CE 473 Open-Channel Flow. Chapter 3 Channel Controls

Transcription

1 CE 473 Open-Channel Flow Chapter 3 Channel Controls

2 3. Introdution A ontrol is an hannel feature, whih fixes a relationship between depth and disharge in its neighborhood. It ma be natural or human-made. Also we ma have: Overflow strutures - spillwas, weirs, free falls Underflow strutures- sluie gates, gates Wh use need ontrol strutures?.flow profile omputation-provides the boundar ondition..to measure disharge; As an engineer we are onerned with the funtioning of the ontrol itself, suh as the abilit of a spillwa to disharge floodwaters at the required rate. We have rapid flow in the viinit of ontrol strutures (setions).sine the streamlines are highl urved near the ontrol setions we an not solve the equations analtiall, Therefore we go to empirial relations based on experiments.

3 Vena Contrata Vena ontrata is the setion where the flow beomes parallel or it is the setion where area of the jet beomes onstant and minimum. Coeffiient of veloit, Coeffiient of ontration

4 3. Flow through orifies and short tubes Tpial examples of orifies and short tubes are shown. As the flow issues out vena ontrata will our. Orifies A knowledge of the laws of the flow through them is neessar in determining the disharge through sluiewas and the entranes to onduits.

5 If the entrane is not properl shaped, a ontration of the jet ours as shown in sketh of vena ontrata. The area of the jet is not as great as the area of the orifie or tube. For properl rounded approahes to orifies, and the onstant diameter short tubes, the diameter of the jet is equal to the area of the orifie or tube. In ase of short tubes without rounded entranes, the ontration does our; but the jet expands again, with ertain exeptions, a partial vauum ourring just inside the entrane. Let H = the head of water on the enter line of a freel flowing orifie or tube, or the differene in water level for a submerged orifie or tube A= the area of the orifie and tube V= the theoretial veloit orresponding to head H; g= the aeleration of gravit

6 R Energ equation between points R and N: H R =H N H datum N z P R R + P = P R γ N + =P U R g atm = z U N R + P N γ = 0, + U and N g z, R = H Therefore H = U N g U N = gh

7 Q= the disharge C =the oeffiient of ontration C f = the oeffiient of frition, the redution in total head due to frition C v = the oeffiient of veloit C d = the oeffiient of disharge The general equation for the veloit of the spouting water is V gh Considering the frition, the atual veloit due to the head H is V atual = C v gh The disharge is equal to the produt of the atual veloit and the area of the jet; and the area of the jet is A j or = C A V A A jet orif atual = C f gh

8 Therefore Q = V at C A = C v C A gh In experiments onduted to determine the disharge through orifies and tubes, the oeffiient of frition and the ontration oeffiient are ombined and the general equation is given as: Q = C d A gh, where C d = disharge oeffiient it depends on the shape of the orifie and tube, it is not greatl affeted b the submergene. If we do not have submergene and enough frition ontrated jet does not expend in tube. It shoots out as ontrated without touhing the walls. C d = C v C

9 Orifies and Their Nominal Coeffiients C d

10 Dimensional Analsis Let s onsider the following figure; V 0 H W L Aording to values of L, W and roundness, it an be a weir or sharp rested weir. The disharge oeffiient, C d, will be a funtion of: C d = φ V ( ) o,h,w,l,g,μ,ρ,σ 8 variables 3 basi dimensions, M,L,T 5 dimensionless quantit

11 If we hoose ρ, V o, H as primar (repeating variables); the dimensional analsis would give us: ρv0h Re = = Re noldsnumber μ C C = d = f H W H, L,R e,f,w Therefore, the disharge oeffiient is a funtion of: r F r = W = V ρv 0 gh σ o H = Froudenumber = Weber number d C d A Q gh = Froudenumber diret measureof C D = f H W H, L R e,w Renolds and Weber numbers beome important at small values of H.

12 3.3 Sharp-Crested Weirs A sharp rested weir normall onsists of a vertial plate mounted at right angles to the flow and having a sharp-edged rest, as shown in the figure V 0 V 0 g H V 0 g A C h B EGL W 45 o D P/g Sharp-rested weirs are ommonl used as means of flow measurement.

