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Estimating Flow Characteristics of Different Weir Types and Optimum Dimensions of Downstream Receiving Pool

Estimating Flow Characteristics of Different Weir Types and Optimum Dimensions of Downstream... J. Hydrol. Hydromech., 58, 2010, 4, 245­260 DOI: 10.2478/v10098-010-0023-z M. EMIN EMIROGLU Firat University, Civil Engineering Department, Hydraulic Division, 23119, Elazig-Turkey; Mailto: memiroglu@firat.edu.tr This paper presents the results of a laboratory study on the flow characteristics of sharp-crested weirs, broad-crested weirs, and labyrinth weirs. The variation of the maximum bubble penetration depth for different weir types is investigated depending on overfall jet expansion, discharge, and drop height. Moreover, most efficient depth, length and width of the downstream receiving pool in an open channel system are studied by considering the penetration depth, overfall jet expansion, jet trajectory and the bubble zone. The results show that overfall jet expansion at the labyrinth weirs is significantly wider than the rectangular sharp-crested weirs and the trapezoidal sharp-crested weir. It is demonstrated that the labyrinth weirs have the lowest values of bubble penetration depth among the weirs tested. Furthermore, it is found that the rectangular and the trapezoidal weirs are observed to have the highest bubble penetration depth among all weirs. Consequently, empirical equations are obtained for predicting the maximum penetration depth of bubbles, trajectory of free overfall nappe, jet expansion of free overfall nappe, and the length of the bubble zone. KEY WORDS: Weir, Penetration Depth, Jet Expansion, Jet Trajectory, Bubble, Air Entrainment, Aeration. M. Emin Emiroglu: URCOVANIE PRÚDOVÝCH CHARAKTERISTÍK ROZDIELNYCH TYPOV PRIEPADOV A OPTIMÁLNY NÁVRH VÝVARU. J. Hydrol. Hydromech., 58, 2010, 4; 28 lit., 13 obr., 4 tab. Príspevok prezentuje výsledky laboratórneho výskumu charakteristík prúdenia cez ostrohranné priepady, cez priepady so sirokou hranou a cez labyrintové priepady. Bola studovaná variácia maximálnej hbky prieniku vzduchových bublín pre rozdielne typy priepadov v závislosti od rozsírenia prúdu, prietoku a výsky priepadu. Okrem toho bola analyzovaná efektívna hbka, sírka a dzka vývaru v systéme otvorených kanálov s uvázením rozsírenia prúdu, prietoku a výsky priepadu a oblasti so vzduchovými bublinami. Z výsledkov vyplýva, ze rozsírenie prúdu pri labyrintových priepadoch je výrazne väcsie, ako pri pravouhlých a lichobezníkových ostrohranných priepadoch. Bolo ukázané, ze labyrintové priepady majú najmensiu hbku prieniku vzduchových bublín spomedzi vsetkých testovaných priepadov. Zistilo sa vsak, ze pravouhlé a lichobezníkové priepady majú najväcsie hbky prieniku vzduchových bublín spomedzi vsetkých testovaných priepadov. Tieto výsledky výskumu viedli k empirickým rovniciam, umozujúcim výpocet maximálnej hbky prieniku vzduchových bublín, trajektórií prepadového lúca, rozsírenia prúdu a dzky prevzdusnenej oblasti. KÚCOVÉ SLOVÁ: priepad, hbka prieniku vzduchových bublín, rozsírenie prúdu, trajektória prúdu, bublina, prevzdusovanie. 1. Introduction The overflow section shape cut with a sharp upstream corner into a thin plate is the weir notch, sometimes called the overflow section. If the notch plate is mounted on the supporting bulkhead such that the water does not contact or cling to the downstream weir plate or supporting bulkhead, but springs clear, the weir is a sharp-crested weir (USBR, 200. A broad-crested weir is a flatcrested structure with a crest length large compared to the flow thickness. The ratio of crest length to upstream head over crest must be typically greater than 3 (Henderson, 1966; Chanson, 2004). The flow control section can have different shapes, such as rectangular, triangular, trapezoidal or circular. Labyrinth weirs are those for which the weir crest is not straight in planform (Wormleaton and Soufiani, 245 1998). The increased crest length of labyrinth weirs gives them the clear advantage of reducing upstream levels for a particular discharge over low head ranges (Wormleaton and Tsang, 2000). The flow over weirs would be classified as a free jet, as shown in Figs. 1a­c) and Figs. 2a, b). Weirs have a large number of uses in engineering applications. Listing the more usual, these are: 1. to measure the flow of water in small stream and constructed channels; 2. to regulate the flow of water in open channel and reservoir; 3. to take the water for irrigation systems; 4. to divert flow during high flow conditions in sewer systems; 5. to increase the dissolved oxygen content of the water by mean of aeration the water; 6. to reduce tastes and odors caused by dissolved gases in the water by mean of aeration the water, such as hydrogen sulphide, which are then released; and also to oxidize and remove organic matter; 7. to decrease the carbon dioxide content of a water by mean of aeration the water and thereby reduce its corrosiveness and raise its pH value; 8. to remove certain volatile organic compounds by mean of aeration the water. An overflow jet that plunges into a channel after passing through a gas phase entrains a substantial amount of air into the receiving pool, and forms a submerged two-phase region with a considerable interfacial area. Almost all the mass transfer substantially takes place in this two-phase region. Therefore, maximum penetration depth of the bubble swarm is one of the most important parameters that characterize the performance of weir aeration system and the design of downstream receiving pool. Moreover, estimating flow characteristics of weirs would be useful for scour depth in the downstream receiving channel because flow characteristics of the weirs are different from each other. Apted and Novak (1973), Avery and Novak (1978), and Nakasone (1987) reported that tailwater depth was an important weir operation parameter, influencing mass transfer. Tailwater depth is also an important parameter for energy dissipation and scouring cases. Grindrod (1962) and Albrecht (1968) pointed out that the aeration efficiency would increase with increasing tailwater depth. However, there should be a limit because the penetrating air bubbles will not go to infinite depths. Actually, for each combination of discharge, drop height, and weir type, there would be an approximate maximum depth to which the bubbles would penetrate, thus limiting the aeration efficiency. Studies by Nakasone (1975, 1976, and 1987) contributed substantially to the experimental investigations of the weir aeration. Nakasone (1987) did not investigate 246 the case whether or not air bubbles reached the floor of the stilling basin. For tailwater depth, Nakasone suggested to adapt as 2/3 of drop height. However, the bubble penetration depth and the overfall jet expansion would be different for each weir type and shape and it would not be true to give the bubble penetration depth as a function of only drop height. Moreover, the bubble penetration depth differs from each weir type. Kobus and Koschitzky (199 presented the empirical correlation for penetration depth 0.39 0.24 D p = 0.00433Re j F j , where Rej ­ jet Reynolds number, Re j = qj and Fj ­ the jet Froude g h3 number, F j = 2q 2 j 1/4 . Nakasone (1987) advised that the receiving pool depth Dp ceases to affect aeration efficiency when it exceeds two-thirds of the drop height h (i.e., D p 2 / 3 h ), and Avery and Novak (1978) gave the relationship as D p 7.5 h 0.58 F-0.53 , where h ­ drop height and Fj j ­ the jet Froude number. Kumagai et al. (1993), Clanet and Lasheras (1997), and Ito et al. (2000) studied depth of penetration of bubbles entrained by a plunging water jet. They presented correlations for predicting the bubble penetration depth of plunging water jet as a function of the operational conditions and nozzle geometry. Baylar and Emiroglu (2002), and Emiroglu and Baylar (2003 and 2005) studied air entrainment rate of the broad crested, sharp crested weirs and labyrinth weirs and they stated that weir shape is very sensitive for flow characteristics. The behavior of water flowing over a weir varies significantly with weir shape and type. It is therefore highly desirable to investigate flow characteristics for different weir types and their shapes. This paper presents an experimental investigation of the bubble penetration depth Dp, trajectory of free overfall nappe from the weirs, the length of the bubble zone, and the overfall jet expansion Je of the sharp crested weirs, broad crested weirs, and labyrinth weirs. In addition, the paper studies most efficient length, width and depth of the downstream receiving pool for especially weir aerators. 2. Dimensional analysis The functional relationship involving the maximum bubble penetration depth and influencing variables can be written as L Bw s p Rectangular weir L Bw s L Bw s b' Triangular weir Trapezoidal weir a) Flow direction Flow direction Bw p L Bw s p L Bw s b' p L b) Flow A Flow d Plan view d A-A cross-section c) Fig. 1. Weirs; a) Sharp-crested weir, b) Broad-crested weir, c) Labyrinth weir. Obr. 1. Priepady; a) ostrohranné; b) so sirokou korunou; c) labyrintové. Flow Direction Flow Direction Je Je (a) Straight Weir (b) Triangular Labyrinth Weir Fig. 2. Sketch of flow patterns of free jet over weirs; a) Rectangular sharp-crested weir; b) Labyrinth weir. Obr. 2. Nácrt voného prúdu nad priepadom; a) pravouhlý, ostrohranný, b) labyrintový priepad. H , Bw , q j , , ,g, , D p = f1 , , Pon , P, z , br , b , bd , bu ( H = h + H w . (2) in which Dp ­ maximum bubble penetration depth, H ­ energy difference between the flow over the weir crest and the receiving pool, Bw ­ water surface width over the crest of the weir, qj ­ discharge through the per unit water surface width at the crest of the weir (qj = Q/Bw; Q ­ total discharge through the weir), ­ density of fluid (water), = ­ a, a ­ density of gas (air), g ­ gravity acceleration, ­ visity of the fluid (water), ­ surface tension of the fluid (water), Pon ­ ambient pressure on the outer nappe of the jet, Pun ­ ambient pressure on the lower nappe of the jet (under the nappe), P=(Pon ­ ­ Pun), z ­ depth of flow in the receiving pool, br , b , bu , bd ­ the distances of the right side, left side, upstream and downstream walls of the receiving pool to the center point of the jet where it impacts the water surface in the receiving pool, respectively as shown in Fig. 3a­c). 