Density, viscosity and humidity measurement

Why it matters

Density tells a plant the concentration of a solution, the solids content of a slurry, the grade of a fuel and the conversion from volume to mass. Viscosity governs pumping, coating, polymer and food quality. Humidity controls dryers, pharmaceutical and electronics clean rooms, compressed-air quality and comfort air-conditioning. These three "physical property" measurements are less glamorous than flow or level, but they are often the quality variables a plant is actually paid for.

Key ideas

Density (ρ = m/V, kg/m³; specific gravity SG = ρ/ρ_water).

• Laboratory: hydrometer (floats deeper in lighter liquids; reads SG directly on its stem), pycnometer (bottle of exactly known volume, weighed full and empty).
• Hydrostatic: two bubbler tubes or DP taps a fixed vertical distance H apart in the liquid; ΔP = ρ·g·H, so ΔP is directly proportional to density and independent of level (as long as both taps are covered).
• Buoyancy: a fully submerged displacer of fixed volume; apparent weight falls as density rises.
• Vibrating element: a U-tube or fork vibrates at a natural frequency that falls as the mass of contained fluid rises; ρ = A·τ² − B, where τ is the period and A, B are calibration constants. Coriolis meters use the same principle.
• Radiation: gamma attenuation through a pipe increases with density; used on slurries and dredging lines.
• Always report the temperature: liquid densities change by roughly 0.1 % per °C, so readings are corrected to a reference temperature.

Viscosity.

• Dynamic viscosity μ (Pa·s): from Newton's law of viscosity, τ = μ·(du/dy). 1 Pa·s = 1000 cP = 10 P. Kinematic viscosity ν = μ/ρ (m²/s; 1 cSt = 10⁻⁶ m²/s).
• Newtonian fluids (water, oils) have μ independent of shear rate; non-Newtonian fluids (paints, slurries, polymer melts) need a rotational instrument that sets a known shear rate.
• Capillary (tube) viscometers: laminar flow, Hagen–Poiseuille; Ostwald and Ubbelohde glass viscometers measure ν from efflux time.
• Falling-ball viscometer: a sphere falls at terminal velocity; Stokes' law gives μ. Valid only for creeping flow (Re < about 0.1–0.5) and with wall corrections in narrow tubes.
• Rotational viscometers: concentric cylinders (Couette), cone-and-plate, or a spindle (Brookfield type); viscosity is proportional to torque at a given speed.
• On-line: vibrating-rod or torsional-resonator sensors whose damping depends on μ·ρ.
• Liquid viscosity falls sharply with temperature; gas viscosity rises. Temperature must be controlled or reported.

Humidity.

• Absolute humidity: mass of water vapour per unit volume. Humidity ratio ω: kg water per kg dry air. Relative humidity RH = p_v/p_s(T) × 100 %, where p_s is the saturation pressure at the dry-bulb temperature. Dew point: temperature at which the air becomes saturated on cooling at constant pressure.
• Psychrometer: dry-bulb and wet-bulb thermometers; evaporation cools the wet bulb, and the depression (T_db − T_wb) gives p_v through the psychrometric equation. Needs adequate air speed (aspirated, about 3 m/s) and clean wick.
• Capacitive sensors: a thin hygroscopic polymer film between electrodes changes permittivity with RH; the most common electronic hygrometer.
• Resistive sensors: lithium chloride or conductive polymer films whose resistance falls as they absorb moisture.
• Aluminium oxide sensors: for trace moisture (low dew points) in gases.
• Chilled-mirror hygrometer: a mirror is cooled until dew forms; its temperature is the dew point. A fundamental, highly accurate method used as a reference.
• Hair hygrometer: hair length changes with RH; old, slow, but needs no power.

Formulas

ρ = ΔP/(g·H) (two taps H apart, fully submerged)

ρ = A·τ² − B (vibrating U-tube; A, B from calibration)

τ_s = μ·(du/dy) (Newton's law of viscosity; τ_s = shear stress)

ν = μ/ρ

μ = 2·r²·(ρ_s − ρ)·g / (9·v) (falling ball, Stokes' law, Re < ~0.1)

μ = π·r⁴·ΔP / (8·Q·L) (capillary, laminar, Hagen–Poiseuille)

RH = (p_v/p_s)·100 %

ω = 0.622·p_v / (p − p_v)

p_v = p_s(T_wb) − A·p·(T_db − T_wb) (psychrometric equation; A ≈ 6.6 × 10⁻⁴ °C⁻¹ for an aspirated psychrometer)

