February | 2017 | Metrology Rules

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 $\displaystyle p = \frac{\left( \sum_i m_i \right) \cdot g \cdot (1-\rho_a / \rho_m) + \tau \cdot \pi d}{A_0 \cdot (1 + \lambda p) \cdot \left(1 + (\alpha + \beta) \cdot (t - 20 \textdegree \mathrm{C}) \right)} + (\rho_{fl} - \rho_a) \cdot g \cdot h$ .

$\displaystyle \sum_i m_i$ is the combined mass of the piston, sleeve (weight carrier) and the weights used. The uncertainty of calibration may be $\displaystyle \frac{U(m)}{m} \approx 10 \cdot 10^{-6}$ (around F2 level). Gravitational acceleration, $\displaystyle g$ , may be measured to better than $\displaystyle 10^{-6}$ , in which case its uncertainty is negligible. However, if it is estimated by a formula, $\displaystyle \frac{U(g)}{g}$ may be $\displaystyle 50 \cdot 10^{-6}$ , which is significant. The certificate typically reports conventional masses, so the buoyancy correction $\displaystyle (1-\rho_a / \rho_m)$ 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 $\displaystyle 150 \cdot 10^{-6}$ : the additional correction $\displaystyle (1.2 - \rho_a) / \rho_m$ for deviation from conventional air density, and its uncertainty, is often negligible ( $\displaystyle \leq 5 \cdot 10^{-6}$ below 300 m altitude).

The surface tension correction, $\displaystyle \tau \cdot \pi d$ , is unimportant for pneumatic systems. For hydraulic ones, the contribution to total weight may be $\displaystyle 14 \cdot 10^{-6}$ , important to correct for, but whose uncertainty has a negligible effect.

The effective area at zero pressure, $\displaystyle A_0$ , has an uncertainty around $\displaystyle 70 \cdot 10^{-6}$ : this is usually the dominant component of total DWT uncertainty, especially at high pressures. The pressure distortion coefficient, $\displaystyle \lambda$ , is ~ $\displaystyle 1 \cdot 10^{-6}$ per MPa, i.e., the correction goes up to $\displaystyle 500 \cdot 10^{-6}$ at 500 MPa. Say its uncertainty is 10% of its value, i.e., up to $\displaystyle 50 \cdot 10^{-6}$ . This may be significant at very high pressures, but is negligible compared to $\displaystyle \frac{U(A_0)}{A_0}$ at lower pressures. The thermal expansion coefficient, $\displaystyle (\alpha + \beta)$ , is ~ $\displaystyle 11 \cdot 10^{-6}$ per °C. So, for $\displaystyle U(t)$ ~1 °C, the contribution to $\displaystyle \frac{U(p)}{p}$ is around $\displaystyle 11 \cdot 10^{-6}$ , which may be significant.

For pneumatic systems, where $\displaystyle \rho_{fl}$ is ~1000 times smaller than for oil, the head correction, $\displaystyle (\rho_{fl} - \rho_a) \cdot g \cdot h$ , is often negligible. The uncertainty in the head correction is typically dominated by $\displaystyle U(h)$ , i.e., $\displaystyle U(\rho_{fl})$ and $\displaystyle U(g)$ are negligible. If h~0.27 m (as in the Fluke P3830 pressure balance) and $\displaystyle U(h)$ is 0.025 m (25 mm, which is not very conservative), the effect on $\displaystyle \frac{U(p)}{p}$ at 4 MPa(g) is $\displaystyle 56 \cdot 10^{-6}$ , one of the two largest contributors. At higher pressures, the $\displaystyle \frac{m}{A_0}$ term dominates and U(head) becomes less important (e.g., $\displaystyle 3 \cdot 10^{-6}$ contribution to $\displaystyle \frac{U(p)}{p}$ at 70 MPa(g)).

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

$\displaystyle U(m)$ (sensitivity coefficient = $\displaystyle p/m$ )
$\displaystyle U(A_0)$ (sensitivity coefficient = $\displaystyle p/A_0$ )
$\displaystyle U(t)$ (sensitivity coefficient = $\displaystyle p \cdot (\alpha + \beta)$ )
for hydraulic systems, $\displaystyle U(h)$ (sensitivity coefficient = $\displaystyle \rho_{fl} \cdot g$ )

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.)

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Monthly Archives: February 2017

Uncertainty of measurement: pressure dead-weight tester

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