Weir Flow Equation

The weir flow equation is one of the most widely used tools in open‑channel hydraulics. Whether you’re designing a SuDS outfall, assessing a spillway, or measuring flow in a channel, understanding how weir equations work is essential for producing reliable, defensible hydraulic calculations.

This guide breaks down the principles behind the weir flow equation, explains how different weir types use different forms of the equation, and provides practical examples to help you apply it correctly.

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🌊 What Is the Weir Flow Equation?

At its core, the weir flow equation expresses the relationship between:
- the head of water upstream of the crest
- the geometry of the weir
- a discharge coefficient that accounts for real‑world behaviour

The general form of the weir flow equation is:

Q = C · L · Hⁿ

Where:
- Q = discharge (m³/s)
- C = discharge coefficient
- L = crest length (m)
- H = head over the crest (m)
- n = exponent depending on weir type (commonly 1.5 or 2.5)

📐 Why Weirs Are So Useful for Flow Measurement

Weirs are popular because they create a predictable head–discharge relationship. Once the crest geometry is fixed, the upstream water level becomes a reliable indicator of flow.

Engineers use weirs for:
- SuDS discharge control
- Flow monitoring
- Spillway design
- Channel capacity assessment
- Laboratory flumes

Types of Weirs and Their Flow Equations

🔺 1. V‑Notch (Triangular) Weir Flow Equation

Q = 1.38 · H^(5/2)

This comes from the general triangular‑weir equation:
Q = C · H^(5/2)
Where C depends on the notch angle.

Why it’s useful:
The H^(5/2) term makes the weir extremely responsive to small changes in head.

▭ 2. Rectangular Weir Flow Equation

Q = C · L · H^(3/2)

Suppressed: Q = 1.84 · L · H^(3/2)

Contracted: L_eff = L - 0.2H

Why it’s useful:
Simple geometry, robust behaviour, and widely validated coefficients.

🧱 3. Broad‑Crested Weir Flow Equation

Q = C · L · H · √(2gH)

Why it’s useful:
Stable at high flows and ideal for rivers, culverts, and spillways.

🧮 Worked Example Using the Weir Flow Equation

Problem:
A rectangular sharp‑crested weir has:
- Crest length L = 0.5 m
- Head H = 0.16 m
- Suppressed geometry

Solution:
Q = 1.84 · 0.5 · (0.16)^(3/2)
Q = 1.84 · 0.5 · 0.064 = 0.0589 m³/s
That’s 59 L/s.

🛠️ Key Factors Affecting Weir Flow Accuracy

1. Submergence – If downstream water levels rise too high, the weir becomes drowned and the equation overestimates flow.
   2. Crest condition – Sharp‑crested weirs require clean edges, vertical upstream faces, and smooth approach flow.
   3. Approach velocity – High approach velocity adds kinetic energy and may require correction.
   4. Head measurement location – Measure at least 4 × H upstream of the crest.
   5. Debris and maintenance – Especially important in SuDS features.

📊 Which Weir Flow Equation Should You Use?

Situation	Best Weir Type	Equation
Low flows	V‑notch	Q = 1.38H^(5/2)
Medium flows	Rectangular	Q = C L H^(3/2)
High flows	Broad‑crested	Q = C L H √(2gH)
Limited space	Cipolletti	Trapezoidal equation

🧭 Conclusion

The weir flow equation is a fundamental tool in hydraulic engineering. By understanding how head, geometry, and discharge coefficients interact, you can select the right weir type and apply the correct equation for accurate flow estimation.

For SuDS, drainage design, and open‑channel hydraulics—especially in UK site work—mastering these equations gives you a strong foundation for reliable, defensible design.

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