Flow Measurement — Venturimeter, Orifice, Notches & Weirs

1. Classification of Flow Measurement Devices

2. Venturimeter — Derivation & Formula

A venturimeter consists of three sections: a converging inlet cone (half-angle ≈ 10°–25°), a cylindrical throat (minimum diameter), and a diverging diffuser (half-angle ≈ 5°–7°, gradual to minimise losses). Pressure tappings at the inlet and throat are connected to a differential manometer.

2.1 Derivation

Apply Bernoulli between inlet (section 1) and throat (section 2), and continuity A₁V₁ = A₂V₂:

2.2 Differential Head from Manometer

2.3 Key Values

3. Orifice Meter

An orifice meter is a flat plate with a concentric circular hole (sharp-edged orifice) inserted perpendicular to the pipe axis. The pressure is measured at an upstream tapping (typically D upstream) and at the vena contracta or a downstream tapping (D/2 downstream from the plate — “flange taps” or “corner taps” also used).

3.1 Comparison: Venturimeter vs Orifice Meter

4. Flow Nozzle

A flow nozzle is an intermediate device — it has a streamlined converging entry like a venturimeter throat but discharges directly into the pipe (no diverging section). This gives higher C_d than an orifice plate (≈ 0.95–0.99) but higher permanent pressure loss than a venturimeter.

5. Orifice in a Tank — Coefficients C_c, C_v, C_d

When fluid discharges from a small orifice in the wall or base of a tank, three hydraulic coefficients characterise the real-fluid behaviour:

5.1 Experimental Determination of C_v

The jet from a horizontal orifice in a tank follows a parabolic trajectory under gravity. If the jet travels a horizontal distance x and falls a vertical distance y from the orifice:

6. Mouthpiece — Types & Discharge

A mouthpiece is a short tube (length ≈ 2–3 times diameter) attached to the orifice opening. It changes the flow pattern and affects the discharge coefficients.

Key fact for GATE: The external cylindrical mouthpiece has C_d = 0.82 — higher than a sharp-edged orifice (0.62) because the jet completely fills the tube at exit (C_c = 1), even though there are more internal losses (lower C_v). The pressure at the vena contracta inside the mouthpiece drops significantly below atmospheric — the minimum pressure point can be found by applying Bernoulli from tank surface to the internal vena contracta.

7. Notches & Weirs — Open Channel Measurement

A notch is an opening in the side wall of a tank or channel through which flow occurs. A weir is an overflow structure built across a channel to measure or regulate flow. Both work on the same principle: the depth of water above the sill (the crest) is measured, and the discharge is computed from a formula derived by integrating velocity over the flow cross-section using Bernoulli.

7.1 General Derivation Approach

Consider a horizontal strip of width b(y) at height y above the sill, where the total head is H (water depth above sill). The theoretical velocity at that strip (by Bernoulli from the still water upstream):

The shape of b(y) (width variation with y) determines whether the result is proportional to H^3/2 (rectangular) or H^5/2 (triangular).

8. Rectangular Notch & Sharp-Crested Weir

For a rectangular notch of crest length L, the width b(y) = L = constant for all y.

8.1 End Contraction Correction (Francis Formula)

When the weir does not span the full channel width, horizontal flow contractions reduce the effective crest length:

8.2 Velocity of Approach Correction

If the upstream channel velocity V_a is significant, the effective head increases:

9. Triangular (V-notch) Weir

For a triangular notch with vertex angle θ (half-angle θ/2), the width at height y above the vertex is:

b(y) = 2y tan(θ/2)

9.1 Why Use a V-notch for Low Flows?

Because Q ∝ H^5/2 (steeper than H^3/2 for rectangular), a small change in H at low flows produces a relatively larger change in Q — making it more sensitive for measuring small discharges. For a rectangular notch at low H, the narrow water section makes the head measurement imprecise. The V-notch’s width reduces to zero at the vertex, so even low heads give a measurable flow width.

10. Trapezoidal Notch — Cipolletti Weir

A trapezoidal notch combines a rectangular base with inclined sides. Its discharge formula is the sum of the rectangular and triangular components:

11. Broad-Crested Weir

A broad-crested weir has a horizontal crest wide enough (width > H/2 approximately) that critical flow conditions develop on the crest. This makes the discharge independent of downstream conditions (drowned flow does not occur as easily).

