Open channel flow equation

Open Channel Flow
Equation
Through Sluice Gate
By Dr.Mrinmoy Majumder, Founnding and Honorary Editor
www.baipatra.ws

Content
› Sluice Gates
› Flow Equation through sluice gate

Sluice Gate
SLUICE GATES ARE USED FOR CONTROLLING AND MEASURING FLOW
RATES IN OPEN CHANNELS AND RIVERS, MAINLY IN CONNECTION TO
HYDRO POWER PLANTS

Flow Equation through Sluice Gate
The sluice gate flow rate measurement is based on the Bernoulli Equation and can
be expressed as:
1
2
𝜌𝑣1
2
+ 𝜌𝑔ℎ1 =
1
2
𝜌𝑣2
2
+ 𝜌𝑔ℎ2
(1)
where
h = elevation height (m)
ρ = density (kg/𝑚3)
v = flow velocity (m/s)
The pressure components in the equation are in general irrelevant since pressure
upstream and downstream are the same (p1 - p2 = 0).

Assuming uniform upstream and downstream velocity profiles - the
Continuity Equation gives:
q = v1 A1= v2 A2 (2)
where q = flow rate (m3/s), A = flow area (m2)
(2) can be modified to:
q = v1 h1 b = v2 h2 b (3)
Where
b = width of the sluice (m),h1 = upstream height (m),h2 = downstream
height (m)

CONTD.
Combining (1) and (3), gives the "ideal"
equation:
𝑞 = ℎ2 𝑏
2𝑔 ℎ1 − ℎ2
1 − ℎ2 ℎ1
(4)
Assuming h1 >> h2 (4) can be modified to:
Q = h2 b [2 g h1]1/2 (5)

This is approximately true when the depth ratio h1 / h2 is large, the kinetic energy
upstream is negligible (v1 is small) and the fluid velocity after it has fallen the
distance (h2 - h1) ≈ h1 - is:
v2 = [2 g h1]1/2
(6)
The ideal equation (3) can be modified with a discharge coefficient::
q = cd h2 b [2 g h1]1/2
(7)
where
cd = discharge coefficient
The discharge coefficient depends on different parameters - such as upstream
and tail-water depths, gate opening, contraction coefficient of the gate and the
flow condition.
In practice the typical discharge coefficient is approximately 0.61 for free flow
conditions and depth ratios ho / h1 < 0.2.

Sluice Gate Specifications
The most commonly used specification for sluice gates in water and wastewater
treatment plants is ANSI/AWWA C560-00. This specification should be used as a
guidance for gates selection and operating equipment and associated hardware.
Example - Flow Rate through a Sluice Gate
Water flows under a sluice gate with an opening height of 0.4 m. The width of the
sluice is 3 m and the height from the water surface to the bottom of the sluice is
10 m.
Since ℎ1 >> ℎ2 and the depth ratio 0.4/10 < 0,2 - the contraction coefficient can
be set to 0.61 - and equation (7) can be used for flow calculation:
q = 0.61 (0.4 m) (3 m) [2 (9.81 m/𝑠2) (10 m)]
= 10.25 𝑚3/s
Specification and Examples

Flow Rate Calculated with the Velocity-Area Principle

Calculate flow rate or discharge in an open conduit, channel or river based on the
velocity-area principle
Velocities and depths across the stream are measured as indicated in the figure above.
A partial discharge in a section of the stream can be calculated as
qn = vn an
(1)
where
qn = flow rate or discharge in section n (m3/s, ft3/s)
vn = measured velocity in section n (m/s, ft/s)
an = area of section n (m2, ft2)
One simple way to express the section area is
an = dn (ln+1 - ln-1) / 2
(2)
The total flow in the stream can be summarized to
Q = Σ1
n vn an
(3)
where
Q = summarized flow rate or discharge in the conduit (m3/s, ft3/s)
The accuracy of estimate depends on the profile of the conduit and the number of
measurements. For conduits with regular shapes like rectangular channels a limited
number of measurements are required. For irregular shapes - like natural rivers or
similar - higher accuracy requires more measurements both horizontal and vertical.

