open channel flow sluices and venturis - civil engineering channel flow... · 3 channel terminology...

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Civil Engineering Hydraulics

Open Channel Flow

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Channel Terminology

Open Channel Flow 2

¢ No matter what the shape of the open channel, we use the same terminology in working with the characteristics of the channel

¢ We will start with a rectangular channel

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Channel Terminology

Open Channel Flow 3

Channel Terminology

Open Channel Flow 4

¢ Hydraulic Radius (Rh) ¢ The ratio of the cross sectional area to the

wetted perimeter

¢ Mean depth (zm) ¢ The ratio of the cross sectional area to the

top width (bt)

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Channel Terminology

Open Channel Flow 5

¢ Section factor (Z) ¢ Product of the cross sectional area and the

square root of the mean depth

mZ A z=

Channel Terminology

Open Channel Flow 6

¢ Centroid Depth (zc) ¢ Depth from the water surface to the

centroid of the cross sectional area

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Channel Terminology

Open Channel Flow 7

For the section shown, calculate the hydraulic radius, the centroid depth, the section factor, and the mean depth. The rightmost dimension should be b.

Froude Number

Open Channel Flow 8

¢  In full pipe flow, the dimensionless ratio of the Reynolds number was used to describe the flow characteristics.

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Froude Number

Open Channel Flow 9

¢  In open channel flow, another dimensionless ratio known as the Froude number is used.

¢ The Froude number is a dimensionless number defined as the ratio of a characteristic velocity to a gravitational wave velocity.

Froude Number

Open Channel Flow 10

¢  It may also be defined as the ratio of a body's inertia to gravitational forces.

¢  In fluid mechanics, the Froude number is used to consider the resistance of an object moving through water or of water moving through an object.

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Froude Number


¢ For rectangular cross-sections with uniform depth d, the Froude number can be simplified to:

Fr = v

gzm

= vgd

Froude Number


¢ For Fr < 1 the flow is called a subcritical flow, further for Fr > 1 the flow is characterized as supercritical flow.

¢ When Fr ≈ 1 the flow is denoted as critical flow.

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Sluice Gate


¢ Sluice gates are used to control flow typically out of an impounded body of water

¢ The depth of water upstream of the gate is typically much deeper than the water downstream.

Sluice Gate


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Sluice Gate


¢ The cross sectional area upstream is Aupstream=(bt)(z1)

¢ The cross sectional area downstream is Adownstream=(bt)(z2)

¢ For conservation of mass (continuity) the flow upstream must be equal to the flow downstream so Qupstream = Qdownstream

Sluice Gate


¢ From this continuity

1 2

upstream downstream

upstream upstream downstream downstream

upstream t downstream t

Q Qv A v Av b z v b z

=

=

=

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Sluice Gate


¢ We can also consider the energy on both sides of the gate neglecting any losses due to friction

2 21 1 2 2

1 22 2I have adopted the subscripts of the text where 1 denotes upstream and 2 denotes downstream.

p v p vz g z gρ ρ+ + = + +

Sluice Gate


¢ Our reference plane is the bottom of the channel and the points are on the surface of the flows

2 21 1 2 2

1 22 2I have adopted the subscripts of the text where 1 denotes upstream and 2 denotes downstream.

p v p vz g z gρ ρ+ + = + +

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Sluice Gate


¢ This means that the pressure at points 1 and 2 are both atmospheric and can be removed from the expression

2 21 2

1 22 2v vz g z g+ = +

Sluice Gate


¢  If we make the assumption that the downstream velocity is significantly greater than the upstream velocity in the direction of the flow, we can neglect the upstream velocity and the expression reduces to

22

1 2

22

1 2

2

2

vz g z g

vz zg

= +

= +

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Sluice Gate


¢ Using the downstream area and the volumetric flow rate to calculate the velocity downstream

¢ Notice that z2 has a power of 2.

¢ This would indicate that there are two solutions for z2.

