Electromagnetic, ultrasonic and vortex flow meters

Why it matters

Electromagnetic, ultrasonic and vortex meters have no moving parts and little or no pressure loss, so they have replaced many orifice plates and turbine meters. Magnetic meters dominate water, effluent, slurry and chemical-dosing service; ultrasonic meters handle large pipes, custody transfer of gas and temporary clamp-on surveys; vortex meters are the workhorse for steam and clean gases. Choosing between them depends on fluid conductivity, cleanliness, Reynolds number and pipe size.

Key ideas

Electromagnetic (magnetic) flow meter.

• Faraday's law: a conductor of length D moving at velocity v through a field B perpendicular to both generates E = B·D·v. Here the conductor is the liquid itself; two electrodes on a diameter, at right angles to the field, pick up E.
• For an axisymmetric velocity profile and uniform field, E depends on the mean velocity, so the meter reads volumetric flow regardless of density, viscosity or (laminar or turbulent) profile.
• Requirements: the liquid must be conductive (typically at least about 5 µS/cm); the pipe section must be non-magnetic and lined with an insulator (PTFE, rubber, ceramic) so the signal is not short-circuited; the meter and liquid must be properly earthed (grounding rings for plastic or lined pipes) and the pipe must run full.
• Excitation: pulsed DC is standard, to reject electrochemical electrode potentials and drift; AC excitation suffers from quadrature (transformer) pick-up.
• Cannot measure hydrocarbons, gases or steam (non-conductive). No obstruction, so ideal for slurries and dirty water. Bidirectional.

Ultrasonic flow meters.

• Transit-time: two transducers on a path of length L at angle θ to the pipe axis send pulses both ways. Downstream travel is faster: t_d = L/(c + v·cosθ), upstream t_u = L/(c − v·cosθ). The exact relation v = L·(1/t_d − 1/t_u)/(2·cosθ) does not involve the sound speed c, so temperature and composition changes cancel. Needs reasonably clean fluid (bubbles and solids scatter the beam). Multi-path meters integrate the profile for high accuracy (custody transfer); clamp-on versions mount outside the pipe.
• Doppler: a beam reflects from particles or bubbles moving with the flow; the frequency shift is Δf = 2·f₀·v·cosθ / c. Needs scatterers, so it suits dirty or aerated liquids; accuracy is lower and depends on c.
• A single path measures the average along the path, not the area-average; a profile (meter) factor corrects for this.

Vortex flow meter.

• A bluff body (shedder bar) across the pipe sheds vortices alternately from each side (a Kármán vortex street). The shedding frequency is f = St·v/d, where the Strouhal number St is nearly constant (about 0.2 for many shapes) over a wide Reynolds-number range (roughly 10⁴ to 10⁶).
• Frequency is linear in velocity, so the output is a pulse train with a K-factor (pulses per m³) set by geometry and largely independent of density, viscosity and pressure within that range.
• Sensors: piezoelectric or capacitive sensors detect the alternating force or pressure.
• Limits: below a minimum Reynolds number (and velocity) shedding becomes irregular, so there is a low-flow cut-off; viscous liquids raise that cut-off; pipe vibration can give false pulses; needs straight runs. Widely used on steam, gases and clean liquids.

Formulas

E = B·D·v̄ (electromagnetic meter)

Q = (π·D²/4)·v̄ = π·D·E / (4·B)

t_d = L/(c + v·cosθ), t_u = L/(c − v·cosθ) (ultrasonic transit-time)

v = L·(1/t_d − 1/t_u) / (2·cosθ) (exact, c-independent)

Δt = t_u − t_d ≈ 2·L·v·cosθ / c² so v ≈ c²·Δt / (2·L·cosθ) (for v ≪ c)

Δf = 2·f₀·v·cosθ / c (Doppler)

f = St·v / d (vortex shedding)

Symbols: E = electrode voltage (V); B = magnetic flux density (T); D = pipe inner diameter (m); v̄ = mean velocity (m/s); Q = volumetric flow (m³/s); L = acoustic path length (m); θ = angle between acoustic path and pipe axis; c = speed of sound in the fluid (m/s); t_d, t_u = downstream and upstream transit times (s); f₀ = transmitted frequency, Δf = Doppler shift (Hz); f = vortex frequency (Hz); St = Strouhal number; d = bluff-body width (m). For a diagonal path across the full pipe, L·cosθ = D/tanθ.

