Fluid statics, manometers and pressure measurement

Why it matters

Pressure is the most measured variable in a chemical plant: tank levels, filter and column pressure drops, flow-meter readings and pump suction conditions are all read as pressures. Fluid statics tells you how pressure varies with depth and how a manometer converts a liquid-column height into a pressure, which is the basis of every pressure gauge calibration and every orifice or venturi reading.

Key ideas

Pressure at a point. In a fluid at rest there are no shear stresses, so the only surface force is the normal pressure. Pressure at a point is the same in all directions (Pascal's law).

Hydrostatic equation. A force balance on a small vertical fluid element gives dp/dz = −ρg, with z measured upward. For a liquid of constant density this integrates to p = p₀ + ρgh, where h is the depth below a surface at pressure p₀. Consequences:

• Pressure depends only on depth, not on the shape of the vessel (the "hydrostatic paradox").
• All points at the same level in the same continuous fluid at rest are at the same pressure. This is the rule used to write every manometer equation.
• For gases, ρ varies with p, so the integration must use the gas law (isothermal atmosphere: p = p₀·exp(−Mgz/RT)); over a few metres the variation is negligible.

Absolute, gauge and vacuum pressure.

• p_abs = p_atm + p_gauge. A vacuum is a negative gauge pressure: p_vac = p_atm − p_abs.
• Standard atmosphere: 101.325 kPa = 760 mm Hg = 10.33 m of water.
• Pressure head h = p/(ρg): the height of a column of a given liquid that the pressure can support.

Pressure-measuring devices.

• Piezometer: a vertical open tube; measures moderate positive gauge pressure in a liquid. Cannot measure gas pressure or vacuum.
• U-tube manometer: a bent tube with a heavier, immiscible manometric liquid (mercury, or water for gas service). Measures gauge pressure (one limb open) or vacuum.
• Differential manometer: both limbs connected to two points; reads the pressure difference.
• Inverted U-tube: a lighter fluid (air or oil) at the top; used for small differences between two liquid points.
• Inclined manometer: one limb inclined at angle θ, so a small vertical rise is read as a longer length L along the tube (h = L·sinθ); more sensitive for small gas pressure differences.
• Micromanometer / well-type manometer: a large reservoir on one side so only one level needs to be read.
• Bourdon gauge, diaphragm gauge and electronic transducers: mechanical or electrical elements, calibrated against manometers or dead-weight testers; used for high pressures where liquid columns would be impractically tall.
• Barometer: an inverted mercury tube with vacuum (mercury vapour) at the top; measures absolute atmospheric pressure.

Writing a manometer equation. Start at one point with its pressure, move through the fluids adding ρgΔh when going down and subtracting when going up, and end at the other point. Equate. Always mark the fluid in each leg.

Forces on submerged surfaces and buoyancy. The hydrostatic force on a plane surface equals the pressure at its centroid times its area, and acts at the centre of pressure, which lies below the centroid. A floating or submerged body experiences an upward buoyant force equal to the weight of displaced fluid (Archimedes).

Formulas

• dp/dz = −ρ·g — hydrostatic equation (z upward). p pressure (Pa), ρ density (kg/m³), g = 9.81 m/s².
• p = p₀ + ρ·g·h — constant-density liquid, h depth (m) below the surface at p₀.
• p_abs = p_atm + p_gauge — all in Pa.
• p_A − p_B = (ρ_m − ρ)·g·R — differential manometer between two points at the same elevation, R manometer reading (m), ρ_m manometric liquid, ρ flowing fluid.
• (p_A/ρg + z_A) − (p_B/ρg + z_B) = R·(ρ_m/ρ − 1) — general differential manometer: the reading gives the difference in piezometric head, whatever the elevations of A and B.
• Δp = ρ_m·g·L·sinθ — inclined manometer with a large reservoir; L reading along the tube (m), θ inclination from horizontal.
• F = ρ·g·h_c·A and h_cp = h_c + I_G/(A·h_c) — force (N) on a plane vertical surface; h_c centroid depth (m), A area (m²), I_G second moment of area about the centroidal axis (m⁴).
• F_B = ρ·g·V_displaced — buoyant force (N).

