A very wide rectangular channel is designed to carry a discharge of 8...
Given data, q = 8 m
3 /s/meter width
By using Strickler’s equation
y = 1.0027 meter
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A very wide rectangular channel is designed to carry a discharge of 8...
Solution:
Given data:
Discharge (Q) = 8 m3/s/m
Normal Depth (y) = 2 m
Step 1: Calculation of hydraulic radius (R)
The hydraulic radius is defined as the ratio of the cross-sectional area of the flow to the wetted perimeter. It is given by the formula:
R = A/P
Where,
A = Cross-sectional area of flow
P = Wetted perimeter
For a rectangular channel, the cross-sectional area and wetted perimeter are given by:
A = y*B
P = 2*y+B
Where,
B = Width of the channel
Substituting the values, we get:
A = 2* B
P = 2*y+ B
R = (2* B) / (2*y+ B)
Step 2: Calculation of Manning’s roughness coefficient (n)
The Manning’s roughness coefficient depends on the grain size of the channel. It can be calculated using Strickler’s equation, which relates the roughness coefficient to the grain diameter. The equation is given by:
n = (k* D84) / (R1/6)
Where,
k = Constant (depends on the units used)
D84 = Particle size for which 84% of the sediment is finer
R = Hydraulic radius
The value of k depends on the units used for diameter (m, cm, or mm). For m, k = 1.0; for cm, k = 1.49; and for mm, k = 1.486.
In this case, the engineer used the grain diameter in mm instead of in metre. Therefore, we need to convert the diameter to metres before using Strickler’s equation.
Step 3: Calculation of correct normal depth (y')
The normal depth can be calculated using Manning’s equation, which relates the flow rate, channel slope, and roughness coefficient to the normal depth. The equation is given by:
Q = (1/n)*A*(R2/3)*S1/2
Where,
S = Channel slope
Substituting the values, we get:
8 = (1/n)*(2* B)*[(2* B) / (2*y+ B)]2/3*(S1/2)
We need to solve this equation for y.
Step 4: Calculation of D84
The particle size for which 84% of the sediment is finer can be obtained from the following equation:
D84 = 6.8*(k* n* R1/6)
Substituting the values, we get:
D84 = 6.8*(1.486* n* R1/6)
Step 5: Calculation of correct roughness coefficient (n')
Using the correct value of D84, we can calculate the correct value of Manning’s roughness coefficient using Strickler’s equation:
n' = (k* D84) / (R1/6)
Substituting the values, we get:
n' = (1.486* D84) / (R1/6)
Step 6: Calculation of correct normal depth (y')
Using the corrected values of n and R, we can solve Manning’s equation to obtain the correct normal depth:
8 = (1/n')*(2* B)*[(2* B) / (2*y'+ B)]2/3*(S1/2)
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