13 Sharp-rested weir has an importane in Open Channel Hdraulis, simpl beause, its theor forms a basis for the design of spillwa. Beause, the edge is sharp. no boundar laer development onl on the vertial surfae on whih the veloities are small Therefore no visous effets no energ dissipation Let s onsider the simplest form of weir; onsisting of a plate set perpendiular to the flow in a retangular hannel, its horizontal upper edge running the full width of the hannel dimensionalit no lateral ontration effets. Sine the lateral ontration effets are suppressed b the hannel sidewalls, this tpe of weir is sometimes termed the SUPPRESSED weir.

14 Let s make an elementar analsis b assuming; no ontration over the weir pressure it atmospheri aross the whole setion AB. The total head at a point C is: h = P γ =0 + V g V gh The disharge per unit width then; = q v / g H o v o / g vdh v o v / gh o / g ghdh

15 The disharge per unit width beomes; Here h is measured downwards from the EGL, not from the free surfae. Now, the effet of the flow ontration ma be expressed b a ontration C, leading to the result: 3/ 3/ 3 g v H g v g q o o 3/ 3/ 3 g v H g v g q o o

16 We an make this expression more ompat b introduing a disharge oeffiient C d ; where; d q 3 d g H 3 / 3/ 3/ v o gh vo gh we should expet both C and the ratio of Vo gh to be dependent on the boundar geometr alone, in partiular on the ratio of H/W; when W is large V o is small o V C gh d C

17 REHBOCK found experimentall that C d H = W 000H Shows the visous and the surfae tension effets. Neglet unless H is ver small. H is in meter. As W beomes ver large C d =0.6, and sine V o / g beomes ver small, C C d 0.6. This happens to be the numerial value of shown last entur b Kirhoff to be the ontration oeffiient of a jet issuing without energ loss, and with negligible defletion b gravit from a long retangular slot in a large tank, a two-dimensional problem. C = π π + This is the onl fluid flow problem in ideal fluid for whih the ontration oeffiient is obtained theoretiall.

18 B man investigators the value 0.6 have been onformed for high weirs. On the other hand, when W beomes small, H/W beomes large, and this formula annot be true for high values of H/W. In fat, experimental work has shown that it is true onl for the values of H/W up to approximatel 5. Therefore: C d = H W + 000H H W 5 For, 5 H W 0 C d begins to diverge from the value given b the formula, reahing a value of.35 when H/W=0. If W vanishes ompletel, so that H/W beomes infinite, we have the ase of free over fall. In this ase H=, ritial depth, and q is determined aordingl.

19 In fat, it has been shown experimentall that ritial flow also ours just upstream of a ver low weir. In this ase the weir is alled a sill, i.e in the range H/W > 0. V 0 g EGL V 0 H h W = H + W V = g q = V q = 3 / ( ) ( ) ( ) 3 / W 3 / H + W g H + W = g H + W = + H H

20 q = 3 C d gh 3 / = g + W H 3 / 3 / The range 0 0 has not et been ompletel explored. W H But Kanda Swan and Rouse showed that: H C d =.06 + W H 3 / C d H/W W/H Max. C d ours at H/W=0, whih is the border line for weir and sill.

21 For ompletel free overfull, W = 0, the equation for C d beomes d gh 3/ 3/ v o vo gh Cd =.484C d.06 W H 3/ C =0.75 will be disussed later at free overfall.

22 Two important features of the pressure distribution down the vertial setion ABC; Pressure distribution is nonhdrostati, beause of distint urvature of the streamlines in the vertial plane, Pressure is not atmospheri, ontrar to the assumption in the elementar analsis that : the pressure > the atmospheri pressure Consistent with the ontration and aeleration downstream of AB. Clearl a pressure fore is neessar to ause this aeleration in the region of atmospheri pressure.

23 It has been assumed throughout that the pressure is atmospheri along the lower surfae of the jet, or nappe as well as upper surfae if this jet is just onfined between parallel walls downstream of the weir, air will be trapped and this all will be graduall removed b the flow, pressure in this region is redued. For a given head as Q inreases, there is risk of avitation, then ventilation is neessar. An other tpe of weir will be three dimensional beause of the lateral ontration from the sides, as well as in the vertial plane. Tpial examples are the ontrated retangular weir and the triangular weir.

