Liquid head correction

The pressure difference between a reference level (a deadweight tester, a reference gauge) and the device under test at a different height. Pure water by temperature (Tanaka density) or any fluid by its density; gauge or absolute; air density defaults to the conventional 1.2 kg/m³. Gas lines are computed too, with a note that the head is usually negligible.

Head correction · Δp = (ρ_f − ρ_a)·g·Δh

v0.1.0 · EURAMET cg-3

Δh = h_ref − h_device (m)ModeFluid in the lineWater temperature (°C)Air density ρ_a (kg/m³)Local gravity gLatitude (°, south negative)Show in

Add to the pressure at the reference level2443.38 Paρ_f = 998.2067 kg/m³ · g = 9.802861 m/s² (estimate)

Pure (SMOW) water. Tap or deionized water can differ; cg-19 says to adjust for the water actually used.

Estimate only: GRS80 normal gravity on the ellipsoid (height 0). Real gravity at a site differs from it because of height and local geology.

Not corrected for height: g falls as you go up, so for a lab above sea level this estimate is high. The free-air correction is not included until its source can be verified.

For calibration work use a surveyed value, or a predicted one: NOAA NGS Surface Gravity Prediction (US) or the PTB Gravity Information System (SIS). Enter it as g; it takes precedence.

Sign: Δh = h_ref − h_point. A device below the reference level sees a higher pressure (Δp > 0 for Δh > 0 when ρ_f > ρ_a).

Gauge mode: the air column outside the line balances part of the fluid column, hence ρ_f − ρ_a (cg-3 eq. 3.6).

Results are full precision. Round for display or a certificate separately.

ShowFiguresRounding

Show full precision

gauge Δp = (ρ_f − ρ_a)·g·Δh · absolute Δp = ρ_f·g·Δh · Δh > 0 when the reference is above the device

How this is calculated

Equation: gauge: Δp = (ρ_f − ρ_a)·g·Δh; absolute: Δp = ρ_f·g·Δh; Δh = h_ref − h_point (positive when the reference level is above the point where the pressure is wanted)
Source: EURAMET cg-3 v2.1 (02/2025) clause 3.12, eqs. (3.6) and (3.7), sign convention Δh = h₁ − h₂. Default air density 1.2 kg/m³: OIML D 28:2004 clause 4 (conventional value; also used by cg-3 C4.6). Water density: Tanaka et al. 2001 via EURAMET cg-19 v4.1 eq. (2).
Valid range: Fluid density > 0; air density 0 to 2 kg/m³; g 9.7 to 9.9 m/s² (or latitude −90° to 90° for the estimate); water preset 0 to 40 °C
Rounding: Shown to 6 significant figures by default, rounded half up; full precision is always available.
Version: v0.1.0 · changelog
NCSLI MII taxa: Measure.Pressure.Differential.Static Measure.Pressure.Hydraulic.Static Measure.Pressure.Pneumatic.Absolute.Static Measure.Pressure.Pneumatic.Gage.Static Source.Pressure.Hydraulic.Static Source.Pressure.Pneumatic.Absolute.Static Source.Pressure.Pneumatic.Differential.Static Source.Pressure.Pneumatic.Gage.Static

Calculated value, not a certified result. Accredited laboratories must validate any software they use (ISO/IEC 17025 clause 7.11).

Test vectors for this tool

The reference cases this calculator is tested against on every change. Use them to check your own software. All test vectors · Download JSON

ID	Inputs	Expected	Tolerance	Source
HEAD-01	temperature = 3.983035	999.97495 kg/m³	± 1e-9	EURAMET cg-19 v4.1 (12/2025) eq. (2): a₅ = 0.999 974 950 g/mL is the maximum density of SMOW water, at t = −a₁ = 3.983 035 °C.
HEAD-02	temperature = 20	998.2067455596168 kg/m³	± 1e-10	Hand-derived with mpmath (50 digits) from the Tanaka et al. (2001) formula with the coefficients printed in EURAMET cg-19 v4.1 (12/2025) eq. (2) (and eq. (3) for air saturation).
HEAD-03	temperature = 20, airSaturated = true	998.2042535596166 kg/m³	± 1e-10	Hand-derived with mpmath (50 digits) from the Tanaka et al. (2001) formula with the coefficients printed in EURAMET cg-19 v4.1 (12/2025) eq. (2) (and eq. (3) for air saturation).
HEAD-04	temperature = 45	OutOfRangeError per rule	exact	Metrology Bench rule: refuse rather than extrapolate
HEAD-05	waterTemperature = 20, deltaH = 1, gravity = {"g":9.80665}	9777.296201342217 Pa	± 1e-8	Hand-derived with mpmath (50 digits) from EURAMET cg-3 v2.1 (02/2025) eqs. (3.6) gauge and (3.7) absolute, Δh = h₁ − h₂.
HEAD-06	waterTemperature = 20, deltaH = 1, gravity = {"g":9.80665}, mode = absolute	9789.064181342215 Pa	± 1e-8	Hand-derived with mpmath (50 digits) from EURAMET cg-3 v2.1 (02/2025) eqs. (3.6) gauge and (3.7) absolute, Δh = h₁ − h₂.
HEAD-07	fluidDensity = 1.1404, airDensity = 1.1854, deltaH = 1, gravity = {"g":9.80616}, medium = gas	-0.4412772 Pa	± 1e-12	Hand-derived with mpmath (50 digits) from EURAMET cg-3 v2.1 (02/2025) eqs. (3.6) gauge and (3.7) absolute, Δh = h₁ − h₂. Gas and air densities and g are the inputs of the IMEKO PWC-2006 TC16-013 worked example (Agarwal and Jacques, NRC).
HEAD-08	fluidDensity = 915, airDensity = 1.2, deltaH = -0.2, gravity = {"g":9.81}	-1792.8756 Pa	± 1e-9	Hand-derived with mpmath (50 digits) from EURAMET cg-3 v2.1 (02/2025) eqs. (3.6) gauge and (3.7) absolute, Δh = h₁ − h₂. Oil density 915 kg/m³ as in cg-3 clause C4.8.
HEAD-09	fluidDensity = 915, airDensity = 1200, deltaH = 1, gravity = {"g":9.81}	OutOfRangeError per rule	exact	Metrology Bench rule: refuse rather than extrapolate

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