Topic 3 Specific Energy and Control Section

Transcription

1 Tpi Speifi nerg and Cntrl Setin Prepared b: Tan Lai Wai et al.

2 Learning Outes At the end f this tpi, students shuld be able t: i. Appl speifi energ nept in deterining ritial flw nditins ii. iii. Analse flw ver brad-rested weir Analse flw thrugh width nstritin

3 1.1 Cnept f Speifi nerg Speifi energ (intrdued b Bakheteff) is the energ f flw easured with respet t the hannel btt. V1 g V g 1 h f 1 V g z 1 Datu z

4 The nept f speifi energ is useful in defining ritial depth and in the analsis f flw prbles. Fr nstant, ga Variatin f with is represented b a ubi parabla, () 4 sub super 1 45 V g V1 g in 1 = subritial, Fr < 1 ritial, Fr = 1 superritial, Fr >

5 1. Alternate Depths and Critial Depth Fr a speifi (exept in ), there are tw flw depths 1 and, i.e. Subritial Alternate depths Superritial If there is energ lss, e.g. during hdrauli jup, 1 and are knwn as njugate (r seuent) depths Critial flw urs when speifi energ is iniu, in with = ritial depth Nte: Negative flw depth is nt pssible.

6 Speifi energ ga Differentiating d d 1 ga da d At iniu speifi energ in, = and A = A T A ga 1 1 ga T da d T da d V gd V gd 1 1 Fr 1

7 Speifi nerg f Flw in Retangular Setin Sine B g Rearranging g Variatin f with is represented 6b the fllwing urve, 5 4 sub subritial, Fr < 1 1 () 1 super 1 = ritial, Fr = 1 superritial, Fr > 1 1 ax 4 5

8 Fr a speifi (exept ax ), there are tw flw depths 1 and, i.e. Subritial Cnjugate depths Superritial Critial flw urs when disharge per unit width is axiu, i.e. ax A g Keeping nstant, d da g d d ga T A T 1 ga A Fr 1 ga g

9 State f flw an be established b paring with. Charateristis Fr = 1 = Fr < 1 > Fr > 1 < Flw nditin Critial flw Subritial flw Superritial flw

10 Ativit.1 The rate f flw in a - wide retangular hannel is 1 /s. Calulate the speifi energ if the depth f flw is (a) ; and (b) 1.. Given = 1 /s and B =. When =, ga When = 1., ga

11 1. Calulatin f Critial Depth Critial depth an be deterined b: i. Trial and errr; r ii. Graphiall 1..1 Critial Depth fr Trial-and-rrr Fr all hannel setins, during ritial flw ( in ) T 1 ga is usuall prvided Rewritten as a funtin A f ritial depth, T g

12 Fr retangular hannel, T = B and A = B A T g bees B B g B g Sine B g (nl applies t retangular hannel)

13 Critial depth als urs when is axiu g g Differentiating with respet t d d gives d g in d in in (nl applies t retangular hannel)

14 Critial slpe S is used t ategrize the tpe f hannel slpe Cnditin f S S = S S < S S > S Tpe f slpe Critial slpe Mild slpe Steep slpe Critial slpe S an be alulated b euating Manning resistane flw euatin t ritial flw nditin T 1 ga At ritial slpe, S = S S n ga T ga T R 4 1 AR n S 1

15 Ativit. A 4. wide retangular hannel nves water t a reservir. If the disharge in the hannel = 5 /s and Manning effiient n =., find (a) (b) () Critial depth Critial velit Critial slpe Given = 5 /s, B = 4., n =. (a) g

16 (b) V gd 1 V g 1 (fr retangular setin, D = ) V g /s () S S S n ga 4 TR n g 4 (fr retangular setin, T = B) R

17 Ativit. (a) (b) An infinitel wide and straight river has a disharge f 5. /s/. Calulate: (i) Critial depth (ii) Frude nuber f the flw when the flw depth is 6. and deterine the tpe f flw (iii) Critial slpe f the hannel if Manning effiient n =.44. Based n the river harateristis given in (a), find the pssible depth f flw fr the sae speifi energ and the rrespnding Frude nuber. Given = 5. /s/, = 6., n =.44, Fr infinitel wide hannel R

