This article will discuss the calibration of a 1 kg weight of class by comparison with a 1 kg weight of class (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 of a standard weight that balances this body under “conventional” conditions, namely, ambient temperature = 20 °C, air density and standard weight density [D 28 section 4, p 5]. The conditions have been chosen such that mass, , and conventional mass, , 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 weight, which is [R 111-1 section 5.2 and Table 1, p 11-12], or, equivalently, a relative standard uncertainty of , that typical deviations of weight density from the conventional can cause significant differences between and .
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 [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 , with the largest calibration uncertainty allowed by R 111-1 for class weights. Relevant data are presented in the table below.
|[R 111-1 Table B.7, p 58]|
|[R 111-1 Table 1, p 12]|
|[R 111-1 Table B.7, p 58]|
|(see below for calculation)||[R 111-1 equation (E.3-1), p 76]|
1.The conventional mass is lower than : this difference is large (compared to the 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 , the sensitivity coefficient of will be calculated using the worst-case scenario: as for 1 kg weights and for 1 kg weights [R 111-1 Table 5, p 17], the sensitivity coefficient will be used. (The calculation is performed below.)
3. The air pressure , and, consequently, air density , are unusually low, because the measurements were performed at an altitude of 2400 m. The even-more-approximate formula for air density, [R 111-1 equation (E.3-2), p 76], yields a similar air density, .
The uncertainty in air density is calculated as follows: . Sensitivity coefficients come from R 111-1 section C.6.3.6 (p 68), with and being converted from to 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:
[D 28 equation (9), p 8],
where is the measured difference in apparent masses.
The air buoyancy correction [D 28 equation (10), p 8] is applied, because the air density differs from by more than 10 % [R 111-1 section 10.2.1, p 18].
To determine the uncertainty , we’ll need the sensitivity coefficients , and .
Then, , where (the uncertainty of the weighing process and the balance, combined) and , in the notation of R 111-1 section C.6, p 66-70.
Expressed as a relative uncertainty, .
From the table above, .
From equation (C.6.3-1) of R 111-1 (p 67), ,
where is the air density during the last calibration of the reference weight.
Using , + . So, , with the terms in and dominating over the term in .
The term contains the uncertainty of the weighing process, , which is an ESDM calculated as 0.01 mg, the sensitivity of the balance, , and the display resolution of the balance, . (Eccentric loading of the balance is included in , and magnetism is negligible, according to p 69 of R 111-1.) In total, let’s suppose that , so that .
Finally, . So, we have achieved our goal of a relative standard uncertainty smaller than .
(Contact the author at lmc-solutions.co.za.)