Weirs and Notches: A Complete Guide to Discharge Measurement

Measuring the flow of water is a fundamental task in hydraulic engineering. It is essential for managing water resources, designing irrigation systems, and operating treatment plants. While complex electronic meters exist, one of the simplest and most reliable methods involves a class of structures known as weirs and notches. These devices provide an elegant, low-cost way to determine the discharge in an open channel simply by measuring a height of water.

This comprehensive guide will walk you through everything you need to know about weirs and notches. We will explore the different types, from rectangular to triangular. You will learn the derivations of their discharge formulas. We will also tackle practical considerations like end contractions and velocity of approach. Finally, with solved numericals aimed at exams like GATE and SSC JE, this article will serve as your ultimate resource for mastering these essential hydraulic structures.

---

What are Weirs and Notches? The Fundamental Difference

The terms “weir” and “notch” are often used interchangeably, but there is a subtle distinction based on scale.

• A Notch is typically a small structure. It is an opening in the side of a tank or a small channel. Its primary purpose is to measure the discharge of a small flow. The bottom edge of the notch, over which water flows, is called the sill or crest.
• A Weir is a much larger structure. It is usually a concrete or masonry barrier built across a river or an open channel. While weirs can be used for discharge measurement, they also serve other purposes. They can raise the upstream water level or divert water for irrigation.

In essence, a notch is a small-scale weir, and a weir is a large-scale notch. The hydraulic principles and discharge formulas for both weirs and notches are identical.

---

Classifying Weirs and Notches

We can classify weirs and notches based on several criteria. Understanding this classification is the first step toward selecting the right type and formula.

Classification Based on Shape

The shape of the opening is the most common way to classify these structures.

1. Rectangular Weir/Notch: The opening is rectangular. It is simple to construct and widely used.
2. Triangular Weir/Notch (or V-Notch): The opening is triangular, with the apex pointing down. It is particularly useful for measuring low flow rates accurately.
3. Trapezoidal Weir/Notch: The opening is a trapezoid. The most famous type is the Cipolletti weir, which has specific side slopes designed to simplify the discharge calculation.

Classification Based on Nappe Condition

The sheet of water flowing over the weir is called the nappe.

1. Free Weir: The nappe flows clear of the downstream face of the weir. The space below the nappe is fully ventilated to the atmosphere. This is the standard condition for accurate measurement.
2. Drowned or Submerged Weir: The downstream water level is above the crest of the weir. This “drowns” the nappe and significantly affects the discharge. It requires a more complex formula to analyze.

Classification Based on Crest Width

1. Sharp-Crested Weir: The crest is very thin or beveled to a sharp edge. The water springs clear of the crest. This type is used primarily for flow measurement. Our discussion will focus mainly on sharp-crested weirs.
2. Broad-Crested Weir: The crest has a significant width. The water flows with a depth nearly parallel to the crest. These are more robust structures, often used for flow control in larger channels.

---

The Heart of Flow Measurement: The Rectangular Weir/Notch

The rectangular weir is the most common type of weir. It is simple to build and its theory is straightforward.

Deriving the Discharge Formula for a Rectangular Weir

To find the discharge, we consider a small horizontal strip of water flowing over the weir.

Basic Principle: Discharge (dQ) through the strip is its area (dA) times its velocity (V).

• Let the height of water above the crest be H.
• Consider a strip of thickness dh at a depth h below the free surface.
• The length of the strip is the crest length, L.
• The area of the strip is dA = L * dh.
• The theoretical velocity of water through this strip, by Torricelli’s law, is V = √(2gh).

Now, we calculate the discharge through this small strip:
dQ = V * dA = √(2gh) * L * dh

To get the total discharge (Q), we integrate this expression from the bottom of the nappe (h=0 at the surface) to the top (h=H at the crest).
Q = ∫₀ᴴ L * √(2g) * h^(1/2) dh
Q = L * √(2g) * [h^(3/2) / (3/2)] from 0 to H
Q = (2/3) * L * √(2g) * H^(3/2)

This is the theoretical discharge. In reality, due to friction and contraction of the nappe, the actual discharge is lower. We introduce an empirical factor called the Coefficient of Discharge (Cd) to account for this.

Final Discharge Formula:
Q = (2/3) * Cd * L * √(2g) * H^(3/2)

The value of Cd for a rectangular weir is typically around 0.62.

![alt text](placeholder_for_generated_image.jpg)

Accounting for End Contractions: Francis’ Formula

When the sides of a rectangular weir are not flush with the channel walls, the nappe contracts laterally. This is called end contraction. This reduces the effective length of the weir crest.

James Francis conducted experiments and found that each end contraction reduces the effective length by 0.1H.

• If a weir has contractions at both ends (n=2), the effective length becomes L_eff = (L – 0.2H).
• If one end is suppressed (n=1), L_eff = (L – 0.1H).
• If both ends are suppressed (flush with the channel walls), there is no end contraction (n=0).

