Evaluation of expanded uncertainty at glass thermometer calibration

A method for calibrating a glass thermometer is investigated and a procedure for measurement uncertainty evaluation based on the kurtosis method is developed. The correlation between the indication of the reference and calibrated thermometers at uncertainty evaluation is taken into account. The effectiveness of the reduction method applying in calculating the uncertainty of correlated measurements is demonstrated. Uncertainty budgets have been drawn up, which can be used as the basis for developing software tools to automate the uncertainty evaluation. A real example of the measurement uncertainty evaluation at glass thermometer calibration is considered. It is shown that taking into account the correlation between the measurement results of the calibrated and reference thermometers allows to reduce the values of the combined and expanded measurement uncertainty by almost 1.5 times. The coincidence of the results obtained by the proposed method and the Monte Carlo method is shown.


Introduction
Glass thermometers are widespread in laboratory and industrial practice due to the high accuracy, cheapness, ease of use [1]. Their measurement range, depending on the thermometric fluid used (mercury, toluene, ethyl alcohol, kerosene, petroleum ether, pentane), extends from -200 to +750 °C.
Glass thermometers, like other measuring instruments, need periodic calibration. In this case, in accordance with the requirements of the standard ISO/ IEC 17025 [2], it is necessary to evaluate the measurement uncertainty. The main method for calibrating these thermometers is to compare them with a reference thermometer using a transfer device (thermostat). In the process of calibrating a thermometer, the difference Δ between the indications of the calibrated thermometer and the reference thermometer is estimated, thus determining the systematic error of the calibrated thermometer at the calibration point [3].
Since the measurements by both thermometers are carried out simultaneously under the same conditions, the instability of the temperature of the thermostat causes a statistical interrelation (correlation) between their indications, which must be taken into account when developing the procedure of measurement uncertainty evaluation.

Analysis of literary data and problem statement
Currently, the efforts of Working Group 1 (WG-1) of the Joint Committee for Guides in Metrology (JCGM) are focused on revision the Guide to the Expression of Uncertainty in Measurement (GUM) [4]. The reason for the revision is the inconsistency of the uncertainty estimates obtained by the GUM method [5] and estimates obtained by the Monte Carlo Method (MCM) in accordance with Supplement 1 to the GUM [6]. Since [6] is based on the Bayesian approach to the measurement uncertainty evaluation, this approach should also be used in the revised Guide (NewGUM). In this case, it is necessary to consider the issues of taking into account correlation in the measurement uncertainty evaluation [7].

The purpose and objectives of the study
The article considers the procedure for measurement uncertainty evaluation based on the Bayesian approach [4], which implements the kurtosis method proposed by the authors [8]. To determine the reliability of the developed procedure, one should compare its results with the results obtained by the Monte Carlo method [9].

Measurement Uncertainty Evaluation Algorithm
The error of indication E X of the calibrated thermometer obtained from the relation: UDC 006.91:053.088

Evaluation of expanded uncertainty at glass thermometer calibration
where T c is the temperature indicated by the calibrated thermometer; T s is the temperature indicated by the reference thermometer; ∆ s is the correction due to calibration error of the reference thermometer; ' c is the correction due to the finite resolution of the calibrated thermometer; ∆ T is the correction due to the temperature unevenness inside the thermostat. (2) where T ci , T si are indications of calibrated and reference thermometers; n is the number of indications. All corrections in expression (1) are centered values, so their estimates are zero: -standard uncertainty due to the finite resolution of the calibrated thermometer: where d is the resolution of the calibrated thermometer; -standard uncertainty of the reference thermometer: where U s and k s , respectively, the expanded uncertainty and coverage factor taken from the reference thermometer calibration certificate; -standard uncertainty associated with the temperature unevenness in the thermostat: where T T is the limit of temperature unevenness in the thermostat.

Determination of pairwise correlation of input quantities
The change in temperature in the thermostat leads to a correlation between the indications of the reference and the calibrated thermometers. Estimate of coefficient of correlation is made by the formula: The standard uncertainty of the measurand is calculated according to the formula: If there is a correlation between the reference and calibrated thermometer indications, the reduction method and a simpler expression to calculate u E ( ) X can be used: where u( ) ∆ is standard uncertainty of observed dispersion of the varying indications of thermometers will be determined like that: 9. Calculating expanded uncertainty: where the k coverage factor for the confidence level is 0.95 calculated by the formula [7]: 0 1085  0 1 2   3  3 2   3   0 95 6  4 , , , , . ; and η is the kurtosis of the distribution of the measurand, determined as: or when evaluating by the reduction method: where the kurtosis values of input quantities are taken from Table 1 in accordance with their distribution laws.
T T n c s 6 5 10. Uncertainty budget All above obtained information on input quantities and measurand is summarized in Table 2, which is an uncertainty budget.
The uncertainty budget is convenient to use as a basis for building a software tool to automate the process of measurement uncertainty evaluation. The uncertainty budget with applying the reduction method is shown in Table 3.

An example of measurement uncertainty evaluation
Indications of calibrated and reference glass thermometers are shown in Table 4.  °C is taken from the passport of the thermostat. The uncertainty budget for these data is listed in Table 5.
The uncertainty budget for the reduction method is presented in Table 6. The results of the measurement uncertainty evaluation for this example, obtained by the Monte Carlo method, are presented in Table 7 and completely coincide with the results of Tables 5 and 6.

Measurand
The value of the measurand Combined standard uncertainty Coverage factor Expanded uncertainty E X -0.01667 0.005227 1.974 0.01033

Conclusions
1. The procedure of the measurement uncertainty evaluation at glass thermometer calibration, based on the Bayesian approach and the kurtosis method, is described.
2. Applying the reduction method to process correlated measurements makes it easier to calculate the measurement uncertainty.
3. The procedure is illustrated by a specific example, the results of which showed a complete agreement with the results of the calculation by the Monte Carlo method.
4. Taking into account the correlation between the measurement results of calibrated and reference thermometers allows to reduce the values of combined and expanded measurement uncertainties by almost 1.5 times.