2 Note that H = h + H w + vo / 2 g , in which h ­ vertical height of the weir crest with respect to the water surface level in the receiving pool (i.e., drop height), Hw ­ the vertical difference between the water surface in the upstream pool and the weir crest, vo ­ average flow velocity in the upstream pool. It is well known that if Hw is measured at a distance sufficiently for upstream of the weir (i.e., It should be mentioned here that the air entrainment in a jet highly depends on the shear hence velocity, between the surface of the free jet (nappe) and air. Also, the fluid and flow conditions about the bubble cloud within the receiving pool are very complex. Due to these reasons, the average flow velocity may not be that good influencing factor to be chosen the dimensional analysis. The average flow velocity may cause under estimation of the quantities. This is why, the discharge per unit water surface width (qj = Q/Bw) is preferred to the average flow velocity as an influencing variable in the dimensional analysis. The dimensional analysis in Eq. ( yields, z br F j , Re j ,W , Eu , B , B , Dp w w = f1 , b bd bu P H , , , Bw Bw Bw Pon g h3 in which, Fj ­ jet Froude number = 2q 2 j Rej ­ jet Reynolds number = W ­ jet Weber number qj Bw . qj (3) 1/4 ( = / ), q2 j Bw and 2 more than 4Hw), vo / 2g is generally negligible. Therefore one can take Eu = jet Euler number = Jet Upstream pool Weir bu bd Receiving b r pool (a) Plan 2 vo / 2g Hw Weir crest y o h Nappe jet P on P un A H = h + H w + v o / 2g Upstream pool Receiving pool bu Dp Lb bd z (b) Cross-Section Bw Flow at the crest of the weir Weir crest (c) Trapezoidal Weir, Cross-Section Fig. 3. Receiving pool; a) Plan, b) Cross-section, c) Trapezoidal weir, Cross-section. Obr. 3. Vývar; a) nákres; b) priecny rez, c) lichobezníkový priepad, priecny rez. In this study, since the jet nappe is freely ventilated and open into atmosphere and no clinging of the nappe onto the weirs, P/Pon has no physical importance and keeping in mind the scale effects, the effect of Weber and Euler number are negligible in air entrainment problems (Kells and Smith, 199. Thus, under these reasons Eq. ( becomes, Lb z br b bd bu , , , , = f 2 F j , Re j , , Bw Bw Bw Bw Bw H (6) (7) (8) (9) Lb = f 2 F j , Re j , H bu = f3 F j , Re j , H Je = f 4 F j , Re j . H z br b bd bu = f1 F j , Re j , , , , , , H Bw Bw Bw Bw Bw Dp Dp H = f1 F j , Re j . (4) (5) If one conducts the similar dimensional analysis for Lb ­ length of the bubble zone; bu ­ horizontal distance of the jet impact point at the water surface in the receiving pool and to the upstream back wall, one gets Note that since Dp, Lb, bu and Je are just before the impact of the jet at the water surface they are independent of z , br , b , bd and bu as shown in Eqs. (5), (7), (8) and (9). 3. Air Entrainment mechanisms-classification and break-up length of the jets Lbr Bi Fi 0.85 (1.07 Tu Fi2 )0.82 (10) Tsang (1987) described four basic air entrainment mechanisms with reference to a free falling jet from a weir. Tsang's four air entrainment mechanisms are smooth, rough, oscillating, and disintegrated. For small drop heights, a water jet with a relatively smooth surface issues from the weir. The major source of air supply is visualized as a thin layer surrounding the jet and carried into the water upon impact, and therefore the air entrainment capacity is limited. The water surface in the receiving pool is relatively undisturbed. As the drop height increases, the surface of the jet becomes roughened. The air supply can be considered as coming largely from small air pockets entrapped between the jet surface roughness and the receiving water. At impact, the jet produces ripples on the pool surface. Compared with the smooth jet under similar conditions, this mechanism results in shallower bubble penetration but increased entrainment rate, because the bubbles are more densely packed in the biphasic zone. As the drop height is increased further, the jet begins to oscillate during the fall. The primary air source originates from large air pockets entrapped between the undulating jet and the pool surface. The pool surface is considerably agitated, and air may also be entrained by surface roller action and splashing. Large air pockets are transported from the surface into the water depth and broken down due to turbulence. With an even larger drop height, the jet breaks up into discrete droplets. The pool surface is intensely agitated, and air is entrained by the action of surface rollers and by the engulfing of air pockets as jet fragments hit the pool surface. The bubbles are generally only transported to relatively shallow depths. Disintegrated jets have the advantage over solid jets of greater surface area; however, air entrainment rate QA and bubble penetration are significantly reduced because of energy loss to the surrounding atmosphere during the fall (Wormleaton and Tsang, 2000). The break-up length Lbr at which the jet begins to disintegrate can be determined using Horeni's 0 Equation for rectangular jets Lbr = 31.19 q w.319 , where Lbr is the break-up length [cm] and qw is the specific flow over the weir in cm3 s-1 cm-1. On the other hand, the equation of the break-up length for rectangular jet, established by Castillo (2007) is where Bi and Fi are the jet thickness and Froude number at issuance conditions, respectively. Tu is the initial turbulence intensity when the flow passes on spillway ( 0 Tu 3% ). 4. Experimental equipment and procedure Fig. 4 shows the laboratory arrangement for the current study. The experiments have been conducted in a rectangular flume with glass sides and well painted steel bed. The flume is 3.40 m in length, 0.6 m in width, and 0.6 m in depth and the bed has an average Manning's roughness coefficient n of 0.0099. A digital point gauge has been mounted on rails along the channel allowed the upstream water surface profile, then the nappe profile, and Hw have been measured. Furthermore, a digital point gauge has been mounted on the received pool to measure h. The water in the experimental channel has been re-circulated by a pump. The water jet from the test weir has plunged into a downstream water pool, whose height is adjusted using a release valve. The plan-view dimensions of the downstream water pool made of glasswall are 1.75 m in length, 1.50 m in width, and 2 m in height. The experiments reported here are carried out with rectangular sharp-crested weirs (Bw = 0.10, 0.20 and 0.30 m), trapezoidal sharp-crested weir (Cipolletti weir with 1H/4V, b' = 0.20 m and, Lw = 0.40 m), triangular sharp-crested weirs ( = 30°, 60°, 90° and 120°), triangular labyrinth weirs ( = 45°, 90° and 135°) all having the same total sill length of 0.30 m, rectangular broad-crested weirs (Bw = 0.10, 0.20 and 0.30 m and, Lw = 0.40 m), trapezoidal broad-crested weir (1H/4V, b' = 0.20 and, Lw = 0.40 m), triangular broad-crested weirs ( = 30°, 60°, 90° and 120° and, Lw = 0.40 m), as can be seen in Figs. 1a­c). The difference between crest and top in all weirs s is 0.25 m and crest height p is 0.35 m. Each weir configuration has been tested under flow rates Q varying from 1.0 to 5.0 L s-1 in 1 L s-1 steps. The drop height h, defined as the difference between the water levels upstream and downstream of the weir, is varied between 0.20 to 1.00 m in 0.20 m steps. Penetration depth, Dp, of the bubbles produced by the jet, which is defined as the vertical distance from the water surface to the lower end of the submerged biphasic region in the water. Penetration depth is measured by a movable scale from the receiving pool wall. This scale is capable of reading to the nearest 1.0 mm. All measurements are repeated three times and then averaged, especially for penetration depth measurements. In all of the experiments, the tailwater depth is selected greater than the maximum bubble penetration depth so that bubbles cannot reach the floor of the downstream water pool. The expansion measured at impact point on the liquid pool of free overfall jet is named as jet expansion, Je. And, it is measured by a scale which is capable of reading to the nearest 0.5 mm. The length, Lb, shown in Fig. 3a) is named as the length of the bubble zone. Also, it is measured by a scale which is capable of reading to the nearest 1.0 mm. A total of 675 tests are performed in this study. Grid Electromagnetic flowmeter Flowrate adjustment Point Gauge Test weir Overfall jet Point Gauge Water channel Air bubbles Downstream Receiving Pool Tailwater depth control Water feed line Pump Fig. 4. Experimental apparatus. Obr. 4. Aparatúra. 