Capacitive, ultrasonic, radar and radiation level gauges

Why it matters

Many vessels cannot use floats or pressure taps: corrosive acids, sticky polymers, powders and grain in silos, molten material, or vessels where nothing may enter. Capacitance probes, ultrasonic and radar (microwave) transmitters and nuclear (radiation) gauges fill that gap. They are now the most common choices for new installations, and each has a physical weakness (dielectric changes, temperature, foam, licensing) that decides where it can be used.

Key ideas

Capacitance level gauges.

• A probe (rod or cable) and the vessel wall, or a concentric outer tube, form a capacitor. The material in the gap has a relative permittivity εr greater than that of the vapour above (≈ 1), so capacitance rises with level.
• For a non-conducting liquid with a bare probe, C is linear in level: C = C₀ + (2π·ε₀·(εr − 1)/ln(b/a))·h.
• For a conducting liquid (water, acids) the probe is insulated (PTFE sheath); the liquid then acts as the outer plate and the insulation is the dielectric, so the reading is set mainly by the sheath and is nearly independent of the liquid's εr.
• Calibration depends on εr, which varies with temperature, composition and moisture; coating of the probe by conductive material causes false high readings (modern RF admittance designs reject this). Used for liquids, interfaces and granular solids; also as point level switches.

Ultrasonic level gauges.

• A transducer at the top emits pulses (typically 10–70 kHz) that reflect from the surface; the echo time t gives distance d = c·t/2, and level = (mounting height) − d.
• The speed of sound in air depends on temperature (≈ 331.3·√(1 + T/273.15) m/s, about 0.17 % per °C near room temperature), so a built-in temperature sensor compensates. Vapour composition and pressure also change c.
• Problems: foam absorbs sound, dust and vapour attenuate it, turbulence scatters it, and there is a blanking distance near the transducer where it cannot measure. Non-contact, cheap, good for open tanks, sumps and open channels.

Radar (microwave) level gauges.

• Electromagnetic waves (about 6–80 GHz) travel at c ≈ 3 × 10⁸ m/s and are almost unaffected by temperature, pressure, vapour and dust, so radar suits reactors, hot tanks and high-accuracy inventory (custody) gauging.
• Pulse radar measures nanosecond echo times (5 m corresponds to 33 ns round trip). FMCW radar sweeps frequency linearly; the beat frequency between transmitted and received signals is proportional to distance: d = c·f_b/(2·S), where S is the sweep rate.
• Guided-wave radar (TDR) sends pulses down a rod or cable probe; it works with low-εr liquids, foam and interfaces.
• The reflection strength depends on εr: water (εr ≈ 80) reflects strongly; hydrocarbons (εr ≈ 2) weakly. Heavy foam, condensation on the antenna and internal obstructions cause false echoes.

Radiation (nuclear) level gauges.

• A sealed gamma source (Cs-137 or Co-60) on one side of the vessel and a detector (scintillator or Geiger–Müller tube) on the other. Material in the beam attenuates the radiation: I = I₀·e^(−μ·x).
• Point gauges act as switches; continuous gauges use a long source or a long detector. Nothing is mounted inside the vessel, so they work through thick walls on molten, toxic, highly corrosive, very high-pressure or abrasive contents.
• Drawbacks: regulatory licensing and safety procedures, source decay (Cs-137 half-life about 30 years requires periodic compensation), and density-dependent reading. Used only where other methods fail.

Formulas

C = 2π·ε₀·[εr·h + (L − h)] / ln(b/a) (coaxial probe, non-conducting liquid, vapour εr ≈ 1)

ΔC/Δh = 2π·ε₀·(εr − 1) / ln(b/a)

d = c·t / 2 and level = H − d (ultrasonic or pulse radar)

c_air ≈ 331.3·√(1 + T/273.15) (m/s, T in °C)

d = c·f_b / (2·S), S = ΔF/T_s (FMCW radar)

I = I₀·e^(−μ·x) (gamma attenuation)

Symbols: C = capacitance (F); ε₀ = 8.854 × 10⁻¹² F/m; εr = relative permittivity of the liquid; h = liquid height on the probe (m); L = active probe length (m); a = probe radius, b = inner radius of outer tube (m); d = distance from sensor to surface (m); c = wave speed (m/s); t = round-trip time (s); H = sensor height above tank bottom (m); f_b = beat frequency (Hz); ΔF = swept bandwidth (Hz); T_s = sweep time (s); I, I₀ = detected and unattenuated intensity; μ = linear attenuation coefficient of the material (m⁻¹, from data tables); x = path length in the material (m).

