Question

Water moves through a constricted pipe in steady, ideal flow. At the lower point shown in the figure below, the pressure is 1.65 ✕ 105 Pa and the pipe radius is 2.70 cm. At the higher point located at y = 2.50 m, the pressure is 1.30 ✕ 105 Pa and the pipe radius is 1.40 cm. (a) Find the speed of flow in the lower section. (b) Find the speed of flow in the upper section. (c) Find the volume flow rate through the pipe.

Answer 1

Density of water = = 1000 kg/m³

Gravitational acceleration = g = 9.81 m/s²

Pressure in the lower section of the pipe = P₁ = 1.65 x 10⁵ Pa = 165000 Pa

Radius of the lower section of the pipe = R₁ = 2.7 cm = 0.027 m

Speed of flow at the lower section of the pipe = V₁

Area of the pipe at the lower section = A₁

A₁ = R₁²

Pressure in the upper section of the pipe = P₂ = 1.3 x 10⁵ Pa = 130000 Pa

Radius of the upper section of the pipe = R₂ = 1.4 cm = 0.014 m

Speed of flow at the upper section of the pipe = V₂

Height of the upper section compared to the lower section = y = 2.5 m

Area of the pipe at the upper section = A₂

A₂ = R₂²

By continuity equation,

A₁V₁ = A₂V₂

R₁²V₁ = R₂²V₂

R₁²V₁ = R₂²V₂

(0.027)²V₁ = (0.014)²V₂

V₂ = 3.719V₁

By bernoulli's equation,

6415.48V₁² = 10475

V₁ = 1.278 m/s

V₂ = 3.719V₁

V₂ = (3.719)(1.278)

V₂ = 4,752 m/s

Volume flow rate = Q

Q = A₁V₁

Q = R₁²V₁

Q = (0.027)²(1.278)

Q = 2,927 x 10^-3 m³/s

a) Speed of flow in the lower section = 1.278 m/s

b) Speed of flow in the upper section = 4.752 m/s

c) Volume flow rate through the pipe = 2.927 x 10^-3 m³/s