Determine the slope required in $$ \mathrm{m} / \mathrm{m} $$ (accurate to three decimal places) for a 750 mm diameter PVC wastewater sewer, to achieve a full flow velocity of $$ 1.92 \mathrm{~m} / \mathrm{s} $$. Manning's roughness coefficient for PVC is 0.010 .

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To determine the slope required for a 750 mm diameter PVC wastewater sewer to achieve a full flow velocity of $$ 1.92 \, \mathrm{m/s} $$, we can use Manning's equation, which is given by:

$$
V = \frac{1}{n} R^{2/3} S^{1/2}
$$

where:
- $$ V $$ is the flow velocity (in m/s),
- $$ n $$ is Manning's roughness coefficient,
- $$ R $$ is the hydraulic radius (in m),
- $$ S $$ is the slope of the energy grade line (dimensionless).

### Step 1: Calculate the Hydraulic Radius $$ R $$

For a circular pipe flowing full, the hydraulic radius $$ R $$ is given by:

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

where:
- $$ A $$ is the cross-sectional area of flow,
- $$ P $$ is the wetted perimeter.

For a full circular pipe, the area $$ A $$ and wetted perimeter $$ P $$ can be calculated as follows:

- Diameter $$ D = 750 \, \mathrm{mm} = 0.75 \, \mathrm{m} $$
- Radius $$ r = \frac{D}{2} = 0.375 \, \mathrm{m} $$

The area $$ A $$ is:

$$
A = \pi r^2 = \pi (0.375)^2 \approx 0.4418 \, \mathrm{m}^2
$$

The wetted perimeter $$ P $$ for a full circular pipe is:

$$
P = \pi D = \pi (0.75) \approx 2.3562 \, \mathrm{m}
$$

Now, we can calculate the hydraulic radius $$ R $$:

$$
R = \frac{A}{P} = \frac{0.4418}{2.3562} \approx 0.187 \, \mathrm{m}
$$

### Step 2: Rearranging Manning's Equation to Solve for Slope $$ S $$

Now we can rearrange Manning's equation to solve for the slope $$ S $$:

$$
S = \left( n V \right)^2 \frac{1}{R^{4/3}}
$$

Substituting the known values:

- $$ n = 0.010 $$
- $$ V = 1.92 \, \mathrm{m/s} $$
- $$ R \approx 0.187 \, \mathrm{m} $$

Calculating $$ S $$:

$$
S = \left( 0.010 \times 1.92 \right)^2 \frac{1}{(0.187)^{4/3}}
$$

Calculating $$ 0.010 \times 1.92 $$:

$$
0.010 \times 1.92 = 0.0192
$$

Now squaring it:

$$
(0.0192)^2 = 0.00036864
$$

Next, we need to calculate $$ (0.187)^{4/3} $$:

$$
(0.187)^{4/3} \approx 0.187^{1.3333} \approx 0.079
$$

Now substituting back into the equation for $$ S $$:

$$
S = \frac{0.00036864}{0.079} \approx 0.00466
$$

### Step 3: Final Result

Converting to $$ \mathrm{m/m} $$:

$$
S \approx 0.00466 \, \mathrm{m/m}
$$

Rounding to three decimal places:

$$
S \approx 0.005 \, \mathrm{m/m}
$$

Thus, the required slope for the sewer is approximately:

$$
\boxed{0.005}
$$
The required slope for the sewer is approximately 0.005 m/m.

Determine the slope required in (accurate to three decimal
places) for a 750 mm diameter PVC wastewater sewer, to achieve a
full flow velocity of . Manning’s roughness coefficient for
PVC is 0.010 .

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Determine the slope required in (accurate to three decimal places) for a 750 mm diameter PVC wastewater sewer, to achieve a full flow velocity of . Manning’s roughness coefficient for PVC is 0.010 .