Question

A 30 cm diameter pipe carries oil of sp. gr. 0.8 at a velocity of 2 m/s. At another section the diameter is 20 cm. Find the velocity at this section and also mass rate of flow of oil.

Answer 1

Diameter of pipe 1 $d_1=30 cm=0.3\ m$

C/S Area of pipe 1 $A_1 =\dfrac{\pi(d_1)^2}{4}=0.0225\pi\ m^2$

Diameter of pipe 2 $d_2=20\ cm=0.2\ m$

C/S Area of pipe 2 $A_2 =\dfrac{\pi(d_2)^2}{4}=0.01\pi\ m^2$

Velocity at pipe 1 $v_1=2\ m/s$

Velocity at pipe 2 $v_2=?$

Discharge through pipe 1 $Q_1=A_1v_1=0.0225\pi\ m^2 (2\ m/s)$

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

From continuity $Q_1=Q_2+Q_3$

Discharge through pipe 2 $Q_2=A_2v_2=0.01\pi\ m^2 (v_2)$

$=Q_1=0.045\pi\ m^3/s$

Velocity at pipe 2 $v_2=\dfrac{Q_2}{A_2}=\dfrac{0.045\pi\ m^3/s}{0.01\pi\ m^2 }=4.5\ m/s$

$v_2=4.5\ m/s$

Mass rate of flow 

$$\dfrac{\Delta m}{\Delta t}=\rho Av=0.8\cdot1000kg/m^3(0.045\pi\ m^3/s)$$

$$=113.097 kg/s$$

$\dfrac{\Delta m}{\Delta t}=113.097 kg/s$