Deadweight tester / piston gauge

The pressure a deadweight tester or piston gauge generates from the masses on it, or the mass to load for a target pressure. Local gravity, air buoyancy matched to true or conventional mass, thermal expansion and pressure distortion of the effective area, oil surface tension and the head to the device under test, gauge or absolute, per EURAMET cg-3.

Deadweight tester · piston gauge

v0.1.0 · EURAMET cg-3 v2.1 · OIML D 28

FindTotal mass incl. piston (kg)Mass basis (certificate)Weight density ρ_m (kg/m³)ModeAir density ρ_a (kg/m³)Local gravity gLatitude (°, south negative)Effective area A₀ at t_r (mm²)Distortion λ (× 10⁻⁶ /MPa)α piston (× 10⁻⁶ /K)α cylinder (× 10⁻⁶ /K)Piston-cylinder temperature t (°C)Reference temperature t_r (°C)Oil surface tension σ (N/m, optional)Piston circumference c (mm, optional)Head Δh = h_balance − h_device (m, optional)Fluid density in the line ρ_f (kg/m³)Show in

Pressure at the device99.950870 MPaAt the balance reference level 99.95087 MPa; head 0 MPa; A_p = 1.9612063 mm²; g = 9.802861 m/s² (estimate); 4 iterations

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.

Conventional mass (OIML D 28): buoyancy factor 1 − ρ₀ₐ/ρ₀ + (ρ₀ₐ − ρ_a)/ρ_m with ρ₀ₐ = 1.2 kg/m³, ρ₀ = 8000 kg/m³ (cg-3 eq. 3.1a). At ρ_a = 1.2 kg/m³ it is 0.99985 whatever the weights are made of.

The pressure is solved by fixed-point iteration: p₀ from λ = 0, then p_{n+1} = F / (A₀(1 + λp_n)(1 + (α_p + α_c)(t − t_r))) (+ μ), stopping when |p_{n+1} − p_n| ≤ 1 × 10⁻¹⁴·|p_{n+1}|; it is refused after 50 iterations. cg-3 3.7(b) allows a nominal p in A₀(1 + λp); solving the equation exactly differs from that by about λ·(p − p_nominal) relative.

Not modelled: piston buoyancy from a non-cylindrical piston end (cg-3 3.11, eqs. 3.4 and 3.5), tilt of the piston, second-order distortion λ′.

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

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Show full precision

p = [Σm·g·B + σ·c] / [A₀(1 + λp)(1 + (α_p + α_c)(t − t_r))] + (ρ_f − ρ_a)·g·Δh · B per mass basis · solved by iteration