18 5 (a) (i) 1.66 g 9.81 (a) (ii) Fr retangular setin, D = Fr V g g Fr subritial flw (a) (iii) Fr retangular setin, T = B S n R g 4 n g

19 (b) Speifi energ at 1 = 6 g The alternate depth f 1 = 6 with = 1 = 6.5 is g At =.4789, Fr V g g Fr superritial flw 4.817

20 Ativit.4 Fr a trapezidal hannel with btt width B = 6 and side slpe z =, find the ritial flw depth if the disharge is 17 /s using trial-and-errr ethd. Given = 17 /s, B = 6, z = A T g = 17 /s 1 z = Fr trial-and-errr,.8468 B = 6

21 1.. Graphial Methd Critial depth f flw an be slved b pltting against A T r A D Ativit.5 Fr a trapezidal hannel with btt width B = 6 and side slpe z =, find the ritial flw depth if the disharge is 17 /s graphiall. Given = 17 /s, B = 6, z =

22 Als, A T A T A T g 9.46 () A /T () T A Fr the graph,.84

23 1.4 Cntrl Setins A ntrl setin is where fr a given disharge, the flw depth and velit V are fixed. The ritial depth is als a ntrl pint sine at this setin Fr = 1, effetive when subritial flw hanges t superritial flw. When superritial flw hanges t subritial flw, a hdrauli jups usuall bpass the ritial depth as ntrl pint. A ntrl setin 'ntrls' the upstrea r dwnstrea flw.

24 xaples f ntrl setins: (a) Flw fr a ild hannel t steep hannel (b) A ild-slpe hannel disharging int a pl M ntrl M ntrl S Drp Mild Steep Pl

25 () Free verflw (sudden drp) (d) Reservir water flws n a steep slpe H ntrl ntrl S Hrizntal bed Reservir Steep

26 (e) Flw thrugh sluie gate (f) Flw ver spillwa M 1 ntrl M 1 ntrl Mild ntrl M Jup Mild

27 (g) Flw ver brad-rested weir ntrl (h) Flw thrugh nstrited hannel width ntrl H Hup Cnstritin B Plan view

28 1.5 Flw Over Brad-Crested Weir Flw in a prisati pen hannel is unifr if there is n bstrutin e.g. f a hdrauli struture. If brad-rested weir is installed, unifr flw hanges t nnunifr flw. Changes t the water surfae prfile is influened b the weir height H and the flw nditin befre the weir (upstrea flw), i.e. either superritial r subritial. 1 H Weir = nral depth f flw 1 = depth f flw just befre weir = depth f flw n the weir = depth f flw just after weir

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30 Miniu Height f Weir H in Height f weir H deterines the depth f flw abve the weir, i.e. whether = r nt. H in = iniu height f weir whih will start t prdue ritial flw depth abve the weir ( starts t hange t ) Generall, depth f flw abve the weir is If H H in If H H in If H H in Therefre, = and = in if H H in

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32 Case 1 Case Case H H H Cnditin f upstrea flw H H in r in H Suberged weir 1 H H H in r in H Rarel ur 1 in H H in r in H Cntrl weir 1 1 = 1 = 1, in H in 1 and Subritial H 1 in H 1 Bakwater 1 in H 1 1 and Superritial H 1 in H 1 Jup 1 in H 1

33 Steps in Analsing Flw Over Brad-Crested Weir 1. Calulate and Deterine state f upstrea flw, i.e. either subritial r superritial b paring with. If subritial upstrea If superritial upstrea. Calulate H in B paring height f weir H with H in, the nditin f flw ver weir an be established, i.e. If H H in Case 1 If H H in Case If H H in Case. Deterine 1, and.