Worked examples

Example 1 (standard). A pipe carries water. A U-tube mercury manometer has one limb connected to the pipe centre A and the other open to the atmosphere. The mercury surface in the left limb is 0.20 m below A, and the mercury in the open limb stands 0.35 m above that surface. Take ρ_Hg = 13 600 kg/m³ and ρ_w = 1000 kg/m³. Find the gauge pressure at A.

1. Start at A and move down through water to the left mercury surface: p_A + ρ_w·g·0.20.
2. Same level in mercury in the right limb; move up 0.35 m through mercury to the open surface: subtract ρ_Hg·g·0.35.
3. p_A + ρ_w·g·h₁ − ρ_Hg·g·h₂ = 0 (gauge).
4. p_A = 13 600 × 9.81 × 0.35 − 1000 × 9.81 × 0.20 = 46 695.6 − 1962 = 44 733.6 Pa. p_A ≈ 44.7 kPa (gauge).

Example 2 (GATE level). Water flows upward in a vertical pipe. Point B is 0.5 m above point A. A mercury differential manometer connected between A and B reads R = 0.10 m. Find p_A − p_B.

1. The manometer measures piezometric head difference: (p_A/ρg + z_A) − (p_B/ρg + z_B) = R·(ρ_m/ρ − 1).
2. R·(ρ_m/ρ − 1) = 0.10 × (13.6 − 1) = 1.26 m of water.
3. p_A − p_B = ρg[1.26 + (z_B − z_A)] = 1000 × 9.81 × (1.26 + 0.50) = 12 360.6 + 4905 = 17 265.6 Pa. p_A − p_B ≈ 17.3 kPa. Note that 4.9 kPa of this is simply the static head of the water column; only the 12.4 kPa piezometric part is lost to friction or converted to kinetic energy.

Example 3 (inclined manometer). An inclined manometer uses oil of density 800 kg/m³, the tube at 20° to the horizontal, and the reservoir is large. The meniscus moves 80 mm along the tube. Find the pressure difference.

1. Vertical rise h = L·sinθ = 0.080 × sin 20° = 0.02736 m.
2. Δp = ρ_m·g·h = 800 × 9.81 × 0.02736 = 214.7 Pa. Δp ≈ 215 Pa — a reading 2.9 times longer than the vertical rise, which is why the inclined tube is more sensitive.

Example 4 (force on a gate). A vertical rectangular gate 2 m wide and 3 m high has its top edge 1 m below a water surface. Centroid depth h_c = 2.5 m, A = 6 m². F = 1000 × 9.81 × 2.5 × 6 = 147 150 N. I_G = 2 × 3³/12 = 4.5 m⁴, so h_cp = 2.5 + 4.5/(6 × 2.5) = 2.8 m. F ≈ 147 kN acting 2.8 m below the surface.

Common mistakes

• Using ρ_m·g·R instead of (ρ_m − ρ)·g·R when the manometer limbs above the mercury are filled with the flowing liquid.
• Forgetting the elevation term when the two tappings are at different heights; the manometer reads piezometric head, not pressure.
• Mixing gauge and absolute pressure in the same equation.
• Equating pressures at the same level across two different fluids or across a gas–liquid interface.
• Using the length along an inclined tube instead of the vertical height.

For GATE CH

Typical questions are manometer numericals (U-tube, differential, inverted, inclined), conversion between heads of different liquids, absolute versus gauge pressure, and occasionally hydrostatic force on a tank wall or buoyancy. Practise writing the manometer equation step by step from one point to the other, and recognise that a differential manometer on a pipe measures the piezometric head difference.

Quick check

1. What is the gauge pressure 4 m below the surface of an open water tank?
2. Why can a piezometer not measure the pressure of a gas?
3. A mercury–water differential manometer reads 50 mm with both tappings at the same level. What is Δp?
4. What is the absolute pressure if a vacuum gauge reads 30 kPa and p_atm = 101.3 kPa?

Answers: 1. 1000 × 9.81 × 4 = 39.2 kPa. 2. A gas cannot support a free liquid column in an open tube; it would simply escape. 3. 12 600 × 9.81 × 0.05 = 6.18 kPa. 4. 71.3 kPa.