How this is calculated

Equation: gauge: p = [Σm·g·B + σ·c] / [A₀(1 + λp)(1 + (α_p + α_c)(t − t_r))] + (ρ_f − ρ_a)·g·Δh, B = 1 − ρ_a/ρ_m (true mass) or 1 − ρ₀ₐ/ρ₀ + (ρ₀ₐ − ρ_a)/ρ_m (conventional mass); absolute: p = Σm·g / [A₀(1 + λp)(1 + (α_p + α_c)(t − t_r))] + μ + ρ_f·g·Δh. Solved for p by fixed-point iteration.
Source: EURAMET Calibration Guide No. 3, Guideline on the Calibration of Pressure Balances, v2.1 (02/2025), clause 3: eqs. (3.1), (3.1a), (3.2), (3.2a), (3.3), (3.6), (3.7) and A_p = A₀(1 + λp) from 3.7(b). Conventional mass: OIML D 28:2004, ρ₀ₐ = 1.2 kg/m³, ρ₀ = 8000 kg/m³, eq. (1) to convert conventional to true mass in absolute mode.
Valid range: Temperature of the piston-cylinder 0 to 40 °C; |α_p|, |α_c| ≤ 1 × 10⁻³ /K; weight density 1000 to 25 000 kg/m³; air density 0 to 2 kg/m³; |λ·p| ≤ 0.01; g 9.7 to 9.9 m/s²; A₀ > 0; total mass > 0.
Rounding: Shown to 8 significant figures by default, rounded half up; solved and inverted at full precision.
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
DWT-01	weights = [{"mass":3.4263,"density":7800}], massBasis = true, gravity = {"g":9.80616}, area = 0.00033592, alphaPiston = 0.0000112, alphaCylinder = 0.0000112, temperature = 22.85, airDensity = 1.1854	99998.79541994532 Pa	± 1e-8	Hand-derived with mpmath from EURAMET cg-3 v2.1 eq. (3.1), using the inputs of the IMEKO PWC-2006 TC16-013 worked example (pneumatic deadweight tester, NRC): m = 3.4263 kg, ρ_m = 7800 kg/m³, g = 9.806 16 m/s², ρ_a = 1.1854 kg/m³, A₀ = 335.920 mm², α_p = α_c = 11.2 × 10⁻⁶ /K, t = 296 K.
DWT-02	weights = [{"mass":10,"density":8000}], massBasis = conventional, gravity = {"g":9.80665}, area = 0.0001, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 20, airDensity = 1.2	980517.90025 Pa	± 1e-8	Hand-derived from EURAMET cg-3 v2.1 eq. (3.1a) with OIML D 28 conventional values ρ₀ₐ = 1.2 kg/m³, ρ₀ = 8000 kg/m³.
DWT-03	weights = [{"mass":10,"density":7920}], massBasis = conventional, gravity = {"g":9.80665}, area = 0.0001, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 20, airDensity = 1.2	980517.90025 Pa	± 1e-8	Hand-derived from EURAMET cg-3 v2.1 eq. (3.1a).
DWT-04	massBasis = true, gravity = {"g":9.80665}, area = 0.000001961004, lambda = 1.032e-12, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 22, airDensity = 1.2, weights = [{"mass":20,"density":7920}]	99989352.30697627 Pa	± 0.00001	Hand-derived with mpmath (50 digits) from EURAMET cg-3 v2.1 (02/2025) clause 3; A_p = A₀(1 + λp) solved for p as the positive root of the quadratic λθA₀p² + θA₀p − F = 0 (independent of the library’s iteration). A₀ = 1.961 004 mm² and λ = 1.032 × 10⁻⁶ /MPa are the cg-3 Appendix B3 results; ρ_m = 7920 kg/m³ and α_p + α_c = 9 × 10⁻⁶ /K are its B4.6 and B4.3 example values.
DWT-05	massBasis = true, gravity = {"g":9.80665}, area = 0.000001961004, lambda = 1.032e-12, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 22, airDensity = 1.2, weights = [{"mass":20,"density":7920}], surfaceTension = 0.031, circumference = 0.004964, head = {"deltaH":0.1,"fluidDensity":915}	99990326.89309752 Pa	± 0.00001	Hand-derived with mpmath (50 digits) from EURAMET cg-3 v2.1 (02/2025) clause 3; A_p = A₀(1 + λp) solved for p as the positive root of the quadratic λθA₀p² + θA₀p − F = 0 (independent of the library’s iteration). Surface tension added per eq. (3.2), head per eq. (3.6).
DWT-06	massBasis = true, gravity = {"g":9.80665}, area = 0.000001961004, lambda = 1.032e-12, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 22, airDensity = 1.2, weights = [{"mass":20,"density":7920}], surfaceTension = 0.031, circumference = 0.004964, head = {"deltaH":0.1,"fluidDensity":915}, output = pressureAtReference	99989430.76142052 Pa	± 0.00001	Hand-derived with mpmath (50 digits) from EURAMET cg-3 v2.1 (02/2025) clause 3; A_p = A₀(1 + λp) solved for p as the positive root of the quadratic λθA₀p² + θA₀p − F = 0 (independent of the library’s iteration).
