HYDROLOGY - TUTORIAL 1 UNIFORM FLOW IN CHANNELS

Transcription

1 HYDROLOGY - TUTORIAL 1 UNIFORM FLOW IN CHANNELS In ti tutrial yu will Derive frmula fr flw trug nte. Slve prlem invlving flw trug nte. Define unifrm annel flw. Derive frmulae relating annel dimenin and flw rate. Define te Frude Numer. Define u-ritial and uper ritial flw. Te tudent i advied t tudy Tutrial 1 frm te Fluid meani D0 etin efre tarting ti tutrial. D.J.DUNN 1

2 1. FLOW THROUGH NOTCHES A nt i plaed in a annel t meaure te flw y retriting it. Te flw rate i related t te dept f water eind te nt and a alirated dept gauge i all tat i needed t indiate te flw rate. RECTANGULAR NOTCH Te velity f water due t a preure ead nly i u g. Ti aume tere i negligile velity appraing te nt. Te flw trug te elementary trip i d u B d H H 1/ B / ud B g d gh Figure B Were te flw apprae te edge f a nt, tere i a ntratin eaue te velity at te edge i nt nrmal t te plane f te nt. Ti prdue a redutin in te r etin f flw and me fritin in te flw. Depending n te deign f te edge a effiient f diarge C d i needed t rret te frmula. B / Cd g H Furter tudy will yield frmula fr C d aed n te variu ape f te edge. SUBMERGED RECTANGULAR NOTCH and SLUICE GATE If te nt i a retangular le, te integratin mut e etween te tw dept H 1 and H yielding B / / C d g( H H1 ) If te ttm f te nt i te flr f te dwntream annel, we ave a luie gate and te ame frmula applie. VEE NOTCH Figure Te widt f te elementary trip varie dept u tat (H - ) tan(θ/) H θ ud g tan ( H - ) 0 θ gtan H D.J.DUNN H 0 1/ / ( H ) θ / g tan H H 8 θ / Cd g tan H 1 VELOCITY OF APPROACH 0 d / 1/ d Figure θ 4 / g tan H 1 and intrduing C d we ave If te velity appraing te nt i nt negligile ay u 1 ten te velity trug te elementary trip i u ( u g). If a nt i fitted int a annel nt mu igger tan te 1 + nt, te velity f te water appraing te nt i nt negligile and a rretin need t e made.

3 WORKED EXAMPLE N.1 Te dept f water ave te ill f a retangular nt i 0. m and te nt i 0. m wide. Te effiient f diarge i 0.6. Calulate te flw rate f water. B / x 0. / Cd gh 0.6 g m / WORKED EXAMPLE N. Te dept f water ave te ill f a vee nt i 0.4 m and a an inluded angle f 90. Te effiient f diarge i 0.6. Calulate te flw rate f water. 8 θ / 8 / Cd g tan H 0.6 x g tan4 x m 1 1 / WORKED EXAMPLE N. Te dept f water eind a luie gate in a rizntal retangular annel i m and te luie i 0.8 m ig. Te effiient f diarge i 0.7. Calulate te flw rate f water in te annel dwntream. B / / x / / C g( H H ) 0.7 g(. ) 1.84 m d SELF ASSESSMENT EXERCISE N.1 1 Te dept f water ave te ill f a retangular nt i 0.4 m and te nt i 0.7 m wide. Te effiient f diarge i 0.6. Calulate te flw rate f water. (0.47 m /) Te dept f water ave te ill f a Vee nt i 0. m and a an inluded angle f 60. Te effiient f diarge i 0.6. Calulate te flw rate f water. (0.8 m /). A luie ntrl te flw in a retangular annel. m wide. Te dept eind te luie i m and te luie i 0. m ig. Wat i te diarge? Take C d 0.8. (.8 m /) D.J.DUNN

4 . UNIFORM FLOW IN CHANNEL Cannel flw i arateried y ntant preure (uually atmpere) at all pint n te urfae. Ti mean tat flw an nly e indued y gravity te ed f te annel mut lpe dwnward. Tere i n preure gradient in te fluid puing it alng. If te r etin i unifrm and te dept i unifrm ten te flw rate i unifrm at all pint alng te lengt. Ti an nly ur if te ange f ptential eigt i alaned y te fritin le. Ti i UNIFORM FLOW. DEFINITIONS Flw rate (m /) Flw rate per unit widt q m / Cr etinal area A (m ) Wetted perimeter P (m) Mean velity u /A (m/) Slpe f ed S wi i terwie alled te energy gradient. Te ydrauli gradient i i and ti i te fritin ead l per unit lengt f te ed. Te ydrauli gradient i te ame a te lpe if te flw a a ntant dept (unifrm flw). Te ydrauli radiu i defined a R A/P and ti i al ften alled te ydrauli mean dept wit yml m. Te wetted area i A w PL τ w i te wall ear tre. Ti i te fre per unit urfae area reiting flw at te urfae f ntat etween te fluid and te wall. CHEZY FORMULA Cnider part f a flw f regular r etin A and lengt L. Figure 4 If te lpe i mall te weigt f te etin nidered i W ρgal Relving te weigt parallel t te ed te fre auing flw i F W in(s) If S i mall in S S radian F W S ρgals If te flw i teady tere i n inertia invlved te fre reiting mtin mut e equal t ti fre. Te reiting fre per unit urfae area F/A w τ w F/PL ρgals /PL ρgas /P ρg R S Cezy tugt tat Te Cezy frmula i τ w u and τ w C 1 u Hene C 1 u ρg R S u C (R S) ½ C (ρg/c 1 ) ½ and C i te Cezy ntant. D.J.DUNN 4

