Flow measurement: orifice, venturi, pitot tube, rotameter

Why it matters

Every material balance in a plant depends on measured flows. Orifice plates are the most common industrial flow element, venturis are used where pressure loss is costly, pitot tubes survey velocity profiles in ducts and stacks, and rotameters are the everyday meter on lab and utility lines. Each works by converting a flow into a pressure difference or a force, so each is a direct application of Bernoulli, continuity and the momentum balance.

Key ideas

Obstruction (variable-head) meters. A constriction accelerates the fluid; continuity and Bernoulli between the upstream section (1) and the constriction (2) give the ideal flow, and an empirical coefficient corrects for friction and contraction: Q = C·A₂·√[2Δp/(ρ(1 − β⁴))], with β = d₂/d₁ the diameter ratio. The factor 1/√(1 − β⁴) is the velocity-of-approach factor. Because Q ∝ √Δp, these meters have a limited turndown (about 3:1 to 4:1 for good accuracy) — a 10:1 flow range needs a 100:1 Δp range.

Venturi meter.

• Converging cone (about 21°), throat, then a long diverging cone (about 5–15°) that recovers most of the pressure.
• Coefficient of discharge C_v ≈ 0.95–0.99.
• Permanent pressure loss only about 10–15 % of the measured Δp; low wear, handles slurries; but long and expensive.

Orifice meter.

• A thin plate with a sharp-edged concentric hole (bevelled edge faces downstream) clamped between flanges; taps at flange, corner or vena-contracta positions.
• The jet contracts to a vena contracta downstream of the plate, so C_o is low, about 0.61–0.62 for sharp-edged plates (it depends on β, Re and tap location; take it from the standard or data book).
• Permanent loss is large: roughly (1 − β²)Δp, i.e. 60–75 % of the measured Δp for usual β.
• Cheap, compact, easy to change; needs straight upstream pipe runs (often 10–40 D depending on fittings) for a developed profile.

Pitot and pitot-static tubes.

• A tube facing the flow brings fluid to rest and reads the stagnation pressure p₀ = p + ρV²/2; a static tap reads p. Their difference gives the local velocity V = C·√(2(p₀ − p)/ρ), C ≈ 0.98–1.0.
• Measures point velocity; to get flow rate, traverse the duct and integrate (or use the centreline velocity with V_avg/V_max ≈ 0.5 laminar, ≈ 0.8–0.85 turbulent).
• Negligible pressure loss; unsuitable for dirty fluids (plugging) or very low velocities (tiny Δp). Must be aligned with the flow.

Rotameter (variable-area meter).

• A float in a vertical tapered tube rises until the annular area between float and tube is large enough that the pressure difference across the float balances its buoyancy-corrected weight.
• The pressure drop across the float is constant: Δp = V_f(ρ_f − ρ)g/A_f. Flow varies with annular area, so the float height is nearly linear in flow; turndown about 10:1.
• Calibrated for a particular fluid; a correction for a different density is Q₂/Q₁ = √[(ρ_f − ρ₂)ρ₁/((ρ_f − ρ₁)ρ₂)].
• Must be vertical; usually glass or with magnetic coupling for opaque tubes; not for high pressure or dirty fluids.

Other meters (brief). Magnetic (conductive liquids, no obstruction), turbine, vortex-shedding, ultrasonic, Coriolis (direct mass flow), positive-displacement and notches/weirs for open channels.

Formulas

• Q = C·A₂·√[2·Δp/(ρ·(1 − β⁴))] — venturi or orifice; Q (m³/s), A₂ throat or orifice area (m²), Δp (Pa), ρ (kg/m³), β = d₂/d₁.
• Δp = (ρ_m − ρ)·g·R — Δp from a differential manometer reading R (m), tappings at equal level.
• V = C·√[2·(p₀ − p)/ρ] — pitot-static tube, local velocity (m/s).
• Δp_loss ≈ (1 − β²)·Δp — approximate permanent loss of an orifice.
• Δp = V_f·(ρ_f − ρ)·g/A_f — rotameter pressure drop; V_f float volume (m³), A_f float cross-section (m²), ρ_f float density.
• Q = C_R·A_a·√[2·g·V_f·(ρ_f − ρ)/(A_f·ρ)] — rotameter flow; A_a annular area (m²), C_R discharge coefficient.

