The Francis formula for the discharge over cippoletti weir is
1.84 LH 3/2
Weir Discharge and the Francis Formula
Understanding the flow of water over a weir is crucial in hydraulic engineering. A weir is a barrier across a river or channel that alters the flow characteristics and can be used to measure discharge. Different types of weirs exist, each with specific formulas for calculating the discharge or flow rate.
Francis Formula for Discharge Calculation
The Francis formula is an empirical formula widely used to calculate the discharge over rectangular weirs. The general form of the Francis formula for a suppressed rectangular weir is given by:
$$\text{Q} = 1.84 \text{L} \left( \text{H} + \text{h}_v \right)^{3/2}$$
Where:
- \(\text{Q}\) is the discharge in cubic meters per second (\(\text{m}^3/\text{s}\)).
- \(\text{L}\) is the length of the weir crest in meters (\(\text{m}\)).
- \(\text{H}\) is the head over the weir crest in meters (\(\text{m}\)).
- \(\text{h}_v\) is the head due to the velocity of approach (\(\text{V}_a^2 / (2\text{g})\)). For many practical applications, especially when the velocity of approach is negligible, this term is often omitted, simplifying the formula to:
$$\text{Q} = 1.84 \text{L} \text{H}^{3/2}$$
This simplified version is commonly referred to as the basic Francis formula.
Cipoletti Weir and its Unique Design
The Cipoletti weir is a specific type of trapezoidal weir designed to automatically compensate for the effect of end contractions. In a rectangular weir, end contractions reduce the effective length of the weir crest, requiring adjustments in discharge calculations. The Cipoletti weir addresses this by incorporating side slopes (1 horizontal to 4 vertical) such that the increase in flow area due to these sloping sides exactly offsets the reduction in flow caused by end contractions.
This ingenious design allows the simplified Francis formula, which is typically applied to weirs with suppressed end contractions, to be used directly for the Cipoletti weir without needing to account for effective length reductions. Therefore, the discharge formula for a Cipoletti weir is accurately represented by the simplified Francis formula:
$$\text{Q} = 1.84 \text{L} \text{H}^{3/2}$$
Let's break down the components of this formula as it applies to the Cipoletti weir:
- 1.84: This is an empirical coefficient, typically used when L and H are measured in meters and Q is in cubic meters per second.
- L: Represents the crest length of the Cipoletti weir.
- H3/2: Represents the head of water over the weir crest, raised to the power of 3/2. This specific power is characteristic of weir flow equations.
Analyzing the Options
Let's examine the given options in the context of the Francis formula for the Cipoletti weir:
| Option | Formula | Analysis |
|---|---|---|
| 1 | \(1.84 \text{ LH}^{1/2}\) | The exponent of H is incorrect. Weir discharge equations typically feature H raised to the power of 3/2. |
| 2 | \(1.84 \text{ LH}\) | The exponent of H is incorrect. This formula does not represent the characteristic discharge relationship for weirs. |
| 3 | \(1.84 \text{ LH}^{3/2}\) | This formula correctly represents the discharge over a Cipoletti weir, which utilizes the simplified Francis formula with H raised to the power of 3/2. |
| 4 | \(1.84 \text{ LH}^{5/2}\) | The exponent of H is incorrect. This power is not characteristic of standard weir discharge formulas. |
Based on the analysis, the formula \(1.84 \text{ LH}^{3/2}\) is the correct representation of the Francis formula for discharge over a Cipoletti weir.
Important Questions from Open Channel Flow
- A trapezoidal channel carries $5\text{ m}^3\text{.s}^{-1}$ of water under uniform flow condition. The channel has a bottom width of $2\text{ m}$, side slope of $2:1$ (horizontal: vertical), and a bed slope of $1\%$. If the Manning roughness coefficient is $0.03$, the conveyance of the channel, in $\text{m}^3\text{.s}^{-1}$, is
- Two rectangular channels, A and B, join to form a large rectangular channel, C. Each rectangular channel is lined with the same material and has the same bottom slope. The bottom widths of Channels A and B are $2\text{ m}$ and $1\text{ m}$, respectively. If the flow depth in each channel is $2\text{ m}$, the bottom width of Channel C (in m) is ________. (Rounded off to two decimal places)
- Water is flowing at the rate of 15 m$^3$/s through a rectangular channel of 5 m width.
If the acceleration due to gravity ($g$) is 9.81 m/s$^2$, the critical velocity for the flow is ______ m/s (rounded off to three decimal places). A Hydraulic Jump is said to be Oscillating Jump if its Froude number is ________.
The Cippoleti weir is a ______ weir.
