Answer:
D ≈ 8.45 m
L ≈ 100.02 m
Explanation:
Given
Q = 350 m³/s (volumetric water flow rate passing through the stretch of channel, maximum capacity of the aqueduct)
y₁ - y₂ = h = 2.00 m (the height difference from the upper to the lower channels)
x = 100.00 m (distance between the upper and the lower channels)
We assume that:
- the upper and the lower channels are at the same pressure (the atmospheric pressure).
- the velocity of water in the upper channel is zero (v₁ = 0 m/s).
- y₁ = 2.00 m (height of the upper channel)
- y₂ = 0.00 m (height of the lower channel)
- g = 9.81 m/s²
- ρ = 1000 Kg/m³ (density of water)
We apply Bernoulli's equation as follows between the point 1 (the upper channel) and the point 2 (the lower channel):
P₁ + (ρ*v₁²/2) + ρ*g*y₁ = P₂ + (ρ*v₂²/2) + ρ*g*y₂
Plugging the known values into the equation and simplifying we get
Patm + (1000 Kg/m³*(0 m/s)²/2) + (1000 Kg/m³)*(9.81 m/s²)*(2 m) = Patm + (1000 Kg/m³*v₂²/2) + (1000 Kg/m³)*(9.81 m/s²)*(0 m)
⇒ v₂ = 6.264 m/s
then we apply the formula
Q = v*A ⇒ A = Q/v ⇒ A = Q/v₂
⇒ A = (350 m³/s)/(6.264 m/s)
⇒ A = 55.873 m²
then, we get the diameter of the pipe as follows
A = π*D²/4 ⇒ D = 2*√(A/π)
⇒ D = 2*√(55.873 m²/π)
⇒ D = 8.434 m ≈ 8.45 m
Now, the length of the pipe can be obtained as follows
L² = x² + h²
⇒ L² = (100.00 m)² + (2.00 m)²
⇒ L ≈ 100.02 m