Worked examples

Example 1 (standard): magnetic flow meter. Given: D = 150 mm, B = 0.01 T, electrode voltage E = 2.4 mV.

1. v̄ = E/(B·D) = 2.4 × 10⁻³ / (0.01 × 0.15) = 1.60 m/s.
2. Q = (π × 0.15²/4) × 1.60 = 0.017 67 × 1.60 = 0.0283 m³/s.
3. Answer: v̄ = 1.6 m/s, Q ≈ 0.0283 m³/s ≈ 102 m³/h. Note that the signal is only millivolts, which is why electrode noise rejection and earthing matter.

Example 2 (GATE level): transit-time ultrasonic meter. Given: water, c = 1480 m/s; pipe D = 200 mm; diagonal path at θ = 45° crossing the full pipe; measured Δt = t_u − t_d = 0.365 µs. Find the path-average velocity and flow (meter factor 1).

1. Path length L = D/sinθ = 0.2/0.7071 = 0.2828 m; L·cosθ = D/tanθ = 0.200 m.
2. v ≈ c²·Δt / (2·L·cosθ) = 1480² × 0.365 × 10⁻⁶ / (2 × 0.200).
3. c² = 2.1904 × 10⁶ m²/s²; numerator = 2.1904 × 10⁶ × 0.365 × 10⁻⁶ = 0.7995 m²/s.
4. v = 0.7995 / 0.400 = 2.00 m/s.
5. Q = (π × 0.2²/4) × 2.00 = 0.0314 × 2.00 = 0.0628 m³/s.
6. Answer: v ≈ 2.0 m/s, Q ≈ 0.063 m³/s (≈ 226 m³/h). The transit times themselves are about 191 µs, so the meter must resolve a 0.2 % difference: this is why timing electronics resolve to fractions of a nanosecond.

Example 3 (vortex). A shedder bar of width d = 30 mm with St = 0.2 sheds at f = 100 Hz. v = f·d/St = 100 × 0.03/0.2 = 15 m/s. In practice the manufacturer's K-factor (pulses per m³), which allows for the blockage of the bar, converts the frequency to flow.

Common mistakes

• Using a magnetic flow meter on oil, demineralised water below the conductivity limit, or a partly full pipe.
• Writing Q = E·πD²/(4B): the correct result is Q = πD·E/(4B), since v = E/(B·D).
• Forgetting the cosθ factor in transit-time and Doppler formulas.
• Assuming a Doppler meter works on clean water; it needs scatterers. Conversely, transit-time meters fail with heavy bubbles or solids.
• Using a vortex meter at low Reynolds number (viscous liquid, low flow) where shedding is irregular.
• Ignoring installation: straight runs for ultrasonic and vortex meters, earthing and full pipe for magnetic meters.

For GATE IN

Numericals: velocity and flow from the Faraday voltage, transit-time difference or Doppler shift, and vortex frequency through the Strouhal number. Conceptual questions: which meter suits which fluid (conductive slurry, clean gas, steam, aerated liquid), why pulsed-DC excitation is used, why transit-time measurement is independent of c, and the low-Reynolds-number limit of vortex meters. Practise the geometry of the acoustic path (L, θ, D).

Quick check

1. Why can a magnetic flow meter not measure diesel flow?
2. In a transit-time meter, which way is the pulse faster: with or against the flow?
3. A vortex meter's frequency doubles. What happens to the flow?
4. B = 0.02 T, D = 0.1 m, E = 4 mV: what is the mean velocity?
5. Which ultrasonic type suits a sewage line with suspended solids?

Answers: 1. Diesel is non-conductive, so no measurable voltage is induced. 2. With the flow (downstream). 3. It doubles (f ∝ v). 4. v = 4 × 10⁻³/(0.02 × 0.1) = 2 m/s. 5. Doppler.