24 Contrated Weirs A ontrated weir involves a 3-dimensional flow problem, beause of: the ontration from the sides as well as in the vertial plane. Tpial examples: ontrated retangular weirs triangular weir Suh weirs are ommonl used for flow measurement, normall in tanks large enough to be effetivel infinite, so that the ontration oeffiient, C,has its minimum value. In this ase, general equations are available relating disharge to head; but for small tanks weirs should be alibrated for eah partiular problem.

25 Contrated Retangular Weirs Franis found experimentall that the amount of lateral ontration at eah end of the ontrated weirs was equal to 0.0 H, provided that the length L of the weir was greater than 3H. On the basis of this result, it is ommonl aepted that the disharge Q is given b; Q = 3 C 3 / ( L 0.H) gh, C = 0. 6 L s L s W L > 3H, L s > 4H and W > 3H

26 The triangular weir (V-Noth) The triangular weir an be analzed in the same elementar wa as the suppressed retangular weir leading to the result: Q = 8 5 C α tan gh 5 / The most ommonl used value of the noth angle α is 90 o. For this ase C =0.585 somewhat less than for retangular weir. a

27 Although the value of C is near to 0.59 in general, it is affeted b visosit, surfae tension, and weir plate roughness; A omprehensive stud of triangular weir flow has been made b Lenz, He used man liquids in order to disover the effets of visosit and surfae tension on weir oeffiients, thus extending the utilit of the triangular weir as a reliable measuring deree. For α = 90 o, Lenz proposed that C 0.56 R W 0.70 Appliable to all liquids providing that the failing sheet of liquid does not ding to the weir plate and that H > 0.06 m, Re > 300, W > 300. As Re and Weber number derease, C will inrease.

28 V-noth weir A V-noth weir with stilling wells is shown in figure below

29 Compound Weirs Unusual situations ma require speial weirs. For example, a V-noth weir might easil handle the normal range of disharges at a struture; but oasionall, muh larger flows would require a retangular weir. A ompound weir, onsisting of a retangular noth with a V-noth ut into the enter of the rest, might be used in this situation. The ompound weir, as desribed, has a disadvantage. When the disharge begins to exeed the apait of the V-noth, thin sheets of water will begin to pass over the wide horizontal rests. This overflow auses a disontinuit in the disharge urve (Bergmann, 963). Therefore, the size and elevation of the V-noth should be seleted so that disharge measurements in the transition range will be those of minimum importane.

30 Determining disharges over ompound weirs has not been full investigated either in the laborator or in the field. However, an equation has been developed on the basis of limited laborator tests on a -ft-deep, 90-degree V-noth ut into retangular nothes, 4, and 6 ft wide to produe horizontal extensions of L=0, L=, and L=4 ft, respetivel (Bergmann, 963). The weirs were full ontrated, and heads up to.8 ft above the noth point were used. The equation is as follows: Q = 3.9h Lh where: Q = disharge in ft3/s h = head above the point of the V-noth in ft L = ombined length of the horizontal portions of the weir in ft h = head above the horizontal rest in ft When h is ft or less, the flow is onfined to onl the V-noth portion of the weir, and the standard V-noth weir equation is used. Further testing is needed to onfirm this equation before it is used for weirs beond the sizes for whih it was developed.

31 Compound weir with 90-degree noth and suppressed retangular rest used b U.S. Forest Servie.

32 North Fork 0-degree ompound V-noth weir and sampling bridge

33 The finish of the edge and upstream surfae of a weir is important, sine the roughness of the surfae or rounding of the edge tends to suppers the lateral omponents of flow, inrease C and hene inrease the disharge. H should be measured 3-4H upstream of weir.

34 3.4 Free Fall Atual pressure distribution 0.5 b EGL C b A 3-4 B In this situation, flow takes plae over a drop whih is sharp enough that the lowermost streamline separates from the hannel bed.

