Understanding Proportionate Wetted Perimeter in Circular Sewers

When a circular sewer pipe is not flowing full, only a portion of its inner surface is in contact with the fluid. This contact surface is called the wetted perimeter. The hydraulic efficiency of a sewer depends on parameters related to flow depth, including the wetted perimeter.

The question asks for the proportionate wetted perimeter. This is the ratio of the wetted perimeter when the pipe is flowing partially full ($p$) to the wetted perimeter when the pipe is flowing completely full ($P$). Mathematically, it is given by $\frac{p}{P}$.

Calculating Wetted Perimeter

Consider a circular sewer with diameter $D$. Let the fluid surface subtend an angle of $2\alpha$ at the center of the circle. The angle $\alpha$ is often used in formulas, representing half of the central angle subtended by the water surface. For this question, let's assume $\alpha$ in the options refers to the angle of the wetted portion, which is the arc length. A full circle is $360^\circ$. The wetted perimeter is the arc length corresponding to the angle subtended by the water at the center.

• When the sewer is flowing full, the wetted perimeter ($P$) is the circumference of the circle.
• The circumference of a circle is given by $\pi D$. So, $P = \pi D$.

When the sewer is flowing partially full, let the angle subtended by the w wetted arc at the center be $\theta$ (in degrees). The length of an arc is given by the formula: Arc Length = $\frac{\theta}{360^\circ} \times \text{Circumference}$.

• So, the wetted perimeter ($p$) when partially full is $p = \frac{\theta}{360^\circ} \times \pi D$.

The question uses the symbol $\alpha$ in the options. If we interpret $\alpha$ as the angle of the wetted portion in degrees (i.e., $\theta = \alpha$), then the wetted perimeter when partially full is $p = \frac{\alpha}{360^\circ} \times \pi D$.

Deriving Proportionate Wetted Perimeter

Now, we can find the proportionate wetted perimeter by taking the ratio of the partial wetted perimeter to the full wetted perimeter:

Proportionate Wetted Perimeter = $\frac{p}{P}$

Substitute the formulas for $p$ and $P$:

$\frac{p}{P} = \frac{\frac{\alpha}{360^\circ} \times \pi D}{\pi D}$

We can cancel out the $\pi D$ term from both the numerator and the denominator:

$\frac{p}{P} = \frac{\alpha}{360^\circ}$

This formula shows that the proportionate wetted perimeter is directly proportional to the angle (in degrees) subtended by the wetted surface at the center, divided by $360^\circ$.

Checking the Options

Let's compare our derived formula with the given options:

• Option 1: $\frac{p}{P}=\frac{D}{4}$ - This does not match our derivation.
• Option 2: $\frac{p}{P}=\frac{\alpha}{360^0 }$ - This matches our derived formula exactly.
• Option 3: $\frac{p}{P}=\frac{\alpha}{\pi D}$ - This does not match our derivation.
• Option 4: $\frac{p}{P}=\frac{\alpha}{180^o}$ - This does not match our derivation.

Based on the derivation, the correct formula for the proportionate wetted perimeter is $\frac{p}{P}=\frac{\alpha}{360^0 }$, where $\alpha$ is the angle in degrees subtended by the wetted surface at the center.