Measurement of temperature, pressure, flow and level

Why it matters

A control loop can never be better than its measurement: a controller drives the measured value to set point, so a biased or slow sensor drives the real process somewhere else. Temperature, pressure, flow and level are the four variables on almost every P&ID, and a plant engineer is expected to choose the sensor, read its signal and spot when it is lying.

Key ideas

Static and dynamic characteristics. Every instrument has a range (lower to upper limit), a span (upper minus lower limit), accuracy (closeness to the true value), precision/repeatability, sensitivity (output change per unit input change), resolution, dead band and hysteresis. Its dynamics matter as much: most sensors behave like a first-order lag with a time constant τ, which adds to the loop's dynamics.

Temperature.

• Thermocouple: two dissimilar metals joined at a measuring (hot) junction. The Seebeck effect gives an emf that depends on the temperature difference between the hot junction and the reference (cold) junction, so the reference must be held at a known temperature or compensated electronically (cold-junction compensation). Wide range (type K to about 1250 °C, types R/S/B higher), rugged, fast, but low output (tens of µV/K) and modest accuracy.
• RTD (usually Pt100, 100 Ω at 0 °C): resistance rises almost linearly with temperature. More accurate and stable than a thermocouple, but slower, more expensive and limited to about 600 °C. Three- or four-wire connection cancels lead resistance.
• Thermistor: semiconductor with a large, strongly non-linear (usually negative) temperature coefficient; very sensitive over a narrow range.
• Filled-system and bimetallic thermometers for local indication.
• A thermowell protects the sensor and allows removal under pressure, but adds a thermal lag.

Pressure.

• Manometers (U-tube, inclined, well type) balance pressure against a liquid column; they are primary standards for low differential pressures.
• Bourdon tube: a curved flattened tube straightens under pressure and moves a pointer; diaphragms and bellows serve lower ranges.
• Electronic transmitters: strain-gauge, capacitance or piezo-resistive cells convert deflection of a diaphragm into an electrical signal. Gauge pressure is measured relative to atmosphere, absolute pressure relative to vacuum, differential pressure between two ports.

Flow.

• Differential-pressure (head) meters: orifice plate, venturi, flow nozzle, pitot tube. Bernoulli's equation gives Q ∝ √ΔP, so the signal must be square-root extracted. The orifice is cheap but loses much of its ΔP permanently; the venturi recovers most of it.
• Rotameter (variable area): a float rises in a tapered tube until the annular area makes the pressure drop across the float balance its weight; the pressure drop is nearly constant and the area varies.
• Magnetic flowmeter (conducting liquids only, Faraday's law), turbine, vortex, ultrasonic and Coriolis (direct mass flow, independent of density) meters.
• Most meters need a straight pipe run upstream and downstream to give a developed velocity profile.

Level.

• Sight glass, float and displacer (Archimedes) gauges.
• Hydrostatic (DP) level: ΔP = ρgh, so the reading depends on density; a closed (pressurised) vessel needs a DP cell whose low-pressure side is connected to the vapour space.
• Bubbler (purge), capacitance, ultrasonic and radar (non-contact), and nuclear gauges for very hostile service.

Transmission. Field transmitters send a standard 4–20 mA signal (or 20–100 kPa, i.e. 3–15 psig, pneumatic). The "live zero" of 4 mA lets a broken wire (0 mA) be told apart from a genuine zero reading.

Formulas

Span = URV − LRV

• URV, LRV: upper and lower range values (in the measured unit).

Signal (mA) = 4 + 16·(x − LRV)/(URV − LRV)

• x: measured value; applies to a linear 4–20 mA transmitter.

R_T = R₀·(1 + α·T)

• R_T: RTD resistance at T (Ω); R₀: resistance at 0 °C (Ω); α: temperature coefficient (1/°C), about 0.00385 /°C for platinum; T in °C. Linear approximation valid over a moderate range.

ΔP = (ρ_m − ρ)·g·h_m

• Differential manometer: ρ_m manometer-fluid density, ρ flowing-fluid density (kg/m³), g = 9.81 m/s², h_m reading (m), ΔP in Pa.

Q = C_d·A₀·√(2·ΔP / (ρ·(1 − β⁴)))

• Orifice or venturi: Q volumetric flow (m³/s), C_d discharge coefficient (about 0.61 for a sharp-edged orifice, about 0.98 for a venturi), A₀ throat/orifice area (m²), β = d₀/D diameter ratio, ΔP in Pa, ρ in kg/m³. Incompressible flow.

