Understanding Cipoletti Weir Side Slope

A Cipoletti weir is a specific type of trapezoidal weir commonly used for measuring the discharge or flow rate of water in open channels. It is designed with sloping sides to account for the effect of end contractions.

Standard rectangular weirs experience 'end contractions', which reduce the effective width of the flow over the weir crest. This makes the calculation of flow rate slightly more complex, often requiring correction factors or formulas like the Francis formula with contraction adjustments.

The unique feature of the Cipoletti weir is its trapezoidal shape with outward sloping sides. This slope is specifically chosen to add flow area on the sides as the head increases, precisely compensating for the reduction in flow due to end contractions. The goal is to simplify the discharge calculation, allowing the use of a formula similar to that for a rectangular weir, but applied to the crest length (bottom width of the trapezoid) as if there were no end contractions.

Cipoletti Weir Horizontal to Vertical Slope Ratio

The sides of a Cipoletti weir are tapered outwards. The slope is defined as the ratio of the horizontal distance to the vertical distance. For a Cipoletti weir, this standard ratio is 1 horizontal to 4 vertical.

• This means for every 4 units of vertical height, the sides slope outwards by 1 unit horizontally.
• This specific slope angle is approximately 14 degrees from the vertical.

The reason for this precise 1:4 slope is rooted in empirical observations and theoretical considerations related to flow over weirs. It was determined that this particular taper effectively cancels out the reduction in flow caused by the two end contractions in a rectangular weir of the same crest length, based on certain standard assumptions (like the Francis formula for rectangular weirs).

Let's look at the ratio options provided:

• 1 : 1 (This represents a 45-degree slope, much steeper than a Cipoletti weir.)
• 1 : 3 (A steeper slope than Cipoletti.)
• 1 : 2 (A steeper slope than Cipoletti.)
• 1 : 4 (This is the standard slope for a Cipoletti weir.)

Therefore, the horizontal to vertical side slope in the case of a Cipoletti weir is 1 : 4.

Comparing Weir Types and Slopes

Revision Table: Key Cipoletti Weir Features

Additional Information: Weir Flow Measurement

Weirs are hydraulic structures used to measure the volumetric flow rate of water. They are essentially barriers placed across an open channel with a notch or opening of a specific shape over which the water flows.

Different types of weirs are used depending on the expected range of flow rates, accuracy requirements, and channel conditions:

• Rectangular Weirs: Have a rectangular notch. Can be with or without end contractions. Formulas like the Francis formula are used to calculate flow rate. If end contractions are present, the effective length of the weir crest is reduced in the formula.
• Triangular Weirs (V-notch Weirs): Have a triangular notch, typically with a 90-degree angle. They are particularly useful for measuring low flow rates accurately because the flow area changes significantly with the head. The flow rate formula for a V-notch weir depends on the head and the notch angle.
• Trapezoidal Weirs: Have a trapezoidal notch. The Cipoletti weir is a specific type of trapezoidal weir designed for compensation. The general formula for a trapezoidal weir is the sum of the flow over a central rectangular section and the flow over two triangular sections.

The Cipoletti weir simplifies this. The 1:4 slope ensures that the flow through the triangular sections added by the slope exactly equals the head loss caused by the end contractions of the rectangular part, assuming the Francis formula applies. The discharge formula for a Cipoletti weir is often given as:

\( Q = C_d \frac{2}{3} \sqrt{2g} L H^{3/2} \)

Where:

• \( Q \) is the discharge
• \( C_d \) is the discharge coefficient (often taken around 0.63)
• \( g \) is the acceleration due to gravity
• \( L \) is the length of the weir crest (bottom width of the trapezoid)
• \( H \) is the head over the weir crest

This formula is the same form as the Francis formula for a rectangular weir *without* end contractions, which is the key benefit of the 1:4 Cipoletti slope.