Question

A 7.0-cm-diameter horizontal pipe gradually narrows to 5.0 cm . When water flows through this pipe at a certain rate, the gauge pressure in these two sections is 33.0 kPa and 23.0 kPa , respectively.

What is the volume rate of flow?

Answer 1

Solution-

density of water = 1000 kg/m³
D1 = 0.070 m A1 = πD1²/4 = 3.35E-3 m²
D2 = 0.050 m A2 = πD2²/4 = 1.96E-3 m²

P/1000 = fluid energy due to pressure
E1 = 33E3/E3 = 33 J/kg
E2 = 23E3/E3 = 23 J/kg
E1 - E2 = 10 J/kg

assuming pipe is level (doesn't change height):
(V2² - V1²)/2 = 10
V2² - V1² = 20
V1(A1) = V2(A2)
V1 = V2(A2/A1) = (1.96/3.35)V2 = 0.58V2
V2² - (0.58V2)² = 20
V2² - 0.35V2² = 20
0.65V2² = 20
V2² = 30.769
V2 =5.55 m/s
V2A2 = (5.55)(1.96E-3) = 10.9E-3 m³/s