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# Chezy Equation

Mean flow velocity within pipes with turbulent flow
Contents

## Overview

The Chezy equation applied to pipes with turbulent flow is

$v=C\sqrt{mi}$
where
• i is $\inline&space;\displaystyle&space;\frac{h_f}{l}$ or head loss due to fiction over the pipe length,
• m is $\inline&space;\displaystyle&space;\frac{A}{P}$ or wetted area divided by the wetted perimeter,
• and C is $\inline&space;\displaystyle&space;\sqrt{\frac{2g}{f}}$ 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".

Frictional resistance is proportional $\inline&space;\frac{1}{2}\:\rho\:v^2$ and the wetted area around the circumference of the pipe. Therefore,
$H_l=f\:P\:L\:\frac{1}{2}\rho&space;v^2$
where
• P is the perimeter of the pipe $\inline&space;=h_1\:A-h_2\:A$,
• L is the length of the pipe section
• and f is the coefficient of friction

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

$\therefore\;\;\;v&space;=&space;C\:\sqrt[]{\mu\:i}$
where:
$C&space;=&space;\sqrt[]{\frac{2\:w}{f\:\rho}}&space;=&space;\sqrt[]{\frac{2g}{f}}$

$m&space;=\frac{A}{P}$
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