5. Evaluation of experimental results For different weir types, the bubble penetration depth Dp, the overfall jet expansion Je, the length of the bubble zone Lb, and the horizontal distance to overfall jet impact point from the weir bu are determined for a range of discharges and drop heights. The following sections present and discuss the experimental results. To examine the effect of weir types on the overfall jet expansion, Je values are plotted against drop height. The experimental data are presented in Fig. 5. The overfall jet expansion of the weirs is highly different from each other. The experimental data indicate that the overfall jet expansion increases with an increasing water discharge. As shown in Fig. 5, the labyrinth weirs have the highest values of overfall jet expansion among weirs tested. At labyrinth weirs, the overfall jet expansion increases as the included angle in the triangular labyrinth weirs decreases. Overfall jet expansion at the labyrinth weirs is significantly more than the trapezoidal sharp-crested weir, the rectangular sharp- crested weirs, and circular sharp-crested weir. This is due to the fact that the labyrinth weirs have more crest length than the other weirs tested. The overfall jet expansion of the labyrinth weirs show an increasing tendency when drop height increases. But, the tendency of overfall jet expansion for the trapezoidal sharp-crested weir, the rectangular sharpcrested weirs slightly show a decrease as drop height increases, as illustrated in Fig. 5. The results indicate that the trapezoidal sharp-crested weir and the rectangular sharp-crested weirs have the lowest overfall jet expansion values. At the rectangular sharp-crested weirs, the overfall jet expansion increases with an increasing crest width Bw. It can be seen from the Fig. 5 that overfall jet expansion for the rectangular broad-crested weirs is more than the rectangular sharp-crested weirs. For triangular sharp-crested weirs, the overfall jet expansion increases as the angle of triangular weirs decreases. Fig. 5 shows that overfall jet expansion for the triangular broad-crested weirs is significantly less than the triangular sharp-crested weirs. 251 Q=5 L/s Rectangular sharp-crested weir (b=10 cm) Rectangular sharp-crested weir (b=20 cm) Rectangular sharp-crested weir (b=30 cm) Trapezoidal sharp-crested weir (Cipolletti) 30 Triangular sharp-crested weir 60 Triangular sharp-crested weir o o Water jet expansion (m) 90 Triangular sharp-crested weir 120 Triangular sharp-crested weir Labyrinth weir ( =45 ) o o o Labyrinth weir ( =90 ) Labyrinth weir ( =135 ) Rectangular broad-crested weir (b=10 cm) Rectangular broad-crested weir (b=20 cm) Rectangular broad-crested weir (b=30 cm) Trapezoidal broad-crested weir (Cipolletti) 30 Triangular broad-crested weir 60 Triangular broad-crested weir o o 90 Triangular broad-crested weir 120 Triangular broad-crested weir 0.00 0.00 0.20 0.40 0.60 Drop height (m) 0.80 1.00 1.20 Fig. 5. Variation in overfall jet expansion of different weir types with drop height for Q = 5 L s-1. Obr. 5. Variácie rozsírenia prepadového prúdu rozdielnych typov priepadov s výskou Q = 5 L s-1. Q=5 L/s Rectangular sharp-crested weir (b=10 cm) Rectangular sharp-crested weir (b=20 cm) Rectangular sharp-crested weir (b=30 cm) Trapezoidal sharp-crested weir (Cipolletti) Bubble penetration depth (m) 30 Triangular sharp-crested weir 60 Triangular sharp-crested weir 90 Triangular sharp-crested weir 120 Triangular sharp-crested weir Labyrinth weir ( =45 ) o o o o Labyrinth weir ( =90 ) Labyrinth weir ( =135 ) Rectangular broad-crested weir (b=10 cm) Rectangular broad-crested weir (b=20 cm) Rectangular broad-crested weir (b=30 cm) Trapezoidal broad-crested weir (Cipolletti) 30 Triangular broad-crested weir 60 Triangular broad-crested weir 90 Triangular broad-crested weir 120 Triangular broad-crested weir o o o o 0.25 0.00 0.20 0.40 0.60 Drop height (m) 0.80 1.00 1.20 Fig. 6. Variation in penetration depth of different weir types with drop height for Q = 5 L s-1. Obr. 6. Variácie hbky prieniku vzduchových bublín rozdielnych typov priepadov s výskou Q = 5 L s-1. The residence time of entrained air bubbles in a water body directly affects the oxygen mass transfer. This residence time is related to the bubble flow path and hence Dp into the receiving water. It is observed from experimental data that bubble penetration depth is closely related to the jet shape and expansion that is unique to each weir type. The bubble penetration depth of the weirs is also highly different from each other. It would not be true to give the bubble penetration depth as a function of only drop height. Fig. 6 shows variation in Dp of nineteen different weir types with drop height while the change in discharge is constant. Experiments with all nineteen weirs indicate that the weir type and shape, the water discharge, and the drop height are important factors influencing Dp. It is found from these results that Dp tends to increase with water discharge Q. It is clear from the results in Fig. 6 that Dp generally decreases with increasing drop height. The experimental results show that Dp decreases with increasing overfall jet expansion Je. The decrease in Dp with increasing in Je is might be due mainly to the increased buoyant forces of the entrained bubbles as a result of the increase in the air entrainment rate, which may have caused by the increase of the jet surface roughness. As is seen from Fig. 6, the labyrinth weirs have the lowest values of bubble penetration depth among the weirs tested. It can be seen from the experimental results that Dp values for triangular labyrinth weir with included angle of 45° are observed the lowest at all discharge, from 1 to 5 L s-1. The rectangular and trapezoidal sharp-crested weirs are observed to have the highest bubble penetration depth among all weirs tested (Fig. 6). Especially, the trapezoidal sharp-crested weir has the highest bubble penetration depth, Dp, values. Moreover, at both triangular sharp-crested weirs and triangular broad-crested weirs, it is demonstrated that Dp increased as the angle of in triangular weirs increased. It may be concluded from Fig. 5 and Fig. 6 that Dp decreases while the overfall jet expansion increases. Thus, the scour depth in the downstream channel would decrease with increasing the overfall jet expansion because the bubble penetration depth would decrease. Moreover, overfall jet's momentum is sensitive for aeration. It can be seen from these two figures that if drop height is highly increased, then the overfall jet is disintegrated and the bubble penetration depth is much decreased. At this state, the aeration performance of the weirs would be extremely little. Fig. 7 shows the relationship between Dp and h with Kobus and Koschitzky's results; Avery and Novak's results and present study data when the weir type is rectangular broad crested weir. Good agreement between the measured values and the values computed from the Kobus and Koschitzky's, and Avery and Novak's equations are obtained. Scimeni (1937) expressed the shape of the nappe in co-ordinates x and y, measured from an origin at the highest point, for a unit value of H as y = Kx n H 1- n , where K = 0.5 and n = 1.85. These values are taken as K = 0.47 and n = 1.80 for Creager profile. Thus, the lower jet trajectory can define with y = 0.50 x1.85 .H -0.85 and y = 0.47 x1.80 .H -0.80 (Scimemi, 1930; Scimemi, 1937; Creager et al., 1945). Where x, y coordinates and H is equal to Hw + vo2/2g. Nowadays most crests have an ogee shape, (e.g. Scimemi profile, Creager profile). 1.4 1.2 1.0 D (m) p 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 h (m) 0.6 0.8 1.0 Present study Kobus and Koschitzky (199 Avery and Novak (1978) Fig. 7. Comparison of the Dp values of rectangular broad crested weir with the equations of Kobus and Koschitzky (199, and Avery and Novak (1978). Obr. 7. Porovnanie hodnôt Dp pravouhlého priepadu so sirokou hranou s výsledkami poda rovníc Kobusa a Koschitzkého (199, a Avery, Novaka (1978). Fig. 8 shows the jet trajectories for different weir types tested. Equations for the jet trajectory of weirs other than rectangular sharp-crested weir do not exist in literature as far as I know. But, free jet trajectory equations for an open channel, a flip bucket, an orifice through a dam and an overtopping dam crest exist in literature (Davis et al., 1999; Dey, 2002; Wahl et al., 2008). Therefore, present study data are compared with Scimemi and Creager profiles (Fig. 9). Good agreement between the present study bu values and the values computed from the Scimemi and Creager equations is obtained. The nature of the nappe affects significantly the aeration mechanism due to bubble behavior. There253 fore, the jet break-up was considered in this experimental study. The break-up length Lbr was calculated using Horeni's and Castillo's equations and, the Lbr values for the experimental data were obtained between 120 cm to 250 cm. Castillo stated that the Horeni's formula gives correct values for flows smaller than qw = 0.25 m2 s-1. The jet breakup was not observed in the study because maximum drop height was taken as 100 cm. This situation was confirmed with Horeni's and Castillo's equations. Empirical correlations predicting the overfall jet expansion Je, the bubble penetration depth Dp, the distance between the channel intake and the overfall jet impact point bu, and the length of the bubble zone Lb are developed for the weirs. The resulting correlations are given in Eqs. (1­(14) and the values of constants in these correlations are shown in Tabs. 1­4. a3 Dp a a = A. ln B.Re j1 .F j 2 , H a3 Lb a a = A. ln B.Re j1 .F j 2 , H ( ( ) ) (1 (12) bu (m) a2 Je a = A. ln B.Re j1 .F j H a 3 , (13) a3 bu a a = A. ln B.Re j1 .F j 2 , H (14) 0.80 h (m) Rectangular sharp-crested Trapezoidal sharp-crested o 30 Triangular sharp-crested o Labyrinth weir =135 Rectangular broad-crested Trapezoidal broad-crested o 120 Triangular broad-crested where Je ­ the overfall jet expansion [m], Dp ­ the penetration depth [m], bu ­ the distance between the channel intake and the overfall jet impact point [m], Lb ­ the length of the bubble zone [m], H ­ the drop height [m], Q ­ the water discharge [m3 s-1], Bw ­ the crest width in rectangular sharp and broadcrested weirs [m], ­ the angle in triangular sharp and broad-crested weirs [°], ­ the included angle in triangular labyrinth weirs [°]. Fig. 8. Jet trajectories for the different weir types. Obr. 8. Trajektórie prúdu rozdielnych typov priepadov. T a b l e 1. Values of constants (Dp) in Eq. (10). T a b u k a 1. Hodnoty konstánt (Dp) v rov. (10). Weir Type Sharp-crested triangular Sharp-crested rectangular2) Sharp-crested trapezoidal3) Broad-crested triangular4) Broad-crested rectangular5) Broad-crested trapezoidal6) Triangular labyrinth7) B 2.353 68.90 15.805 0.72 14.649 exp(7.15 0.945 a1 0.009 ­0.264 ­0.163 0.100 ­0.140 ­0.406 0.064 a2 0.032 ­0.583 ­0.392 0.108 ­0.311 ­1.000 0.241 a3 ­48.10 1.523 1.629 ­5.046 5.933 1.871 ­5.717 F-0.612 0.706 j n F1.977 j 0.724 Estimating flow characteristics of different weir types and optimum dimensions of downstream receiving pool T a b l e 2. Values of constants (Lb) in Eq. (1. T a b u k a 2. Hodnoty konstánt (Lb) v rov. (1. Weir Type Sharp-crested triangular Sharp-crested rectangular2) Sharp-crested trapezoidal3) Broad-crested triangular4) Broad-crested rectangular5) Broad-crested trapezoidal6) Triangular labyrinth7) A Re0.394 0.414 j Re0.304 j n Re0.165 j B 1.979 1.580 a1 ­0.035 ­0.030 0.059 ­0.001 ­0.075 0.046 ­0.033 a2 ­0.097 ­0.061 0.243 ­0.258 ­0.216 0.193 0.466 a3 2.233 1.000 ­6.248 5.719 5.866 ­6.067 ­2.580 ( ( ) ) 0.449 n Re0.028 j 1.079 T a b l e 3. Values of constants (Je) in Eq. (12). T a b u k a 3. Hodnoty konstánt (Je) v rov. (12). Weir Type Sharp-crested triangular Sharp-crested rectangular2) Sharp-crested trapezoidal3) Broad-crested triangular4) Broad-crested rectangular5) Broad-crested trapezoidal6) Triangular labyrinth7) B 2.223 2.101 a1 0.039 0.028 0.043 ­0.004 0.585 ­0.118 0.018 a2 ­0.418 0.086 0.107 ­0.307 ­1.148 ­0.480 0.133 a3 1.000 ­28.998 ­19.246 3.049 23.904 2.486 ­8.208 n Re0.115 j 0.394 ( n Re j / F j )-22.097 1.445 The calculated penetration depth Dpo is compared with the observed penetration depth Dpc to yield the average percent error as 100 N D po - D pc , N i =1 D po (15) in which N is the number of data points. The average percent error is a function of the constant in the penetration depth equation. Eq. (15) can be also written for Lb, Je and bu. The average percent error values belong to Eqs. (1­(14) are given in Tabs. (1­4). Moreover, the measured bubble penetration depths are compared with those predicted with Eq. (1. Good agreement between the measured bubble penetration depth and the values computed from the predictive equations is obtained. The values of Lb measured are also compared with those predicted with Eq. (12). Good agreement between the measured Lb and the values computed from the predictive equations is obtained. The measured Je are compared with those predicted with Eq. (13). Good agreement between the measured Je and the values computed from the predictive equations is obtained. The values of bu measured are compared with those predicted with Eq. (14). Excellent agreement between the measured bu and the values computed from the predictive equations is obtained. Further 255 T a b l e 4. Values of constants (bu) in Eq. (13). T a b u k a 4. Hodnoty konstánt (Je) v rov. (13). Weir Type Sharp-crested triangular Sharp-crested rectangular2) Sharp-crested trapezoidal3) Broad-crested triangular4) Broad-crested rectangular5) Broad-crested trapezoidal6) Triangular labyrinth7) B 5.371 3.961 3.642 2.288 4.004 5.781 0.930 a1 ­0.044 ­0.03 ­0.018 ­0.006 ­0.051 0.082 ­0.042 a2 0.164 0.12 0.093 ­0.190 0.146 ­0.343 1.388 a3 ­2.177 ­4.600 ­5.787 0.867 ­2.652 2.112 ­1.035 1.054 1.007 confidence in the correlations is seen in Figs. 10, 11, 12, and 13. The following equations can be written for the determination of the minimum width, length and depth of the downstream receiving pool. Wd J e , Ld bu + Lb , (16) (17) (18) In this study, tested discharges are smaller than that of some prototype applications. But, tested drop heights are similar to a lot of prototype scale. Scaling of bubble penetration depth data to prototype size is virtually impossible, largely due to the relative invariance of bubble size. Clearly, tests at higher discharges should be carried out to see if this trend extrapolates. zd D p , +10% bu (m) Dp calculated Present study Scimemi profile Creager profile h (m) Fig. 9. Comparison of the bu values of rectangular broad crested weir with the equations of Scimemi and Creager. Obr. 9. Porovnanie hodnôt bu pravouhlého priepadu so sirokou korunou a výsledkami rovníc Scimemi a Creagera. Dp measured where Wd ­ minimum width of downstream receiving pool [m], Ld ­ minimum length of downstream receiving pool [m], and zd ­ minimum tailwater depth [m]. 256 Fig. 10. Comparison of observed Dp values with those calculated from Eq. (1. Obr. 10. Porovnanie pozorovaných hodnôt Dp s vypocítanými poda rov. (1. -10% +12% -12% trajectory, overfall jet expansion, the maximum penetration depth, and the length of the bubble zone for different weir types. Downstream receiving pool using equations presented in this study have properly been designed as the optimum case. The concluding remarks obtained are as follows. Lb calculated +5% -5% bu calculated 0.00 0.00 0.20 0.40 0.60 0.80 1.00 Lb measured Fig. 11. Comparison of observed Lb values with those calculated from Eq. (12). Obr. 11. Porovnanie pozorovaných hodnôt Lb s vypocítanými poda rov. (12). bu measured +10% -10% Fig. 13. Comparison of observed bu values with those calculated from Eq. (14). Obr. 13. Porovnanie pozorovaných hodnôt bu s vypocítanými poda rov. (14). Je measured Fig. 12. Comparison of observed Je values with those calculated from Eq. (13). Obr. 12. Porovnanie pozorovaných hodnôt Je s vypocítanými poda rov. (13). 6. Conclusions The present experimental investigation estimates the free jet trajectory, the overfall jet expansion, and bubble penetration depth for different weir types and so dimensions of receiving pool. The correlations are developed for predicting the jet 1. The values of overfall jet expansion Je, the bubble penetration depth Dp, jet trajectory, and the length of the bubble zone Lb are different for each weir type and shape. 2. The overfall jet expansion Je increases with an increasing water discharge. The labyrinth weirs have the highest values of Je. At the labyrinth weirs, Je increases as the included angle in the triangular labyrinth weirs decreases. The overfall jet expansion Je at the labyrinth weirs is significantly wider than the trapezoidal sharp-crested weir and the rectangular sharp-crested weirs. Je of the labyrinth weirs has shown an increasing tendency when drop height increased. The results indicate that the trapezoidal sharp-crested weir and the rectangular sharp-crested weirs have the lowest Je. At the rectangular sharp-crested weirs, Je increases with an increasing crest width Bw. Furthermore, Je for the rectangular broad-crested weirs is wider than the rectangular sharp-crested weirs. For triangular sharp-crested weirs, Je increases as the angle of triangular weirs decreases. The overfall jet expansion Je for the tri257 Je calculated angular broad-crested weirs is significantly smaller than the triangular sharp-crested weirs. 3. Dp is closely related to the jet shape and expansion that is different for each weir type. The bubble penetration depth Dp increases with increasing the water discharge. At the same time, the bubble penetration depth Dp decreases with increasing drop height. The bubble penetration depth Dp decreased with increasing Je. The labyrinth weirs have the lowest values of Dp among the weirs tested. The bubble penetration depth Dp values for triangular labyrinth weir with included angle of 45° are observed as the lowest one. The rectangular and trapezoidal sharp-crested weirs are observed to have the highest Dp among all weirs tested. Especially, the trapezoidal sharpcrested weir has the highest Dp values. At both triangular sharp-crested weirs and triangular broad-crested weirs, Dp increases as the angle of in triangular weirs increases. 4. The labyrinth weirs and triangular sharp-crested weirs with small angle are not appropriate due to enormous overfall jet expansion, if the downstream receiving pool width is very small. List of symbols Bw ­ water surface width over the crest of the weir [m], bd ­ distance of the downstream wall of the receiving pool to the center point of the jet where it impacts the water surface in the receiving pool [m], b ­ distance of the left side wall of the receiving pool to the center point of the jet where it impacts the water surface in the receiving pool [m], bL ­ half-crest length in triangular labyrinth weir [m], br ­ distance of the right side wall of the receiving pool to the center point of the jet where it impacts the water surface in the receiving pool [m], bu ­ distance of the upstream wall of the receiving pool to the center point of the jet where it impacts the water surface in the receiving pool [m], d ­ breadth in triangular labyrinth weir [m], Dp ­ bubble penetration depth [m], Dpc ­ calculated penetration depth [m], Dpo ­ observed penetration depth [m], Eu ­ jet Euler number, Fi ­ Froude number at the issuance, Fj ­ jet Froude number, g ­ acceleration due to gravity [m s-2], h ­ drop height [m], Hw ­ the vertical difference between the water surface in the upstream pool and the weir crest [m], Je ­ overfall jet expansion [m], L ­ the experimental channel width [m], Lb ­ length of the bubble zone [m], Ld ­ minimum length of the downstream pool [m], Lbr ­ jet break-up length [m], LW ­ length in triangular broad-crested weir [m], Pon ­ ambient pressure on the outer nappe of the jet [N m-2], Pun ­ ambient pressure on the lower nappe of the jet [N m-2], Rej ­ jet Reynolds number, s ­ difference between crest and top in all weirs [m], Tu ­ initial turbulence intensity, Q ­ water discharge [m3 s-3], QA ­ air entrainment rate [m3 s-3], qj ­ discharge through the per unit water surface width at the crest of the weir [m2 s-1], vo ­ average flow velocity in the upstream pool [m s-1], W ­ jet Weber number, w ­ width in triangular labyrinth weir [m], Wd ­ minimum width of the downstream receiving pool [m], z ­ tailwater depth [m], zd ­ minimum depth of the downstream pool [m], ­ angle in triangular sharp and broad-crested weir [°], ­ average percent error, ­ included angle in triangular labyrinth weir [°]; ­ visity of the fluid (water) [kg s-1 m-1], ­ density of fluid (water) [kg m-3], a ­ density of gas (air) [kg m-3], ­ surface tension of the fluid (water) [N m-1]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Hydrology and Hydromechanics de Gruyter

Estimating Flow Characteristics of Different Weir Types and Optimum Dimensions of Downstream Receiving Pool

