PLAIN AND CIVIL: 6.4. THE HYDRAULIC RADIUS

Friday, 19 April 2019

The hydraulic radius is one of the parameters needed for Manning equation calculations. Hydraulic radius is defined as the ratio of the area with the wetted perimeter.

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

where: R = hydraulic radius
A = cross area of the stream flow (not the pipe or section)
P = wetted perimeter

The wetted perimeter of any conduit is the line of intersection of its wetted surface with a cross-sectional plane.

Let's take a look at the hydraulic radius of different pipe cases:

1. Pipe flowing full:

$A=\pi r^{2}$

$P=2\pi r$

For hydraulic radius, $R=\frac{A}{P}=\frac{\pi r^{2}}{2\pi r}$

${\color{Blue} \mathbf{R=\frac{r}{2}\simeq \frac{D}{4}}}$

2. Pipe flowing half full:

$A=\frac{\pi r^{2}}{2}$

$P=\frac{2\pi r}{2}=\pi r$

For hydraulic radius, $R=\frac{A}{P}=\frac{\frac{\pi r^{2}}{2}}{\pi r}$

${\color{Blue} \mathbf{R=\frac{r}{2}\simeq \frac{D}{4}}}$

3. Pipe flowing less than half full:

Taking the triangular area above the fluid to the center of the pipe and using trigonometry, the angle can be determined as:

$\theta =2arccos\left ( \frac{r-h}{r} \right )$

$A=\frac{r^{2}\left ( \theta -sin\theta \right )}{2}$

$P=r\theta$

$R=\frac{A}{P}=\frac{\frac{r^{2}\left ( \theta -sin\theta \right )}{2}}{r\theta }$

${\color{Blue} \mathbf{R=\frac{r\left ( \theta -sin\theta \right )}{2\theta }}}$

where $\theta$ is in radians.

4. Pip flowing more than half full:

$\theta =2arccos\left ( \frac{r-h}{r} \right )$
$A=\pi r^{2}-\frac{r^{2}\left ( \theta -sin\theta \right )}{2}$

$P=2\pi r-r\theta$

$R=\frac{A}{P}=\frac{\pi r^{2}-\frac{r^{2}\left ( \theta -sin\theta \right )}{2}}{2\pi r-r\theta }$

${\color{Blue} \mathbf{R=\frac{\pi r-\frac{r\left ( \theta -sin\theta \right )}{2}}{2\pi -\theta }}}$

EXAMPLE 6.4.1. HYDRAULIC RADIUS FOR PIPES FLOWING IN FULL AND HALF FULL

EXAMPLE 6.4.2. HYDRAULIC RADIUS FOR PIPES FLOWING LESS THAN HALF FULL

EXAMPLE 6.4.3. HYDRAULIC RADIUS FOR PIPES FLOWING LESS THAN HALF FULL

EXAMPLE 6.4.4. HYDRAULIC RADIUS FOR PIPES FLOWING MORE THAN HALF FULL

RELATED TOPIC:
6.0. PIPES

PLAIN AND CIVIL

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Friday, 19 April 2019

6.4. THE HYDRAULIC RADIUS

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