34 8 Case 1: H H in 6 Subritial upstrea H 1 GL () in H () 6 H in Superritial upstrea H 1 GL () 4 1, in H H in ()

35 Fr Case 1, H H in H < in 1 Steps All setins Useful euatins 1. Calulate n and Manning: AR 1 Manning: S. Calulate H in. Deterine 1, & A T g 1 = H Retangular setin R g ga g in in ga H in in n S ga g 1

36 8 Case : H H in 6 Subritial upstrea in GL () 4 1 H in () HH in 6 Superritial upstrea in GL () 4 H 1 1, in HH in ()

37 Fr Case, H H in H in in 1 in Steps All setins Useful euatins 1. Calulate n and Manning: AR 1 Manning: S. Calulate H in. Deterine 1, & A T g 1 = Retangular setin R g ga g in in ga H in in n S 1

38 8 Subritial upstrea 1 and Bakwater 1 in GL () H in 4 6 H in 1, () Superritial upstrea 1 and in Jup 1 GL () HH in 1 H 1 in H () in HH in

39 Fr Case, H H in H in 1 Steps All setins Useful euatins 1. Calulate n and Manning: AR 1 Manning: S. Calulate H in. Deterine 1, & in A T ga ga H g in 1 1, = in H Retangular setin in in R g g n S 1, 1, 1, 1, ga 1, g 1, 1

40 Ativit.6 1 /s f flw is nveed in a retangular hannel f 4 width, n =.15 and S =.75. If a weir with height.9 is built in the hannel, deterine the depth f flw n the weir. Given = 1 /s, B = 4, n =.15, S =.75, and H =.9 Step 1. Deterine and 4 4 R n S B

41 g superritial flw Step. Calulate H in g H in in in

42 Step. Deterine Sine H =.9 H in =.78 Case Hdrauli jup &.865

43 Ativit.7 A retangular hannel nves flw at = 1.6 and R =.77 n S = 1/ and Manning n.1. (a) (b) () What is the iniu height f weir t ntrl the flw in the hannel? Calulate depth f flw upstrea, dwnstrea and abve the weir in (a). Calulate depth f flw upstrea, dwnstrea and abve the weir if the height f weir is (i).4, and (ii).6.

44 Given = 1.6, R =.77, S = 1/, n.1 (a) Step 1. Deterine and 1.6 n g R S /s/ subritial flw Step. Calulate H in g H in in in

45 (b) Step. Deterine 1, &. In (a), H H in Case. Fr Case, () (i) If H.4 H in.445 Case 1 Fr Case 1, H g Thrugh trial-and-errr, 1. r.785 Sine is subritial,, 1.

46 () (ii) If H =.6 H in.445 Case : Bakwater Fr Case, , g H 1, , in Thrugh trial-and-errr, and , 1, 1, sine 1 and

47 1.6 Flw Thrugh Cnstrited Channel If width f a prisati hannel is redued/enlarged at a setin, unifr flw hanges t nn-unifr flw. Changes t the water surfae prfile is influened b the width f nstritin B and the flw nditin befre the nstritin, i.e. either superritial r subritial. Plan view B g B 1 Bed elevatins at 1 and are the sae, 1 = g 1 Side view GL () 1 Sine B < B, > V g gb ax in ( /s) B g 1 V g gb 1

48 Maxiu Width f Cnstritin B ax Width f nstritin B deterines the depth f flw at the nstrited setin, i.e. whether = r nt. B ax = axiu width f nstritin whih will start t prdue ritial flw depth at the nstritin ( starts t hange t ) r 1 = ritial depth f flw alng the unnstrited setin = ritial depth f flw at the nstrited setin. Generall, depth f flw at nstritin is If B B ax If B B ax If B B ax > Therefre, = r and = in if B B ax

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50 Case 1 Case Case B B B Cnditin f upstrea flw B B ax r in r ax 1 B B ax r in r ax 1 in = B B ax r in r ax Cntrl nstritin 1 1 = 1 = 1, ' in ' in 1 and Subritial in Bakwater in and Superritial in 1 in Jup 1 1 BFC11 Hdraulis Tan et al. (laiwai@uth.edu.) 1

51 Steps in Analsing Flw Thrugh Cnstritin 1. Calulate and Deterine state f upstrea flw, i.e. either subritial r superritial b paring with. If subritial upstrea If superritial upstrea. Calulate, ax and B ax When width f a hannel is being nstrited, an be btained sine in =. One B ax is alulated, the nditin f flw thrugh the nstritin an be established, i.e. If B B ax Case 1 If B B ax Case If B B ax Case. Deterine 1, and.