Journal of Hydrology and Hydromechanics , Volume 58 (4) – Dec 1, 2010

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10.2478/v10098-010-0023-z
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Abstract

J. Hydrol. Hydromech., 58, 2010, 4, 245­260 DOI: 10.2478/v10098-010-0023-z M. EMIN EMIROGLU Firat University, Civil Engineering Department, Hydraulic Division, 23119, Elazig-Turkey; Mailto: memiroglu@firat.edu.tr This paper presents the results of a laboratory study on the flow characteristics of sharp-crested weirs, broad-crested weirs, and labyrinth weirs. The variation of the maximum bubble penetration depth for different weir types is investigated depending on overfall jet expansion, discharge, and drop height. Moreover, most efficient depth, length and width of the downstream receiving pool in an open channel system are studied by considering the penetration depth, overfall jet expansion, jet trajectory and the bubble zone. The results show that overfall jet expansion at the labyrinth weirs is significantly wider than the rectangular sharp-crested weirs and the trapezoidal sharp-crested weir. It is demonstrated that the labyrinth weirs have the lowest values of bubble penetration depth among the weirs tested. Furthermore, it is found that the rectangular and the trapezoidal weirs are observed to have the highest bubble penetration depth among all weirs. Consequently, empirical equations are obtained for predicting the maximum penetration depth of bubbles, trajectory of free overfall nappe, jet expansion of free overfall nappe, and the length of the bubble zone. KEY WORDS: Weir, Penetration Depth, Jet Expansion, Jet Trajectory, Bubble, Air Entrainment, Aeration. M. Emin Emiroglu: URCOVANIE PRÚDOVÝCH CHARAKTERISTÍK ROZDIELNYCH TYPOV PRIEPADOV A OPTIMÁLNY NÁVRH VÝVARU. J. Hydrol. Hydromech., 58, 2010, 4; 28 lit., 13 obr., 4 tab. Príspevok prezentuje výsledky laboratórneho výskumu charakteristík prúdenia cez ostrohranné priepady, cez priepady so sirokou hranou a cez labyrintové priepady. Bola studovaná variácia maximálnej hbky prieniku vzduchových bublín pre rozdielne typy priepadov v závislosti od rozsírenia prúdu, prietoku a výsky priepadu. Okrem toho bola analyzovaná efektívna hbka, sírka a dzka vývaru v systéme otvorených kanálov s uvázením rozsírenia prúdu, prietoku a výsky priepadu a oblasti so vzduchovými bublinami. Z výsledkov vyplýva, ze rozsírenie prúdu pri labyrintových priepadoch je výrazne väcsie, ako pri pravouhlých a lichobezníkových ostrohranných priepadoch. Bolo ukázané, ze labyrintové priepady majú najmensiu hbku prieniku vzduchových bublín spomedzi vsetkých testovaných priepadov. Zistilo sa vsak, ze pravouhlé a lichobezníkové priepady majú najväcsie hbky prieniku vzduchových bublín spomedzi vsetkých testovaných priepadov. Tieto výsledky výskumu viedli k empirickým rovniciam, umozujúcim výpocet maximálnej hbky prieniku vzduchových bublín, trajektórií prepadového lúca, rozsírenia prúdu a dzky prevzdusnenej oblasti. KÚCOVÉ SLOVÁ: priepad, hbka prieniku vzduchových bublín, rozsírenie prúdu, trajektória prúdu, bublina, prevzdusovanie. 1. Introduction The overflow section shape cut with a sharp upstream corner into a thin plate is the weir notch, sometimes called the overflow section. If the notch plate is mounted on the supporting bulkhead such that the water does not contact or cling to the downstream weir plate or supporting bulkhead, but springs clear, the weir is a sharp-crested weir (USBR, 200. A broad-crested weir is a flatcrested structure with a crest length large compared to the flow thickness. The ratio of crest length to upstream head over crest must be typically greater than 3 (Henderson, 1966; Chanson, 2004). The flow control section can have different shapes, such as rectangular, triangular, trapezoidal or circular. Labyrinth weirs are those for which the weir crest is not straight in planform (Wormleaton and Soufiani, 245 1998). The increased crest length of labyrinth weirs gives them the clear advantage of reducing upstream levels for a particular discharge over low head ranges (Wormleaton and Tsang, 2000). The flow over weirs would be classified as a free jet, as shown in Figs. 1a­c) and Figs. 2a, b). Weirs have a large number of uses in engineering applications. Listing the more usual, these are: 1. to measure the flow of water in small stream and constructed channels; 2. to regulate the flow of water in open channel and reservoir; 3. to take the water for irrigation systems; 4. to divert flow during high flow conditions in sewer systems; 5. to increase the dissolved oxygen content of the water by mean of aeration the water; 6. to reduce tastes and odors caused by dissolved gases in the water by mean of aeration the water, such as hydrogen sulphide, which are then released; and also to oxidize and remove organic matter; 7. to decrease the carbon dioxide content of a water by mean of aeration the water and thereby reduce its corrosiveness and raise its pH value; 8. to remove certain volatile organic compounds by mean of aeration the water. An overflow jet that plunges into a channel after passing through a gas phase entrains a substantial amount of air into the receiving pool, and forms a submerged two-phase region with a considerable interfacial area. Almost all the mass transfer substantially takes place in this two-phase region. Therefore, maximum penetration depth of the bubble swarm is one of the most important parameters that characterize the performance of weir aeration system and the design of downstream receiving pool. Moreover, estimating flow characteristics of weirs would be useful for scour depth in the downstream receiving channel because flow characteristics of the weirs are different from each other. Apted and Novak (1973), Avery and Novak (1978), and Nakasone (1987) reported that tailwater depth was an important weir operation parameter, influencing mass transfer. Tailwater depth is also an important parameter for energy dissipation and scouring cases. Grindrod (1962) and Albrecht (1968) pointed out that the aeration efficiency would increase with increasing tailwater depth. However, there should be a limit because the penetrating air bubbles will not go to infinite depths. Actually, for each combination of discharge, drop height, and weir type, there would be an approximate maximum depth to which the bubbles would penetrate, thus limiting the aeration efficiency. Studies by Nakasone (1975, 1976, and 1987) contributed substantially to the experimental investigations of the weir aeration. Nakasone (1987) did not investigate 246 the case whether or not air bubbles reached the floor of the stilling basin. For tailwater depth, Nakasone suggested to adapt as 2/3 of drop height. However, the bubble penetration depth and the overfall jet expansion would be different for each weir type and shape and it would not be true to give the bubble penetration depth as a function of only drop height. Moreover, the bubble penetration depth differs from each weir type. Kobus and Koschitzky (199 presented the empirical correlation for penetration depth 0.39 0.24 D p = 0.00433Re j F j , where Rej ­ jet Reynolds number, Re j = qj and Fj ­ the jet Froude g h3 number, F j = 2q 2 j 1/4 . Nakasone (1987) advised that the receiving pool depth Dp ceases to affect aeration efficiency when it exceeds two-thirds of the drop height h (i.e., D p 2 / 3 h ), and Avery and Novak (1978) gave the relationship as D p 7.5 h 0.58 F-0.53 , where h ­ drop height and Fj j ­ the jet Froude number. Kumagai et al. (1993), Clanet and Lasheras (1997), and Ito et al. (2000) studied depth of penetration of bubbles entrained by a plunging water jet. They presented correlations for predicting the bubble penetration depth of plunging water jet as a function of the operational conditions and nozzle geometry. Baylar and Emiroglu (2002), and Emiroglu and Baylar (2003 and 2005) studied air entrainment rate of the broad crested, sharp crested weirs and labyrinth weirs and they stated that weir shape is very sensitive for flow characteristics. The behavior of water flowing over a weir varies significantly with weir shape and type. It is therefore highly desirable to investigate flow characteristics for different weir types and their shapes. This paper presents an experimental investigation of the bubble penetration depth Dp, trajectory of free overfall nappe from the weirs, the length of the bubble zone, and the overfall jet expansion Je of the sharp crested weirs, broad crested weirs, and labyrinth weirs. In addition, the paper studies most efficient length, width and depth of the downstream receiving pool for especially weir aerators. 2. Dimensional analysis The functional relationship involving the maximum bubble penetration depth and influencing variables can be written as L Bw s p Rectangular weir L Bw s L Bw s b' Triangular weir Trapezoidal weir a) Flow direction Flow direction Bw p L Bw s p L Bw s b' p L b) Flow A Flow d Plan view d A-A cross-section c) Fig. 1. Weirs; a) Sharp-crested weir, b) Broad-crested weir, c) Labyrinth weir. Obr. 1. Priepady; a) ostrohranné; b) so sirokou korunou; c) labyrintové. Flow Direction Flow Direction Je Je (a) Straight Weir (b) Triangular Labyrinth Weir Fig. 2. Sketch of flow patterns of free jet over weirs; a) Rectangular sharp-crested weir; b) Labyrinth weir. Obr. 2. Nácrt voného prúdu nad priepadom; a) pravouhlý, ostrohranný, b) labyrintový priepad. H , Bw , q j , , ,g, , D p = f1 , , Pon , P, z , br , b , bd , bu ( H = h + H w . (2) in which Dp ­ maximum bubble penetration depth, H ­ energy difference between the flow over the weir crest and the receiving pool, Bw ­ water surface width over the crest of the weir, qj ­ discharge through the per unit water surface width at the crest of the weir (qj = Q/Bw; Q ­ total discharge through the weir), ­ density of fluid (water), = ­ a, a ­ density of gas (air), g ­ gravity acceleration, ­ visity of the fluid (water), ­ surface tension of the fluid (water), Pon ­ ambient pressure on the outer nappe of the jet, Pun ­ ambient pressure on the lower nappe of the jet (under the nappe), P=(Pon ­ ­ Pun), z ­ depth of flow in the receiving pool, br , b , bu , bd ­ the distances of the right side, left side, upstream and downstream walls of the receiving pool to the center point of the jet where it impacts the water surface in the receiving pool, respectively as shown in Fig. 3a­c). 