coeffcent and the factors whch affect t. It was concluded that n practce NFIS model can be used as a sutable and effectve method and general hdraulc problems whch are mostl based on laborator tests can be analzed wth NFIS model. M.N. Knanu et al. [14] nvestgated on modelng flud flow n open channel wth crcular cross-secton. Ths stud consdered unstead non-unform open channel flow n a closed condut wth crcular cross-secton. The man obectves were to nvestgate the effects of the flow depth the cross secton area of flow channel radus slope of the channel roughness coeffcent and energ coeffcent on the flow veloct as well as the depth at whch flow veloct s mamum. The obtaned Sant-enant partal dfferental equatons of contnut and momentum governng free surface flow n open channels are nonlnear and therefore do not have analtcal solutons. The Fnte Dfference ppromaton method s used to solve these equatons because of ts accurac stablt and convergence. Patel [15] descrbed a hdraulc model testng of varous mportant structures of open channel as well as rver flow wth analtcal soluton of the flow dstrbuton along the open channel. The stud consdered dfferent roughness constant and dfferent flow rate to eamne the flow pattern n an open channel. The research enabled the predcton of the veloct profle of open channel flow that would be applcable for a wde range of Renolds number b changng the dscharge flow rate. Governng equatons II. MTHEMTICL NLYSIS The Contnut and the Momentum or the energ equatons ma be used to analze open channel flow. Ecept for the momentum coeffcent and the energ coeffcent the Momentum and energ equatons are equvalent; Cunge et al. [16] provded that the flow depth and veloct are contnuous.e. there are no dscontnutes along the channel flow. However the Momentum should be used f there are no dscontnutes snce unlke the energ equaton t s not necessar to know the amount of losses n the dscontnutes n the applcaton of momentum equaton. Ths stud consders the Contnut and Momentum equatons wth veloct and flow depth as the two flow varables. The statement of the problem s a crcular condut that runs partl flled at certan tmes. The governng equatons dscussed heren are borrowed from conservaton of mass and momentum prncple gven b Yen [4]; The cross-sectonal area s taken from the volume element shown below where the dsplacement s n the man flow drecton() gven b where the flow flud enters from volume element at secton U at a mass transfer rate of and leaves at secton D at a mass transfer rate of. 336
ISO 9001:008 Certfed Internatonal Journal of Engneerng Scence and Innovatve Technolog (IJESIT) olume 3 Issue 5 September 014 337 In ths stud the flow s unstead and non-unform. From M.N. Knanu et al. [14] the contnut equaton s gven b q t Q (1) whch s epanded to 0 T q T t () The momentum equaton s gven b Yen 1973 [4]; 0 0 S f g S g Q t Q (3) The cross-sectonal area of ths stud s a unform crcular cross-secton thus the value of momentum coeffcent s equal to one. Replacng the contnut equaton for unstead flow the momentum equaton becomes: 0 0 f S S g g t q (4) The soluton procedure The governng equatons; equaton () and (4) are solved usng a dffusng fnte dfference scheme as proposed b essman et al.[17].these equatons are gven as: T q h T h k 0.5 1 1 (5) R n S g h g h k q 3 4 0 1 0.5 (6) subect to the ntal condtons 0 > all (0)= 4.5for = 0 t) (0 and boundar condtons Fg 1: olume element
ISO 9001:008 Certfed Internatonal Journal of Engneerng Scence and Innovatve Technolog (IJESIT) (0t) = 0(0)= 4.5for all olume 3 Issue 5 September 014 t > 0. III. RESULTS ND DISCUSSION The values for veloct aganst those of the depth were plotted gvng the curve n fgure below. Furthermore b varng the specfed parameters the curves fgures 3 to 6 were obtaned. Fg : eloct profle versus depth Fg 3: eloct profles for varng and S. 338
ISO 9001:008 Certfed Internatonal Journal of Engneerng Scence and Innovatve Technolog (IJESIT) olume 3 Issue 5 September 014 Fg 4: eloct profle for varng r and n Fg 5: eloct profles for varng q. 339
