Uncertainty of measurement: conventional mass of a weight

This article will discuss the calibration of a 1 kg weight of class $\displaystyle F_1$ by comparison with a 1 kg weight of class $\displaystyle E_2$ (weight classes are described in OIML R 111-1), using a balance of resolution 0.1 mg. In particular, it will focus on the estimation of uncertainty of measurement (UoM) for this calibration, for which the result is intended to be reported as a conventional mass (as discussed in OIML D 28).

The 80-page document R 111-1 describes many characteristics of weights (for example, construction) that are of limited relevance to the calibration laboratory: we will use mostly the sections on Maximum Permissible Errors (section 5, p 11-12), Density (sections 10, p 17-18 and B.7.9.3, p 58) and, especially, Calibration (Annex C, p 61-70). The 12-page document D 28, “Conventional value of the result of weighing in air”, presents a convenient summary of the information most relevant to calibration.

The conventional mass of a body is equal to the mass $\displaystyle m_c$ of a standard weight that balances this body under “conventional” conditions, namely, ambient temperature $\displaystyle t_{ref}$ = 20 °C, air density $\displaystyle \rho_0 = 1.2 \ \mathrm{kg.m^{-3}}$ and standard weight density $\displaystyle \rho_c = 8000 \ \mathrm{kg.m^{-3}}$ [D 28 section 4, p 5]. The conditions have been chosen such that mass, $\displaystyle m$ , and conventional mass, $\displaystyle m_c$ , of a weight do not differ “much” [D 28 section 0, p 4]. However, we will see, when we try to achieve the required calibration uncertainty for a 1 kg $\displaystyle F_1$ weight, which is $\displaystyle U(m_{ct}) \le \frac{\delta m}{3} = \frac{5 \ \mathrm{mg}}{3} = 1.7 \ \mathrm{mg}$ [R 111-1 section 5.2 and Table 1, p 11-12], or, equivalently, a relative standard uncertainty of $\displaystyle \frac{u_c(m_{ct})}{m_{ct}} \le 0.8 \cdot 10^{-6}$ , that typical deviations of weight density from the conventional $\displaystyle \rho_c = 8000 \ \mathrm{kg.m^{-3}}$ can cause significant differences between $\displaystyle m$ and $\displaystyle m_c$ .

Note: The subscript “c” indicates “conventional”, “r” refers to the reference weight (in other words, the measurement standard) and “t” refers to the test weight (or Unit Under Test).

The relation between conventional mass and mass is given by $\displaystyle m_c = m \frac{1-\rho_0 / \rho}{1-\rho_0 / \rho_c}$ [D 28 equation 1, p 6].

In our example, both the reference weight and the test weight are made of stainless steel. The mass of the reference weight is chosen to be $\displaystyle m_r = 1.000 \ 000 \ 00 \ \mathrm{kg}$ , with the largest calibration uncertainty allowed by R 111-1 for class $\displaystyle E_2$ weights. Relevant data are presented in the table below.

Quantity	Uncertainty	Reference
$\displaystyle m_r = 1.000 \ 000 \ 00 \ \mathrm{kg}$
$\displaystyle \rho_r = 7950 \ \mathrm{kg.m^{-3}}$	$\displaystyle u(\rho_r) = 70 \ \mathrm{kg.m^{-3}}$	[R 111-1 Table B.7, p 58]
$\displaystyle m_{cr} = m_r \frac{1-\rho_0 / \rho_r}{1-\rho_0 / \rho_c} = (1-0.94 \cdot 10^{-6}) \ \mathrm{kg}$	$\displaystyle u(m_{cr}) = \frac{\delta m (E_2)}{6} = 0.000 \ 000 \ 27 \ \mathrm{kg}$	[R 111-1 Table 1, p 12]
$\displaystyle \rho_t = 7950 \ \mathrm{kg.m^{-3}}$	$\displaystyle u(\rho_t) = 70 \ \mathrm{kg.m^{-3}}$	[R 111-1 Table B.7, p 58]
Air pressure $\displaystyle p = 750 \ \mathrm{mbar}$	$\displaystyle u(p) = 5 \ \mathrm{mbar}$
Humidity $\displaystyle hr = 50 \ \mathrm{\% rh}$	$\displaystyle u(hr) = 5 \ \mathrm{\% rh}$
Air temperature $\displaystyle t = 19.0 \ \mathrm{^{\circ}C}$	$\displaystyle u(t) = 0.5 \ \mathrm{^{\circ}C}$
$\displaystyle \rho_a = \frac{0.34848 \cdot p - 0.009 \cdot hr \cdot e^{0.061t}}{273.15+t}$ $\displaystyle = 0.890 \ \mathrm{kg.m^{-3}}$	$\displaystyle u(\rho_a) = 0.005 \ \mathrm{kg.m^{-3}}$ (see below for calculation)	[R 111-1 equation (E.3-1), p 76]

