Continuity Equation

  • Recognize continuity equation as a form of conservation of mass.

\[Q= A_1 \times v_1 = A_2 \times v_2\] Where: Q = the volumetric flow rate (m\(^3\)/s) A = the cross sectional area of flow (m\(^2\)) v = the mean velocity (m/s)

Solved Example:

62-1-01

10 m$^3$/h of water flows through a pipe with 100 mm inside diameter. The pipe is reduced to an inside dimension of 80 mm. Using continuity equation, calculate the velocity in the 100 mm pipe.

Solution:

(10 m$^3$/h) (1 / 3600 h/s) = $v_{100}$ (3.14 (0.1 m)$^2$ / 4)

\[v_{100} = (10 m^3/h) (1 / 3600 h/s) / (3.14 (0.1 m)^2 / 4) = 0.35 m/s\]

Using equation the velocity in the 80 mm pipe can be calculated

(10 m$^3$/h) (1 / 3600 h/s) = $v_{80}$ (3.14 (0.08 m)$^2$ / 4)

or \[v_{80} = (10 m^3/h) (1 / 3600 h/s) / (3.14 (0.08 m)^2 / 4) = 0.55 m/s\]

Correct Answer: C

Solved Example:

62-1-02

A fluid flowing through a pipe of diameter 450 mm with velocity 3 m/s is divided into two pipes of diameters 300 mm and 200 mm. The velocity of flow in 300 mm diameter pipe is 2.5 m/s, then the velocity of flow through 200 mm diameter pipe will be: (ESE Mechanical 2014)

Correct Answer: D

Solved Example:

62-1-03

When 0.1 m$^3$/s water flows through a pipe of area 0.25 m$^2$, which later reduces to 0.1 m$^2$, what is the velocity of flow in the reduced pipe? (SSC JE CE Oct 2020 Morning)

Correct Answer: C

Solved Example:

62-1-04

If an incompressible fluid enters a pipe with a velocity of 4 cm/s and moves out with a velocity of 2 cm/s, calculate the cross sectional area of the inlet if the diameter of the pipe at the outlet is 7 cm. (UPPCL JE CE 2016)

Correct Answer: B

Solved Example:

62-1-05

Water is coming out from a tap and falls vertically downwards. At the tap opening, the stream diameter is 20 mm with a uniform velocity of 2 m/s. Acceleration due to gravity is 9.81 m/s^2. Assuming steady, inviscid flow, constant atmospheric pressure everywhere and neglecting curvature and surface tension effects, the diameter in mm of the stream 0.5 m below the tap is approximately. (GATE ME 2013)

Correct Answer: B