ISO 9001:008 Certfed Internatonal Journal of Engneerng Scence and Innovatve Technolog (IJESIT) olume 3 Issue 5 September 014 Fg 6: eloct profles for varng depth Fgure shows that veloct ncreases wth ncrease n depth. The free surface occurs at a depth of 4.5m and the veloct of the flud at ths depth s 10m/s. It s also observed that the mamum veloct occurs at a depth of 3.5m and begns to decrease n veloct as t approaches the free surface. The flow veloct n a channel vares from one pont to another as a result of shear stress at the bottom and the sdes of the channel. The free surface does not ehbt mamum veloct due to surface tenson caused b strong cohesve forces among the lqud molecules. Each lqud molecule s pulled equall n ever drecton b neghborng lqud molecules resultng nto a conservatve net force. t the free surface occurs a gaseous-lqud nterface whereb the lqud molecules are attracted nwards to other lqud molecules rather than outward to the gaseous molecules. Ths creates nternal pressure among the lqud molecules that forces the lqud surface to contract leadng to a reducton n veloct at the free stream. Mllons of collsons take place between the atmosphere and lqud molecules. s a result the atmospherc pressure eerts some surface force on the water surface and gravt creates nternal pressure that causes contracton of the lqud surface resultng to reduced veloctes. Furthermore the wnd blowng over the free surface also affects the veloct due to frctonal resstance partcularl when wnd blows over the free surface at hgh veloctes and n the opposte drecton to the man flow drecton. Fgure 3 shows that ncrease n the slope from 0.0 to 3.000 leads to ncrease n the flow veloct.e. from Curve I to Curve III. It also shows that ncrease n the cross-sectonal area from 56.7648 to 99.197 leads to decrease n the flow veloct.e. Curve I to Curve II. Increase n the slope causes the center of gravt to move up causng nstablt n the water molecules. The drect relatonshp of the flow s show n equaton.11 where an ncrease n slope causes an ncrease n the flow veloct. Increase n the cross-sectonal area of flow leads to ncrease n the wetted permeter that leads to ncrease n the shear stress between the sdes and bottom of the channel wth the flud partcles whch results to decrease n the flow veloct. 340
ISO 9001:008 Certfed Internatonal Journal of Engneerng Scence and Innovatve Technolog (IJESIT) olume 3 Issue 5 September 014 Fgure 4 shows that an ncrease n the radus of the channel from 8.5m to 18.5m leads to a decrease n the flow veloct.e. from Curve I and Curve II. It also shows that an ncrease n the roughness coeffcent from 0.09 to 0.9 leads to a decrease n the flow veloct. n ncrease n the radus of the channel results to an ncrease n the wetted permeter as a result of the flud spreadng more n the condut leadng to a larger cross-sectonal area. large wetted permeter results to large shear stresses at the sdes and bottom of the channel leadng to reducton n veloct. n ncrease n the roughness coeffcent results to large shear stress at the sdes of the channel. Ths means that the moton of the flud partcles at or near the surface of the condut wll be reduced. The veloct of the neghborng molecules wll also be lowered due to constant bombardment wth the flow movng molecules leadng to an overall reducton to the flow veloct. Fgure 5 shows that a decrease n the lateral nflow rate per unt length of the channel from 0.04 to 0.004 leads to an ncrease n the channel veloct.e. from Curve I to Curve II. n ncrease n the lateral nflow rate per unt length of the channel means an ncrease n the amount of flow per unt tme. Ths leads to ncrease n the crosssectonal area of whch leads to ncrease n the wetted permeter that ncreases the shear stress between the sdes and bottom of the channel wth the flud partcles leadng to decrease n the flow veloct. Fgure 6 shows that an ncrease n the channel depth from 4.5m to 5.5m to 6.5m.e. Curve I to Curve II to Curve III leads to a decrease n the channel veloct. n ncrease n the flow depth leads to ncrease n the wetted permeter whch results to ncrease n the shear stresses at the sdes of the channel that results to decrease n the flow veloct. I. CONCLUSION crcular channel flow has been developed wth the resultng partal dfferental equatons solved to obtan the veloct profles usng the veloct and depth as the flow varables. There s a reasonable agreement between the results of our model and those publshed b dfferent authors. Ths proves the fdelt of our equatons and the numercal scheme used. CKNOWLEDGEMENT The authors gratefull acknowledge the support of Pure and ppled Mathematcs department of Jomo Kenatta Unverst of grculture and Technolog Mr. P.R. Kogora and Dr. Kang ethe for ther techncal assstance. : Cross-sectonal area (m ) P: Wetted Permeter (m) : Mannng roughness coeffcent (sm -1/3 ) Q: Dscharge (m 3 s -1 ) T: Top wdth NOMENCLTURE : Lateral nflow rate per unt length of the channel (m s -1 ) R: Hdraulc radus (m) S f : Frctonal slope S o : Longtudnal slope of the channel : Tme(s) : eloct of flow : Depth of flow 341
