Solution: Calculating Open Channel Slope

This solution details the calculation of the channel slope using Manning's equation for an open rectangular channel.

1. Determine Channel Area

The flow rate ($Q$) is defined as the product of the cross-sectional area ($A$) and the flow velocity ($V$):

$$Q = A \times V$$

Given $Q = 2 \text{ m}^3/\text{s}$ and $V = 1 \text{ m/s}$, the area ($A$) is calculated as:

$$A = \frac{Q}{V} = \frac{2 \text{ m}^3/\text{s}}{1 \text{ m/s}} = 2 \text{ m}^2$$

2. Calculate Channel Dimensions

For a rectangular channel, the area is given by $A = B \times D$, where $B$ is the width and $D$ is the depth.

The problem states that the depth is one-half of the width, so $D = 0.5 B$. Substituting this relationship into the area formula:

$$A = B \times (0.5 B) = 0.5 B^2$$

Using the calculated area $A = 2 \text{ m}^2$:

$$2 \text{ m}^2 = 0.5 B^2 \\ B^2 = \frac{2 \text{ m}^2}{0.5} = 4 \text{ m}^2 \\ B = \sqrt{4 \text{ m}^2} = 2 \text{ m}$$

The corresponding depth ($D$) is:

$$D = 0.5 B = 0.5 \times 2 \text{ m} = 1 \text{ m}$$

3. Calculate Wetted Perimeter and Hydraulic Radius

The wetted perimeter ($P$) for a rectangular channel is the sum of the channel width and twice the depth: $P = B + 2D$.

$$P = 2 \text{ m} + 2 \times (1 \text{ m}) = 2 \text{ m} + 2 \text{ m} = 4 \text{ m}$$

The hydraulic radius ($R_h$) is defined as the ratio of the cross-sectional area to the wetted perimeter:

$$R_h = \frac{A}{P} = \frac{2 \text{ m}^2}{4 \text{ m}} = 0.5 \text{ m}$$

4. Apply Manning's Equation to Find Slope

Manning's equation for flow velocity ($V$) in an open channel is:

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

where $n$ is Manning's roughness coefficient and $S$ is the channel slope.

Rearranging the equation to solve for the slope ($S$):

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

Substitute the given values: $V = 1$ m/s, $n = 0.02$, and $R_h = 0.5$ m:

$$S = \left( \frac{1 \text{ m/s} \times 0.02}{(0.5 \text{ m})^{2/3}} \right)^2$$

First, calculate $(0.5)^{2/3} \approx 0.630$. Then, substitute this value:

$$S \approx \left( \frac{0.02}{0.630 \text{ m}^{2/3}} \right)^2 \approx (0.03175)^2 \approx 0.001008$$

5. Convert Slope to Percentage

The calculated slope ($S$) is a decimal value. To express it as a percentage, multiply by 100:

$$\text{Slope \%} = S \times 100 \approx 0.001008 \times 100 \approx 0.1008\%$$

Rounded off to 2 decimal places, the channel slope is 0.10%.