Measured Values Calculated Values
n
v
(m/s)
d
(m)
l
(m)
a
(m2)
q
(m3/s)
0 0 0 0
1 3 1 2 2 6
2 4 1.5 4 3 12
3 3 0.9 6 1.8 5.4
4 0 0 8
Summarized 23.4
From a conduit we have three measurements:

Example - Computing Flow Rate in a Channel
The section areas can be calculated like
a1 = (1 m) ((4 m) - (0 m)) / 2
= 2 m2
a2 = (1.5 m) ((6 m) - (2 m)) / 2
= 3 m2
a3 = (0.9 m) ((8 m) - (4 m)) / 2
= 1.8 m2
q1 = (3 m/s) (2 m2) = 6 m3/s
q2 = (4 m/s) (3 m2)= 12 m3/s
q3 = (3 m/s) (1.8 m2) = 5.4 m3/s
The total flow can be summarized as
Q = (6 m3/s) + (12 m3/s) + (5.4 m3/s)
= 23.4 m3/s
Note - there are alternative ways to calculate the section flow rates.

Simple Average Method
Using the simple average of two successive vertical depths, their mean velocity,
and the distance between them can be expressed as:
qn to n+1 = [(vn + vn+1) / 2] [(dn + dn+1 ) / 2] (ln+1 - ln) (4)
Midsection Method
With the midsection method, the depth and mean velocity are measured for
each number of verticals along the cross section. The depth at a vertical is
multiplied by the width, which extends halfway to the preceding vertical and
halfway to the following vertical, to develop a cross-sectional area. The section
flow rate can be expressed as
qn = vn [((ln - ln-1) + (ln+1 - ln)) / 2] dn (5)

Weirs - Open Channel Flow Rate Measurement
Weirs can be used to measure flow rates in open channels and rivers - common for
water supply and sewage plants

Weirs are structures consisting of an obstruction
such as a dam or bulkhead placed across the open
channel with a specially shaped opening or notch.
The flow rate over a weir is a function of the head on
the weir.
Common weir constructions are the rectangular weir, the triangular
or v-notch weir, and the broad-crested weir.
Weirs are called sharp-crested if their crests are constructed of thin
metal plates, and broad-crested if they are made of wide timber or
concrete. 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 or thin-plate weir.
Water level-discharge relationships can be applied and meet
accuracy requirements for sharp-crested weirs if the installation is
designed and installed consistent with established ASTM and ISO
standards.

Rectangular weirs and triangular or v-notch weirs are often used in water supply,
wastewater and sewage systems. They consist of a sharp edged plate with a
rectangular, triangular or v-notch profile for the water flow.
Broad-crested weirs can be observed in dam spillways where the broad edge is
beneath the water surface across the entire stream. Flow measurement installations
with broad-crested weirs will meet accuracy requirements only if they are calibrated.
Rectangular weirs

Example
The flow rate measurement in a rectangular weir is based on the Bernoulli
Equation principles and can be expressed as:
q = 2/3 cd b (2 g)1/2 h3/2 (1)
where
q = flow rate (m3/s)
h = elevation head on the weir (m)
b = width of the weir (m)
g = 9.81 (m/s2) - gravity
cd = discharge constant for the weir - must be determined
cd must be determined by analysis and calibration tests. For standard weirs -
cd - is well defined or constant for measuring within specified head ranges.
The lowest elevation (h = 0) of the overflow opening of the sharp-crested
weirs or the control channel of broad-crested weirs is the head
measurement zero reference elevation.

The Francis Formula - Imperial Units
Flow through a rectangular weir can be expressed in imperial
units with the Francis formula
q = 3.33 (b - 0.2 h) h3/2
where
q = flow rate (ft3/s)
h = head on the weir (ft)
b = width of the weir (ft)

Triangular or V-Notch Weir
The triangular or V-notch, thin-plate weir is an accurate flow
measuring device particularly suited for small flows.
For a triangular or v-notch weir the flow rate can be expressed as:
q = 8/15 cd (2 g)1/2 tan(θ/2) h5/2
where
θ = v-notch angle

Broad-Crested Weir
For the broad-crested weir the flow rate can be expressed as:
q = cd h2 b ( 2 g (h1 - h2) )1/2 (3)

Measuring the Levels
For measuring the flow rate it's obviously necessary to measure the flow
levels, then use the equations above for calculating. It's common to measure
the levels with:
•ultrasonic level transmitters, or
•pressure transmitters
Ultrasonic level transmitters are positioned above the flow without any direct
contact with the flow. Ultrasonic level transmitters can be used for all
measurements. Some of the transmitters can even calculate a linear flow
signal - like a digital pulse signal or an analog 4 - 20 mA signal - before
transmitting it to the control system.
Pressure transmitters can be used for the sharp-crested weirs and for the
first measure point in broad-crested weir. The pressure transmitter outputs a
linear level signal - typical 4-20 mA - and the flow must be calculated in the
transmitter or the control system.

Open channel flow equation

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