# Uncertainty of measurement: pressure dead-weight tester

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 $Latex formula$.

$Latex formula$ is the combined mass of the piston, sleeve (weight carrier) and the weights used. The uncertainty of calibration may be $Latex formula$ (around F2 level). Gravitational acceleration, $Latex formula$, may be measured to better than $Latex formula$, in which case its uncertainty is negligible. However, if it is estimated by a formula, $Latex formula$ may be $Latex formula$, which is significant. The certificate typically reports conventional masses, so the buoyancy correction $Latex formula$ 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 $Latex formula$: the additional correction $Latex formula$ for deviation from conventional air density, and its uncertainty, is often negligible ($Latex formula$ below 300 m altitude).

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

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

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

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

• $Latex formula$ (sensitivity coefficient = $Latex formula$)
• $Latex formula$ (sensitivity coefficient = $Latex formula$)
• $Latex formula$ (sensitivity coefficient = $Latex formula$)
• for hydraulic systems, $Latex formula$ (sensitivity coefficient = $Latex formula$)

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.)

## 2 thoughts on “Uncertainty of measurement: pressure dead-weight tester”

1. JASSI says:

How to calculate uncertainty of pressure distortion coefficient ?

1. Hans Liedberg says:

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.