Flow measurement: orifice, venturi, flow nozzle, pitot tube

Why it matters

Head-type (differential pressure) flow meters are still the most widely installed flow meters in refineries, power plants and water systems. An orifice plate costs little, has no moving parts and can be sized from standards without a flow calibration. Venturis and nozzles are chosen where energy loss or erosion matters, and Pitot tubes give point velocities in ducts, stacks and aircraft. Every one of them rests on the same equations, so getting β, the discharge coefficient and the square-root law right is essential.

Key ideas

Principle. A restriction speeds the fluid up; by Bernoulli's equation the static pressure falls. Combining Bernoulli with continuity between the upstream section (area A₁) and the throat or vena contracta (area A₂) gives the ideal flow. Real flow is smaller, and the discharge coefficient Cd corrects for friction and for the vena contracta being smaller than the orifice bore.

Square-root law. Q ∝ √ΔP. Doubling the flow quadruples the DP. This limits practical turndown of a single DP meter to about 3:1 to 4:1 and means the DP signal must be square-rooted to read flow.

Orifice plate.

• A thin, sharp-edged plate with a concentric bore of diameter d in a pipe of diameter D; β = d/D, typically 0.2 to 0.75.
• Cd ≈ 0.6 (standards give Cd as a function of β, Reynolds number and tap location).
• Taps: flange taps, corner taps, D and D/2 taps; Cd depends on the tap arrangement.
• High permanent pressure loss: roughly (1 − β^1.9) of the measured DP, so about 73 % at β = 0.5.
• Needs long straight upstream runs (or a flow conditioner); the sharp edge wears and rounds with abrasive fluids, making the meter read low. Eccentric and segmental plates are used for liquids carrying solids or gases carrying liquid.

Venturi tube.

• Converging cone (about 21°), cylindrical throat, then a gentle diverging cone (about 7–15°) that recovers most of the pressure.
• Cd ≈ 0.98; permanent loss only about 10–15 % of the DP.
• Expensive and long, but ideal for large water mains, slurries and where pumping energy matters.

Flow nozzle.

• A smooth, elliptical inlet with a cylindrical throat but no diverging cone; Cd ≈ 0.95–0.99.
• Permanent loss between orifice and venturi. Very robust at high velocity and temperature, so it is the classic choice for high-pressure steam lines; also the standard for critical-flow (sonic) nozzles.

Pitot tube.

• A tube facing upstream brings the fluid to rest and senses the stagnation (total) pressure p₀; a static tap senses static pressure p. The difference is the dynamic pressure ½ρv².
• Measures point velocity only; volumetric flow needs a traverse across the duct (or an averaging Pitot, e.g. a multi-port bar).
• Very low pressure loss; poor at low velocities because the DP is tiny; ports clog with dirty fluids.

Compressible fluids. For gases an expansibility factor ε (less than 1, from the standard) multiplies the equation when ΔP is not small compared with line pressure.

Formulas

Q = Cd·A₂·√(2·ΔP/ρ) / √(1 − β⁴) (orifice, nozzle, venturi)

β = d/D, E = 1/√(1 − β⁴) (velocity-of-approach factor)

ṁ = ρ·Q = Cd·ε·E·A₂·√(2·ρ·ΔP) (mass flow; ε = 1 for liquids)

ΔP = (ρm − ρ)·g·h (manometer under the process fluid)

v = √(2·(p₀ − p)/ρ) (Pitot tube, incompressible; multiply by the probe coefficient if given)

Permanent loss (orifice) ≈ (1 − β^1.9)·ΔP (approximation)

Symbols: Q = volumetric flow (m³/s); ṁ = mass flow (kg/s); Cd = discharge coefficient; A₂ = bore or throat area (m²); ΔP = differential pressure between taps (Pa); ρ = fluid density (kg/m³); d = bore/throat diameter, D = pipe diameter (m); β = diameter ratio; ε = expansibility factor; ρm = manometer liquid density (kg/m³); h = manometer reading (m); v = local velocity (m/s); p₀ = stagnation pressure, p = static pressure (Pa). Valid for steady, single-phase, fully developed flow with the specified straight runs.