35 It is a speial ase of sharp-rested weir; W = 0 but it needs a speial treatment, beause of its use as a form of spillwa a means of flow measurement beause of unique relationship between the brink depth and Q. Clearl, an important feature of the flow is the strong departure from hdrostati pressure distribution whih exist near the brink, indued b strong vertial omponents of aeleration in the neighborhood. The form of this pressure distribution at the brink B will evidentl be somewhat as shown in the figure, with a mean pressure onsiderable less than hdrostati. At some short distane bak from the brink, the vertial aelerations will be small and the pressure will be hdrostati. This is experimentall verified that from A to B there is pronouned aeleration and redution in depth as shown in figure.

36 Consider a long hannel of two setions, one of mild upstream and one of steep slope downstream. If the upstream hannel is steep, the flow at A will be superritial and determined b the upstream onditions. If on the other hand the hannel slope is mild, horizontal or adverse, the flow will be subritial at A. Reall that the transition from mild (horizontal or adverse) slope to a steep slope. NDL CDL M A 0 S f > S 0 O S S f = S 0 S f < S 0 B

37 The flow will graduall hange from subritial at a great distane upstream to superritial at a great distane downstream passing through ritial state at same intermediate point. In the transition region upstream of (0), < 0 V > V0 S f > S0 Similarl, downstream of (0) > V < V S 0 0 f S0 If we onsider the point 0 to be a short urve joining two long slopes, < d dx ( F ) r = S0 Sf there must be some point on this urve at whih S f =S 0 and sine d dx 0 in this neighborhood, it follows that F r =. Therefore flow is ritial.

38 Imagine now that in this ase the steep slope is graduall made even steeper, until the lower streamline separates and the overfall ondition is reahed. The ritial setion annot disappear, it simpl retreats upstream into a region of hdrostati pressure i.e to A. The loal effets of the brink is onfined to the region AB; experiments shows this setion to quite short, of the order 3-4 times.

39 The Head of the free overfall The simplest ase is that of a retangular hannel with side walls ontinuing downstream on either side of the free jet, so that we have the effet of atmosphere onl on the upper & lower streamlines, not to the sides. This is a -D ase and it is onl in this form of the problem that man investigators worked on for a theoretial solution. Consider the setion C, a vertial setion thru the jet far enough downstream for the pressure throughout the jet to be atmospheri, and the horizontal veloit to be onstant. Assume that -the hannel bottom is horizontal and -no resistane

40 C b V x A 3-4 B a Appl momentum eq. b/w A and C, along the x diretion. γb + ρqv = ρqvx, Q = VA γ b + ρv A = ρva Vx

41 g b V b V Vx From Equation of Continuit: b Eq.() V b = V b " = os α V = V osα = Vx V x = V V x Substituting into Eq.() Dividing b + ρ V ρ + V = V ρ = V

42 Or F F F rearranging we get or F F F if the flow is ritial at setion A, then =, then we have 3 sets a lower, limit on the brink depth b : F sine there is some residual pressure at the brink, b must be greater than ; it follows that 3 b The experiment of Rouse showed that the brink setion has a depth of and

43 Rouse also pointed out that ombination of the weir Eq. q = 3 C g g + H with the ritial flow equation: V o 3 / V o g 3 / q = V = g w = 0 H =, V o = V q q C = = 3 C g V g g + 3 / V g 3/ 3 / 3 / 3/ 3/ 3 /.5 g Therefore b C = 0.75

44 There have been ma researh on this and the onlusion suggested b all these investigations is that a brink depth b =0.75 an safel be used for flow measurements in retangular hannels with a likel error of onl to or %. We have seen that the ontrol strutures are neessar in design of hdrauli strutures, beause the fix a relationship between the depth and the disharge. We have seen weirs and free-overfall as ontrol strutures. But in ertain hdrauli problems, the ma not be useful due to ertain disadvantages the have.

