A sump pump used to drain water from the basement of houses

A sump pump (used to drain water from the basement of houses built below the water table) is draining a flooded basement at the rate of 0.900 L/s, with an output pressure of 4.00 10⁵ N/m². You may neglect frictional losses in both parts of this problem.

(a)

The water enters a hose with a 3.00 cm inside diameter and rises 3.10 m above the pump. What is its pressure (in N/m²) at this point?

  N/m²

(b)

The hose goes over the foundation wall, losing 1.80 m in height, and widens to 4.00 cm diameter. What is the pressure now (in N/m²)?

  N/m²

Solution

a) P₂ = P₁ + p g (h₁ - h₂)

= (4.00 * 10⁵) + (1000 * 9.8 * 3.1)

= 4.30 * 10⁵ N/m²

b) V = v₁ A₁

v₁ = 0.900 * 10^-3 / pi * 0.015² = 1.27 m/s

v₂ = 0.900 * 10^-3 / pi * 0.02² = 0.716 m/s

P₂ = P₁ + 1/2 p (v₁² - v₂²) + p g (h₁ - h₂)

= (4.00 * 10⁵) + 1/2 * 1000 (1.27² - 0.716²) + 1000 * 9.8 (0 - (3.1 - 1.8))

= 3.88 * 10⁵ N/m²