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Continuity

The continuity equation says the same volume (or mass) must pass every cross-section of a pipe each second, since the fluid can neither be created nor destroyed along the way. Where the pipe narrows, the speed must therefore increase, and where it widens, the speed decreases accordingly.

Q=AvQ = Avvolumetric flow rate
A1v1=A2v2A_1v_1 = A_2v_2continuity along a pipe

Symbols

QQvolumetric flow ratem³/s
AAcross-sectional aream²
vvspeedm/s

Example

A pipe narrows from d1=100d_1 = 100 mm to d2=60d_2 = 60 mm with v1=2v_1 = 2 m/s. v2=v1(d1/d2)2=2⋅(100/60)2≈5.56v_2 = v_1(d_1/d_2)^2 = 2\cdot(100/60)^2 \approx 5.56 m/s.

Use area, not diameter, in the continuity equation: A=πd2/4A = \pi d^2/4.
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Part of Fluid Mechanics: Continuity and Bernoulli.