Hydraulic Characteristics of Circular Pipe Section

Note: This theory is specifically for gravity flow (Open Channel Flow). It is not applicable to Drinking Water pipelines flowing under pressure.

In Bridge construction, Culverts, and Sewerage systems, understanding the hydraulic characteristics of partial flow in circular sections is essential for accurate design and discharge calculation.

Common Applications:

• Pipe Culverts: Calculating the number of pipes required based on catchment discharge.
• Diversions: Designing temporary passage for water during foundation works.
• Sewer Lines: Essential for self-cleansing velocity and peak flow capacity.

1. Depth of Flow

The depth of water at partial flow ($d$) is determined by the central angle ($\alpha$) subtended by the wetted perimeter:

$$ d = \frac{D}{2} - \frac{D}{2}\cos\left(\frac{\alpha}{2}\right) = \frac{D}{2}\left(1 - \cos\frac{\alpha}{2}\right) $$

Proportionate Depth:

$$ \frac{d}{D} = \frac{1}{2}\left(1 - \cos\frac{\alpha}{2}\right) $$

2. Area of Cross-Section

The area of flow ($a$) when running partially full is the area of the sector minus the area of the triangle:

$$ a = \frac{\pi D^2}{4}\left(\frac{\alpha}{360^\circ} - \frac{\sin\alpha}{2\pi}\right) $$

Proportionate Area ($a/A$):

$$ \frac{a}{A} = \left(\frac{\alpha}{360^\circ} - \frac{\sin\alpha}{2\pi}\right) $$

3. Wetted Perimeter and Hydraulic Mean Depth

Wetted Perimeter ($P$): The length of the arc in contact with water.

$$ P = \pi D \left(\frac{\alpha}{360^\circ}\right) \implies \frac{P}{P_{full}} = \frac{\alpha}{360^\circ} $$

Hydraulic Mean Depth ($r$):

$$ \frac{r}{R} = \left(1 - \frac{360^\circ \sin\alpha}{2\pi\alpha}\right) $$

4. Velocity and Discharge (Manning's Formula)

Assuming the rugosity coefficient ($n$) remains constant regardless of depth:

Proportionate Velocity ($v/V$):

$$ \frac{v}{V} = \left(\frac{r}{R}\right)^{2/3} = \left[1 - \frac{360^\circ \sin\alpha}{2\pi\alpha}\right]^{2/3} $$

Proportionate Discharge ($q/Q$):

$$ \frac{q}{Q} = \frac{a \cdot v}{A \cdot V} = \left(\frac{\alpha}{360^\circ} - \frac{\sin\alpha}{2\pi}\right) \cdot \left[1 - \frac{360^\circ \sin\alpha}{2\pi\alpha}\right]^{2/3} $$

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Design Tip: Maximum velocity occurs at $d/D \approx 0.81$ and maximum discharge occurs at $d/D \approx 0.95$.