Question

A 40 cm diameter pipe, conveying water, branches into two pipes of diameters 30 cm and 20 cm respectively. If the average velocity in the 40 cm diameter pipe is 3 m/s. Find the discharge in this pipe. Also determine the velocity in 20 cm pipe if the average velocity in 30 cm diameter pipe is 2 m/s.

Answer 1

Diameter of pipe 1 $d_1=40\ cm=0.4\ m$

C/S Area of pipe 1 $A_1 =\dfrac{\pi(d_1)^2}{4}=0.04\pi\ m^2$

Diameter of pipe 2 $d_2=30\ cm=0.3\ m$

C/S Area of pipe 2 $A_2 =\dfrac{\pi(d_1)^2}{4}=0.0225\pi\ m^2$

Diameter of pipe 3 $d_3=20\ cm=0.2\ m$

C/S Area of pipe 3 $A_3 =\dfrac{\pi(d_3)^2}{4}=0.01\pi\ m^2$

Discharge through pipe 1 $Q_1=A_1v_1=0.04\pi\ m^2 (3\ m/s)$

$=0.12\pi\ m^3/s=0.3770\ m^3/s$

$Q_1=0.3770\ m^3/s$

Velocity at pipe 2 $v_2=2\ m/s$

Velocity at pipe 3 $v_3=?$

From continuity $Q_1=Q_2+Q_3$

Discharge through pipe 2 $Q_2=A_2v_2=0.0225\pi\ m^2 (2\ m/s)$

$=0.045\pi\ m^3/s=0.1414\ m^3/s$

Discharge through pipe 3 $Q_3=Q_1-Q_2=0.12\pi\ m^3/s-0.045\pi\ m^3/s$

$=0.075\pi\ m^3/s$

Velocity at pipe 3 $v_3=\dfrac{Q_3}{A_3}=\dfrac{0.075\pi\ m^3/s}{0.01\pi\ m^2 }=7.5\ m/s$

$v_3=7.5\ m/s$