HYD-06

Pipe continuity

Q = A1 V1 = A2 V2 for incompressible flow.

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ContinuityEuler continuity

Governing equation

Q=A_1 V_1=A_2 V_2,\quad A=\pi D^2/4

where

D_1: Upstream diameter (m)
V_1: Upstream velocity (m/s)
D_2: Downstream diameter (m)
Q: Discharge (m³/s)
V_2: Downstream velocity (m/s)

Lecture brief

Historical brief

Open-channel and pipe flow were written by Chezy, Manning, Darcy and Weisbach in the nineteenth century, then Bakhmeteff and Bélanger on the hydraulic jump. The sheets compute conveyance, head loss and gradually varied profiles. This sheet (HYD-06 — Pipe continuity) is the form associated with Euler continuity. Working symbols: $D_1$ , $V_1$ , $D_2$ $\rightarrow$ $Q$ , $V_2$ . Mass conservation in a stream tube of constant density collapses to area–velocity continuity.

Purpose

Purpose: compute

Q

V_2

from

D_1

V_1

D_2

in Hydraulics via

Q=A_1 V_1=A_2 V_2,\quad A=\pi D^2/4

Q = A1 V1 = A2 V2 for incompressible flow. Use it when a real hydraulics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

D_1 = 0.400\,\mathrm{m}

V_1 = 2.000\,\mathrm{m/s}

D_2 = 0.250\,\mathrm{m}

, the governing relation

Q=A_1 V_1=A_2 V_2,\quad A=\pi D^2/4

yields

Q = 0.251\,\mathrm{m^{3}/s}

V_2 = 5.120\,\mathrm{m/s}

. Circular sections, steady, incompressible. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Upstream diameter D_10.4 mUpstream velocity V_12 m/sDownstream diameter D_20.25 m

Outputs

Discharge Q0.251 m³/s
Downstream velocity V_25.120 m/s

Reading speed

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Narration of this film

Circular sections, steady, incompressible.

Mass conservation in a stream tube of constant density collapses to area–velocity continuity.

Reading speed

Watch on YouTube