A 6.0-cm-diameter horizontal pipe gradually narrows to 4.5 cm. When water flows through this pipe at a certain rate, the gauge pressure in these two sections is 31.0 kPa and 24.0 kPa, respectively. What is the volume rate of flow?

0.0072 m³/s

Explanation:

Using Bernoulli's law

P₁ + 1/2ρv₁² = P₂ + 1/2ρv₂ since the pipe is horizontal

1/2ρv₂² - 1/2ρv₁² = P₁ - P₂

flow rate is constant

A₁v₁ = A₂v₂

A₁ = πr₁² = π (0.06/2) ² = 0.0028278 m²

A₂ = πr₂² = π (0.0225) ² = 0.00159 m²

v₁ = (A₂ / A₁) v₂

v₁ = (0.00159 m² / 0.0028278 m²) v₂ = 0.562 v₂

substitute v₁ into the Bernoulli's equation

1/2ρv₂² - 1/2ρv₁² = P₁ - P₂

500 (1 - 0.3161) v₂² = (31.0 - 24) * 10³ Pa

341.924 v₂² = 7000

v₂² = 20.472

v₂ = √ 20.472 = 4.525 m/s

volume follow rate = 0.00159 m² * 4.525 m/s = 0.0072 m³/s