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
\(m = \rho _{i 1} \ v _{i 1} \ A _{i 1} + \rho _{i 2} \ v _{i 2} \ A _{i 2} + ….. + \rho _{i n} \ v _{i n} \ A _{i m}\)
\(m = \rho _{o 1} \ v _{o 1} \ A _{o 1} + \rho _{o 2} \ v _{o 2} \ A _{o 2} + ….. + \rho _{o n} \ v _{o n} \ A _{o m}……….. (1)\)
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?
Where is the Equation of Continuity used?
What is the importance of the continuity equation?
What does the continuity equation in fluid dynamics describe?
When can the Continuity equation be used?
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