5 WORKED EXAMPLE N. 4 An pen annel a a retangular etin m wide. Te flw rate i 0.0 m / and te dept i 0.4 m. Calulate te lpe f te annel uing te Cezy frmula fr teady flw. Take te ntant C 0 m ½ / A x m P m R A/P 0.87 m u /A 0.0/ m/ u 0.06 C (R S) ½ u (0.87 S) ½ S.469 x 10-6 SELF ASSESSMENT EXERCISE N. 1. An pen annel a a triangular etin wit ide at 4 t te vertial. Te flw rate i 0.04 m / and te dept i 0. m. Calulate te lpe f te annel uing te Cezy frmula fr teady flw. Take te ntant C 49 m ½ / (Anwer ). A annel wit a etin a wn arrie 1.1 m / f water wit te dept a wn. Te lpe f te ed i 1/000. Calulate te ntant C in te Cezy frmula. (Anwer 1.44) Figure D.J.DUNN

6 THE CHEZY - MANNING FORMULA 1/6 R Manning extended Cezy' frmula. Baed n reear e tated tat C n n i a dimeninle ntant aed n te urfae rugne f te annel. Sutituting ti int te Cezy frmula yield / 1/ R S u Ti i te Cezy - Manning frmula. n WORKED EXAMPLE N. An pen annel a a retangular etin m wide. Te flw rate i 1. m / and te dept i 1.4 m. Calulate te lpe f te annel uing te Manning frmula fr teady flw. Take te ntant n m ½ / A x m P m R 7/ m u /A m/ / 1/ R S u n rearrange nu x S / R 1.6 x 10 / SELF ASSESSMENT EXERCISE N. 1. A retangular annel i m wide and run 1. m deep. Te lpe f te ed i 1/4000. Uing te Manning frmula wit n 0.0, alulate te flw rate. (Anwer 1.4 m /). An pen annel a a retangular etin m wide. Te flw rate i 1.4 m / and te dept i 0.8 m. Calulate te lpe f te annel uing te Manning frmula fr teady flw. Take te ntant n 0.0 m ½ / (Anwer 9. x 10-6 ). Water flw dwn a alf full irular pipeline f diameter 1.4m. Te pipeline i laid at a gradient if 1/0. If te ntant n in te Manning frmula i n 0.01 wat i te diarge. (1.61 m /) D.J.DUNN 6

7 DARCY FORMULA APPLIED TO CHANNELS Te Cezy frmula may e related t te Dary frmula fr flw in rund pipe. Te Dary frmula (nt derived ere) i f 4Cf Lu gd f Lu Smetime ti i tated a f were 4C f f gd f i te fritin ead and C f i te fritin effiient wi i related t te Reynld' numer and te relative urfae rugne. If a rund pipe run full ut wit ntant preure alng te lengt, ten te Cezy and Dary frmulae may e equated. Frm te Dary frmula we ave Fr ntant preure, f /L S Frm te Cezy frmula we ave gdf u 4Cf L gds u 4Cf u C R S Fr a rund pipe diameter d running full R d/4 u C Sd/4 Equating we ave Frm te Cezy equatin we ave C Sd gds 4 4Cf g C f C C Rf u L Lu Cf Lu f C R gr Ti verin f te Dary frmula may e ued fr pipe and annel f any ape wit n preure gradient. Diuin f te Dary frmula w tat C f i related t te urfae rugne and ti mpare wit Manning' wrk. In te ae f LAMINAR FLOW Pieuille' equatin i al relevant and ti give te fritin ead a µ L u f ρgd Equating ti t te Dary frmula give: µ L u ρgd 4Cf Lu gd 16µ ene C f ρu d Te mplete relatinip etween te Reynld' numer R e and te relative urfae rugne i given n te Mdy Cart. Te art a everal regin, laminar flw, turulent flw and a regin etween were it i in tranitin. Te turulent flw varie etween mt urfae and fully rug urfae tat prdue fully develped turulent flw. Relative urfae rugne i defined a ε k/d were k i te mean urfae rugne and D te re diameter. Te art i a plt f C f vertially againt R e rizntally fr variu value f ε. In rder t ue ti art yu mut knw tw f te tree -rdinate in rder t pik ut te pint n te art and ene pik ut te unknwn tird -rdinate. 16 R e D.J.DUNN 7

