# Continuity Equation

The continuity equation describes the transport of some quantities like fluid or gas. The equation explains how a fluid conserves mass in its motion. Many physical phenomena like energy, mass, momentum, natural quantities, and electric charge are conserved using the continuity equations.

This equation provides very useful information about the flow of fluids and their behaviour during its flow in a pipe or hose. The hose, a flexible tube, whose diameter decreases along its length has a direct consequence. The volume of water flowing through the hose must be equal to the flow rate on the other end. The know flow rate formula visit BYJU’S.

Continuity Equation is applied on tubes, pipes, rivers, ducts with flowing fluids or gases and many more. Continuity equation can be expressed in an integral form and is applied in the finite region or differential form which is applied at a point.

### The Equation of Continuity and can be expressed as:

$m = \rho _{i 1} \ v _{i 1} \ A _{i 1} + \rho _{i 2} \ v _{i 2} \ A _{i 2} + ….. + \rho _{i n} \ v _{i n} \ A _{i m}$

$m = \rho _{o 1} \ v _{o 1} \ A _{o 1} + \rho _{o 2} \ v _{o 2} \ A _{o 2} + ….. + \rho _{o n} \ v _{o n} \ A _{o m}……….. (1)$

Where,

$m$ = Mass flow rate

$\rho$ = Density

$v$ = Speed

$A$ = Area

With uniform density equation (1) it can be modified to:

$q = v _{i 1} \ A _{i1} + v _{i2} \ A _{i2} + …. + v _{i n} \ A _{i m}$

$q = v _{o 1} \ A _{o1} + v _{o2} \ A _{o2} + …. + v _{o n} \ A _{o m}………..(2)$

Where,

$q$ = Flow rate

$\rho _{i 1} = \rho _{i 2} .. = \rho _{i n} = \rho _{o 1} = \rho _{o 2} = …. = \rho _{o m}$

## Fluid Dynamics

The continuity equation in fluid dynamics describes that in any steady state process, the rate at which mass leaves the system is equal to the rate at which mass enters a system.

The differential form of the continuity equation is:

$\frac{\partial \rho}{\partial t} + \bigtriangledown \cdot \left (\rho u \right) = 0$

Where,

$t$ = Time

$\rho$ = Fluid density

$u$ = flow velocity vector field.

## Continuity Equation Example

Question: Calculate the velocity if $\small 10 \ m^{3}/h$ of water flows through a 100 mm inside diameter pipe. If the pipe is reduced to 80 mm inside diameter.

Solution

Velocity of 100 mm pipe

Using the equation (2), to calculate the velocity of 100 mm pipe

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

Or

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

$= 0.35 \ m/s$

Velocity of 80 mm pipe

Using the equation (2), to calculate the velocity of 80 mm pipe

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

Or

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

$= 0.55 \ m/s$

Video below helps you to understand the Continuity Equation in detail.

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