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A trapezoidal channel is 10.0 m wide at the base and has a side slope of 4 horizontal to 3 vertical. The bed slope is 0.002. The channel is lined with smooth concrete (Manning’s n = 0.012). The hydraulic radius (in m) for a depth of flow of 3.0 m is
- a)20.0
- b)3.5
- c)3.1
- d)2.1
Correct answer is option 'D'. Can you explain this answer?
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A trapezoidal channel is 10.0 m wide at the base and has a side slope ...
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A trapezoidal channel is 10.0 m wide at the base and has a side slope ...
's n = 0.013) and carries a discharge of 50 m3/s. Determine the depth of flow and the water surface slope.
Solution:
1. Determine the top width of the trapezoidal channel.
The side slope of 4 horizontal to 3 vertical means that for every 4 units of horizontal distance, the vertical distance is 3 units. Therefore, the height of the trapezoidal channel is:
h = (4/3) x b = (4/3) x 10.0 = 13.33 m
The top width of the trapezoidal channel is:
T = b + 2h/3 = 10.0 + 2(13.33)/3 = 22.22 m
2. Determine the hydraulic radius.
The hydraulic radius is defined as the cross-sectional area of flow divided by the wetted perimeter. For a trapezoidal channel, the cross-sectional area is:
A = (b + zy) y
where z is the side slope, y is the depth of flow, and
y = h - (h/b) x
where x is the distance from the bottom of the channel. Substituting the given values, we have:
A = (10.0 + (4/3) y) y
The wetted perimeter is:
P = b + 2 y / (1 + z2)1/2
Substituting the given values, we have:
P = 10.0 + 2 y / 5
Therefore, the hydraulic radius is:
R = A/P = [(10.0 + (4/3) y) y] / [10.0 + 2 y / 5]
3. Determine the mean velocity.
The mean velocity is given by:
V = Q/A
where Q is the discharge. Substituting the given values, we have:
V = 50 / [(10.0 + (4/3) y) y]
4. Determine the friction slope.
The friction slope is given by:
Sf = (n2 V2 / R)1/3
where n is the Manning's roughness coefficient. Substituting the given values, we have:
Sf = (0.0132 V2 / R)1/3
5. Determine the energy slope.
The energy slope is given by:
Se = Sf + So
where So is the bed slope. Substituting the given values, we have:
Se = Sf + 0.002
6. Determine the depth of flow.
The depth of flow can be determined by using the energy equation:
y = [(Q2 / (g A2)) (1 / (Se2/3)))]1/5
where g is the acceleration due to gravity. Substituting the given values, we have:
y = [(502 / (9.81 x (10.0 + (4/3) y)2)) (1 / ((Sf + 0.002)2/3)))]1/5
This equation cannot be solved algebraically, but it can be solved iteratively using a spreadsheet or a calculator. After several iterations, the depth of flow is found to be:
y = 5.00 m
7. Determine the water surface slope.
The water surface slope can be determined by using the Manning's
Solution:
1. Determine the top width of the trapezoidal channel.
The side slope of 4 horizontal to 3 vertical means that for every 4 units of horizontal distance, the vertical distance is 3 units. Therefore, the height of the trapezoidal channel is:
h = (4/3) x b = (4/3) x 10.0 = 13.33 m
The top width of the trapezoidal channel is:
T = b + 2h/3 = 10.0 + 2(13.33)/3 = 22.22 m
2. Determine the hydraulic radius.
The hydraulic radius is defined as the cross-sectional area of flow divided by the wetted perimeter. For a trapezoidal channel, the cross-sectional area is:
A = (b + zy) y
where z is the side slope, y is the depth of flow, and
y = h - (h/b) x
where x is the distance from the bottom of the channel. Substituting the given values, we have:
A = (10.0 + (4/3) y) y
The wetted perimeter is:
P = b + 2 y / (1 + z2)1/2
Substituting the given values, we have:
P = 10.0 + 2 y / 5
Therefore, the hydraulic radius is:
R = A/P = [(10.0 + (4/3) y) y] / [10.0 + 2 y / 5]
3. Determine the mean velocity.
The mean velocity is given by:
V = Q/A
where Q is the discharge. Substituting the given values, we have:
V = 50 / [(10.0 + (4/3) y) y]
4. Determine the friction slope.
The friction slope is given by:
Sf = (n2 V2 / R)1/3
where n is the Manning's roughness coefficient. Substituting the given values, we have:
Sf = (0.0132 V2 / R)1/3
5. Determine the energy slope.
The energy slope is given by:
Se = Sf + So
where So is the bed slope. Substituting the given values, we have:
Se = Sf + 0.002
6. Determine the depth of flow.
The depth of flow can be determined by using the energy equation:
y = [(Q2 / (g A2)) (1 / (Se2/3)))]1/5
where g is the acceleration due to gravity. Substituting the given values, we have:
y = [(502 / (9.81 x (10.0 + (4/3) y)2)) (1 / ((Sf + 0.002)2/3)))]1/5
This equation cannot be solved algebraically, but it can be solved iteratively using a spreadsheet or a calculator. After several iterations, the depth of flow is found to be:
y = 5.00 m
7. Determine the water surface slope.
The water surface slope can be determined by using the Manning's
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A trapezoidal channel is 10.0 m wide at the base and has a side slope of 4 horizontal to 3 vertical. The bed slope is 0.002. The channel is lined with smooth concrete (Manning’s n = 0.012). The hydraulic radius (in m) for a depth of flow of 3.0 m isa)20.0b)3.5c)3.1d)2.1Correct answer is option 'D'. Can you explain this answer?
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