8 Fr te laminar regin C f 16 R e Fr mt pipe, (te ttm urve n te diagram), variu frmulae ave een derived u a te y Blaiu and Lee. BLASIUS C f R e 0. LEE C f R e 0.. Te Mdy diagram w tat te fritin effiient redue wit Reynld numer ut at a ertain pint, it eme ntant. Wen ti pint i reaed, te flw i aid t e fully develped turulent flw. Ti pint ur at lwer Reynld numer fr rug pipe. A frmula tat give an apprximate anwer fr any urfae rugne i tat given y Haaland ε.6 lg10 + Cf R e. 71 Figure 6 SELF ASSESSMENT EXERCISE N Te Dary - Weia frmula fr a rund pipe running full tate tat f 4 C f Lu /gd were L i te lengt, d te diameter and u te mean velity. a. Sw tat fr laminar flw C f 16/R e. Relate te Cezy frmula u C (RS) 1/ and te Manning frmula u (R / S 1/ )/n t te Dary - Weia frmula and lit te range f appliaility f all tree frmula.. Sket te relatinip etween C f and R e fr te range R e 10 0 t R e 10 6 in a pipe f irular r etin fr typial value f urfae rugne k. d. If ageing aue te urfae rugne f a pipe t inreae, wat affet wuld ti ave n te flw arrying apaity f te pipe? D.J.DUNN 8

9 . CRITICAL FLOW SPECIFIC ENERGY HEAD - At any pint in te lengt f te annel te fluid a tree frm f energy relative t te ed, kineti, gravitatinal (ptential) and flw (preure) energy. Figure 7 Stritly, all energy term uld e te mean value. Te mean dept i and te mean gravitatinal (ptential) ead i y (te ditane t te entrid). Te dept at te ttm i + y and te mean velity i u u u Frm te Bernulli Equatin + y + + g g Text k jump traigt t ti frmula wrngly giving a te preure ead. Rearrange te frmula and u { g( )} 1/ Cnider a annel wit an unpeified r etin f area A. Au A{ g( )} 1/ CRITICAL DEPTH C It will e wn tat fr a given value f tere i a dept tat prdue maximum flw rate ut te value f depend n te ape f te annel ine te widt i a funtin f dept and ene te area i a funtin f dept. Let' examine a retangular r etin. RECTANGULAR SECTION 1/ { g( )} B { g( } 1/ A ) B g {( )} 1/ Figure 8 If we plt fr a given value f B and we get figure 9a and if we plt fr a given value f B and we get figure 9. Figure 9a Figure 9 Te plt reveal me intereting ting. Pint C i alled te ritial pint and ti give te minimum energy ead fr a given flw rate r a maximum flw rate fr a given energy ead. Fr a flw rate ter tan te ritial value, tere are tw pile dept f flw. Ti i lgial ine fr a given amunt f energy te flw an e lw and deep r fat and allw. Flw at te allw dept i uper-ritial and flw at te larger dept i u-ritial. Te ritial dept i dented. D.J.DUNN 9

10 T find te ritial dept we ue max and min tery. At pint C d/d 0 Differentiate and we get: d d Sine 1/ 1/ B ( g) 0 ( ) 1/ 1/ u { g( )} ten utituting fr will prdue te ritial velity. u g g It fllw tat te ritial flw rate i Au B g Here i an alternative derivatin fr te retangular annel. A B u /(A) /(B ) + g B Fr a given flw rate te minimum value f i fund y differentiating. d ( ) 1 ( B ) g( B ) g 1 Fr a minimum value equate t zer. g 0 1 g ( B ) g ( B ) Tee are te ritial value it fllw tat / B g u g r B B u + g u u + g g u + g u r u g u g + + g g Te ritial flw in term f i B g / / gb B Te ritial velity in term f i u g FROUDE NUMBER B 1/ g / g g r / B gb Yu may ave tudied ti in dimeninal analyi. Te Frude Numer i a dimeninle numer imprtant t annel flw a well a t urfae wave. It i defined a : u u g F r Fr ritial flw F r Sutitute u g int ti and Fr 1 g g g Te Frude numer i alway 1 wen te flw i ritial in a RECTANGULAR CHANNEL ut nt fr ter ape. Anter name fr uper-ritial flw i SHOOTING r RAPID FLOW and u ritial i alled TRANUIL FLOW. Summary fr a retangular annel Te ritial dept i Te ritial velity i u g g Te ritial flw i g 8 7 1/ / / 8 / B g B g Frude Numer F r 1 7 D.J.DUNN 10

