Water at a gauge pressure of 3.8 atm at street level flows in to an office building at a speed of 0.06 m/s through a pipe 5.0 cm in diameter. The pipes taper down to 2.6cm in diameter by the top floor, 20 m above. Calculate the flow velocity and the gauge pressure in such a pipe on the top floor. Assume no branch pipe and ignore viscosity.

a) Flow velocity = = 0.222 m/s

b) Gauge pressure = 1.84 atm

Explanation:

P₁ = 3.8 atm = 3.8 * 10⁵ Pa

v₁ = 0.06 m/s

d₁ = 5.0 cm

r₁ = 5/2 = 2.5 cm

At the gauge pressure, the water was at street level, h₁ = 0 m

d₂ = 2.6 cm

r₂ = 2.6/2 = 1.3 cm

h₂ = 20 m

Density of water,

Assumptions: Flow is steady and laminar, no viscosity, no branch pipe

a) calculate the flow velocity

To calculate the flow velocity, use the continuity equation

A₁v₁ = A₂v₂.............(1)

A₁ = πr²₁ = π(2.5)² = 6.25 π

A₂ = πr²₂ = π(1.3)² = 1.69π

Substituting the appropriate values into equation (1)

6.25 π * 0.06 = 1.69π * v₂

v₂ = 0.375/1.69

v₂ = 0.222 m/s

b) Calculate the gauge pressure

Using the Bernoulli equation:

pythagoras' theorem

square root (-5^2+9^2)

square root (25+81)

square root (106)

angle is tan^-1(9/5)

neg x, pos y