A trapezoidal channel with side slopes of 1 to 1 has to be designed to convey 9m3/s at a velocity of 1.5 m/s so that the amount of concrete lining for the bed and sides is the minimum. Calculate the area of lining required for one meter length of canal.?

Question

A trapezoidal channel with side slopes of 1 to 1 has to be designed to convey  9m3/s at a velocity of 1.5 m/s so that the amount of concrete lining for the bed and sides is the minimum. Calculate the area of lining required for one meter length of canal.?

Answer

Design of Trapezoidal Channel

Given Data

Discharge (Q) = 9 m³/s

Velocity (V) = 1.5 m/s

Side Slopes = 1:1

Calculations

Step 1: Determine the Area of Cross-Section (A)

Since we know the discharge and velocity, we can calculate the area of cross-section using the formula:

A = Q / V

Substituting the given values, we get:

A = 9 / 1.5 = 6 m²

Step 2: Determine the Bottom Width (B)

Assuming a trapezoidal channel with side slopes of 1:1, we can use the formula for the area of a trapezoid to calculate the bottom width:

A = (B + b) * h / 2

where b = side slope height, h = depth of water

Since the side slopes are 1:1, b = h. Therefore, we can simplify the formula to:

A = B * h

Substituting the values of A and h, we get:

B = A / h = 6 / h

Step 3: Determine the Depth of Water (h)

We can use Manning's formula to calculate the depth of water:

V = (1 / n) * R^2/3 * S^1/2

where n = Manning's roughness coefficient, R = hydraulic radius, S = slope

Assuming a value of n = 0.015 for concrete lining, we can calculate the hydraulic radius using the formula:

R = A / P

where P = wetted perimeter

Since we know the bottom width and side slopes, we can calculate the wetted perimeter as:

P = B + 2 * h * sqrt(2) = 6 / h + 2 * h * sqrt(2)

Substituting the values of A, n, and P in Manning's formula, we get:

1.5 = (1 / 0.015) * (A / P)^2/3 * S^1/2

Solving for S, we get:

S = (1.5 * 0.015 / A^2/3)²

Substituting the value of S in Manning's formula, we get:

h = (A / (0.015 * R * S^1/2))^3/5

Step 4: Determine the Area of

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