11 WORKED EXAMPLE N. 6 A retangular annel 1.6 m wide mut arry water at dept f 1 m. Wat wuld e te maximum pile flw rate and wat wuld e te mean velity? Fr maximum flw rate te dept mut e te ritial dept 1m. Te ritial velity i u (g ) ½ (9.81 x 1) ½.1 m/ Te ritial flw i A u 1.6 x 1 x.1.01 m / Cek te Frude numer Fr u.1 1 g g x 1 If te ntant n in te Manning frmula i m ½ / wat mut te lpe f te ed e fr ntant dept at maximum flw rate? A 1.6 x m P m R 1.6/ m u u.1 m/ / 1/ R S u n nu x.1 S / R / WORKED EXAMPLE N. 7 Water flw in a retangular annel m wide wit a mean velity f 1. m/ and a dept f 1. m. Determine weter te flw i tranquil r ting. Calulate te fllwing. Te atual flw rate Te peifi energy ead Te ritial dept Te maximum flw pile u 1. Fr 0.47 It fllw tat te flw i tranquil. g g x 1. Atual flw rate A u ( x 1.) x 1.. m / Energy Head + u /g /g 1.1 m / x 1.1/ m Fr maximum flw rate F r 1 u F r 1 u g 9.81 x m/ g A x m Au.69 x m / If te dept anged t te ritial dept, te flw rate wuld inreae. D.J.DUNN 11

12 SELF ASSESSMENT EXERCISE N. 1. A retangular annel i. m wide and mut arry m / f water wit te minimum peifi ead. Wat wuld te dept and mean velity e? (1.6 m and.91 m/). If te annel in quetin 1 mut arry flw at a ntant dept and n in te manning frmula i 0.0, wat i te lpe f te ed? (0.01). Te flw in a rizntal, retangular annel, 6m wide i ntrlled y a luie gate. Te dept f flw uptream and dwntream f te gate are 1.m and 0.00 m repetively. Determine: (a) te diarge () te peifi energy f te flw () te ritial dept. VEE OR TRIANGULAR SECTION { ( )} 1/ ( ){ ( )} 1/ tan θ/ g 4 tan ( θ/ ) g ( A g A ½ x tan(θ/) d d 1/ { )} 1/ 4 1/ ( θ/)( g) ( 4 tan ) Figure 10 4 Fr maximum ( ) Sine u u g 4 D.J.DUNN ( 4 ) 1/ { ( ) g } ten utituting fr will prdue te ritial velity. 1/ ( g) 1/ 1/ 4 1/ g 1/ 1/ u 4 g g g ( ) ( ) / Au tan θ/ tan θ/ FROUDE NUMBER g u u 1 F Fr ritial flw F r Sutitute fr u r Fr g g g In term f 1/ 1/ / / g / g 4 g 4 tan( θ/) tan( θ/) tan( θ/) 1/ 1/ g g 4 u x Summary fr triangular etin Te ritial dept i Te ritial velity i g 4 g u g Te ritial flw i g tan( θ/) / tan( θ/) 1/ Frude Numer F r / g 4 /

13 WORKED EXAMPLE N. 8 A triangular annel m wide wit an inluded angle f 90 mut arry water wit a dept f m. Wat wuld e te maximum pile flw rate te mean velity at ti flw rate? Fr maximum flw rate te dept mut e te ritial dept m. g Te ritial velity i u A tan(θ/) tan(4) 9 m 1/ 1/ g.86 m/ Te ritial flw i A u 9 x m / u.86 Cek te Frude numer Fr g g x If te flw mut remain at ntant dept and n in te manning frmula i 0.0, alulate te lpe f te ed. P /(θ/) 8.48 R A/P nu S 0.0 x.86 / R / WORKED EXAMPLE N. 9 A triangular annel m wide wit an inluded angle f 10 mut arry 0.7 m / wit te minimum peifi ead. Wat wuld e te maximum flw rate te mean velity? Fr minimum peifi ead, te flw rate and velity mut e te ritial value. 1/ g tan( θ/) / rearranging / / m / / 1/ 1/ g tan( θ/) g tan( 60) A tan θ m u /A 1.6 m SELF ASSESSMENT EXERCISE N.6 1. A unifrm annel a a vee r etin wit a ymmetrial inluded angle f 100. If it arrie 1. m / f water wit minimum peifi energy ead, wat wuld e te dept and mean velity. (0.74 m and m/). Te ame annel deried in quetin 1 mut arry te flw at a ntant dept. If n in te Manning frmula i 0.0, wat mut e te lpe f te ed. (0.0094) D.J.DUNN 1