Symbols: ρ = fluid density (kg/m³); ρ_s = sphere density; ΔP = pressure difference (Pa); g = 9.81 m/s²; H = vertical tap separation (m); τ = oscillation period (s); μ = dynamic viscosity (Pa·s); du/dy = shear rate (s⁻¹); ν = kinematic viscosity (m²/s); r = sphere or capillary radius (m); v = terminal velocity (m/s); Q = volumetric flow (m³/s); L = capillary length (m); p_v = partial pressure of water vapour (Pa); p_s = saturation pressure (Pa, from steam tables); p = total pressure (Pa); T_db, T_wb = dry- and wet-bulb temperatures (°C); ω = humidity ratio (kg/kg dry air).

Worked examples

Example 1 (standard): falling-ball viscometer. Given: steel ball of diameter 2.0 mm (r = 1.0 mm), ρ_s = 7800 kg/m³, oil ρ = 900 kg/m³; the ball falls 0.20 m at constant speed in 10.0 s.

1. v = 0.20/10.0 = 0.020 m/s.
2. μ = 2·r²·(ρ_s − ρ)·g/(9·v) = 2 × (1.0 × 10⁻³)² × 6900 × 9.81 / (9 × 0.020).
3. Numerator = 2 × 10⁻⁶ × 6900 × 9.81 = 0.1354; denominator = 0.18.
4. μ = 0.1354/0.18 = 0.752 Pa·s.
5. Check Re = ρ·v·d/μ = 900 × 0.020 × 0.002/0.752 = 0.048, so Stokes' law is valid.
6. Answer: μ ≈ 0.75 Pa·s (752 cP).

Example 2 (GATE level): psychrometer. Given: aspirated psychrometer, p = 101.325 kPa, T_db = 30 °C, T_wb = 22 °C. Steam-table data (given): p_s(22 °C) = 2.645 kPa, p_s(30 °C) = 4.246 kPa; A = 6.6 × 10⁻⁴ °C⁻¹.

1. p_v = p_s(T_wb) − A·p·(T_db − T_wb) = 2.645 − 6.6 × 10⁻⁴ × 101.325 × 8.
2. Correction = 6.6 × 10⁻⁴ × 101.325 × 8 = 0.535 kPa; p_v = 2.645 − 0.535 = 2.110 kPa.
3. RH = 2.110/4.246 × 100 = 49.7 %.
4. ω = 0.622 × 2.110/(101.325 − 2.110) = 1.3124/99.215 = 0.0132 kg/kg dry air.
5. Dew point: the temperature where p_s = 2.110 kPa, about 18.4 °C from steam tables.
6. Answer: RH ≈ 50 %, ω ≈ 0.0132 kg/kg, dew point ≈ 18 °C.

Example 3 (density by DP). Two bubbler tubes 1.5 m apart vertically in a brine tank show ΔP = 15.9 kPa. ρ = 15 900/(9.81 × 1.5) = 1081 kg/m³ (SG ≈ 1.08), whatever the level above the upper tube.

Common mistakes

• Using the dew-point or wet-bulb temperature's saturation pressure as the denominator of RH; it must be p_s at the dry-bulb temperature.
• Mixing up dynamic and kinematic viscosity (cP versus cSt): ν = μ/ρ.
• Applying Stokes' law when Re is not small, or ignoring wall effects in a narrow tube.
• Using a single-speed viscosity reading for a non-Newtonian fluid.
• Reporting density or viscosity without the temperature.
• Letting the wet-bulb wick dry out or using an unventilated psychrometer, which overstates RH.

For GATE IN

Questions use the falling-ball and capillary formulas, unit conversion of viscosity, hydrostatic density by two-point DP, and relative humidity from partial pressures or psychrometer readings (with steam-table values given). Conceptual questions cover the principles of hygrometers (capacitive, chilled mirror, psychrometer) and density sensors (vibrating tube, buoyancy, radiation). Practise the definitions of RH, humidity ratio and dew point until they are automatic.

Quick check

1. p_v = 1.2 kPa, p_s = 2.0 kPa: what is RH?
2. Convert 0.5 Pa·s to cP.
3. Why is ΔP between two submerged taps independent of level?
4. What does a chilled-mirror hygrometer measure directly?
5. A ball's terminal speed doubles in a new oil (same ball). What happened to μ?

Answers: 1. 60 %. 2. 500 cP. 3. Both taps see the same gas pressure and liquid above the upper tap; only the column between them, ρ·g·H, differs. 4. The dew point. 5. It halved (μ ∝ 1/v), assuming similar density.