52 8 Case 1: B B ax 6 Subritial upstrea 1 GL () 1 4 B r B r 1 8 in in B ax r ax 6 () Superritial upstrea 1 GL () 4 B r 1 in in B r () B ax r ax

53 Fr Case 1, B B ax in Steps Useful euatins 1. Calulate n and Manning: AR 1 r S. Calulate, ax and B ax. Deterine 1, & g 1 = R n S 1 ; in g g ax ; g ax B ax

54 8 Case : B B ax Subritial upstrea 1 GL () in in () B ax r ax B r Superritial upstrea 1 GL () 4 1 in 4 in 6 8 () B ax r ax B r

55 Fr Case, B B ax in in Steps Useful euatins 1. Calulate n and Manning: AR 1 r S. Calulate, ax and B ax. Deterine 1, & g 1 = R n S 1 ; in g g ax ; ax B ax

56 8 Case : B B ax Subritial upstrea Bakwater 1 in 1 GL () in in 4 in 6 8 () B B ax B ax B r Superritial upstrea Jup 1 in 1 GL () in in 4 in 6 8 () B B ax B ax r ax B r

57 Fr Case, B B ax in in Steps Useful euatins 1. Calulate n and Manning: AR 1 r S. Calulate, ax and B ax. Deterine 1, & g R n S 1 ; in g ax g ax B in ; ; ; 1, ax B 1, ax ax g g ax 1, 1 ; =

58 Ativit.8 A bridge is t be built arss a 5- wide retangular hannel arring flw f /s at depth 4.. Fr reduing the span f the bridge, what is the iniu width f hannel suh that the upstrea water level will nt be influened b the nstritin? Given = /s, = 4, B = 5 Step 1. Deterine and /s/ B 5 g is influened b. When hanges, varies as well. Therefre, at nstritin where, exists (alulated in Step ). subritial flw

59 Step. Calulate and B ax gb At width B ax, in and ax With n energl lss, in, therefre 4.51 in Als,.71 g ax rearranging gives ax g /s Sine ax B ax B ax ax

60 Ativit.9 A bridge is t be built arss a 5- wide retangular hannel arring flw f /s at depth 4.. The nstrutin has aused the width f the hannel t be redued t -. Deterine the depth f flw upstrea, dwnstrea and under the bridge. Given = /s, = 4, B = 5 Step 1. Deterine and (siilar t the slutin in Ativit.8) /s/ B 5 g subritial flw

61 Step. Calulate and B ax (siilar t slutin in Ativit.8) gb in At width B ax, in and ax With n energl lss, in, therefre Als, g ax rearranging gives ax g /s Sine ax B ax B ax ax

62 Step. Deterine 1, and Sine B B ax 14.9 Case At B, gb Fr trial-and-errr,.899 r.9 Sine, thus. Therefre, =.9

63 Ativit.1 A retangular hannel f. width is reuired t nve /s f flw. The nral depth is.8. At dwnstrea f the hannel, the width f the hannel is t be redued. (a) (b) Deterine the width f the axiu nstritin fr ritial depth t ur. Calulate the depth f flw upstrea, dwnstrea and at the nstritin if the nstrited width is 1.. Given = /s, B =., =.8

64 (a) Step 1. Deterine and B g /s/ Sine subritial flw Step. Calulate and B ax g B in ax ax g ax /s

65 (b) If B = 1., B Bax Case, where new ax, i.e. ax is reuired Step. Calulate 1, and ax B.5 1. /s g ax.865 in

66 1 in 1, g 1, 1, , 1, 1.91 Fr trial-and-errr, and.489

67 Ativit.11 Flw inside a retangular hannel f. width has a velit f. /s at. depth. The hannel is experiening a step f.61 high at the hannel btt. What is the nstritin t be ade t the hannel width in rder t ensure the depth f flw upstrea des nt hange. Given V = /s, B =, =, and H =.61 Thus, = V = = 9 /s Step 1. Deterine and. g Sine subritial flw

68 Step. Calulate H in 9 g 9.81 H in in.1. in Step. Deterine 1, and Sine H.61 H in.47 Case : Bakwater upstrea f weir H..61 1, in.64 In rder t aintain the sae speifi energ and redue 1 t, has t be inreased, i.e. via width nstritin.