2 Note that H = h + H w + vo / 2 g , in which h ­ vertical height of the weir crest with respect to the water surface level in the receiving pool (i.e., drop height), Hw ­ the vertical difference between the water surface in the upstream pool and the weir crest, vo ­ average flow velocity in the upstream pool. It is well known that if Hw is measured at a distance sufficiently for upstream of the weir (i.e., It should be mentioned here that the air entrainment in a jet highly depends on the shear hence velocity, between the surface of the free jet (nappe) and air. Also, the fluid and flow conditions about the bubble cloud within the receiving pool are very complex. Due to these reasons, the average flow velocity may not be that good influencing factor to be chosen the dimensional analysis. The average flow velocity may cause under estimation of the quantities. This is why, the discharge per unit water surface width (qj = Q/Bw) is preferred to the average flow velocity as an influencing variable in the dimensional analysis. The dimensional analysis in Eq. ( yields, z br F j , Re j ,W , Eu , B , B , Dp w w = f1 , b bd bu P H , , , Bw Bw Bw Pon g h3 in which, Fj ­ jet Froude number = 2q 2 j Rej ­ jet Reynolds number = W ­ jet Weber number qj Bw . qj (3) 1/4 ( = / ), q2 j Bw and 2 more than 4Hw), vo / 2g is generally negligible. Therefore one can take Eu = jet Euler number = Jet Upstream pool Weir bu bd Receiving b r pool (a) Plan 2 vo / 2g Hw Weir crest y o h Nappe jet P on P un A H = h + H w + v o / 2g Upstream pool Receiving pool bu Dp Lb bd z (b) Cross-Section Bw Flow at the crest of the weir Weir crest (c) Trapezoidal Weir, Cross-Section Fig. 3. Receiving pool; a) Plan, b) Cross-section, c) Trapezoidal weir, Cross-section. Obr. 3. Vývar; a) nákres; b) priecny rez, c) lichobezníkový priepad, priecny rez. In this study, since the jet nappe is freely ventilated and open into atmosphere and no clinging of the nappe onto the weirs, P/Pon has no physical importance and keeping in mind the scale effects, the effect of Weber and Euler number are negligible in air entrainment problems (Kells and Smith, 199. Thus, under these reasons Eq. ( becomes, Lb z br b bd bu , , , , = f 2 F j , Re j , , Bw Bw Bw Bw Bw H (6) (7) (8) (9) Lb = f 2 F j , Re j , H bu = f3 F j , Re j , H Je = f 4 F j , Re j . H z br b bd bu = f1 F j , Re j , , , , , , H Bw Bw Bw Bw Bw Dp Dp H = f1 F j , Re j . (4) (5) If one conducts the similar dimensional analysis for Lb ­ length of the bubble zone; bu ­ horizontal distance of the jet impact point at the water surface in the receiving pool and to the upstream back wall, one gets Note that since Dp, Lb, bu and Je are just before the impact of the jet at the water surface they are independent of z , br , b , bd and bu as shown in Eqs. (5), (7), (8) and (9). 3. Air Entrainment mechanisms-classification and break-up length of the jets Lbr Bi Fi 0.85 (1.07 Tu Fi2 )0.82 (10) Tsang (1987) described four basic air entrainment mechanisms with reference to a free falling jet from a weir. Tsang's four air entrainment mechanisms are smooth, rough, oscillating, and disintegrated. For small drop heights, a water jet with a relatively smooth surface issues from the weir. The major source of air supply is visualized as a thin layer surrounding the jet and carried into the water upon impact, and therefore the air entrainment capacity is limited. The water surface in the receiving pool is relatively undisturbed. As the drop height increases, the surface of the jet becomes roughened. The air supply can be considered as coming largely from small air pockets entrapped between the jet surface roughness and the receiving water. At impact, the jet produces ripples on the pool surface. Compared with the smooth jet under similar conditions, this mechanism results in shallower bubble penetration but increased entrainment rate, because the bubbles are more densely packed in the biphasic zone. As the drop height is increased further, the jet begins to oscillate during the fall. The primary air source originates from large air pockets entrapped between the undulating jet and the pool surface. The pool surface is considerably agitated, and air may also be entrained by surface roller action and splashing. Large air pockets are transported from the surface into the water depth and broken down due to turbulence. With an even larger drop height, the jet breaks up into discrete droplets. The pool surface is intensely agitated, and air is entrained by the action of surface rollers and by the engulfing of air pockets as jet fragments hit the pool surface. The bubbles are generally only transported to relatively shallow depths. Disintegrated jets have the advantage over solid jets of greater surface area; however, air entrainment rate QA and bubble penetration are significantly reduced because of energy loss to the surrounding atmosphere during the fall (Wormleaton and Tsang, 2000). The break-up length Lbr at which the jet begins to disintegrate can be determined using Horeni's 0 Equation for rectangular jets Lbr = 31.19 q w.319 , where Lbr is the break-up length [cm] and qw is the specific flow over the weir in cm3 s-1 cm-1. On the other hand, the equation of the break-up length for rectangular jet, established by Castillo (2007) is where Bi and Fi are the jet thickness and Froude number at issuance conditions, respectively. Tu is the initial turbulence intensity when the flow passes on spillway ( 0 Tu 3% ). 4. Experimental equipment and procedure Fig. 4 shows the laboratory arrangement for the current study. The experiments have been conducted in a rectangular flume with glass sides and well painted steel bed. The flume is 3.40 m in length, 0.6 m in width, and 0.6 m in depth and the bed has an average Manning's roughness coefficient n of 0.0099. A digital point gauge has been mounted on rails along the channel allowed the upstream water surface profile, then the nappe profile, and Hw have been measured. Furthermore, a digital point gauge has been mounted on the received pool to measure h. The water in the experimental channel has been re-circulated by a pump. The water jet from the test weir has plunged into a downstream water pool, whose height is adjusted using a release valve. The plan-view dimensions of the downstream water pool made of glasswall are 1.75 m in length, 1.50 m in width, and 2 m in height. The experiments reported here are carried out with rectangular sharp-crested weirs (Bw = 0.10, 0.20 and 0.30 m), trapezoidal sharp-crested weir (Cipolletti weir with 1H/4V, b' = 0.20 m and, Lw = 0.40 m), triangular sharp-crested weirs ( = 30°, 60°, 90° and 120°), triangular labyrinth weirs ( = 45°, 90° and 135°) all having the same total sill length of 0.30 m, rectangular broad-crested weirs (Bw = 0.10, 0.20 and 0.30 m and, Lw = 0.40 m), trapezoidal broad-crested weir (1H/4V, b' = 0.20 and, Lw = 0.40 m), triangular broad-crested weirs ( = 30°, 60°, 90° and 120° and, Lw = 0.40 m), as can be seen in Figs. 1a­c). The difference between crest and top in all weirs s is 0.25 m and crest height p is 0.35 m. Each weir configuration has been tested under flow rates Q varying from 1.0 to 5.0 L s-1 in 1 L s-1 steps. The drop height h, defined as the difference between the water levels upstream and downstream of the weir, is varied between 0.20 to 1.00 m in 0.20 m steps. Penetration depth, Dp, of the bubbles produced by the jet, which is defined as the vertical distance from the water surface to the lower end of the submerged biphasic region in the water. Penetration depth is measured by a movable scale from the receiving pool wall. This scale is capable of reading to the nearest 1.0 mm. All measurements are repeated three times and then averaged, especially for penetration depth measurements. In all of the experiments, the tailwater depth is selected greater than the maximum bubble penetration depth so that bubbles cannot reach the floor of the downstream water pool. The expansion measured at impact point on the liquid pool of free overfall jet is named as jet expansion, Je. And, it is measured by a scale which is capable of reading to the nearest 0.5 mm. The length, Lb, shown in Fig. 3a) is named as the length of the bubble zone. Also, it is measured by a scale which is capable of reading to the nearest 1.0 mm. A total of 675 tests are performed in this study. Grid Electromagnetic flowmeter Flowrate adjustment Point Gauge Test weir Overfall jet Point Gauge Water channel Air bubbles Downstream Receiving Pool Tailwater depth control Water feed line Pump Fig. 4. Experimental apparatus. Obr. 4. Aparatúra. 