The quantity of water discharged over a rectangular weir is given by the formula:

$Q = k \times L \times H^{3/2}$

Where:

• $Q$ is the discharge rate (m³/second),
• $k$ is a proportionality constant,
• $L$ is the crest length (m),
• $H$ is the head (height of water above the crest, m).

Step 1: Find the proportionality constant $k$

From the given data, we have:

• Crest length $L = 3 \, \text{m}$,
• Head $H = 0.70 \, \text{m}$,
• Discharge rate $Q = 3.23 \, \text{m}^3/\text{s}$.

Substitute the values into the formula:

$3.23 = k \times 3 \times (0.70)^{3/2}$

First, calculate $(0.70)^{3/2}$:

$(0.70)^{3/2} = 0.70 \times \sqrt{0.70} \approx 0.70 \times 0.8367 = 0.5857$

Now, substitute:

$3.23 = k \times 3 \times 0.5857$

Solve for $k$:

$k = \frac{3.23}{3 \times 0.5857} = \frac{3.23}{1.7571} \approx 1.84$

So, the proportionality constant $k$ is approximately $1.84$.

Step 2: Find the discharge rate for the new weir

Now, we need to find the discharge rate for a weir with:

• Crest length $L = 6 \, \text{m}$,
• Head $H = 1.4 \, \text{m}$.

Substitute the known values and the constant $k$ into the equation:

$Q = 1.84 \times 6 \times (1.4)^{3/2}$

First, calculate $(1.4)^{3/2}$:

$(1.4)^{3/2} = 1.4 \times \sqrt{1.4} \approx 1.4 \times 1.1832 = 1.6565$

Now, substitute:

$Q = 1.84 \times 6 \times 1.6565$

$Q \approx 1.84 \times 9.939 = 18.28 \, \text{m}^3/\text{s}$

Final Answer:

The rate of discharge over the weir with a crest length of 6 m and a head of 1.4 m is approximately 18.28 m³/second.

Would you like further details on the steps or have any questions? Here are a few related questions:

1. How do we calculate the discharge rate for different types of weirs, such as sharp-crested or broad-crested?
2. What other factors can affect the discharge over a weir?
3. How can we determine the value of $k$ for different weir conditions?
4. Can the equation be applied to non-rectangular weirs? If so, how?
5. What is the relationship between head and flow rate in a weir?

Tip: Always check if the units are consistent (e.g., length in meters, discharge in cubic meters per second) before plugging values into formulas!