69 1 and GL Bakwater in. () () = =.1 = and GL Bakwater in. () =.459 =.1 () 1 in 4 1, () H in =.47 () H=.61 =.459 =.1 1 = =.459 =.1 () in 4 1, 1, 6 8 H in =.47 () H=.61

70 in 1, in.48 ax g /s B ax ax

71 =.459 Plan view =.1 1 = in.64 = GL 1 () 8 () = =.459 =.1 () in 4 1, 6 8 H in =.47 () H=.61 B= B =

72 Ativit.1 A retangular hannel f.6 width with Manning n =.15, and lngitudinal slpe f.8 is nveing flw at 9.8 /s. If a nstritin is ade b reduing hannel width t.4, alulate depth f flw upstrea and dwnstrea f the nstritin. Sketh the flw surfae prfile. Given = 9.8 /s, B =.6, B =.4, n =.15, S =.8 Step 1. Deterine and n AR S

73 Thrugh trial-and-errr, g Sine subritial flw Step. Calulate and B ax g B ax is when = ax, where in = in

74 B ax ax g ax /s When B =.4, B Bax 1.56 Case 1, where in < = Step. Calulate 1, and 1 = = = B.4 /s g

75 g =.411 =.7 = GL =.4 = 1.67 Thrugh trial-and-errr,.4 r.759 Sine it is subritial upstrea,.4

76 1.7 Chking Chking f flw urs when H H in B B ax f a brad-rested weir in an pen hannel at the nstrited width in an pen hannel i.e. when the speifi energ r depth f flw iediatel upstrea f the weir r nstritin inreases r is being ntrlled. Chked nditins are undesirable in the design f ulverts and ther surfae drainage features invlving hannel transitins.

77 Assignent # 1. (a) Critial depth urs in an pen hannel when the speifi energ is iniu. Sketh the rrespnding flw depth versus speifi energ graph. Fr this nept, derive the general euatin used t deterine ritial flw depth in an pen hannel. (b) A retangular hannel.5 wide arries.4 /s unifr flw at a depth f.6. A. -high weir is plaed arss the hannel. (i) Des the weir ause hdrauli jup upstrea f the weir? Prvide reasn wh. (ii) Calulate the flw depth abve the weir, and just upstrea f the weir. Classif the surfae prfile f flw upstrea f the weir. Sketh the resulting flw-surfae prfile and energ line, shwing the ritial depth and nral depth.

78 . (a) An engineer is t analze flw in an pen hannel in whih the hannel is designed t be nstrited b plaing bridge ebankent at bth sides f the hannel. xplain the nseuenes due t the nstritin. (b) An 8- wide retangular hannel is nveing flw unifrl at a rate f 18.6 /s and depth f 1.. A teprar shrt span bridge is t be built arss the hannel in whih bridge ebankent is needed at bth sides f the hannel ausing the hannel t be nstrited under the prpsed bridge. (i) Calulate the axiu hannel width under the prpsed bridge whih will nt ause bakwater upstrea. (ii) (iii) If the hannel width under the prpsed bridge is 4 due t the unavidable nditin, alulate the expeted flw depth under the bridge, at just upstrea and just dwnstrea f the bridge. If the flw depth just upstrea f the prpsed bridge is t be liited t. higher than the nral depth, alulate the hannel width under the bridge. - nd f uestin -

79 THANK YOU

80 Leturers Dr. Tan Lai Wai Dr. Mhd Adib Mhaad Razi Dr. Hartini Kasin Dr. Mhd. Shalahuddin Adnan Dr. Mhd Ariff Ahad Nazri Dr. Siti Nazahiah Rahat Md. Zarina Md Ali Md. Nr Aliza Ahad