Notes:

1.The conventional mass $\displaystyle m_{cr}$ is $\displaystyle 0.94 \cdot 10^{-6}$ lower than $\displaystyle m_r$ : this difference is large (compared to the $\displaystyle 0.8 \cdot 10^{-6}$ relative uncertainty goal), although the assumed weight density is not far from ideal.

2. As the densities of the reference and test weights are not known, they are “assumed” according to R 111-1 Table B.7 (for stainless steel). Although they are assumed to be the same in most of the data analysis below, the possible difference between the densities must be taken into account in the uncertainty analysis. In particular, during calculation of the uncertainty in the air buoyancy correction $\displaystyle C$ , the sensitivity coefficient of $\displaystyle u(\rho_a)$ will be calculated using the worst-case scenario: as $\displaystyle 7810 \ \mathrm{kg.m^{-3}} \le \rho_r \le 8210 \ \mathrm{kg.m^{-3}}$ for 1 kg $\displaystyle E_2$ weights and $\displaystyle 7390 \ \mathrm{kg.m^{-3}} \le \rho_t \le 8730 \ \mathrm{kg.m^{-3}}$ for 1 kg $\displaystyle F_1$ weights [R 111-1 Table 5, p 17], the sensitivity coefficient $\displaystyle \left( \frac{1}{\rho_t} - \frac{1}{\rho_r} \right) = \left( \frac{1}{7390 \ \mathrm{kg.m^{-3}}} - \frac{1}{8210 \ \mathrm{kg.m^{-3}}} \right) = 14 \cdot 10^{-6} \mathrm{m^3.kg^{-1}}$ will be used. (The calculation is performed below.)

3. The air pressure $\displaystyle p$ , and, consequently, air density $\displaystyle \rho_a$ , are unusually low, because the measurements were performed at an altitude of 2400 m. The even-more-approximate formula for air density, $\displaystyle \rho_a = \rho_0 \cdot e^{-\rho_0 \cdot g \cdot h / p_0}$ [R 111-1 equation (E.3-2), p 76], yields a similar air density, $\displaystyle 0.91 \ \mathrm{kg.m^{-3}}$ .

The uncertainty in air density is calculated as follows: $\displaystyle u(\rho_a) = \sqrt{\left( \frac{\partial \rho_a}{\partial p} u(p)\right)^2 + \left( \frac{\partial \rho_a}{\partial hr} u(hr)\right)^2 + \left( \frac{\partial \rho_a}{\partial t} u(t)\right)^2} = \rho_a \sqrt{\left( 10^{-3} \mathrm{mbar}^{-1} \cdot u(p) \right)^2 + \left( -10^{-4} \cdot u(hr) \right)^2 + \left( -3.4 \cdot 10^{-3} \mathrm{K}^{-1} \cdot u(t) \right)^2}$ $\displaystyle = 0.890 \ \mathrm{kg.m^{-3}} \cdot \sqrt{0.0050^2 + 0.0005^2 + 0.0017^2} = 0.005 \ \mathrm{kg.m^{-3}}$ . Sensitivity coefficients come from R 111-1 section C.6.3.6 (p 68), with $\displaystyle \frac{\partial \rho_a}{\partial p}$ and $\displaystyle \frac{\partial \rho_a}{\partial hr}$ being converted from $\displaystyle \mathrm{Pa^{-1}}$ to $\displaystyle \mathrm{mbar^{-1}}$ and from fractional relative humidity to %rh, respectively. (Both change by a factor of 100, though in opposite directions.) It is clear that air pressure has the largest effect on air density, followed by temperature, with the effect of relative humidity being negligible.

The conventional masses of test and reference weights are related as follows:

$\displaystyle m_{ct} = m_{cr}(1+C) + \Delta m_w$ [D 28 equation (9), p 8],

where $\displaystyle \Delta m_w$ is the measured difference in apparent masses.

