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A hydraulically efficient trapezoidal channel section has a uniform flow depth of 2m. The bed width (expressed in m) of the channel is
- a)2.31
- b)4
- c)2
- d)1.154
Correct answer is option 'A'. Can you explain this answer?
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To determine the bed width of a trapezoidal channel section with a uniform flow depth, we can use the concept of hydraulic efficiency. The hydraulic efficiency of a channel section refers to how well it utilizes the available cross-sectional area to convey flow.
Given:
Uniform flow depth (y) = 2m
Let's determine the bed width using the Manning's equation, which relates the flow rate (Q) to the channel geometry and Manning's roughness coefficient (n):
Q = (1/n) * A * R^(2/3) * S^(1/2)
Where:
Q = flow rate
A = cross-sectional area of flow
R = hydraulic radius
S = slope of the channel bed
Since the channel section is trapezoidal, the cross-sectional area can be expressed as:
A = y * ((b1 + b2) / 2)
Where:
b1 = width of the channel at the water surface
b2 = width of the channel at the bed
The hydraulic radius can be calculated as:
R = A / P
P = wetted perimeter = b1 + b2 + 2 * √(y^2 + (b1 - b2)^2)
Assuming a uniform flow and a trapezoidal channel section, the slope of the channel bed (S) can be approximated as:
S = (b1 - b2) / L
L = length of the channel section
Let's substitute these equations into the Manning's equation:
Q = (1/n) * y * ((b1 + b2) / 2) * (A / P)^(2/3) * ((b1 - b2) / L)^(1/2)
Since the flow is uniform, the flow rate (Q) remains constant. We can simplify the equation by canceling out common terms and rearranging:
(b1 + b2) / (b1 - b2) = (2/3) * (n^2) * (y / S)^(1/2)
Now, substituting the given values:
(b1 + b2) / (b1 - b2) = (2/3) * (n^2) * (2 / S)^(1/2)
To maximize hydraulic efficiency, we need to minimize the ratio (b1 + b2) / (b1 - b2). This occurs when (b1 + b2) is as close to 2 times (b1 - b2) as possible.
In this case, the bed width (b2) is unknown. However, we can assume that the trapezoidal channel section is hydraulically efficient. This implies that the side slopes are typically 1:1 (vertical:horizontal). Therefore, the ratio of (b1 + b2) / (b1 - b2) can be approximated as 2.
So, we have:
2 = (2/3) * (n^2) * (2 / S)^(1/2)
Simplifying the equation:
1 = (2/3) * (n^2) * (2 / S)^(1/2)
We can rearrange this equation to solve for the slope (S):
S = (4/3) * (n^2)
Since the slope (S) is constant for a given channel, we
Given:
Uniform flow depth (y) = 2m
Let's determine the bed width using the Manning's equation, which relates the flow rate (Q) to the channel geometry and Manning's roughness coefficient (n):
Q = (1/n) * A * R^(2/3) * S^(1/2)
Where:
Q = flow rate
A = cross-sectional area of flow
R = hydraulic radius
S = slope of the channel bed
Since the channel section is trapezoidal, the cross-sectional area can be expressed as:
A = y * ((b1 + b2) / 2)
Where:
b1 = width of the channel at the water surface
b2 = width of the channel at the bed
The hydraulic radius can be calculated as:
R = A / P
P = wetted perimeter = b1 + b2 + 2 * √(y^2 + (b1 - b2)^2)
Assuming a uniform flow and a trapezoidal channel section, the slope of the channel bed (S) can be approximated as:
S = (b1 - b2) / L
L = length of the channel section
Let's substitute these equations into the Manning's equation:
Q = (1/n) * y * ((b1 + b2) / 2) * (A / P)^(2/3) * ((b1 - b2) / L)^(1/2)
Since the flow is uniform, the flow rate (Q) remains constant. We can simplify the equation by canceling out common terms and rearranging:
(b1 + b2) / (b1 - b2) = (2/3) * (n^2) * (y / S)^(1/2)
Now, substituting the given values:
(b1 + b2) / (b1 - b2) = (2/3) * (n^2) * (2 / S)^(1/2)
To maximize hydraulic efficiency, we need to minimize the ratio (b1 + b2) / (b1 - b2). This occurs when (b1 + b2) is as close to 2 times (b1 - b2) as possible.
In this case, the bed width (b2) is unknown. However, we can assume that the trapezoidal channel section is hydraulically efficient. This implies that the side slopes are typically 1:1 (vertical:horizontal). Therefore, the ratio of (b1 + b2) / (b1 - b2) can be approximated as 2.
So, we have:
2 = (2/3) * (n^2) * (2 / S)^(1/2)
Simplifying the equation:
1 = (2/3) * (n^2) * (2 / S)^(1/2)
We can rearrange this equation to solve for the slope (S):
S = (4/3) * (n^2)
Since the slope (S) is constant for a given channel, we
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A hydraulically efficient trapezoidal channel section has a uniform flow depth of 2m. The bed width (expressed in m) of the channel isa)2.31b)4c)2d)1.154Correct answer is option 'A'. Can you explain this answer?
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A hydraulically efficient trapezoidal channel section has a uniform flow depth of 2m. The bed width (expressed in m) of the channel isa)2.31b)4c)2d)1.154Correct answer is option 'A'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A hydraulically efficient trapezoidal channel section has a uniform flow depth of 2m. The bed width (expressed in m) of the channel isa)2.31b)4c)2d)1.154Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A hydraulically efficient trapezoidal channel section has a uniform flow depth of 2m. The bed width (expressed in m) of the channel isa)2.31b)4c)2d)1.154Correct answer is option 'A'. Can you explain this answer?.
A hydraulically efficient trapezoidal channel section has a uniform flow depth of 2m. The bed width (expressed in m) of the channel isa)2.31b)4c)2d)1.154Correct answer is option 'A'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A hydraulically efficient trapezoidal channel section has a uniform flow depth of 2m. The bed width (expressed in m) of the channel isa)2.31b)4c)2d)1.154Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A hydraulically efficient trapezoidal channel section has a uniform flow depth of 2m. The bed width (expressed in m) of the channel isa)2.31b)4c)2d)1.154Correct answer is option 'A'. Can you explain this answer?.
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