open channel weirs
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Hydraulic Structures
Hydraulic structures play an integral role in the design and analysis of open chan-nel flows. Weirs and dams are used to store water in reservoirs, gates are usedto regulate the flow within structures, and culverts are used to discharge water
under embankments or roads. In this section well discuss:
Flow measuring structures
Sharp-crested weirs Broad-crested weirs Parshall flumes
In the following discussion, well only consider the design and operation of weirs.Weirs have simple designs, but can cause high head losses and have the potential
for upstream sedimentation. For cases where the sedimentation or head losses area concern (i.e. Wastewater treatment plants and irrigation channels), a Parshallflume can be used. Please see your textbook for the design of Parshall flumes.
Regulation structures
Gates
Discharge structures
Culverts
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Weirs and Flumes
Weirs are used to measure flow and/or control outflow elevations from basinsand channels
The vocabulary well be using includes
sharp-crested (or thin plate) weirs - thin plastic or metal plate that is setvertically across a channel
rectangular or v-notch weirs - two types of sharp crested weirs which describetheir geometry
tailwater - flow downstream of the weir
submerged weir - a weir that has its tailwater elevation at least as high as theweir crest
nappe - jet of water flowing over crest
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Rectangular Sharp Crested Weirs
A rectangular weir has a rectangular opening and it can be either suppressed orunsuppressed.
A suppressed (uncontracted) weirs has a rectangular opening spans channel
width a vent is often needed to maintain atmospheric pressure. An unsuppressed(contracted) weirs has a rectangular opening which only spans part of thechannel.
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To model the flow through a rectangular weir, the energy equation betweenupstream of the weir crest and at the weir crest is defined with
assumptions: no head losses (E1 = E2) (the means no significant turbulence) water surface elevation remains constant
note: this is physically impossible, but actual errors are small pressure at crest (station 2) = patmo throughout depth
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Rectangular Sharp Crested Weirs (continued)
Estimate the theoretical flowrate with the conservation of mass
where the velocity is determined with the conservation of energy equation
substituting, we find the flowrate is only a function of the geometry and theelevation of the water surface above the weir
Discrepancies in the estimated flowrate arise from:
1) pressure distribution at crest is not uniformly atmospheric
2) water surface does not remain uniform as it approaches the crest
3) energy losses due to viscous effects are not negligible
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To compensate for the discrepancies, we define a discharge coefficient
Cd =Q
Q=
true flow rate
theoretical flow rate
The true flow rate is estimated with
Q =2
3Cd
2gbH3/2
where
Cd = F(Re,We,H
Hw)
W e is the Weber numberHw is the height of the weir crest from the bottom
Empirical estimates of this discharge coefficient were made by Rouse in 1946
Cd = 0.611 + 0.075H
Hwfor
H
Hw< 5 10
A variation of the discharge coefficient is called the weir coefficient that isdesigned to simplify the flowrate equation and is defined with
Cw =2
3Cd
2g
so the flowrate equation becomes
Q = CwbH3/2
Q = 1.83bH3/2 IfH
Hw< .4 (SI units only)
Please note, it is recommended that you measure H at a distance of 4 5 Hupstream of the gate.
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Unsuppressed rectangular weirs
Unsuppressed (or contracted) weirs behave similarly to suppressed weirs with twomajor assumptions
1. venting is not needed2. side contractions recude nappe width
where
Unsuppressed trapezoidal sharp-crested weir (Cipolletti Weir)
for this structure, side contractions do not reduce nappe width
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Example:An end-contracted weir of total length 286 ft and crest height 5 ft is used todischarge water without exceeding a head of 2.5 ft from a tank 300 ft wide. Theweir carries piers that are 10 ft clear distance apart and 2 ft wide, to suport the
footway. Determine the discharge. Cd = 0.6. Solution:
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V-notch weirs
These weirs are typically used in low flow (Q < 0.28 m3/s or 10 cf s) environ-ments in place of rectangular, because they are more accurate.
Recall the weir energy equation
and theoretical flow rate
where
combining
as V-notch weirs are typically applied in low flow environments, we can neglectthe approach velocity
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The actual flow rate for a V-notch weir becomes
Q =8
15Cd
2gtan
2
H5/2
where
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Example:Water passes over a rectangular weir of 10 ft width at a depth of 1 ft. If theweir is replaced by an 80o V-notch, determine the depth of water over the notch.Disregard end contractions. Cd notch = 0.59 and Cd rectangular = 0.63. Solution:
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Broad-Crested weirs
The variety of broad crested weirs have crests that are signigicantly larger thansharp crested weirs and are capable of handling much larger discharges.
Rectangular broad-crested weirs are designed so that the flow above the weiris at critical flow conditions. The theoretical flow rate is given by
Energy equation for a RBC weir
under critical flow conditions
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combining this yields an expression for the critical flow depth
where H is the energy of the upstream flow measured relative to the weir location
the theoretical flow rate over a RBC weir become
To account for the non negligible energy losses over the weir, the actual flow
rate, Q, is given by
Q = Cd
gb
2
3H
32
where the discharge coefficient is estimated with (Chow, 1959)
Cd =0.65
1 + HHw
1/2
These equations are valid 0.08 < h1/L < 0.5. Note that for
h1/L < 0.08 head losses can not be neglectedh1/L > 0.5 the streamlines are not horizontal
The weir discharges freely if
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Example:Determine the discharge over a broad-crested weir of 100 ft length. The up-stream water level over the crest is 2 ft and the crest has a height of 2.25 ft.The width of approach channel is 150 ft. Cd = 0.95 Solution:
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Gates
Gates are used to regulate flow in open channels
Vertical gates are a vertical plate supported by vertical guides on the channelwalls. These gates experience large hydrostatic pressure forces.
Tainter Radial gates more easily resist the hydrostatic forces and are generallymore economical
As with weirs, define the energy equation upstream and downstream of the gate
and the conservation of mass at the same locations
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Combining, the flowrate becomes
Weknow the water depth at the gate, yg, but we dont know y2. As the flowmoves under the gate, the streamline becomes contracted. The coefficient ofcontractions is a coefficient which accounts for the contraction of flow down-stream of the gate
where:
vertical sluice gate:
radial gate:
Given that their will likely be energy losses through the gate, the dischargecoefficient is defined with
combining the flow rate becomes
Q = Cdbyg
2gy1
where
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Supercritical discharge from the gates
In cases where the gate discharge is supercritical and the depth of flow down-stream exceeds the depth at the gate opening then the outflow may be sub-merged. In this case the previous equations are not valid.
To characterize submerged flow through a gate
1) define the energy between 1 - 2
2) define the momentum between 2 - 3
wherey is they1 is they2 is they3 is the
Solve 1) and 2) simultaneously for Q
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Example:Water is ponded behind a vertical gate to a height of 4 m in a rectangular chan-nel of width 7 m. Calculate the gate opening that will release 40 m3/s throughthe gate. How would this discharge be affected by a downstream flow depth of
3.5 m? Solution:
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What happens if downstream depth is 3.5 m?
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