HYD-36

Pipes in series head loss

hf = Σ fi (Li/Di) V²/(2g) with Q the same, Vi = Q / Ai.

Reading speed

Pipe flowDupuit pipes in series

Governing equation

h_f=\sum f_i\dfrac{L_i}{D_i}\dfrac{V_i^2}{2g}

$Q \in \left[1, 200\right]$

where

Q: Discharge (L/s)
D_1: Diameter 1 (mm)
L_1: Length 1 (m)
f_1: f1 (—)
D_2: Diameter 2 (mm)
L_2: Length 2 (m)
f_2: f2 (—)
h_f: Total head loss (m)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

h_f=\sum f_i\dfrac{L_i}{D_i}\dfrac{V_i^2}{2g}

$Q \in \left[1, 200\right]$

$Q = 1$ $h_f = 0.009$

Along the curve

②

Domain of the variable — blue

Q_{\min} = 1,\quad Q_{\max} = 200

$Q \in \left[1, 200\right]$

$Q = 1$ $Q = 1$

Along the curve

③

Response curve — green

h_f=\sum f_i\dfrac{L_i}{D_i}\dfrac{V_i^2}{2g}

$Q \in \left[1, 200\right]$

$Q = 1$ $h_f = 0.009$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}h_f}{\mathrm{d}Q}

$Q \in \left[1, 200\right]$

$Q = 2.252$ $h_f^{\prime} = 0.041$

Along the curve

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-36 — Pipes in series head loss) is the form associated with Dupuit pipes in series. Working symbols: $Q$ , $D_1$ , $L_1$ , $f_1$ , $D_2$ , $L_2$ , $f_2$ $\rightarrow$ $h_f$ . Continuity forces a common discharge. Energy adds the friction (and minor) drops along the chain.

Purpose

Purpose: compute

h_f

from

Q

D_1

L_1

f_1

D_2

L_2

f_2

in Hydraulics via

h_f=\sum f_i\dfrac{L_i}{D_i}\dfrac{V_i^2}{2g}

hf = Σ fi (Li/Di) V²/(2g) with Q the same, Vi = Q / Ai. Use it when a real hydraulics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

Q = 20.000\,\mathrm{L/s}

D_1 = 150.000\,\mathrm{mm}

L_1 = 80.000\,\mathrm{m}

f_1 = 0.020\,\mathrm{—}

D_2 = 100.000\,\mathrm{mm}

L_2 = 40.000\,\mathrm{m}

f_2 = 0.022\,\mathrm{—}

, the governing relation

h_f=\sum f_i\dfrac{L_i}{D_i}\dfrac{V_i^2}{2g}

yields

h_f = 3.605\,\mathrm{m}

. Two pipes, one Q, a stacked piezometric drop. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Discharge Q20 L/sDiameter 1 D_1150 mmLength 1 L_180 mf1 f_10.02 —Diameter 2 D_2100 mmLength 2 L_240 mf2 f_20.022 —

Outputs

Total head loss h_f3.605 m

Reading speed

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Narration of this film

Two pipes, one Q, a stacked piezometric drop.

Continuity forces a common discharge. Energy adds the friction (and minor) drops along the chain.

Reading speed

Watch on YouTube