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A 4 m wide rectangular channel, having bed slope of 0.001 carries a discharge of 16 m3/s. Considering Manning's roughness coefficient = 0.012 and g = 10 m/s2, the category of the channel slope is
- a)horizontal
- b)mild
- c)critical
- d)steep
Correct answer is option 'B'. Can you explain this answer?
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Given:
Width of the channel (B) = 4 m
Discharge (Q) = 16 m^3/s
Bed slope (S) = 0.001
Manning's roughness coefficient (n) = 0.012
Acceleration due to gravity (g) = 10 m/s^2
To find:
The category of the channel slope
Solution:
The category of the channel slope is determined by the value of the hydraulic radius (R) and the critical slope (Sc).
Step 1: Calculate the hydraulic radius (R)
The hydraulic radius is given by the formula:
R = A/P
where A is the cross-sectional area of flow and P is the wetted perimeter.
Step 1.1: Calculate the cross-sectional area (A)
The cross-sectional area of flow is given by the formula:
A = B * y
where B is the width of the channel and y is the depth of flow.
Let's assume the depth of flow (y) as 2 m.
A = 4 * 2 = 8 m^2
Step 1.2: Calculate the wetted perimeter (P)
The wetted perimeter is given by the formula:
P = B + 2y
where B is the width of the channel and y is the depth of flow.
P = 4 + 2(2) = 8 m
Step 1.3: Calculate the hydraulic radius (R)
R = A/P = 8/8 = 1 m
Step 2: Calculate the critical slope (Sc)
The critical slope is given by the formula:
Sc = (n^2 * Q^2)/(g * R^4/3)
where n is Manning's roughness coefficient, Q is the discharge, g is the acceleration due to gravity, and R is the hydraulic radius.
Sc = (0.012^2 * 16^2)/(10 * 1^(4/3))
Sc = 0.0002304/1
Sc = 0.0002304
Step 3: Compare the bed slope (S) with the critical slope (Sc)
If the bed slope (S) is less than the critical slope (Sc), the category of the channel slope is considered mild.
In this case, S = 0.001, which is greater than Sc = 0.0002304.
Therefore, the category of the channel slope is mild (option B).
Width of the channel (B) = 4 m
Discharge (Q) = 16 m^3/s
Bed slope (S) = 0.001
Manning's roughness coefficient (n) = 0.012
Acceleration due to gravity (g) = 10 m/s^2
To find:
The category of the channel slope
Solution:
The category of the channel slope is determined by the value of the hydraulic radius (R) and the critical slope (Sc).
Step 1: Calculate the hydraulic radius (R)
The hydraulic radius is given by the formula:
R = A/P
where A is the cross-sectional area of flow and P is the wetted perimeter.
Step 1.1: Calculate the cross-sectional area (A)
The cross-sectional area of flow is given by the formula:
A = B * y
where B is the width of the channel and y is the depth of flow.
Let's assume the depth of flow (y) as 2 m.
A = 4 * 2 = 8 m^2
Step 1.2: Calculate the wetted perimeter (P)
The wetted perimeter is given by the formula:
P = B + 2y
where B is the width of the channel and y is the depth of flow.
P = 4 + 2(2) = 8 m
Step 1.3: Calculate the hydraulic radius (R)
R = A/P = 8/8 = 1 m
Step 2: Calculate the critical slope (Sc)
The critical slope is given by the formula:
Sc = (n^2 * Q^2)/(g * R^4/3)
where n is Manning's roughness coefficient, Q is the discharge, g is the acceleration due to gravity, and R is the hydraulic radius.
Sc = (0.012^2 * 16^2)/(10 * 1^(4/3))
Sc = 0.0002304/1
Sc = 0.0002304
Step 3: Compare the bed slope (S) with the critical slope (Sc)
If the bed slope (S) is less than the critical slope (Sc), the category of the channel slope is considered mild.
In this case, S = 0.001, which is greater than Sc = 0.0002304.
Therefore, the category of the channel slope is mild (option B).
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A 4 m wide rectangular channel, having bed slope of 0.001 carries a discharge of 16 m3/s. Considering Manning's roughness coefficient = 0.012 and g = 10 m/s2, the category of the channel slope isa)horizontalb)mildc)criticald)steepCorrect answer is option 'B'. Can you explain this answer?
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