: Man flow drecton ISSN: 319-5967 ISO 9001:008 Certfed Internatonal Journal of Engneerng Scence and Innovatve Technolog (IJESIT) : Momentum coeffcent α: Energ coeffcent olume 3 Issue 5 September 014 REFERENCES [1] Chow. T. Open Channel hdraulcs. McGraw Hll Book Compan. New York: pp1-40 (1959). [] Yen B. C and Wenzel H. G. "Dnamc Equatons for Stead Spatall ared Flow" Journal of the Hdraulcs Dvson SCE ol.96 No. 3 pp801-814 (1970). [3] Strelkoff T. One-dmensonal equatons of open-channel flow Journal of the Hdraulcs Dvson SCE ol. 95(HY3) pp 861 876 (1969). [4] Yen B. C. Open-channel flow equatons revsted Journal of the Engneerng Mechancs Dvson SCE ol. 99(EM5) pp 979 1009 (1973). [5] Snha P.C. and Meena ggarwal Entr flow n a straght crcular ppe Journal of ustralan Mathematcs (Seres BG) pp59-66 (198). [6] Mansour M.. Hassanen J. ND Baker Y. The nfluence of lateral mass flu on free convecton flow past a vertcal flat plate embedded n a saturated porous medum Indan Journal of Pure ppl. Math. ol. 1(3) pp 38-45 (1990). [7] Hager. H. and olkert P. U. "Dstrbuton Channels" Journal of Hdraulc Engneerng SCEol.11 No. 10 pp 935-95 (1986). [8] Crossle manda Jane ccurate and effcent numercal solutons for the Sant enant equatons of open channel flow Ph.D thess Unverst of Nottngham Unted Kngdom (1999). [9] Nallur C. and depou B.. Shape effects on resstance to flow n smooth channels of crcular cross secton. Journal of Hdraulc Research ol. 3. Issue 1 Januar pp 37-46 (1985). [10] Carlos L. and Santos C. n adont for the non-lnear potental flow equaton ppl. Math. Comp.ol. 108 pp11-1 (000). [11] Tutoek D.K. and Hcks F.E. Modelng of unstead flow n compound channels frcan Journal Cvl Engneerng ol.4 pp. 45-53 (001). [1] Kwanza J.K.Knanu M. and Nkoro J.M. Modelng flud flow n rectangular and trapezodal open channels. dvances and pplcatons n Flud Mechancs ol. pp. 149-158 (007). [13] han Samandar Neutralzed Based Fuzz Reference Sstem (NFIS) for predcton of frcton coeffcent n open channel flows cademc ournals Scentfc Research and Essas ol.6(5) pp 100-107 (010). [14] MM Knanu DP Tsombe JK Kwanza K Gterere Modelng Flud Flow n Open Channel wth Crcular Crosssecton Journal of grculture Scence and Technolog ol.13 No. (011). [15] Kaushal Patel Mathematcal model of open channel flow for estmatng veloct dstrbuton through dfferent surface roughness and dscharge Internatonal Journal of dvanced Engneerng Technolog IJET ol. III Issue I Januar-March pp 5-53 (01) [16] Cune J.. Holl F.M. Jr and erwe. Practcal aspects of rver computatonal hdraulcs Ptman London. (1980). [17] essman W Jr. Knapp J.W. Lews G. L. and Harbaugh T.E. Introducton to hdrolog Second edton Harper and Row New York pp. 1-60 (197). 34
ISO 9001:008 Certfed Internatonal Journal of Engneerng Scence and Innovatve Technolog (IJESIT) olume 3 Issue 5 September 014 UTHOR BIOGRPHY Mss. ona Nakhulo Oambo obtaned her BSc. n ppled Mathematcs and Computer Scence from Jomo Kenatta Unverst of grculture and Technolog (JKUT) Kena n 011. Presentl she s workng as an dunct Teachng ssstant at JKUT. She s a MSc. student n the same unverst. Her area of research s Channel Flows. Professor Mathew Ngug Knanu Obtaned hs MSc. In ppled Mathematcs from Kenatta Unverst Kena n 1989 and a PhD n ppled Mathematcs from Jomo Kenatta Unverst of grculture and Technolog (JKUT) Kena n 1998. Presentl he s workng as a professor of Mathematcs at JKUT where s also the drector of Post Graduate Studes. He has Publshed over Fft papers n nternatonal Journals. He has also guded man students n Masters and PhD courses. Hs Research area s n MHD and Flud Dnamcs. Dr. Davd Mwang Theur Obtaned hs MSc. In ppled Mathematcs from Kenatta Unverst Kena n 1990 and a PhD n ppled Mathematcs from Jomo Kenatta Unverst of grculture and Technolog (JKUT) Kena n 001. Presentl he the Charperson of the Department of Pure and ppled Mathematcs at JKUT. He has Publshed over ten papers n nternatonal Journals. He has also guded man students n Masters and PhD courses. Hs Research area s n MHD and Flud Dnamcs. Mr. Phneas Ro Kogora obtaned hs MSc. n ppled Mathematcs from Jomo Kenatta Unverst of grculture and Technolog (JKUT) Kena n 007. Presentl he s workng as an ssstant Lecturer at JKUT. He s a PhD student n the same unverst. Hs area of research s Hdrodnamc Lubrcaton. Dr. Kang ethe Gterere obtaned hs MSc. In ppled Mathematcs from Jomo Kenatta Unverst of grculture and Technolog (JKUT) Kena n 007 and a PhD n ppled Mathematcs from the same unverst n 01. Presentl he s workng as a Lecturer at JKUT. He has publshed s papers n nternatonal ournals and guded man students n Masters courses. Hs area of research s MHD and Flud Dnamcs. 343