45 Two important disadvantages of weirs are:.the involve fairl large head loss when the available head ma be ver important to be onserved..the existene of a dead-water region behind the weir where silt an aumulate and greatl hange the head-disharge relationship of the weir. Also we have seen that the ritial flow establishes a fixed relationship between the depth and the disharge. But in order to appl this priniple, it is neessar to reate some devie or use some feature that sets up ritial flow at a known setion in its viinit. Then the measurement of the depth at this setion enables the disharge to be alulated. In ase of free-overfall, in whih, we have seen that the ritial setion retreats upstream to some ill-defined loation; at the brink the depth is a well-defined fration of ritial depth but the rapid variation in depth alls for preise loation of the depth-measuring devie.

46 Before onsidering an partiular devie, we first onsider ertain general priniples. We know that the ritial flow ours at the hange of hannel slope at point O, provided that the pressure distribution is hdrostati. CDL NDL M A 0 O S This will be true if the downstream slope, although steep, is not exessivel so-sa of the order 0.0. In this ase, setion O would an ideal ritialdepth meter, the depth here is definitel ritial, and is not hanging so rapidl that the slight errors in loating the depth-measuring devie would give rise to serious errors in estimates of the depth. B

47 However, the long downstream slope is not usuall available in pratie. Then we have make the downstream slope short but steep but this brings the inonvenienes of free-overfall, in whih we have seen that the ritial setion retreats upstream to some ill-defined region; at the brink the depth is a well-defined fration of ritial depth, but the rapid variation in depth alls for preise loation of the depthmeasuring devie.

48 3.5 Broad-Crested Weir A broad rested weir is more robust than a sharp rested weir for open hannel flow measurement. Hene broad rested weirs are widel used for flow measurement and regulation of water depth in rivers, anals and other natural open hannels. Broad rested weir flow rate alulations an be made with a rather simple equation if the weir height is great enough to ause ritial flow over the weir rest. If there is ritial flow, then the unit disharge is: q H 3 3/ q g H 3 3 A disharge oeffiient C q C d 3 g hene 3 g On the other hand if 3 H.705H d and 3/ 3/ H 3/.705C H head lossisnegleted ma be introdued to take into aount the head lossas d Then the total disharge is Q =.705 C d L H 3/, where Q is the open hannel flow rate in m 3 /s, C d is thw disharge oeffiient, L is the weir length (hannel width) in m, and H is the head over the weir in m, as shown in the diagram below.

49 General Broad Crested Weir Configuration A broad rested weir is normall a flat topped obstrution whih extends aross the entire hannel, as shown in the piture below. A longitidunal setion of a tpial broad rested weir is shown in the diagram below. The head over the weir, H, the height of the weir, P, the approah veloit and depth, V and, and the veloit and depth on the weir rest, V and, are all shown in the diagram. A ke feature of a properl operating broad rested weir is ritial flow over the weir rest. This is shown in the diagram b noting that V = V and =. The next setion disusses determination of the minimum weir height to ensure that ritial flow will our over the weir rest. =

50 Minimum Height Requirement for a Broad Crested Weir The water veloit will inrease whenever the flow in an open hannel passes over an obstrution like a broad rested weir, beause of the derease in ross-setional flow area. Within limits, the higher the obstrution, the greater the water veloit will be going over the obstrution. If the approah flow is subritial, then the flow over a broad rested weir will beome ritial at some partiular weir height. That height needed to give ritial flow over the weir rest an be alulated using some basi hdraulis equations, Using the terminolog in the broad rested weir figure above, along with B for the width of the hannel, the energ equation beomes: + V /g = + P + V /g From the definition of average veloit in an open hannel, assuming that the hannel is approximatel retangular: V = Q/ B and V = Q/ B From the fat that the speifi energ is a minimum for ritial flow onditions: = [Q /gb ] /3

51 If the hannel width, B, the flow rate through the hannel, Q, and the approah depth,, are known, the depth required to give ritial flow over the broad rested weir rest an be alulated with the above three equations. If a broad rested weir has the minimum height needed for ritial flow, then the simple equation, Q =.6 L H3/, an be used for flow rate alulations over that broad rested weir. This is illustrated with example alulations in the next setion. In order to ensure ritial flow over the weir for all flow onditions, the maximum antiipated flow rate through the hannel should be used to alulate the required weir height, P. Example Minimum Height Calulation Problem Statement: Consider an open hannel,5 m wide, with a maximum antiipated flow rate of 0 m 3 /s at a flow depth of.5 m. What is the minimum height needed for a broad rested weir to ensure ritial flow over the weir rest?