11.1 Sharp-Crested vs Broad-Crested Weir

12. Worked Examples (GATE CE Level)

Example 1 — Rectangular Notch Discharge with End Contractions (GATE CE type)

Problem: A rectangular notch 1.2 m long has a head of 0.45 m over it. C_d = 0.623. There are two end contractions. Find the actual discharge using the Francis formula.

Answer: Q = 0.616 m³/s

Example 2 — 90° V-notch Discharge (GATE CE 2019 type)

Problem: A 90° triangular notch has a head of 0.30 m over its crest. C_d = 0.62. Calculate the discharge.

Answer: Q = 72.1 litres/s

Example 3 — Orifice Meter vs Venturimeter (Conceptual + Numerical)

Problem: Two flow meters are installed in a 200 mm pipe carrying water. A venturimeter (C_d = 0.98, throat D = 100 mm) and an orifice meter (C_d = 0.62, orifice D = 100 mm) both show the same manometer deflection x = 50 mm of mercury. Find the ratio of discharges Q_venturi/Q_orifice.

Answer: Q_venturi/Q_orifice = 1.58

Example 4 — Coefficient of Velocity from Jet Trajectory

Problem: Water issues from a sharp-edged orifice in the vertical side of a tank under a head of 1.2 m. The jet travels horizontally 2.1 m and falls 0.75 m vertically below the orifice centre. Calculate C_v.

Answer: C_v = x / (2√(yH)) — apply with given data. Typical value ≈ 0.97–0.99 for sharp orifice.

Example 5 — Broad-Crested Weir Discharge

Problem: A broad-crested weir 3.5 m long has a head of 0.6 m over it. C_d = 0.848. Find the discharge.

Answer: Q = 2.35 m³/s

Example 6 — Comparison of Rectangular and V-notch at Same Head (GATE MCQ type)

Problem: A rectangular notch (L = 0.5 m, C_d = 0.62) and a 90° V-notch (C_d = 0.62) are each used to measure flow with H = 0.25 m. Which gives greater discharge?

Answer: Rectangular notch gives greater discharge (0.114 m³/s vs 0.046 m³/s) at H = 0.25 m.

13. Common Mistakes

Mistake 1 — Using H^3/2 Instead of H^5/2 for V-notch

Error: Applying Q = (2/3)C_dL√(2g)H^3/2 (rectangular formula) to a triangular notch.
Root Cause: Both formulas look similar; the exponent (3/2 vs 5/2) is easily confused under exam pressure.
Fix: Derive both mentally from the integration: rectangular → width is constant → one extra integration step → H^3/2. Triangular → width increases linearly with depth → two extra integration steps → H^5/2. Always write the shape first, then choose the formula.

Mistake 2 — Using Area of Pipe Instead of Area of Orifice in the Orifice Meter Formula

Error: Substituting A₁ (pipe area) for A₀ (orifice area) in Q = C_d(A₁A₀/√(A₁²–A₀²))√(2gΔh).
Root Cause: In a venturimeter, A₂ = throat area. In an orifice meter, A₀ = orifice hole area — not the pipe cross-section. They are named differently for a reason.
Fix: For orifice meter: A₀ = π D_orifice²/4; for venturimeter: A₂ = π D_throat²/4. In both cases, A₁ = pipe area at inlet section. Never mix these up.

Mistake 3 — Ignoring End Contractions in Rectangular Notch Problems

Error: Using the full crest length L without applying the Francis end contraction correction (L – 0.1nH) when end contractions are present.
Root Cause: Students skip the correction either because they forget it or because they are unsure whether n = 1 or 2. The problem statement usually specifies “end contractions present” or the geometry (weir against one wall vs freestanding) makes it clear.
Fix: n = 2 if the weir is narrower than the channel and both sides are free; n = 1 if one side is flush with the channel wall; n = 0 if the weir spans the full channel width (suppressed end contractions).

