Continuity Equation: Meaning, Derivation and Calculator

01Understand in 30 seconds

When the pipe narrows, the flow speeds up.For a nearly constant-density fluid such as water, the same volume must pass in the same amount of time.

The continuity equation says fluid does not disappear

When water flows from a wide pipe into a narrow pipe, it moves faster in the narrow section. The mass passing per unit time must stay the same. The continuity equation expresses this idea for any flow.

1
Observe
The outlet flow area is smaller
2
Conserve
The mass passing per unit time does not change
3
Result
At constant density, the outlet speed increases

The three quantities in the diagram

$A$ Flow area: Area perpendicular to the flow
$V$ Mean speed: Velocity averaged over the area
$\rho$ Density: Mass per unit volume

Constant-density flow

A_1V_1=A_2V_2

Mean speed increases by the same factor that flow area decreases.

02Principle and equation

In − out = accumulationThe continuity equation expresses this mass balance for a region of any shape.

Imagine an arbitrary control volume around a flow. Conservation of mass says that the rate of mass accumulation inside plus the net mass flow leaving through its boundary is zero. Because this holds for any shape or location, it is a starting point for fluid analysis.

Start here: steady, one-dimensional

\\dot{m}=\\rho A V=\\mathrm{const.}

Use this form to compare the inlet and outlet of a pipe or nozzle. V denotes cross-sectional mean speed.

Balance over a region: integral form

\\frac{\\mathrm{d}}{\\mathrm{d}t}\\int_\\Omega \\rho\\,\\mathrm{d}V+\\oint_{\\partial\\Omega}\\rho\\mathbf{u}\\cdot\\mathbf{n}\\,\\mathrm{d}A=0

Mass accumulation inside the control volume plus net outward mass flow across its boundary is zero.

General local form: differential form

\\frac{\\partial \\rho}{\\partial t}+\\nabla\\cdot(\\rho\\mathbf{u})=0

Differential (local) form. The first term is local density change; the second is the divergence of mass flux.

03Symbols and units

Symbol	Quantity	SI unit
$\rho$	Density	$\mathrm{kg\,m^{-3}}$
$t$	Time	$\mathrm{s}$
$\mathbf{u}$	Local velocity vector	$\mathrm{m\,s^{-1}}$
$V$	Cross-sectional mean speed	$\mathrm{m\,s^{-1}}$
$A$	Flow area	$\mathrm{m^2}$
$\dot{m}$	Mass flow rate	$\mathrm{kg\,s^{-1}}$

04Derivation

Mass in a control volume

Integrating density over the control volume gives the total mass inside.

m_\\Omega=\\int_\\Omega\\rho\\,\\mathrm{d}V

Flow through the boundary

Integrating the normal component of mass flux over the boundary gives the net outward mass flow.

\\dot{m}_{\\mathrm{out}}=\\oint_{\\partial\\Omega}\\rho\\mathbf{u}\\cdot\\mathbf{n}\\,\\mathrm{d}A

Apply conservation

Set the rate of mass accumulation plus the net outward mass flow equal to zero.

\\frac{\\mathrm{d}}{\\mathrm{d}t}\\int_\\Omega \\rho\\,\\mathrm{d}V+\\oint_{\\partial\\Omega}\\rho\\mathbf{u}\\cdot\\mathbf{n}\\,\\mathrm{d}A=0

Obtain the local form

Apply the divergence theorem and use the fact that the balance holds for any control volume.

\\frac{\\partial \\rho}{\\partial t}+\\nabla\\cdot(\\rho\\mathbf{u})=0

05Conditions of use

✓ Always preserve

• Mass conservation holds for every fluid

• Retain density changes in compressible flow

• Retain accumulation in unsteady flow

! Conditions for simplification

• Constant volume flow assumes steady, 1-D, constant-density flow

• At a junction, balance the sum of all branches

• Do not confuse mean and local velocity

06Worked example

Water flows at 2.0 m·s⁻¹ through a 100 mm diameter pipe that narrows to 50 mm. Assuming incompressible flow, find the downstream mean velocity.

D_1=100\,\mathrm{mm}

V_1=2.0\,\mathrm{m\,s^{-1}}

D_2=50\,\mathrm{mm}

\\rho=\\mathrm{const.}

$A_1V_1=A_2V_2$

V_2=V_1\\frac{A_1}{A_2}=V_1\\left(\\frac{D_1}{D_2}\\right)^2=8.0\\,\\mathrm{m\\,s^{-1}}

07Calculator

Continuity equation calculator

Find the mean velocity at section 2 from the state at section 1 and the density and area at section 2.

Density at section 1

\rho_1

\mathrm{kg\,m^{-3}}

Area at section 1

A_1

\mathrm{m^2}

Mean speed at section 1

V_1

\mathrm{m\,s^{-1}}

Density at section 2

\rho_2

\mathrm{kg\,m^{-3}}

Area at section 2

A_2

\mathrm{m^2}

Mass flow rate

\dot{m}

20.000 $\mathrm{kg\,s^{-1}}$

Mean speed at section 2

V_2

5.0000 $\mathrm{m\,s^{-1}}$

08Common mistakes

Volume flow is always conserved
Mass flow is conserved. Volume flow changes when density changes in compressible flow.

Halving area and halving diameter are the same
If a circular diameter halves, area becomes one quarter and incompressible velocity becomes four times larger.

Ignoring the direction of velocity
The sign of inflow and outflow follows the dot product of velocity with the outward unit normal.

09Applications and connections

Nozzles & diffusersPipe junctionsPump sizingWind tunnelsCompressible nozzlesCFD mass-balance checks

Next: Momentum conservation

Find forces on bodies from changes in fluid momentum.

→

10Sources

Verification status

The text, symbols, units and worked example were checked against the sources below. The calculator reproduces the 8.0 m·s⁻¹ worked-example result. Independent review by an external engineer has not yet been completed.