The air buoyancy correction $\displaystyle C = (\rho_a - \rho_0) \left( \frac{1}{\rho_t} - \frac{1}{\rho_r}\right)$ [D 28 equation (10), p 8] is applied, because the air density differs from $\displaystyle \rho_0$ by more than 10 % [R 111-1 section 10.2.1, p 18].

To determine the uncertainty $\displaystyle u(m_{ct})$ , we’ll need the sensitivity coefficients $\displaystyle \frac{\partial m_{ct}}{\partial m_{cr}} = 1+C \approx 1$ , $\displaystyle \frac{\partial m_{ct}}{\partial C} = m_{cr}$ and $\displaystyle \frac{\partial m_{ct}}{\partial (\Delta m_w)} = 1$ .

Then, $\displaystyle u(m_{ct}) = \sqrt{u(m_{cr})^2 + (m_{cr} \cdot u(C))^2 + u(\Delta m_w)^2}$ , where $\displaystyle u(\Delta m_w)^2 = u_w^2 + u_{ba}^2$ (the uncertainty of the weighing process and the balance, combined) and $\displaystyle m_{cr} \cdot u(C) = u_b$ , in the notation of R 111-1 section C.6, p 66-70.

Expressed as a relative uncertainty, $\displaystyle \frac{u(m_{ct})}{m_{ct}} = \sqrt{\left( \frac{u(m_{cr})}{m_{cr}} \right)^2 + u(C)^2 + \left( \frac{u(\Delta m_w)}{m_{cr}} \right)^2}$ .

From the table above, $\displaystyle \frac{u(m_{cr})}{m_{cr}} = 0.27 \cdot 10^{-6}$ .

From equation (C.6.3-1) of R 111-1 (p 67), $\displaystyle u(C)^2 = \left( \frac{u_b}{m_{cr}} \right)^2$ $\displaystyle = \left( \left( \frac{1}{\rho_t} - \frac{1}{\rho_r}\right) \cdot u(\rho_a) \right)^2 + \left( \frac{(\rho_a - \rho_0)}{\rho_t} \cdot \frac{u(\rho_t)}{\rho_t} \right)^2$ $\displaystyle + \frac{(\rho_a - \rho_0)(\rho_0 + \rho_a - 2 \rho_{al})}{\rho_r^2} \cdot \left( \frac{u(\rho_r)}{\rho_r} \right)^2$ ,

where $\displaystyle \rho_{al}$ is the air density during the last calibration of the reference weight.

Using $\displaystyle \rho_{al} = 1.2 \ \mathrm{kg.m^{-3}}$ , $\displaystyle u(C)^2 = \left( 14 \cdot 10^{-6} \ \mathrm{m^3.kg^{-1}} \cdot 0.005 \ \mathrm{kg.m^{-3}} \right)^2$ $\displaystyle + \left( -39 \cdot 10^{-6} \cdot 0.0088 \right)^2$ + $\displaystyle + \left( -39 \cdot 10^{-6} \cdot 0.0088 \right)^2$ . So, $\displaystyle u(C) = 0.49 \cdot 10^{-6}$ , with the terms in $\displaystyle u(\rho_t)$ and $\displaystyle u(\rho_r)$ dominating over the term in $\displaystyle u(\rho_a)$ .

The term $\displaystyle u( \Delta m_w)$ contains the uncertainty of the weighing process, $\displaystyle u_w$ , which is an ESDM calculated as 0.01 mg, the sensitivity of the balance, $\displaystyle u_s$ , and the display resolution of the balance, $\displaystyle u_d = \frac{0.1 \ \mathrm{mg} /2}{\sqrt{3}} \cdot \sqrt{2} = 0.04 \ \mathrm{mg}$ . (Eccentric loading of the balance is included in $\displaystyle u_w$ , and magnetism is negligible, according to p 69 of R 111-1.) In total, let’s suppose that $\displaystyle u( \Delta m_w) = 0.05 \ \mathrm{mg}$ , so that $\displaystyle \frac{u( \Delta m_w)}{m_{cr}} = 0.05 \cdot 10^{-6}$ .

Finally, $\displaystyle \frac{u(m_{ct})}{m_{ct}} = \sqrt{(0.27 \cdot 10^{-6})^2 + (0.49 \cdot 10^{-6})^2 + (0.05 \cdot 10^{-6})^2} = 0.56 \cdot 10^{-6}$ . So, we have achieved our goal of a relative standard uncertainty smaller than $\displaystyle 0.8 \cdot 10^{-6}$ .

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

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Uncertainty of measurement: conventional mass of a weight

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