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.

The continuity equation provides beneficial information about the flow of fluids and their behaviour during their flow in a pipe or hose. 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.

Deriving the Equation of Continuity

Continuity Equation
\(m = \rho _{i 1} \ v _{i 1} \ A _{i 1} + \rho _{i 2} \ v _{i 2} \ A _{i 2} + ….. + \rho _{i n} \ v _{i n} \ A _{i m}\)

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

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 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 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

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

\(= 0.55 \ m/s\)

Watch the video to learn more about the practical application of the equation of continuity, its derivation, fluid velocity and different types of flow: steady flow and turbulent flow.


Frequently Asked Questions – FAQs

What is the principle of continuity?

The continuity equation describes the transport of some quantities like fluid or gas. For example, 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.

Where is the Equation of Continuity used?

The continuity equation provides very useful information about the flow of fluids and their behaviour during their flow in a pipe or hose. Continuity Equation is applied on tubes, pipes, rivers, ducts with flowing fluids or gases and many more.

What is the importance of the continuity equation?

The continuity equation proves the law of conservation of mass in fluid dynamics.

What does the continuity equation in fluid dynamics describe?

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.

When can the Continuity equation be used?

The continuity equation applies to all fluids, compressible and incompressible flow, Newtonian and non-Newtonian fluids.

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