Q₂ / Q₁ = √(ΔP₂ / ΔP₁)

• Square-root relation for any head meter at a fixed C_d.

ΔP = ρ·g·h

• Hydrostatic level: h liquid height above the lower tap (m).

Worked examples

Example 1 (standard): orifice meter with a mercury manometer. Water (ρ = 1000 kg/m³) flows in a 100 mm pipe through a sharp-edged orifice of 50 mm diameter, C_d = 0.62. A mercury manometer (ρ_m = 13 600 kg/m³) across the orifice reads 150 mm. Find the flow rate.

1. ΔP = (ρ_m − ρ)·g·h_m = (13 600 − 1000)·9.81·0.150 = 18 541 Pa.
2. A₀ = π·d₀²/4 = π·(0.050)²/4 = 1.963 × 10⁻³ m².
3. β = 50/100 = 0.5, so 1 − β⁴ = 1 − 0.0625 = 0.9375.
4. √(2·ΔP / (ρ·(1 − β⁴))) = √(2·18 541 / (1000·0.9375)) = 6.289 m/s.
5. Q = C_d·A₀·6.289 = 0.62·1.963 × 10⁻³·6.289 = 7.66 × 10⁻³ m³/s.

Q ≈ 7.66 × 10⁻³ m³/s (about 27.6 m³/h)

Example 2 (GATE level): square-root signal from a DP flow transmitter. A DP transmitter across an orifice is calibrated 0–40 kPa for a 4–20 mA output. At 40 kPa the flow is 50 m³/h. The transmitter reads 9.0 mA and has no square-root extractor. Find the actual flow, and the flow a technician would wrongly report by reading the signal linearly.

1. ΔP = LRV + (I − 4)/16·span = (9.0 − 4)/16·40 = 12.5 kPa.
2. Head meter: Q = Q_max·√(ΔP/ΔP_max) = 50·√(12.5/40) = 50·0.5590 = 27.95 m³/h.
3. Linear (wrong) reading: Q = 50·(12.5/40) = 15.6 m³/h.

Actual flow ≈ 27.95 m³/h; the linear reading under-reports by about 44 %. Note that at 25 % of full-scale ΔP the flow is already 50 % of maximum, which is why head meters have poor resolution at low flow (turndown of about 3–4 : 1).

Example 3 (short): RTD. A Pt100 (R₀ = 100 Ω, α = 0.00385 /°C) reads 138.5 Ω. T = (R_T/R₀ − 1)/α = (1.385 − 1)/0.00385 = 100 °C.

Common mistakes

• Using ρ_m instead of (ρ_m − ρ) for a manometer filled with the flowing liquid above the mercury; ΔP is then overstated by about 8 % for water over mercury (about 4 % in flow).
• Treating a DP flow signal as linear in flow. Flow goes as the square root of ΔP.
• Forgetting that a thermocouple measures a temperature difference; without cold-junction compensation the reading is wrong by the reference temperature.
• Using hydrostatic level without correcting for liquid density (or temperature changes in density), or ignoring the vapour-space pressure in a closed tank.
• Confusing span with range, and gauge with absolute pressure (absolute = gauge + atmospheric).
• Ignoring the thermowell lag: it can be the slowest element in a temperature loop.

For GATE CH

Expect numericals on orifice/venturi flow from manometer readings, the square-root behaviour of head meters, transmitter signal-to-value conversion, and hydrostatic level. Conceptual questions ask which sensor suits a duty (high temperature, non-contact level, mass flow, conducting liquids) and about the first-order dynamics of a thermometer, which link directly to the next topics. Practise unit conversion between mm Hg, kPa and m of liquid.

Quick check

1. A DP transmitter's ΔP falls to one-quarter of its value. By what factor does the flow fall?
2. Which flowmeter measures mass flow directly?
3. Why is the transmitter signal 4–20 mA instead of 0–20 mA?
4. A 4–20 mA transmitter ranged 0–200 °C reads 12 mA. What is the temperature?

Answers: 1. To one-half (Q ∝ √ΔP). 2. Coriolis meter. 3. The live zero lets a wiring fault (0 mA) be distinguished from a true zero. 4. 100 °C.