Francis incorporated this into a simplified formula, absorbing the Cd and g terms into a single constant.

Francis’ Formula:
Q = 1.84 * (L – 0.1nH) * H^(3/2) (in SI units)

Where ‘n’ is the number of end contractions (0, 1, or 2).

Solved Numerical (GATE/SSC JE Level) – Rectangular Weir

Problem: A rectangular weir with a crest length of 4 m is used to measure the flow in a channel. The head over the crest is 30 cm. If the coefficient of discharge is 0.62, find the discharge. The weir has two end contractions.

Solution using the standard formula:

1. Identify the given data:
   • L = 4 m
   • H = 30 cm = 0.3 m
   • Cd = 0.62
   • n = 2
2. Calculate the effective length (L_eff):
   • L_eff = L – 0.1nH = 4 – 0.1 * 2 * 0.3 = 4 – 0.06 = 3.94 m
3. Apply the discharge formula:
   • Q = (2/3) * Cd * L_eff * √(2g) * H^(3/2)
   • Q = (2/3) * 0.62 * 3.94 * √(2 * 9.81) * (0.3)^(3/2)
   • Q = (2/3) * 0.62 * 3.94 * 4.429 * 0.1643
   • Q ≈ 1.18 m³/s

Solution using Francis’ formula:

1. Apply Francis’ formula directly:
   • Q = 1.84 * (L – 0.1nH) * H^(3/2)
   • Q = 1.84 * (4 – 0.1 * 2 * 0.3) * (0.3)^(3/2)
   • Q = 1.84 * (3.94) * 0.1643
   • Q ≈ 1.19 m³/s

Both methods give very similar results, demonstrating their consistency.

---

The V-Notch (Triangular Weir): Precision at Low Flows

For measuring very small flow rates, a rectangular weir is not ideal. A small change in head (H) results in a very small change in discharge, which is hard to measure accurately. A triangular notch, or V-notch, solves this problem.

In a V-notch, even a small change in discharge causes a large change in head. This makes it much more sensitive and accurate for low flows.

Deriving the Discharge Formula for a V-Notch

The derivation process is similar, but the geometry is different.

• Let θ be the total angle of the notch.
• Consider a horizontal strip of thickness dh at a depth h from the free surface.
• The width of this strip, x, is a function of h.
• From trigonometry, tan(θ/2) = (x/2) / (H-h). So, x = 2(H-h)tan(θ/2).
• Area of the strip, dA = x * dh = 2(H-h)tan(θ/2) * dh.
• Velocity through the strip, V = √(2gh).

Discharge through the strip:
dQ = V * dA = √(2gh) * 2(H-h)tan(θ/2) * dh

To find the total discharge, we integrate from h=0 to h=H.
Q = ∫₀ᴴ 2 * tan(θ/2) * √(2g) * h^(1/2) * (H-h) dh
This integration is more complex, but it results in:
Q_theoretical = (8/15) * √(2g) * tan(θ/2) * H^(5/2)

Introducing the coefficient of discharge (Cd) to get the actual discharge.

Final Discharge Formula:
Q = (8/15) * Cd * √(2g) * tan(θ/2) * H^(5/2)

For a standard 90° V-notch (θ = 90°, so tan(θ/2)=1), the value of Cd is typically around 0.6.

Solved Numerical (GATE/SSC JE Level) – V-Notch

Problem: Find the discharge over a 90° V-notch if the head is 25 cm. Take Cd = 0.6.

Solution:

1. Identify the given data:
   • θ = 90°, so θ/2 = 45° and tan(45°) = 1
   • H = 25 cm = 0.25 m
   • Cd = 0.6
2. Apply the V-notch formula:
   • Q = (8/15) * Cd * √(2g) * tan(θ/2) * H^(5/2)
   • Q = (8/15) * 0.6 * √(2 * 9.81) * tan(45°) * (0.25)^(5/2)
   • Q = (8/15) * 0.6 * 4.429 * 1 * 0.03125
   • Q ≈ 0.044 m³/s or 44 liters/s

---

The Best of Both Worlds: The Trapezoidal Weir

A trapezoidal weir can be seen as a combination of a rectangular weir and a triangular weir. It can handle a wider range of flows than a V-notch.

The Cipolletti Weir: A Special Case of Weirs and Notches

An Italian engineer named Cesare Cipolletti designed a specific type of trapezoidal weir with a very clever feature. He set the side slopes to 1 Horizontal to 4 Vertical.

Why this specific slope? The discharge through a trapezoidal weir is the sum of the discharge through the central rectangular portion and the two triangular portions at the ends.

• The end contractions reduce the discharge of the rectangular portion.
• The sloping sides increase the discharge.

Cipolletti found that a 1H:4V slope provides just enough extra discharge through the triangular sections to exactly compensate for the loss of discharge due to the two end contractions.

