Which end has the higher pressure, the thick end or the fine end, give out the pressure difference between them.

Question

A water pipe has a thick and a fine end. The radius of the thick is 3cm, and the radius of the fine is 1cm. The thick end is 6 meters higher than the fine end. Ask: how about the flowing speed in fine end if the flowing speed is 9m/s at thick end. Which end has the higher pressure, the thick end or the fine end, give out the pressure difference between them. (Acceleration of gravity is 9.8m/s^2).

Answer 1

This problem can be solved using Bernoulli's equation which is

P2 + H2 x ρ x g + 1/2 x ρ x V2² = P1 + H1 x ρ x g +1/2 x ρ x V2²

where P2 and P2 are the pressures of the thick end and fine end respectively, ρ is the density of water in kg /m³, g is the acceleration of gravity 9.8 m/s², and V2 and V1 are the velocities of the water at the thick end and fine end respectively.

We are given the velocity of water at the thick end 9 m/s. We need to determine the velocity at the fine end.

The volume flow of the water can be obtained by V2 x area of thick end. This is 9 m/s x π x 0.03² which calculates to 2.54 x 10^-2 m³ /s. Since water is essentially an incompressible fluid, the volume flow at the fine end must be the same as at the thick end. Use V1 = 2.54 x 10^-2 / π x 0.01² = 80.89 m/s.

Now rearrange the Bernoulli equation to ΔP = P2 - P1 = ρ x g x (H1 - H2) +1/2 x ρ x (V1² - V2²).

Substituting known values ΔP = 997 kg/m³ x 9.8 m/s x (0 - 6 m) + 1/2 x 997 kg/m³ x (80.89² - 9²) = 3.2 x 10⁶ Newtons / m². Since this is a positive quantity, the thick end has a higher pressure.

Please check all calculations.