DWT-07	mode = absolute, massBasis = true, gravity = {"g":9.80665}, area = 0.0001, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 20, residualPressure = 1.5, weights = [{"mass":2,"density":8000}]	196134.5 Pa	± 1e-9	Hand-derived from EURAMET cg-3 v2.1 eq. (3.3), absolute mode.
DWT-08	mode = absolute, massBasis = conventional, gravity = {"g":9.80665}, area = 0.0001, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 20, residualPressure = 1.5, weights = [{"mass":2,"density":7920}]	196134.7972162449 Pa	± 1e-8	Hand-derived from EURAMET cg-3 v2.1 eq. (3.3) with the conventional mass converted to true mass by OIML D 28:2004 eq. (1).
DWT-09	weights = [{"mass":10,"density":8000}], massBasis = conventional, gravity = {"latitude":45}, area = 0.0001, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 20	980472.8272606127 Pa	± 1e-8	Hand-derived from EURAMET cg-3 v2.1 eq. (3.1a) with g from the GRS80 closed formula (Moritz 2000).
DWT-10	massBasis = conventional, gravity = {"g":9.80665}, area = 0.0001, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 20, targetPressure = 980517.90025, massDensity = 8000	10 kg	± 1e-12	Hand-derived inverse of EURAMET cg-3 v2.1 eq. (3.1a).
DWT-11	massBasis = true, gravity = {"g":9.80665}, area = 0.000001961004, lambda = 1.032e-12, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 22, airDensity = 1.2, targetPressure = 100000000, massDensity = 7920	20.00212998514473 kg	± 1e-11	Hand-derived inverse of EURAMET cg-3 v2.1 eq. (3.1) with the Appendix B piston (A₀ = 1.961 004 mm², λ = 1.032 × 10⁻⁶ /MPa).
DWT-12	massBasis = true, gravity = {"g":9.80665}, area = 0.000001961004, lambda = 1.032e-12, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 22, airDensity = 1.2, weights = [{"mass":20,"density":7920}], surfaceTension = 0.031, circumference = 0.004964, head = {"deltaH":0.1,"fluidDensity":915}	20 kg	± 1e-11	Round trip mass → pressure → mass through cg-3 eqs. (3.2) and (3.6), forward by iteration and back in closed form.
DWT-13	mode = absolute, massBasis = conventional, gravity = {"g":9.80665}, area = 0.0001, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 20, residualPressure = 1.5, weights = [{"mass":2,"density":7920}]	2 kg	± 1e-12	Round trip mass → pressure → mass through cg-3 eq. (3.3) and OIML D 28 eq. (1).
DWT-14	weights = [{"mass":10,"density":8000}], massBasis = conventional, gravity = {"g":9.80665}, area = 0.0001, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 45	OutOfRangeError per rule	exact	Metrology Bench rule: refuse rather than extrapolate
DWT-15	massBasis = true, gravity = {"g":9.80665}, area = 0.000001961004, lambda = 0.000001032, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 22, airDensity = 1.2, weights = [{"mass":20,"density":7920}]	OutOfRangeError per rule	exact	Metrology Bench rule: refuse rather than extrapolate
DWT-16	weights = [{"mass":10,"density":7.92}], massBasis = conventional, gravity = {"g":9.80665}, area = 0.0001, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 20	OutOfRangeError per rule	exact	Metrology Bench rule: refuse rather than extrapolate
DWT-17	mode = absolute, massBasis = true, gravity = {"g":9.80665}, area = 0.0001, alphaPiston = 0.0000045, alphaCylinder = 0.0000045, temperature = 20, residualPressure = 1.5, weights = [{"mass":2,"density":8000}], surfaceTension = 0.031, circumference = 0.004964	RangeError per rule	exact	EURAMET cg-3 v2.1 clause 3.9: absolute mode is given for gas-operated balances only.

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