Question

The pressure of water flowing through a 5.9×10^-2 m-radius pipe at a speed of 1.2 m/s is 2.2×10^5 N/m^2.

a) What is the flow rate of the water?

b) What is the pressure in the water after it goes up a 5.8 m-high hill and flows in a 4.2×10^-2 m-radius pipe?

Answer 1

Density of water = = 1000 kg/m³

Gravitational acceleration = g = 9.81 m/s²

Pressure at point 1 = P₁ = 2.2 x 10⁵ N/m² = 220000 N/m²

Radius of the pipe at point 1 = R₁ = 5.9 x 10^-2 m

Speed of flow at point 1 = V₁ = 1.2 m/s

Area of the pipe at point 1 = A₁ = R₁²

Pressure at point 2 = P₂

Radius of the pipe at point 2 = R₂ = 4.2 x 10^-2 m²

Speed of flow at point 2 = V₂

Height of point 2 compared to point 1 = h = 5.8 m

Area of the pipe at point 2 = A₂ = R₂²

Flow rate of water = Q

Q = A₁V₁

Q = R₁²V₁

Q = (5.9x10^-2)²(1.2)

Q = 1.312 x 10^-2 m³/s

By continuity equation,

A₁V₁ = A₂V₂

R₁²V₁ = R₂²V₂

R₁²V₁ = R₂²V₂

(5.9x10^-2)²(1.2) = (4.2x10^-2)V₂

V₂ = 2.368 m/s

By bernoulli's equation,

P₂ = 161018.29 N/m²

a) Flow rate of water = 1.312 x 10^-2 m³/s

b) Pressure in the water after it goes up a 5.8 m high hill and flows in a 4.2x10^-2m radius pipe = 161018.29 N/m²