5. Evaluation of experimental results For different weir types, the bubble penetration depth Dp, the overfall jet expansion Je, the length of the bubble zone Lb, and the horizontal distance to overfall jet impact point from the weir bu are determined for a range of discharges and drop heights. The following sections present and discuss the experimental results. To examine the effect of weir types on the overfall jet expansion, Je values are plotted against drop height. The experimental data are presented in Fig. 5. The overfall jet expansion of the weirs is highly different from each other. The experimental data indicate that the overfall jet expansion increases with an increasing water discharge. As shown in Fig. 5, the labyrinth weirs have the highest values of overfall jet expansion among weirs tested. At labyrinth weirs, the overfall jet expansion increases as the included angle in the triangular labyrinth weirs decreases. Overfall jet expansion at the labyrinth weirs is significantly more than the trapezoidal sharp-crested weir, the rectangular sharp- crested weirs, and circular sharp-crested weir. This is due to the fact that the labyrinth weirs have more crest length than the other weirs tested. The overfall jet expansion of the labyrinth weirs show an increasing tendency when drop height increases. But, the tendency of overfall jet expansion for the trapezoidal sharp-crested weir, the rectangular sharpcrested weirs slightly show a decrease as drop height increases, as illustrated in Fig. 5. The results indicate that the trapezoidal sharp-crested weir and the rectangular sharp-crested weirs have the lowest overfall jet expansion values. At the rectangular sharp-crested weirs, the overfall jet expansion increases with an increasing crest width Bw. It can be seen from the Fig. 5 that overfall jet expansion for the rectangular broad-crested weirs is more than the rectangular sharp-crested weirs. For triangular sharp-crested weirs, the overfall jet expansion increases as the angle of triangular weirs decreases. Fig. 5 shows that overfall jet expansion for the triangular broad-crested weirs is significantly less than the triangular sharp-crested weirs. 251 Q=5 L/s Rectangular sharp-crested weir (b=10 cm) Rectangular sharp-crested weir (b=20 cm) Rectangular sharp-crested weir (b=30 cm) Trapezoidal sharp-crested weir (Cipolletti) 30 Triangular sharp-crested weir 60 Triangular sharp-crested weir o o Water jet expansion (m) 90 Triangular sharp-crested weir 120 Triangular sharp-crested weir Labyrinth weir ( =45 ) o o o Labyrinth weir ( =90 ) Labyrinth weir ( =135 ) Rectangular broad-crested weir (b=10 cm) Rectangular broad-crested weir (b=20 cm) Rectangular broad-crested weir (b=30 cm) Trapezoidal broad-crested weir (Cipolletti) 30 Triangular broad-crested weir 60 Triangular broad-crested weir o o 90 Triangular broad-crested weir 120 Triangular broad-crested weir 0.00 0.00 0.20 0.40 0.60 Drop height (m) 0.80 1.00 1.20 Fig. 5. Variation in overfall jet expansion of different weir types with drop height for Q = 5 L s-1. Obr. 5. Variácie rozsírenia prepadového prúdu rozdielnych typov priepadov s výskou Q = 5 L s-1. Q=5 L/s Rectangular sharp-crested weir (b=10 cm) Rectangular sharp-crested weir (b=20 cm) Rectangular sharp-crested weir (b=30 cm) Trapezoidal sharp-crested weir (Cipolletti) Bubble penetration depth (m) 30 Triangular sharp-crested weir 60 Triangular sharp-crested weir 90 Triangular sharp-crested weir 120 Triangular sharp-crested weir Labyrinth weir ( =45 ) o o o o Labyrinth weir ( =90 ) Labyrinth weir ( =135 ) Rectangular broad-crested weir (b=10 cm) Rectangular broad-crested weir (b=20 cm) Rectangular broad-crested weir (b=30 cm) Trapezoidal broad-crested weir (Cipolletti) 30 Triangular broad-crested weir 60 Triangular broad-crested weir 90 Triangular broad-crested weir 120 Triangular broad-crested weir o o o o 0.25 0.00 0.20 0.40 0.60 Drop height (m) 0.80 1.00 1.20 Fig. 6. Variation in penetration depth of different weir types with drop height for Q = 5 L s-1. Obr. 6. Variácie hbky prieniku vzduchových bublín rozdielnych typov priepadov s výskou Q = 5 L s-1. The residence time of entrained air bubbles in a water body directly affects the oxygen mass transfer. This residence time is related to the bubble flow path and hence Dp into the receiving water. It is observed from experimental data that bubble penetration depth is closely related to the jet shape and expansion that is unique to each weir type. The bubble penetration depth of the weirs is also highly different from each other. It would not be true to give the bubble penetration depth as a function of only drop height. Fig. 6 shows variation in Dp of nineteen different weir types with drop height while the change in discharge is constant. Experiments with all nineteen weirs indicate that the weir type and shape, the water discharge, and the drop height are important factors influencing Dp. It is found from these results that Dp tends to increase with water discharge Q. It is clear from the results in Fig. 6 that Dp generally decreases with increasing drop height. The experimental results show that Dp decreases with increasing overfall jet expansion Je. The decrease in Dp with increasing in Je is might be due mainly to the increased buoyant forces of the entrained bubbles as a result of the increase in the air entrainment rate, which may have caused by the increase of the jet surface roughness. As is seen from Fig. 6, the labyrinth weirs have the lowest values of bubble penetration depth among the weirs tested. It can be seen from the experimental results that Dp values for triangular labyrinth weir with included angle of 45° are observed the lowest at all discharge, from 1 to 5 L s-1. The rectangular and trapezoidal sharp-crested weirs are observed to have the highest bubble penetration depth among all weirs tested (Fig. 6). Especially, the trapezoidal sharp-crested weir has the highest bubble penetration depth, Dp, values. Moreover, at both triangular sharp-crested weirs and triangular broad-crested weirs, it is demonstrated that Dp increased as the angle of in triangular weirs increased. It may be concluded from Fig. 5 and Fig. 6 that Dp decreases while the overfall jet expansion increases. Thus, the scour depth in the downstream channel would decrease with increasing the overfall jet expansion because the bubble penetration depth would decrease. Moreover, overfall jet's momentum is sensitive for aeration. It can be seen from these two figures that if drop height is highly increased, then the overfall jet is disintegrated and the bubble penetration depth is much decreased. At this state, the aeration performance of the weirs would be extremely little. Fig. 7 shows the relationship between Dp and h with Kobus and Koschitzky's results; Avery and Novak's results and present study data when the weir type is rectangular broad crested weir. Good agreement between the measured values and the values computed from the Kobus and Koschitzky's, and Avery and Novak's equations are obtained. Scimeni (1937) expressed the shape of the nappe in co-ordinates x and y, measured from an origin at the highest point, for a unit value of H as y = Kx n H 1- n , where K = 0.5 and n = 1.85. These values are taken as K = 0.47 and n = 1.80 for Creager profile. Thus, the lower jet trajectory can define with y = 0.50 x1.85 .H -0.85 and y = 0.47 x1.80 .H -0.80 (Scimemi, 1930; Scimemi, 1937; Creager et al., 1945). Where x, y coordinates and H is equal to Hw + vo2/2g. Nowadays most crests have an ogee shape, (e.g. Scimemi profile, Creager profile). 1.4 1.2 1.0 D (m) p 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 h (m) 0.6 0.8 1.0 Present study Kobus and Koschitzky (199 Avery and Novak (1978) Fig. 7. Comparison of the Dp values of rectangular broad crested weir with the equations of Kobus and Koschitzky (199, and Avery and Novak (1978). Obr. 7. Porovnanie hodnôt Dp pravouhlého priepadu so sirokou hranou s výsledkami poda rovníc Kobusa a Koschitzkého (199, a Avery, Novaka (1978). Fig. 8 shows the jet trajectories for different weir types tested. Equations for the jet trajectory of weirs other than rectangular sharp-crested weir do not exist in literature as far as I know. But, free jet trajectory equations for an open channel, a flip bucket, an orifice through a dam and an overtopping dam crest exist in literature (Davis et al., 1999; Dey, 2002; Wahl et al., 2008). Therefore, present study data are compared with Scimemi and Creager profiles (Fig. 9). Good agreement between the present study bu values and the values computed from the Scimemi and Creager equations is obtained. The nature of the nappe affects significantly the aeration mechanism due to bubble behavior. There253 fore, the jet break-up was considered in this experimental study. The break-up length Lbr was calculated using Horeni's and Castillo's equations and, the Lbr values for the experimental data were obtained between 120 cm to 250 cm. Castillo stated that the Horeni's formula gives correct values for flows smaller than qw = 0.25 m2 s-1. The jet breakup was not observed in the study because maximum drop height was taken as 100 cm. This situation was confirmed with Horeni's and Castillo's equations. Empirical correlations predicting the overfall jet expansion Je, the bubble penetration depth Dp, the distance between the channel intake and the overfall jet impact point bu, and the length of the bubble zone Lb are developed for the weirs. The resulting correlations are given in Eqs. (1­(14) and the values of constants in these correlations are shown in Tabs. 1­4. a3 Dp a a = A. ln B.Re j1 .F j 2 , H a3 Lb a a = A. ln B.Re j1 .F j 2 , H ( ( ) ) (1 (12) bu (m) a2 Je a = A. ln B.Re j1 .F j H a 3 , (13) a3 bu a a = A. ln B.Re j1 .F j 2 , H (14) 0.80 h (m) Rectangular sharp-crested Trapezoidal sharp-crested o 30 Triangular sharp-crested o Labyrinth weir =135 Rectangular broad-crested Trapezoidal broad-crested o 120 Triangular broad-crested where Je ­ the overfall jet expansion [m], Dp ­ the penetration depth [m], bu ­ the distance between the channel intake and the overfall jet impact point [m], Lb ­ the length of the bubble zone [m], H ­ the drop height [m], Q ­ the water discharge [m3 s-1], Bw ­ the crest width in rectangular sharp and broadcrested weirs [m], ­ the angle in triangular sharp and broad-crested weirs [°], ­ the included angle in triangular labyrinth weirs [°]. Fig. 8. Jet trajectories for the different weir types. Obr. 8. Trajektórie prúdu rozdielnych typov priepadov. T a b l e 1. Values of constants (Dp) in Eq. (10). T a b u k a 1. Hodnoty konstánt (Dp) v rov. (10). Weir Type Sharp-crested triangular Sharp-crested rectangular2) Sharp-crested trapezoidal3) Broad-crested triangular4) Broad-crested rectangular5) Broad-crested trapezoidal6) Triangular labyrinth7) B 2.353 68.90 15.805 0.72 14.649 exp(7.15 0.945 a1 0.009 ­0.264 ­0.163 0.100 ­0.140 ­0.406 0.064 a2 0.032 ­0.583 ­0.392 0.108 ­0.311 ­1.000 0.241 a3 ­48.10 1.523 1.629 ­5.046 5.933 1.871 ­5.717 F-0.612 0.706 j n F1.977 j 0.724 Estimating flow characteristics of different weir types and optimum dimensions of downstream receiving pool T a b l e 2. Values of constants (Lb) in Eq. (1. T a b u k a 2. Hodnoty konstánt (Lb) v rov. (1. Weir Type Sharp-crested triangular Sharp-crested rectangular2) Sharp-crested trapezoidal3) Broad-crested triangular4) Broad-crested rectangular5) Broad-crested trapezoidal6) Triangular labyrinth7) A Re0.394 0.414 j Re0.304 j n Re0.165 j B 1.979 1.580 a1 ­0.035 ­0.030 0.059 ­0.001 ­0.075 0.046 ­0.033 a2 ­0.097 ­0.061 0.243 ­0.258 ­0.216 0.193 0.466 a3 2.233 1.000 ­6.248 5.719 5.866 ­6.067 ­2.580 ( ( ) ) 0.449 n Re0.028 j 1.079 T a b l e 3. Values of constants (Je) in Eq. (12). T a b u k a 3. Hodnoty konstánt (Je) v rov. (12). Weir Type Sharp-crested triangular Sharp-crested rectangular2) Sharp-crested trapezoidal3) Broad-crested triangular4) Broad-crested rectangular5) Broad-crested trapezoidal6) Triangular labyrinth7) B 2.223 2.101 a1 0.039 0.028 0.043 ­0.004 0.585 ­0.118 0.018 a2 ­0.418 0.086 0.107 ­0.307 ­1.148 ­0.480 0.133 a3 1.000 ­28.998 ­19.246 3.049 23.904 2.486 ­8.208 n Re0.115 j 0.394 ( n Re j / F j )-22.097 1.445 The calculated penetration depth Dpo is compared with the observed penetration depth Dpc to yield the average percent error as 100 N D po - D pc , N i =1 D po (15) in which N is the number of data points. The average percent error is a function of the constant in the penetration depth equation. Eq. (15) can be also written for Lb, Je and bu. The average percent error values belong to Eqs. (1­(14) are given in Tabs. (1­4). Moreover, the measured bubble penetration depths are compared with those predicted with Eq. (1. Good agreement between the measured bubble penetration depth and the values computed from the predictive equations is obtained. The values of Lb measured are also compared with those predicted with Eq. (12). Good agreement between the measured Lb and the values computed from the predictive equations is obtained. The measured Je are compared with those predicted with Eq. (13). Good agreement between the measured Je and the values computed from the predictive equations is obtained. The values of bu measured are compared with those predicted with Eq. (14). Excellent agreement between the measured bu and the values computed from the predictive equations is obtained. Further 255 T a b l e 4. Values of constants (bu) in Eq. (13). T a b u k a 4. Hodnoty konstánt (Je) v rov. (13). Weir Type Sharp-crested triangular Sharp-crested rectangular2) Sharp-crested trapezoidal3) Broad-crested triangular4) Broad-crested rectangular5) Broad-crested trapezoidal6) Triangular labyrinth7) B 5.371 3.961 3.642 2.288 4.004 5.781 0.930 a1 ­0.044 ­0.03 ­0.018 ­0.006 ­0.051 0.082 ­0.042 a2 0.164 0.12 0.093 ­0.190 0.146 ­0.343 1.388 a3 ­2.177 ­4.600 ­5.787 0.867 ­2.652 2.112 ­1.035 1.054 1.007 confidence in the correlations is seen in Figs. 10, 11, 12, and 13. The following equations can be written for the determination of the minimum width, length and depth of the downstream receiving pool. Wd J e , Ld bu + Lb , (16) (17) (18) In this study, tested discharges are smaller than that of some prototype applications. But, tested drop heights are similar to a lot of prototype scale. Scaling of bubble penetration depth data to prototype size is virtually impossible, largely due to the relative invariance of bubble size. Clearly, tests at higher discharges should be carried out to see if this trend extrapolates. zd D p , +10% bu (m) Dp calculated Present study Scimemi profile Creager profile h (m) Fig. 9. Comparison of the bu values of rectangular broad crested weir with the equations of Scimemi and Creager. Obr. 9. Porovnanie hodnôt bu pravouhlého priepadu so sirokou korunou a výsledkami rovníc Scimemi a Creagera. Dp measured where Wd ­ minimum width of downstream receiving pool [m], Ld ­ minimum length of downstream receiving pool [m], and zd ­ minimum tailwater depth [m]. 256 Fig. 10. Comparison of observed Dp values with those calculated from Eq. (1. Obr. 10. Porovnanie pozorovaných hodnôt Dp s vypocítanými poda rov. (1. -10% +12% -12% trajectory, overfall jet expansion, the maximum penetration depth, and the length of the bubble zone for different weir types. Downstream receiving pool using equations presented in this study have properly been designed as the optimum case. The concluding remarks obtained are as follows. Lb calculated +5% -5% bu calculated 0.00 0.00 0.20 0.40 0.60 0.80 1.00 Lb measured Fig. 11. Comparison of observed Lb values with those calculated from Eq. (12). Obr. 11. Porovnanie pozorovaných hodnôt Lb s vypocítanými poda rov. (12). bu measured +10% -10% Fig. 13. Comparison of observed bu values with those calculated from Eq. (14). Obr. 13. Porovnanie pozorovaných hodnôt bu s vypocítanými poda rov. (14). Je measured Fig. 12. Comparison of observed Je values with those calculated from Eq. (13). Obr. 12. Porovnanie pozorovaných hodnôt Je s vypocítanými poda rov. (13). 6. Conclusions The present experimental investigation estimates the free jet trajectory, the overfall jet expansion, and bubble penetration depth for different weir types and so dimensions of receiving pool. The correlations are developed for predicting the jet 1. The values of overfall jet expansion Je, the bubble penetration depth Dp, jet trajectory, and the length of the bubble zone Lb are different for each weir type and shape. 2. The overfall jet expansion Je increases with an increasing water discharge. The labyrinth weirs have the highest values of Je. At the labyrinth weirs, Je increases as the included angle in the triangular labyrinth weirs decreases. The overfall jet expansion Je at the labyrinth weirs is significantly wider than the trapezoidal sharp-crested weir and the rectangular sharp-crested weirs. Je of the labyrinth weirs has shown an increasing tendency when drop height increased. The results indicate that the trapezoidal sharp-crested weir and the rectangular sharp-crested weirs have the lowest Je. At the rectangular sharp-crested weirs, Je increases with an increasing crest width Bw. Furthermore, Je for the rectangular broad-crested weirs is wider than the rectangular sharp-crested weirs. For triangular sharp-crested weirs, Je increases as the angle of triangular weirs decreases. The overfall jet expansion Je for the tri257 Je calculated angular broad-crested weirs is significantly smaller than the triangular sharp-crested weirs. 3. Dp is closely related to the jet shape and expansion that is different for each weir type. The bubble penetration depth Dp increases with increasing the water discharge. At the same time, the bubble penetration depth Dp decreases with increasing drop height. The bubble penetration depth Dp decreased with increasing Je. The labyrinth weirs have the lowest values of Dp among the weirs tested. The bubble penetration depth Dp values for triangular labyrinth weir with included angle of 45° are observed as the lowest one. The rectangular and trapezoidal sharp-crested weirs are observed to have the highest Dp among all weirs tested. Especially, the trapezoidal sharpcrested weir has the highest Dp values. At both triangular sharp-crested weirs and triangular broad-crested weirs, Dp increases as the angle of in triangular weirs increases. 4. The labyrinth weirs and triangular sharp-crested weirs with small angle are not appropriate due to enormous overfall jet expansion, if the downstream receiving pool width is very small. List of symbols Bw ­ water surface width over the crest of the weir [m], bd ­ distance of the downstream wall of the receiving pool to the center point of the jet where it impacts the water surface in the receiving pool [m], b ­ distance of the left side wall of the receiving pool to the center point of the jet where it impacts the water surface in the receiving pool [m], bL ­ half-crest length in triangular labyrinth weir [m], br ­ distance of the right side wall of the receiving pool to the center point of the jet where it impacts the water surface in the receiving pool [m], bu ­ distance of the upstream wall of the receiving pool to the center point of the jet where it impacts the water surface in the receiving pool [m], d ­ breadth in triangular labyrinth weir [m], Dp ­ bubble penetration depth [m], Dpc ­ calculated penetration depth [m], Dpo ­ observed penetration depth [m], Eu ­ jet Euler number, Fi ­ Froude number at the issuance, Fj ­ jet Froude number, g ­ acceleration due to gravity [m s-2], h ­ drop height [m], Hw ­ the vertical difference between the water surface in the upstream pool and the weir crest [m], Je ­ overfall jet expansion [m], L ­ the experimental channel width [m], Lb ­ length of the bubble zone [m], Ld ­ minimum length of the downstream pool [m], Lbr ­ jet break-up length [m], LW ­ length in triangular broad-crested weir [m], Pon ­ ambient pressure on the outer nappe of the jet [N m-2], Pun ­ ambient pressure on the lower nappe of the jet [N m-2], Rej ­ jet Reynolds number, s ­ difference between crest and top in all weirs [m], Tu ­ initial turbulence intensity, Q ­ water discharge [m3 s-3], QA ­ air entrainment rate [m3 s-3], qj ­ discharge through the per unit water surface width at the crest of the weir [m2 s-1], vo ­ average flow velocity in the upstream pool [m s-1], W ­ jet Weber number, w ­ width in triangular labyrinth weir [m], Wd ­ minimum width of the downstream receiving pool [m], z ­ tailwater depth [m], zd ­ minimum depth of the downstream pool [m], ­ angle in triangular sharp and broad-crested weir [°], ­ average percent error, ­ included angle in triangular labyrinth weir [°]; ­ visity of the fluid (water) [kg s-1 m-1], ­ density of fluid (water) [kg m-3], a ­ density of gas (air) [kg m-3], ­ surface tension of the fluid (water) [N m-1].

Journal

Journal of Hydrology and Hydromechanicsde Gruyter

Published: Dec 1, 2010

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