Worked examples

Example 1 (standard). A venturi meter with 150 mm inlet and 75 mm throat carries water (ρ = 1000 kg/m³). A mercury–water differential manometer (ρ_m = 13 600 kg/m³) reads 0.2 m. C_v = 0.98. Find Q.

1. Δp = (ρ_m − ρ)gR = 12 600 × 9.81 × 0.2 = 24 721 Pa.
2. β = 75/150 = 0.5; 1 − β⁴ = 0.9375. A₂ = (π/4)(0.075)² = 4.418 × 10⁻³ m².
3. Q = C·A₂·√[2Δp/(ρ(1 − β⁴))] = 0.98 × 4.418 × 10⁻³ × √(2 × 24 721 / 937.5) = 0.98 × 4.418 × 10⁻³ × 7.262 = 0.0314 m³/s. Q ≈ 0.0314 m³/s (31.4 L/s).

Example 2 (GATE level). A sharp-edged orifice of 50 mm diameter is installed in a 100 mm water line (ρ = 1000 kg/m³). The measured pressure difference is 20 kPa and C_o = 0.62. Find (a) the flow rate, (b) the approximate permanent pressure loss, and (c) the pressure difference if the flow doubles.

1. β = 0.5; A₀ = (π/4)(0.05)² = 1.963 × 10⁻³ m².
2. Q = 0.62 × 1.963 × 10⁻³ × √(2 × 20 000 / (1000 × 0.9375)) = 0.62 × 1.963 × 10⁻³ × 6.532 = 7.95 × 10⁻³ m³/s.
3. Permanent loss ≈ (1 − β²)Δp = 0.75 × 20 = 15 kPa.
4. Δp ∝ Q², so doubling Q gives 4 × 20 = 80 kPa. (a) Q ≈ 7.95 L/s; (b) ≈ 15 kPa lost permanently; (c) 80 kPa. A venturi of the same β would read only about 8 kPa at this flow (since C_v ≈ 0.98) and lose permanently only about 1 kPa.

Example 3 (pitot tube). A pitot-static tube in an air duct (ρ = 1.2 kg/m³) is connected to a water manometer reading 50 mm. Δp = 1000 × 9.81 × 0.05 = 490.5 Pa (air column negligible); V = √(2 × 490.5/1.2) = 28.6 m/s at that point.

Common mistakes

• Using the pipe area instead of the throat/orifice area in the meter equation, or omitting the velocity-of-approach factor.
• Using ρ_m·g·R instead of (ρ_m − ρ)·g·R when the manometer limbs contain the flowing liquid.
• Confusing the measured Δp with the permanent pressure loss.
• Treating a pitot reading as the average velocity.
• Using a rotameter calibrated for water on another liquid without a density correction.
• Assuming Q ∝ Δp; it is Q ∝ √Δp for obstruction meters.

For GATE CH

Expect venturi and orifice numericals with manometer readings, comparison of permanent losses, pitot-tube velocity calculations, rotameter force-balance and recalibration problems, and conceptual questions on which meter suits a given duty. Practise combining the manometer equation with the meter equation in one clean chain.

Quick check

1. If the flow through an orifice meter triples, by what factor does Δp change?
2. Which meter has a nearly constant pressure drop across its range?
3. Why does a venturi have a higher discharge coefficient than an orifice?
4. Does a pitot tube measure average velocity?

Answers: 1. 9 times. 2. The rotameter. 3. Its smooth convergence avoids a vena contracta and separation, so the flow fills the throat. 4. No, only the local velocity at its tip.