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Engineering › Fluid Mechanics › Fundamentals ›

Chezy Equation

Mean flow velocity within pipes with turbulent flow

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Contents

Overview
Theory
Page Comments

Key Facts

Gyroscopic Couple: The rate of change of angular momentum ( $\tau$ ) = $I\omega\Omega$ (In the limit).

= Moment of Inertia.
$\omega$ = Angular velocity
$\Omega$ = Angular velocity of precession.

Overview

The Chezy equation applied to pipes with turbulent flow is

$v=C\sqrt{mi}$

where

i is $\displaystyle \frac{h_f}{l}$ or head loss due to fiction over the pipe length,
m is $\displaystyle \frac{A}{P}$ or wetted area divided by the wetted perimeter,
and C is $\displaystyle \sqrt{\frac{2g}{f}}$ where f is the coefficient of friction.

Theory

For the flow of a fluid within a pipe with velocity (v), there will be a reduction in mean pressure with distance, which is usually referred to as "head loss".

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Frictional resistance is proportional $\frac{1}{2}\:\rho\:v^2$ and the wetted area around the circumference of the pipe. Therefore,

$H_l=f\:P\:L\:\frac{1}{2}\rho v^2$

where

P is the perimeter of the pipe $=h_1\:A-h_2\:A$ ,
L is the length of the pipe section
and f is the coefficient of friction

Rearranging

$v^2 = \frac{2}{f\rho}\:\frac{A}{P}\:\left( \frac{h_1\:-\:h_2}{l} \right)$

$\therefore\;\;\;v = C\:\sqrt[]{\mu\:i}$

where:

$C = \sqrt[]{\frac{2\:w}{f\:\rho}} = \sqrt[]{\frac{2g}{f}}$

$$m$ =\frac{A}{P}$

where m is the hydraulic mean depth

$$h$=\frac{m}{l}$

where h is the slope of the hydraulic gradient.

Note: C is not a constant since f is a function of Reynolds number