Worked examples

Example 1 (standard): coaxial capacitance probe. Given: probe of radius a = 5 mm in a stilling tube of inner radius b = 25 mm; active length L = 2.0 m; oil εr = 2.2; level h = 1.2 m.

1. ln(b/a) = ln 5 = 1.609; 2π·ε₀ = 5.563 × 10⁻¹¹ F/m.
2. Empty: C₀ = 5.563 × 10⁻¹¹ × 2.0/1.609 = 69.1 pF.
3. At h = 1.2 m: C = 5.563 × 10⁻¹¹ × (2.2 × 1.2 + 0.8)/1.609 = 5.563 × 10⁻¹¹ × 3.44/1.609 = 118.9 pF.
4. Sensitivity = 5.563 × 10⁻¹¹ × 1.2/1.609 = 41.5 pF/m.
5. Answer: C ≈ 119 pF at 1.2 m; sensitivity ≈ 41.5 pF/m.

Example 2 (GATE level): ultrasonic gauge and temperature error. Given: transducer H = 8.0 m above the tank bottom; echo time t = 30.0 ms. (a) Find the level if the air is at 20 °C. (b) The air is actually at 50 °C but the gauge assumes 20 °C. Find the true level and the error.

1. c(20 °C) = 331.3 × √(1 + 20/273.15) = 343.2 m/s; c(50 °C) = 331.3 × √(1 + 50/273.15) = 360.3 m/s.
2. (a) d = 343.2 × 0.030/2 = 5.148 m; level = 8.0 − 5.148 = 2.852 m.
3. (b) True d = 360.3 × 0.030/2 = 5.405 m; true level = 8.0 − 5.405 = 2.595 m.
4. The uncompensated gauge reads 2.852 m: error = +0.257 m (reads high).
5. Answer: (a) 2.85 m; (b) true level 2.59 m, error ≈ +0.26 m. This is why ultrasonic transmitters carry a temperature sensor.

Example 3 (FMCW radar). A radar sweeps 1 GHz in 1 ms (S = 10¹² Hz/s) and measures a beat of 30 kHz. d = 3 × 10⁸ × 3 × 10⁴ / (2 × 10¹²) = 4.5 m.

Example 4 (gamma attenuation). With μ = 8.57 m⁻¹ for water (given data for Cs-137 gamma), a 0.5 m water path transmits I/I₀ = e^(−4.285) ≈ 0.014, so the detector count falls by about 99 % when liquid enters the beam.

Common mistakes

• Forgetting the factor of 2 in echo-time calculations (the wave goes down and back).
• Reporting distance instead of level (level = H − d).
• Using the same capacitance calibration after the product changes (εr changes).
• Expecting an ultrasonic gauge to work in vacuum, through heavy foam, or within its blanking distance.
• Assuming radar works equally well on low-εr hydrocarbons and on water; low-εr products give weak echoes.
• Neglecting source decay and density changes in radiation gauges.

For GATE IN

Numericals: capacitance of a coaxial probe versus level, echo-time to level for ultrasonic and radar, temperature correction of the speed of sound, and exponential attenuation of gamma rays. Conceptual questions: which method suits foam, vacuum, high temperature, conducting or non-conducting liquids, and why radar is unaffected by vapour temperature while ultrasonic is not.

Quick check

1. An ultrasonic echo returns in 20 ms at c = 343 m/s from a sensor 6 m above the bottom. What is the level?
2. Why is the probe insulated when a capacitance gauge measures a conducting liquid?
3. Which non-contact method works in a vacuum vessel: ultrasonic or radar?
4. What round-trip time corresponds to 3 m for radar?
5. Give one reason radiation gauges are used only as a last resort.

Answers: 1. d = 343 × 0.02/2 = 3.43 m, level = 2.57 m. 2. Otherwise the conducting liquid short-circuits the capacitor; the sheath becomes the dielectric. 3. Radar (sound needs a medium). 4. 2 × 3/(3 × 10⁸) = 20 ns. 5. Licensing and radiation safety requirements (also cost and source decay).