Level measurement: float, displacer and DP methods

Why it matters

Level is measured on almost every vessel: storage tanks for inventory, boiler drums for safety, separators for the oil–water interface, reactors and columns for control. Floats, displacers and differential-pressure (hydrostatic) methods are the classic contact techniques and still make up a large share of installed level instruments. All three depend on the liquid's density in some way, which is the key to using them correctly.

Key ideas

Float methods.

• A float less dense than the liquid rides on the surface; its position is the level.
• Float and tape (or cable) gauge: the float is connected over pulleys to a counterweight and an indicator; used on large storage tanks, with automatic tank gauges for inventory.
• Magnetic float gauges: a float carrying a magnet moves in a side chamber; flags or a reed-switch chain outside follow it, giving local indication and a resistive transmitter signal.
• Float switches: a float tilts or lifts a magnet to operate a switch at one level (high/low alarms, pump control).
• A float's immersion depth depends on liquid density, so its reading shifts slightly when the density changes. Floats suffer from sticking in viscous or coating liquids and from turbulence.

Displacer (buoyancy) method.

• A long, heavy cylinder (denser than the liquid, so it does not float) hangs in the liquid from a torque tube or a spring. By Archimedes' principle the upward buoyant force equals the weight of liquid displaced, ρ·g·A·h, where h is the immersed length.
• The apparent weight W − ρ·g·A·h is sensed by twisting a torque tube (which also forms the pressure seal) or by stretching a range spring; the displacer itself barely moves.
• Range equals the displacer length (typically 0.3–3 m). Calibration depends on density, so a displacer can also measure interface level between two liquids, or density when it is fully submerged.
• Good for high pressure and temperature (torque-tube seal), but long displacers need a side chamber (cage) and the reading is upset by density changes and coating.

Hydrostatic (differential pressure) methods.

• Pressure at depth h: p = p_top + ρ·g·h. A pressure or DP transmitter at the bottom measures ρ·g·h, so h = ΔP/(ρ·g).
• Open tank: L side vented. Closed tank: L side connected to the vapour space with a dry leg (gas-filled, must stay dry) or a wet leg (filled with a seal liquid, giving zero elevation). A transmitter mounted below the bottom tap sees an extra constant head (zero suppression).
• Bubbler (purge) system: air or nitrogen bubbles slowly out of a dip pipe; the back-pressure equals ρ·g·h above the pipe's end. Good for corrosive or slurry liquids because only the dip pipe touches the liquid.
• Interface: with two liquids between two taps, ΔP = g·(ρ₁h₁ + ρ₂h₂), so the interface position follows if the densities are known.
• Hydrostatic methods measure mass per unit area (head); if density changes, the indicated level is in error, though inventory in mass terms is correct.

Formulas

F_b = ρ·g·V_displaced (Archimedes)

W_app = W − ρ·g·A·h (displacer, immersed length h)

h = (W − W_app)/(ρ·g·A)

ΔP = ρ·g·h (open tank or with gas pressure cancelled)

ΔP = ρ·g·h + ρ·g·h₀ (transmitter h₀ below the bottom tap, leg filled with process liquid: zero suppression)

ΔP = g·(ρ₁·h₁ + ρ₂·h₂) (two liquids between taps)

p_bubbler = ρ·g·h (gauge pressure at the dip-pipe outlet)

Symbols: F_b = buoyant force (N); ρ = liquid density (kg/m³); g = 9.81 m/s²; V = displaced volume (m³); W = displacer weight in air (N); W_app = apparent weight (N); A = displacer cross-sectional area (m²); h = immersed length or liquid height (m); h₀ = transmitter offset below the tap (m); h₁, h₂ = thicknesses of the two liquid layers (m); ΔP, p = pressures (Pa).

Worked examples

Example 1 (standard): displacer apparent weight. Given: displacer 80 mm diameter, 1.0 m long, weight in air W = 60 N, liquid ρ = 800 kg/m³; level is 0.6 m above the displacer bottom.

1. A = π × 0.040² = 5.027 × 10⁻³ m².
2. Buoyancy F_b = ρ·g·A·h = 800 × 9.81 × 5.027 × 10⁻³ × 0.6 = 23.67 N.
3. W_app = 60 − 23.67 = 36.33 N.
4. Full range: at h = 1.0 m, F_b = 39.45 N and W_app = 20.55 N, so the torque tube sees a change of 39.45 N over the range.
5. Answer: apparent weight ≈ 36.3 N at 60 % of range.

Example 2 (GATE level): oil–water interface by DP. Given: a separator is always full of liquid between two taps 2.0 m apart; oil (ρ₁ = 800 kg/m³) floats on water (ρ₂ = 1000 kg/m³). A DP transmitter with diaphragm seals (seal-fill heads already zeroed) reads ΔP = 18.0 kPa. Find the height of the interface above the lower tap.

1. Let h_w = water height above the lower tap; oil height = 2.0 − h_w.
2. ΔP = g·(ρ₂·h_w + ρ₁·(2.0 − h_w)) = g·(1600 + 200·h_w).
3. ΔP/g = 18 000/9.81 = 1834.9 kg/m².
4. 200·h_w = 1834.9 − 1600 = 234.9, so h_w = 1.174 m.
5. Check: at h_w = 0, ΔP = 15.70 kPa; at h_w = 2, ΔP = 19.62 kPa. The span for the interface is only 3.92 kPa, so a small density error causes a large interface error.
6. Answer: interface ≈ 1.17 m above the lower tap.

Example 3 (bubbler). In a tank of acid (ρ = 1250 kg/m³) the purge-air back-pressure is 24.5 kPa gauge. h = 24 500/(1250 × 9.81) = 2.00 m of liquid above the dip-pipe end.

Common mistakes

• Treating a displacer like a float: it does not ride on the surface; its apparent weight changes with immersion.
• Using the wrong density: a DP or displacer level instrument calibrated for water reads low on a lighter liquid at the same level, because ρ·g·h (or the buoyancy) is smaller. Always calibrate for the actual density.
• Forgetting the vapour pressure in closed tanks (it must act on both sides) or letting a "dry" leg fill with condensate.
• Ignoring the head between the bottom tap and a transmitter mounted lower (zero suppression).
• Calculating buoyancy on a floating float as ρ·g·V_total; a floating body displaces only its own weight of liquid.

For GATE IN

Expect numericals on buoyancy and displacer apparent weight, hydrostatic level with suppression or elevation, bubbler back-pressure and two-liquid interface by DP. Conceptual questions ask which method suits pressurised, corrosive, high-temperature or interface service and how density affects each method. Practise writing the pressure balance from the bottom tap up through each layer.

Quick check

1. A water tank shows ΔP = 29.43 kPa at the bottom. What is the level?
2. Why does a displacer barely move while its signal changes?
3. A DP transmitter calibrated for water is used on a liquid of SG 0.8 at the same level. Does it read high or low, and by how much?
4. What does a bubbler measure, and why does it suit slurries?
5. In a displacer, what is the apparent weight when it is fully immersed?

Answers: 1. 29 430/(1000 × 9.81) = 3.0 m. 2. The torque tube or spring is stiff; it senses force rather than large movement. 3. Low, by 20 % (it shows 0.8 of the true level). 4. The back-pressure ρgh at the dip-pipe outlet; only the pipe contacts the liquid. 5. W − ρ·g·A·L, where L is the displacer length.