Mistake 4 — Forgetting the C_d = C_v × C_c Relationship for Orifice

Error: Given C_v and C_c, computing discharge as Q = C_v × A₀ × √(2gH) without multiplying by C_c.
Root Cause: The jet velocity is C_v√(2gH), but the jet cross-sectional area is C_cA₀ (not A₀). Discharge = area × velocity = C_cA₀ × C_v√(2gH) = C_dA₀√(2gH).
Fix: Always compute C_d = C_v × C_c first, then use Q = C_d A₀ √(2gH).

Mistake 5 — Confusing Head H in Weir Formulas (Using Total Water Depth Instead of Depth Above Crest)

Error: Using total channel depth d as H in the weir formula instead of the head above the weir crest (H = d – P, where P = height of weir sill above channel bed).
Root Cause: Misreading the problem or not drawing a clear diagram. H in all weir formulas is always measured from the weir crest (sill) to the upstream free surface — not from the channel bed.
Fix: Always draw a sketch showing the channel bed, the weir crest height P, and the water surface. H = total depth – sill height = d – P.

14. Frequently Asked Questions

Q1. Why is C_d of an orifice meter much lower than that of a venturimeter?

The coefficient of discharge C_d accounts for two effects: velocity losses (C_v) and contraction of the jet (C_c). For a venturimeter, the gradual converging profile guides the flow smoothly to the throat without separation, so C_v ≈ 0.995–0.999 and C_c = 1.0 (full bore at throat) → C_d ≈ 0.96–0.99. For a sharp-edged orifice plate, the flow contracts sharply to a vena contracta downstream of the plate (C_c ≈ 0.61–0.64), and the abrupt geometry creates turbulent mixing and energy loss (C_v ≈ 0.97–0.99). The product C_v × C_c ≈ 0.62 gives the low C_d. In practice, this means an orifice meter with a given manometer reading actually discharges much less fluid than a venturimeter with the same reading — a venturimeter more efficiently converts pressure energy to kinetic energy at the throat.

Q2. What is vena contracta and why does it occur?

The vena contracta is the minimum cross-section of a fluid jet issuing from an orifice, located a short distance downstream of the orifice opening. It occurs because the fluid streamlines approaching the orifice converge not just radially but also axially — streamlines from the sides of the tank have inward momentum that continues past the orifice plane. This causes the jet to narrow further before the streamlines become parallel. At the vena contracta, the streamlines are parallel and the pressure equals atmospheric. The coefficient of contraction C_c = A_{vena contracta}/A_orifice ≈ 0.61–0.64 for a sharp-edged orifice, meaning the jet cross-section is about 61–64% of the orifice area at the vena contracta. A rounded (bell-mouthed) orifice eliminates contraction because the streamlines are guided smoothly → C_c = 1.0 and C_d ≈ C_v.

Q3. How does the approach velocity affect weir discharge measurement?

Standard weir formulas assume the upstream velocity is negligible (V_a ≈ 0), which is valid when the weir is installed in a large channel or tank with a large cross-sectional area relative to the weir opening. When the approach channel is narrow (approach velocity V_a is significant), the kinetic energy of the incoming flow adds to the effective head over the weir. The corrected head becomes H_eff = H + V_a²/(2g), and the discharge formula becomes Q = (2/3)C_dL√(2g)[(H + h_a)^3/2 – h_a^3/2] where h_a = V_a²/(2g). For V_a < 0.3 m/s, the correction is typically less than 1% and can be neglected. In irrigation canal measurement (where IS 6966 applies), approach velocity corrections are specified when the velocity exceeds threshold values.

Q4. When should a V-notch be preferred over a rectangular notch for flow measurement?

A V-notch should be chosen when the flow range is small or highly variable, and accuracy at low flows is important. Because Q ∝ H^5/2 for a V-notch (vs H^3/2 for a rectangular notch), the V-notch produces a larger relative change in H for a given change in Q at low flows — making it easier to measure accurately with a staff gauge or water level sensor. For example, a V-notch that passes 5 L/s at H = 0.10 m will only pass 28 L/s at H = 0.20 m — a doubling of H gives a 5.66× increase in Q (2^5/2 = 5.66). The V-notch is also self-cleaning to some extent (no horizontal sill to trap sediment). Its limitation is low capacity — at high heads, a wide V-notch becomes unwieldy and a rectangular weir or broad-crested weir is preferred for large irrigation or spillway discharges.