Worked examples

Example 1 (standard): orifice plate on water. Given: D = 100 mm, d = 50 mm, Cd = 0.61, ΔP = 20 kPa, ρ = 1000 kg/m³.

1. β = 50/100 = 0.5; β⁴ = 0.0625; E = 1/√0.9375 = 1.0328.
2. A₂ = π × 0.050²/4 = 1.9635 × 10⁻³ m².
3. √(2ΔP/ρ) = √(2 × 20 000/1000) = √40 = 6.3246 m/s.
4. Q = 0.61 × 1.0328 × 1.9635 × 10⁻³ × 6.3246 = 7.82 × 10⁻³ m³/s.
5. Answer: Q ≈ 7.82 × 10⁻³ m³/s ≈ 28.2 m³/h. Permanent loss ≈ (1 − 0.5^1.9) × 20 ≈ 14.6 kPa.

Example 2 (GATE level): venturi with a mercury manometer. Given: water in a 150 mm pipe, throat 75 mm, Cd = 0.98, mercury manometer (ρm = 13 600 kg/m³) with water above the mercury reads h = 200 mm.

1. ΔP = (13 600 − 1000) × 9.81 × 0.200 = 24 721 Pa.
2. β = 0.5, so E = 1.0328 (as before).
3. A₂ = π × 0.075²/4 = 4.418 × 10⁻³ m².
4. √(2ΔP/ρ) = √(2 × 24 721/1000) = √49.44 = 7.032 m/s.
5. Q = 0.98 × 1.0328 × 4.418 × 10⁻³ × 7.032 = 0.0314 m³/s.
6. Answer: Q ≈ 0.0314 m³/s ≈ 113 m³/h. Using ρm instead of (ρm − ρ) would overstate ΔP by about 8 % and Q by about 4 %.

Example 3 (Pitot-static tube in a duct). Given: air ρ = 1.2 kg/m³; a water manometer (ρm = 1000 kg/m³) across the total and static ports reads 25 mm.

1. Δp = (1000 − 1.2) × 9.81 × 0.025 = 244.96 Pa.
2. v = √(2 × 244.96/1.2) = √408.3 = 20.2 m/s.
3. Answer: v ≈ 20.2 m/s at the probe tip (not the mean duct velocity).

Common mistakes

• Omitting the velocity-of-approach factor 1/√(1 − β⁴); at β = 0.5 this is a 3.3 % error, at β = 0.7 about 15 %.
• Using pipe area instead of bore area in Q = Cd·A₂·√(2ΔP/ρ).
• Assuming flow is proportional to DP rather than to its square root.
• Treating the Pitot reading as the mean pipe velocity.
• Using ρm instead of (ρm − ρ) when the manometer legs contain process liquid.
• Ignoring straight-run requirements; swirl and distorted profiles give errors of several per cent.
• Confusing measured DP with permanent pressure loss.

For GATE IN

Standard numericals: flow from orifice/venturi data (often with a manometer reading), β and Cd effects, the square-root relation between flow change and DP change, and Pitot velocity. Conceptual questions compare permanent loss, cost and accuracy of orifice, nozzle and venturi, and ask why head meters need square-root extraction. Practise carrying units through √(2ΔP/ρ) and including the 1/√(1 − β⁴) factor.

Quick check

1. Flow through an orifice rises by 50 %. By what factor does ΔP rise?
2. Which head meter has the lowest permanent pressure loss?
3. What is β for a 60 mm bore in a 120 mm pipe, and E?
4. A Pitot tube in water shows a dynamic pressure of 2 kPa. What is the velocity?
5. Why is a flow nozzle preferred for high-pressure steam?

Answers: 1. 1.5² = 2.25. 2. The venturi tube. 3. β = 0.5, E = 1.033. 4. v = √(2 × 2000/1000) = 2.0 m/s. 5. It is robust against erosion at high velocity and temperature and loses less pressure than an orifice, while being shorter and cheaper than a venturi.