Pressure dead-weight testers (DWTs), also known as pressure balances or piston gauges, are used as reference standards for high-accuracy calibration of hydraulic and pneumatic pressure gauges (as well as other DWTs, via the technique of cross-floating). This article estimates the uncertainty of calibration of a pressure gauge by comparison with such a DWT, identifying the uncertainty components that are typically dominant.

The principle of the dead-weight tester is that the pressure exerted by the fluid (usually oil or air) is balanced by a known weight applied to a known surface area (pressure = force / area).

The known weight is provided by calibrated masspieces, together with knowledge of local gravity. The effective area of the piston-cylinder assembly is determined by dimensional measurement, or by comparison with other pressure standards (e.g., cross-floating).

The equation for gauge pressure measured by a DWT is .

is the combined mass of the piston, sleeve (weight carrier) and the weights used. The uncertainty of calibration may be (around F2 level). Gravitational acceleration, , may be measured to better than , in which case its uncertainty is negligible. However, if it is estimated by a formula, may be , which is significant. The certificate typically reports conventional masses, so the buoyancy correction uses conventional air and weight densities (1.2 kg.m^-3 and 8000 kg.m^-3), not the actual ones. The buoyancy correction is around : the additional correction for deviation from conventional air density, and its uncertainty, is often negligible ( below 300 m altitude).

The surface tension correction, , is unimportant for pneumatic systems. For hydraulic ones, the contribution to total weight may be , important to correct for, but whose uncertainty has a negligible effect.

The effective area at zero pressure, , has an uncertainty around : this is usually the dominant component of total DWT uncertainty, especially at high pressures. The pressure distortion coefficient, , is ~ per MPa, i.e., the correction goes up to at 500 MPa. Say its uncertainty is 10% of its value, i.e., up to . This may be significant at very high pressures, but is negligible compared to at lower pressures. The thermal expansion coefficient, , is ~ per °C. So, for ~1 °C, the contribution to is around , which may be significant.

For pneumatic systems, where is ~1000 times smaller than for oil, the head correction, , is often negligible. The uncertainty in the head correction is typically dominated by , i.e., and are negligible. If h~0.27 m (as in the Fluke P3830 pressure balance) and is 0.025 m (25 mm, which is not very conservative), the effect on at 4 MPa(g) is , one of the two largest contributors. At higher pressures, the term dominates and U(head) becomes less important (e.g., contribution to at 70 MPa(g)).

In conclusion, the following DWT uncertainty contributors may be significant, and should be included in the uncertainty budget (UB):

- (sensitivity coefficient = )
- (sensitivity coefficient = )
- (sensitivity coefficient = )
- for hydraulic systems, (sensitivity coefficient = )

The resolution, hysteresis and zero stability of the gauge being calibrated should also be included in the UB, and will often dominate the combined uncertainty (especially at low pressures).

(Contact the author at lmc-solutions.co.za.)

How to calculate uncertainty of pressure distortion coefficient ?

NIST measures the effective area A_e = A_0·(1+lambda·p) versus pressure at 10 pressures and uses linear least squares fitting to determine A_0, the distortion coefficient and their uncertainties. EURAMET Calibration Guide cg-3 (see pages 14 and 19) recommends the same approach.