52 Solution: Using the equations from the previous setion, with Q = 0 m 3 /s, =.5 m, and B = 5 m: q E m Energ Eq. 3 / s / m.m E q g between Setions() and 3 q g 0.74m.59m (): E P E P 0.48 m.

53 3.7 Undeflow Gates Underflow gates are used for man purposes suh as: Controls at the rest of an overflow spillwa, Control at the outlet from a lake to a river or irrigation hannel. Undeflow gates Vertial Gates Radial Gates Drum Gates (Tainter Gates) Flow in underflow gates might be. free outflow or,. submerged (drowned) outflow.

54 U g Free Outflow -Vertial Gates: EGL U g 45 o () ()

55 E = E + q g = + q g Solvingfor q : q = g( ) ( + ) or q = ( g + ) = C d w g q C d w g where C d ( C C w )

56 Drown Outflow -Vertial Gates: Consider the longitudinal setion of flow shown below. The depth is produed b the gate and the depth 3 is produed b some downstream ontrol. If 3 is greater than the depth onjugate to (the depth needed to form a hdrauli jump with ), then the gate outlet must beome drowned as shown in figure. Q=? 3 w () ()

57 The effet is that the jet of water issuing from beneath the gate is overlaid b a mass of water whih, although strongl turbulent, has no net motion in an diretion. While there will be some energ loss between setions () and (), a muh greater proportion of the loss will our in the expanding flow between setions () and (3). We therefore assume as an approximation that all the loss ours between setions () and (3), that is: E E q g Note that the piezometri head term at () is equal to the total depth, not the jet depth. Between () and (3) we an use the momentum equation F =F 3 Noting that at (), the hdrostati thrust term is based on, not. q g F F 3 q g 3 q g 3

58 The Radial Gate (Tainter Gate) The Tainter gate is a tpe of radial arm floodgate used in dams and anal loks to ontrol water flow. It is named for Wisonsinstrutural engineer Jeremiah Burnham Tainter.

59 A side view of a Tainter gate resembles a slie of pie with the urved part of the piee faing the soure or upper pool of water and the tip pointing toward the destination or lower pool. The urved fae or skinplate of the gate takes the form of a wedge setion of linder. The straight sides of the pie shape, the trunnion arms, extend bak from eah end of the linder setion and meet at a trunnion whih serves as a pivot point when the gate rotates. Pressure fores ating on a submerged bod at perpendiular to the bod's surfae. The design of the Tainter gate results in ever pressure fore ating through the entre of the imaginar irle of whih the gate is a setion, so that all resulting pressure fore ats through the pivot point of the gate, making onstrution and design easier.

60 When a Tainter gate is losed, water bears on the onvex (upstream) side. When the gate is rotated, the rush of water passing under the gate helps to open and lose the gate. The rounded fae, long radial arms and trunnion bearings allow it to lose with less effort than a flat gate. Tainter gates are usuall ontrolled from above with a hain/gearbox/eletri motor assembl. A ritial fator in Tainter gate design is the amount of stress transferred from the skinplate through the radial arms and to the trunnion and the resulting frition enountered when raising or lowering the gate. Some older sstems have had to be modified to allow for fritional fores for whih the original design did not plan. The Tainter gate is used in water ontrol dams and loks worldwide. The Upper Mississippi River basin alone has 3 Tainter gates, and the Columbia River basin has 95. A Tainter gate is also used to divert the flow of water to San Fernando Power Plant on the Los Angeles Aquedut. The Tainter gate was invented and first implemented in Menomonie, Wisonsin.

61 r q a w Y =C w () We an write the Energ Equation between setions () and (): () E = E + q g = + q g Solvingfor q : q = g( ) ( + ) or q = ( g + ) = C d w g

62 q C d w g where C d ( C C w ) C 0.75 where q 90 o 0.36 This equation gives results whih are aurate to within provided that: q 5 perent,