This means we can calculate the discharge for a Cipolletti weir using the simple formula for a suppressed rectangular weir (a weir with no end contractions).

Cipolletti Weir Discharge Formula (Francis’ version):
Q = 1.84 * L * H^(3/2) (in SI units)

Here, L is the length of the bottom crest. The formula is beautifully simple because the complex effects of end contractions are designed out of the problem.

Solved Numerical (GATE/SSC JE Level) – Cipolletti Weir

Problem: A Cipolletti weir has a crest length of 2.0 m. Find the discharge when the head over the crest is 40 cm.

Solution:

1. Identify the given data:
   • Weir type is Cipolletti.
   • L = 2.0 m
   • H = 40 cm = 0.4 m
2. Apply the Cipolletti formula:
   • Q = 1.84 * L * H^(3/2)
   • Q = 1.84 * 2.0 * (0.4)^(3/2)
   • Q = 3.68 * 0.253
   • Q ≈ 0.93 m³/s

---

Important Considerations in Weir Design

For accurate measurements using weirs and notches, a few practical factors must be considered.

The Nappe: Aeration and its Importance

The nappe is the sheet of water that flows over the weir. For a sharp-crested weir, there is a space between the nappe and the downstream face of the weir. It is crucial that this space is ventilated so that the pressure inside it remains atmospheric.

If this space is not ventilated, the flowing water will drag the air out. This creates a partial vacuum, causing the atmospheric pressure on top of the nappe to push it down. This “clinging nappe” changes the flow geometry and increases the effective head, leading to an inaccurate (higher) discharge reading. In large weirs, ventilation pipes are often installed to prevent this.

Velocity of Approach

Our derivations assumed that the water in the upstream channel is nearly still. However, the water approaching the weir has some velocity, known as the velocity of approach (Va). This velocity contributes a small amount of kinetic energy, or velocity head (ha = Va²/2g).

If the channel area is large compared to the weir area, Va is negligible. But if the channel is narrow, Va can be significant. To account for this, the velocity head (ha) should be added to the measured static head (H).

The corrected discharge formula for a rectangular weir becomes:
Q = (2/3) * Cd * L * √(2g) * [(H + ha)^(3/2) – ha^(3/2)]

This formula is complex because Va depends on Q, and Q depends on Va. The calculation requires an iterative (trial and error) approach.

---

Field Applications: Where are Weirs and Notches Used?

These simple structures are incredibly versatile and are found in many applications:

• Irrigation Systems: To measure and regulate the amount of water delivered to different farm plots.
• Wastewater Treatment Plants: To measure flow rates entering the plant and between different treatment stages.
• Hydrological Studies: To measure the flow in small streams and rivers to understand watershed runoff.
• Hydraulic Laboratories: As a standard piece of equipment for experiments and calibrating other flow meters.
• Industrial Processes: To measure the flow of water or other fluids in open channels within a plant.

---

Frequently Asked Questions (FAQ)

Q1: What is the main difference between a weir and a notch?
A notch is a small-scale structure, typically on a tank, used for measuring small flows. A weir is a large-scale structure, usually built across a river or channel, used for flow measurement and water level control. The underlying hydraulic principles are the same.

Q2: Why is a triangular notch more accurate than a rectangular notch for low flow rates?
In a V-notch, the flow area decreases rapidly as the head drops. This means that a small change in discharge produces a relatively large, easily measurable change in head. In a rectangular weir, the width is constant, so a small change in discharge produces a tiny change in head, which is difficult to measure accurately.

Q3: What is the coefficient of discharge (Cd) and why is it needed?
The Cd is an empirical correction factor. It accounts for the energy losses due to fluid friction and the contraction of the nappe as it passes over the weir. It adjusts the theoretical discharge formula to match the actual, real-world discharge.

Q4: What happens if a weir becomes submerged?
When the downstream water level rises above the weir crest, the weir is submerged or drowned. This condition significantly hinders the free flow of water. The discharge is no longer a function of the upstream head alone. It becomes a function of both the upstream head and the downstream head, requiring a different, more complex formula (like Villemonte’s formula) to calculate the flow.

---

Conclusion: Simple Structures, Powerful Results

From the basic rectangular shape to the cleverly designed Cipolletti weir, weirs and notches stand as a testament to elegant engineering solutions. They prove that you don’t always need complex technology to achieve accurate, reliable results. By simply creating an obstruction and measuring a height, we can unlock a wealth of information about the flow of water.

A firm grasp of the discharge formulas and the practical considerations for weirs and notches is a fundamental skill for any student or professional in water resources engineering. These structures are not just textbook concepts; they are practical tools used every day to manage our most vital resource.

Do you have any experience using weirs or notches in the field? What challenges have you faced? Share your thoughts or ask a question in the comments below! If this guide was helpful, please share it with your peers.