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Orifice structures: Theory and Examples
1. Introduction: Hydraulic structures in channels.
From the Practical Hydraulics Textbook…
Examples of hydraulic structures: Orifices, gates, weirs, flumes, spillways, stilling basins, dropstructures, siphons, and culverts.
Functions of hydraulic structures in channels:
Measuring discharge
o
Weirs and flumes – better suited to measure discharges in rivers.
o
Orifices (not that useful) – upstream and downstream water levels are usually
required to determine the discharge, and constant attention for opening and
closing gates is needed, due to large variations in flow.
Discharge control (fair and equitable distribution of water, for example in large irrigation
networks; ensure minimum base flow on summer; and measure flows to avoid or control
flooding).
o
Orifices and gates.
o
Weirs and flumes (not that useful).
Controlling water levels (for example on irrigation schemes is used as discharge control).
o
Weirs and flumes
o
Spillways (dams)
o
Orifices and gates (not that useful)
Dissipating unwanted energy (hydraulic jumps).
o
Still basin
o
Drop structure
Some hydraulic structures only carry out one of the functions described above whilst others perform
all three functions at the same time, so a hydraulic structure may be used for discharge
measurement, and at the same time may be performing a water level control function and
dissipating unwanted energy.
From the point of view of measuring discharge and controlling water levels, there are only two types
of structure. Some structures allow water to flow through them and these are called orifice
structures (they include orifices and gates). Others allow water to flow over them and these are
called weirs or flumes. Hydraulically, they behave in quite different ways and so each has certain
applications for which they are best suited. The energy dissipating function can be attached to both
of these structure types.
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2. Orifice structures.
a. Types of orifices and formula.
Figure 1. Left: round-edged orifices (Cc = 1). Right: Sharp-edged orifices (Cc < 1).
As reviewed in hydrodynamics and uses of the total energy equation:
= √ 2ℎ
Where:
is the discharge of the orifice
is the area of the orifice
ℎ is the water depth of the tank
is the gravity constant
is the discharge coefficient calculated as: =
is the coefficient of contraction: = ⁄
is the velocity coefficient and relates the real velocity discharged from an orifice with the
theoretic discharge calculated by Torricelli’s Law: √ 2ℎ.
All coefficients depend on the Reynolds Number, and for > 10, they are independent to the
Reynolds number and there values are:
= 0.99
= 0.605
= 0.60
For Reynolds Number that are less than 10, the following graph applies.
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Figure 2. Variation of the velocity, contraction and discharge coefficients in comparison to the Reynolds
Number.
b. Methods to calculate the real velocity of an orifice.
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EXAMPLE 1. An orifice on a vertical wall is located 2 m from the water surface. The orifice has a
diameter of 100 mm and a discharge of 29.5 lps.
A)
Applying the total energy equation and the trajectory method, calculate the coefficient of
contraction, coefficient of velocity and coefficient of discharge.
B)
Find the Reynolds number of the discharge of the orifice.
C)
Using the Reynolds number definition, identify the water depth at which the previous values
may no longer be used ( < 10).
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c. Partial and total drowned flow
Total drowned flow: = √ 2∆
Where:
is the discharge of the orifice
is the area of the orifice
∆ is the difference between the
water depth of each side of the tank
is the gravity constant
is the discharge coefficient
calculated as: = , and Sotelo
recommends to use the same
discharge coefficient as free-discharge
flow.
Partial drowned flow:
= √ 2
= √ 2
Where:
Discharge: = +
Area of the orifice: = +
is the difference between the water
depth of each side of the tank
= − 2⁄
is the gravity constant
is the discharge coefficient of each
section. Where Schlag proposes the
values:
= 0.70 and
= 0.675.
IMPORTANT: The previous formula is for
orifices that are not located at the bottom of
the tank, as shown in the next images.
EXAMPLE 2. Calculate the discharge of the orifice shown in the picture. (Q = 0.0275 m3/s)
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d. Free flow on gates.
Figure 3. Free flow on a vertical gate (Kay).
According to Munson et al . (1999) if…
< 0.2
Then… (Equation 1)
= √ 2
Where:
is the gate’s opening
is the water depth (upstream)
is the discharge of the channel
is the width of the channel
is the gravity constant, and
is the discharge coefficient that depends on the contraction coefficient: = ⁄ and
water depth: ⁄ (Henderson, 1966), but it also depends on the velocity coefficient
(Sotelo).
NOTE: varies from 0.55 – 0.6 with a vertical sluice gate and free flow. (See Figure 4.)
(NOTE: supercritical flow passing under the sluice gate.)
When…
a > 0.2
Then… (Equation 2)
= 2( − 3)1 − (3 ⁄ )
3
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Figure 4. Discharge coefficient of the vertical gate (Henderson, 1966). Copia en Sotelo, pp. 216.
Submerged discharge
⁄
3 =
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e. Drowned or submerged flow on gates.
Figure 5. Drowned outflow in a vertical sluice gate.
Equation 1 and 2 apply also to drowned flow, and as Figure 4 shows, if = 3, then = 0.
f. Types of gates.
Figure 6. Typical underflow gates (Henderson, 1966).
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Figure 7. Radial or tainter gate.
Where:
is the water depth (upstream)
is the gate’s opening
is the height form the bed of the channel to the pin or bolt.
is the gate’s radius
3 is the water depth (downstream) after the hydraulic jump.
is the discharge coefficient that depends on the contraction coefficient: = ⁄ and
water depth: ⁄ (Henderson, 1966), but it also depends on the velocity coefficient
Figure 8. The discharge coefficient of the radial gate (Henderson, 1966). Copia en Sotelo, pp. 218.
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Figure 9. Discharge coefficient of inclined gates with free discharge (Sotelo, pp. 215).
EXAMPLE 3. A vertical gate with a width of 1 m, discharges 2.6 m3/s, its upstream and downstream
water depth is 4.5 m and 3.45 m, respectively.
a)
Given a submerged discharge, calculate the opening of the gate assuming a Cd = 0.37.
b)
With the answer obtained in (a), calculate the discharge when water flows freely, and the
discharge coefficient.
Bibliography:
Henderson, F.M. 1966. Open Channel Flow. Macmillan, New York, USA.
http://www.lmnoeng.com/TankDischarge.php
Kay, M. 2008. Practical Hydraulics, 2nd Ed. Taylor & Francis, New York, USA.
Mott, R. L. 2006. Mecánica de Fluidos, 6ta Ed. Pearson Educación, México.
Munson, B. R., D. F. Young y T. H. Okiishi. 1999. Fundamentos de mecánica de fluidos. Limusa
Wiley, México D.F., México.
Sotelo, G. 1997. Hidráulica General Vol. 1. Limusa, México.
Streeter, V. L. 1962. Fluid Mechanics, 3d Ed. McGraw-Hill Book Company, Tokyo, Japan.
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