Pipe Slope
Consider water is fully flowing through a 5 cm diameter pipe at a 0.5 m/s. If the pipe’s roughness coefficient is 0.02, what is the pipe’s slope (%)?
Expand Hint
Manning’s equation:
$$$V=\frac{K}{n}R_{H}^{2/3}S^{1/2}$$$
where
$$V$$
is the velocity,
$$K=1$$
for SI units,
$$n$$
is the roughness coefficient,
$$R_H$$
is the hydraulic radius, and
$$S$$
is the slope.
Hint 2
To find hydraulic radius:
$$$R_H=\frac{A}{P}$$$
where
$$A$$
is the area, and
$$P$$
is the perimeter.
Manning’s equation:
$$$V=\frac{K}{n}R_{H}^{2/3}S^{1/2}$$$
where
$$V$$
is the velocity,
$$K=1$$
for SI units,
$$n$$
is the roughness coefficient,
$$R_H$$
is the hydraulic radius, and
$$S$$
is the slope. To find hydraulic radius:
$$$R_H=\frac{A}{P}$$$
where
$$A$$
is the area, and
$$P$$
is the perimeter. Thus, the hydraulic radius is:
$$$R_H=\frac{(\pi/4)(D^2) }{(\pi D)}=\frac{D}{4}=\frac{0.05m}{4}=0.0125\:m$$$
To find the slope:
$$$0.5m/s=\frac{1}{0.02}(0.0125m)^{2/3}(S)^{1/2}$$$
$$$(S)^{1/2}=\frac{0.5m/s}{2.69}$$$
$$$S=\left ( \frac{0.5m/s}{2.69} \right )^2=0.034 \times 100 = 3.4\%$$$
3.4%
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