22

1 2

22

2

21 2

2

1 22 22

2

2

2

t

t

t

vz zgQvb z

Qb z

z zgQz zgb z

= +

=

⎛ ⎞⎜ ⎟⎝ ⎠= +

= +

Problem 7.2


¢ A rectangular channel 8 ft wide conveys water at 300 ft3/s. A partially raised sluice gate in the channel restricts the flow. Determine the downstream height of the flow if the upstream height is 6ft.

2

1 22 222 t

Qz zgb z

= +

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Problem 7.2


Problem 7.2


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Problem 7.4


¢ A rectangular channel 2.3m wide conveys water at a volumetric flow rate of 30 m2/s. A partially raised sluice gate in the channel restricts the flow so that the downstream water height is 0.68m. Determine the upstream water height.

2

1 22 222 t

Qz zgb z

= +

Problem 7.4


¢ A rectangular channel 2.3m wide conveys water at a volumetric flow rate of 30 m2/s. A partially raised sluice gate in the channel restricts the flow so that the downstream water height is 0.68m. Determine the upstream water height.

¢  z1 = 19.439 m

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Venturi Flume


Venturi Flume

¢ Usually a section manufactured to a specification that can be inserted into an open channel allowing the measurement of volumetric flow rate by measuring the depth of flow at two points in the flume.


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Venturi Flume

¢ Consider the Bernoulli expression at points 1 and 2 on the surface of the sections.

¢ Notice that the channel is narrowed at 2 so it will have a different top width than the section at 1


2 21 1 2 2

1 22 2p v p vz g z gρ ρ+ + = + +

Venturi Flume

¢ Once again, our reference plane is the base of the channel and the points we are considering are at the top of the water

¢ So again, we can eliminate pressure from the expression


2 21 2

1 22 2v vz g z g+ = +

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Venturi Flume

¢ And substituting for the velocities


1 2

2 21 2

1 2

2 2

1 21 2

2 2

1 21 2

2 2

2 2

2 2t t

v vz g z g

Q QA Az g z g

Q Qz b z b

z g z g

+ = +

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠+ = +

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠+ = +

Venturi Flume

¢  Isolating and solving for Q


( )

( )

( )

1 2

1 2

1 2

1 2

1

2 2

1 21 2

2 2

1 22 1

2 2

2 11 2

2 2

22 1

1 2

2 12

1

2 2

2 2

2

1 1 2

2

1 1

t t

t t

t t

t t

t

Q Qz b z b

z g z g

Q Qz b z b

z g z g

Q Q z g z gz b z b

Q z g z gz b z b

z g z gQ

z b z

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠+ = +

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠− = −

⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− = −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

−=

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠ 2

2

2 tb

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

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Problem 7.10

¢ A venturi flume is placed in a rectangular channel that conveys water at 250 cfs. The upstream channel width is 12 ft, and the minimum width in the flume is 10 ft. The upstream water depth is 8 ft. What is the expected water depth at the minimum width?


Homework 22-1

¢ A rectangular channel 0.5 m wide conveys water. A partially raised sluice gate in the channel restricts the flow such that the upstream height is 1 m and the downstream height is 0.3 m.

¢ Determine the volume flow rate under the gate.


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Homework 22-2

¢ A venturi flume is placed in a rectangular channel. The upstream channel width is 3 m, and the minimum area section is of width 2 m. The upstream water height is 2 m; the height at the throat is 1 m.

¢ Determine the volume flow rate through the channel.


Homework 22-3

¢ An open-channel-flow apparatus in a hydraulics laboratory is rectangular in cross section and 16 in. wide. A venturi flume is to be placed in the channel to give the volume flow rate. The water height upstream of the flume is 18 in., and at the minimum width it is desired to make the depth about 14 in. Determine the minimum width of the venturi flume for a volume flow rate of 7 ft3/s.


open channel flow sluices and venturis - civil engineering channel flow... · 3 channel terminology...

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