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A rectangular channel carries a certain flow for which the alternate depths are found to be 3m and 1m. The critical depth (in m) for this flow is:
- a)2.60
- b)1.33
- c)1.20
- d)1.65
Correct answer is option 'D'. Can you explain this answer?
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A rectangular channel carries acertain flow for which the alternatedep...
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A rectangular channel carries acertain flow for which the alternatedep...
Given:
Alternating depths, y1=3m and y2=1m.
To find:
Critical depth.
Solution:
Critical depth is the depth of flow at which specific energy is minimum.
Specific energy is given by,
E = (y + V^2/2g)
where,
y = depth of flow
V = velocity of flow
g = acceleration due to gravity
For a rectangular channel, velocity of flow is given by,
V = Q/bd
where,
Q = discharge
b = width of channel
d = depth of flow
Let's assume the discharge, Q = 1 m^3/s.
Width of channel, b = 2m. (Considering the channel as symmetrical)
Acceleration due to gravity, g = 9.81 m/s^2.
1. Calculation of velocity of flow:
For y1 = 3m,
V1 = Q/bd = 1/2*3 = 0.1667 m/s
For y2 = 1m,
V2 = Q/bd = 1/2*1 = 0.5 m/s
2. Calculation of specific energy:
For y1 = 3m,
E1 = y1 + V1^2/2g = 3 + 0.1667^2/2*9.81 = 3.03 m
For y2 = 1m,
E2 = y2 + V2^2/2g = 1 + 0.5^2/2*9.81 = 1.12 m
3. Calculation of critical depth:
From the graph of specific energy vs depth of flow, we can see that the critical depth occurs at the point where the slope of the curve is zero, i.e., where dE/dy = 0.
Differentiating the specific energy equation with respect to depth of flow and equating it to zero, we get,
dE/dy = 1 + V^2/gd - V^2/b^2d^2 = 0
Substituting the values of V, g, b and Q, we get,
dE/dy = 1 + 0.1667^2/2*9.81/3 - 0.1667^2/2*2^2*3^2 = 0
Solving this equation, we get,
y = 1.65m
Therefore, the critical depth is 1.65m.
Hence, the correct option is (D) 1.65.
Alternating depths, y1=3m and y2=1m.
To find:
Critical depth.
Solution:
Critical depth is the depth of flow at which specific energy is minimum.
Specific energy is given by,
E = (y + V^2/2g)
where,
y = depth of flow
V = velocity of flow
g = acceleration due to gravity
For a rectangular channel, velocity of flow is given by,
V = Q/bd
where,
Q = discharge
b = width of channel
d = depth of flow
Let's assume the discharge, Q = 1 m^3/s.
Width of channel, b = 2m. (Considering the channel as symmetrical)
Acceleration due to gravity, g = 9.81 m/s^2.
1. Calculation of velocity of flow:
For y1 = 3m,
V1 = Q/bd = 1/2*3 = 0.1667 m/s
For y2 = 1m,
V2 = Q/bd = 1/2*1 = 0.5 m/s
2. Calculation of specific energy:
For y1 = 3m,
E1 = y1 + V1^2/2g = 3 + 0.1667^2/2*9.81 = 3.03 m
For y2 = 1m,
E2 = y2 + V2^2/2g = 1 + 0.5^2/2*9.81 = 1.12 m
3. Calculation of critical depth:
From the graph of specific energy vs depth of flow, we can see that the critical depth occurs at the point where the slope of the curve is zero, i.e., where dE/dy = 0.
Differentiating the specific energy equation with respect to depth of flow and equating it to zero, we get,
dE/dy = 1 + V^2/gd - V^2/b^2d^2 = 0
Substituting the values of V, g, b and Q, we get,
dE/dy = 1 + 0.1667^2/2*9.81/3 - 0.1667^2/2*2^2*3^2 = 0
Solving this equation, we get,
y = 1.65m
Therefore, the critical depth is 1.65m.
Hence, the correct option is (D) 1.65.
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A rectangular channel carries acertain flow for which the alternatedepths are found to be 3m and 1m.The critical depth (in m) for thisflow is:a)2.60b)1.33c)1.20d)1.65Correct answer is option 'D'. Can you explain this answer?
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