Published at : 04 Apr 2023
Volume : IJtech
Vol 14, No 2 (2023)
DOI : https://doi.org/10.14716/ijtech.v14i2.5549
José Daniel HernándezVásquez  Universidad Antonio Nariño. Faculty of Mechanical, Electronic and Biomedical Engineering (FIMEB) 
Andrés Leonardo JutínicoAlarcon  Universidad Antonio Nariño. Faculty of Mechanical, Electronic and Biomedical Engineering (FIMEB) 
Emanuel Alberto SimancaCeledon  Universidad Antonio Nariño. Faculty of Mechanical, Electronic and Biomedical Engineering (FIMEB) 
Alejandro Romero Acuña  FIMEB / Universidad Antonio Nariño 
Willman OrozcoLozano  FIMEB / Universidad Antonio Nariño 
Mercy VillateFonseca  Universidad Antonio Nariño. Faculty of Mechanical, Electronic and Biomedical Engineering (FIMEB) 
Manuel J. Campuzano  Universidad Antonio Nariño. Faculty of Industrial Engineering 
This paper compares different methods (Kragten, Guide to
the expression of uncertainty in measurement –GUM– and Relative Uncertainty) to
evaluate the measurement uncertainty in bimetallic thermometers. The motivation
for the development of the work is based on a need detected in the industry to
increase the metrological reliability of thermometers for temperature control
in the most diverse measurement processes. The applied methodology is based on
the concepts enshrined in the classical literature about the GUM, Kragten and
Relative methods. The consolidated results confirmed that, for temperature
measurement applications, the GUM method is more appropriate for estimating the
measurement uncertainty. The value obtained was equal to 1.22^{o}C
throughout the calibration range of the instrument. In conclusion, this work
showed that an increase in metrological reliability for the measurement of
temperature, fundamental magnitude in industrial processes, can be obtained by
evaluating different methods for estimating the measurement uncertainty.
GUM; Kragten; Measurement uncertainty; Relative uncertainty; Thermal metrology
The estimation of the uncertainty
associated with the measurement is a fundamental task that produces
technicaleconomic gains for the area of Mechanical Engineering. At an
industrial level, specifically in temperature measurement, it is strategic to
know the parameter that best represents the dispersion of a measurand within a
certain confident level, i.e. measurement uncertainty.
The specialized literature
confirms that the metrological reliability of a measurement system is
intimately related to the estimated value of its uncertainty (Wang et al., 2018). In addition, different studies (GolijanekJ?drzejczyk
et al., 2019; Farrance, Badrick, and Frenkel, 2018; Gnauert et al., 2018; Farrance and Frenkel,
2012) show that an improvement in the factors that
influence the measurement uncertainty tends to increase the availability of
measuring equipment and, consequently, an increase in the quality of
manufactured products, according to guidelines specified in ISO 9001:2015 (International
Organization for Standardization, 2015).
Thus, according to the literature, developing and investigating
different methods for the estimation of measurement uncertainty is a challenge
for engineering and measurement science. Typically, the guidelines defined in
the GUM (Guide for the expression of measurement uncertainty) (JCGM, 2008) are widely applied by several Metrology Laboratories
in calculating measurement uncertainty. Despite its application, this method
presents some inconsistency: (i) it combines different probability
distributions (e.g., normal, rectangular, triangular); (ii) it considers a
Gaussian distribution for the experimental data of the calibration.
In practice, these
limitations contribute to a deviation between the physical nature of the
problem and the mathematical method adopted forits analysis. For the specific
case of the calibration of instruments for the measurement of temperature at an
industrial level, it is not very feasible from a technical and economic point
of view to calibrate all the points along the instrument scale. For example, a
bimetallic thermometer with a scale from 0 to 120^{o}C that carries
out 120 calibration points (assuming that it advances from 1^{o}C to 1^{o}C) and, additionally, perform 10 repetitions at each point, would
generally total 1200 experimental points. This situation would be ideal to
reduce the uncertainty associated with the measurement. However, each
calibration point has a duration of approximately 2 hours due to the thermal
stability of the standard measurement instrument and the medium used for
generating heat. In this way, to carry out the entire process, approximately
2400 hours, which means 100 continuous days of measurement, are required.
Clearly, for the industrial sector, this scenario is unfeasible from a
technical point of view and, logically, from an economic point of view. For
this reason, in practice, a few calibration points (from 1 to 3 experimental
points) are evaluated according to the NT VVS 103 standard (Thermometers, Contact,
Direct Reading: Calibration, 1994) for the calibration of thermal sensors
instruments (RTDs, Thermocouples, and Glass Thermometers). Thus, the hypothesis
is assumed that the experimental data follow a normal distribution of
probability. Under this hypothesis, the GUM method is applied to calculate the
uncertainty associated with the measurement, obtaining results with deviations
in terms of the physical nature of the problem. To counteract the effects of
this situation, the specialized literature presents some alternative methods
for analyzing measurement uncertainty. Among the methods, the ISO GUM
Supplement 1, known as the Monte Carlo method (JCGM,
2008), the Kragten method, and
applications of Bayesian statistics, among others, stand out. The literature
presents various works where the potential of the GUM, Monte Carlo, and Kragten
methods have been compared Zarate et al.
(2022), Aro et al.
(2021), Khan and Ibrayeva (2020), Cremona et al. (2018), Sardjono and Wijonarko (2018), Horsky, Irrgeher, and Prohaska (2016),
Guerrasio et al.
(2013), Theodorou, Zannikou, and Zannikos (2012), resulting in a
robust Monte Carlo and, additionally, similar results between the GUM
and Kragten methods. However, in ??thermal metrology applied to the industrial
sector, the literature does not show a direct comparison between the GUM and
Kragten methods. Additionally, due to its ease of application, the relative
uncertainty method could be adopted without due consideration, obtaining
inbetter results. However, they lack metrological reliability because the
formulation of the method does not represent the physical nature of the
problem.
In this order of ideas, this work seeks to compare three of the most used methods for calculating measurement uncertainty: ISO GUM, the Kragten method, and the relative uncertainty method. Discuss its advantages, disadvantages, limitations, and considerations for its application in the calibration of bimetallictype temperature measurement instruments, which are widely used in the industrial sector.
Theoretical Fundament
To
perform an analysis of the experimentally obtained data, this section
summarizes the main concepts associated with outlier analysis, as well as the
methods adopted for the analysis and calculation of the uncertainty associated
with the measurement, i.e., GUM, Kragten's, and relative method. In relation to
the application of a polynomial model that allows for establishing the
adjustment equation in a measurement system, the reference (Esraa et al., 2022) shows more
details about this procedure. In this order of ideas, this work seeks to
compare three of the most used methods for calculating measurement uncertainty:
ISO GUM, the Kragten method, and the relative uncertainty method. Discuss its
advantages, disadvantages, limitations, and considerations for its application
in the calibration of bimetallictype temperature measurement instruments,
which are widely used in the industrial sector.
2.1. Analysis of outliers
Once the experimental data were consolidated, the results analysis methodology was applied, starting with the application of the Chauvenet method (Wang, Caja, and Gómez, 2018) for the elimination of outliers. According to this criterion, a measure must be eliminated if:


where


where


2.2. Uncertainty Analysis
2.2.1. Uncertainty Analysis: GUM
Method
Following
the GUM methodology described in section 2.1, four sources of uncertainty were
considered:
Equation (4) was used to calculate the combined uncertainty (u_{c}) from the contributing sources of uncertainty:


The coverage factor, determined from a tstudent distribution, corresponds to k = 2 for all the experimental points. Thus, the expanded uncertainty (U_{E}) associated with the temperature measurement for a confidence level (?) of 95.45% was determined by applying Equation (5). Table 6 summarizes the results of uncertainty analysis.


2.2.2. Uncertainty Analysis: Kragten's Method
From
the sources of uncertainties determined by the GUM method, Kragten's method is
applied. Equation (6) denotes the measurand used to determine the corrected
systematic error (E_{c}


The uncertainties associated with the
temperature indicated by the instrument (




Equations (9) and (10) were used to calculate the deviations associated with the temperature indicated by the instrument () and adjusted by the interpolating polynomial ().




In the sequence, Equations (11) and (12) were
applied to estimate the uncertainties associated with the temperature indicated
by the instrument




Finally, the combined uncertainty is calculated by applying Equation (13), and the expanded uncertainty (k = 2) was determined by applying Equation (14).
 


In relation to the relative uncertainty method, Equation (6) shows the expression that represents the measurand. Equations (15) and (16) estimate the measurement uncertainty.




The experimental methodology is described in this section. In the course
of the experiments performed in a metrology laboratory, under controlled
environmental conditions, a bimetallic thermometer was calibrated
(Manufacturer: Rockage; Model: SHB05; Range: 20^{o}C to 120^{o}C;
Resolution: 1^{o}C) using the method of direct comparison. This
calibration was performed using a standard dry block as a means for generating
heat (Manufacturer: Reed BX150, Temperature measurement range: 27.0^{o}C
to 350.0^{o}C) and a thermocouple with a digital temperature indicator
(Manufacturer: Xintest, Model: HT9815 Thermocouple Thermometers, ID: EPS001,
Serial Number: 201703021366, Range: 0.0^{o}C to 99.9^{o}C).
Subsequent sections detail the technical characteristics of the experimental
apparatus and a description of the procedure for collecting experimental data
in the laboratory.
The Rockage bimetallic
thermometer calibration experiments were performed in a Metrology Laboratory
prepared for that purpose, following the guidelines established in the
normative document NTVVS103:1994, Thermometers contact direct Reading:
Calibration. It is important to note that a conventional Metrology Laboratory,
when performing this type of calibration, generally takes of 3 experimental
points along the range of the instrument. In addition, it does not perform
repetitions at each point. As an added value to the research and considering
the relevance of the measurement of temperature in equipment and measurement
processes in the industry, in this work, we chose to be more rigorous with
obtaining the experimental data and its subsequent statistical treatment. In
relation to the calibrated experimental points, 8 of these were taken
throughout the range of indicators of the instrument, varying from 30^{o}C
to 100^{o}C. In addition, for each calibration point 6 repetitions
were taken for the instrument (bimetal thermometer) and 6 for the measurement
pattern, totaling 12 experimental points in each measurement. Throughout the
experiment, 96 points were evaluated in 12 continuous hours of measurement.
Finally, the experimental data allowed to evaluate the metrological
reliability of the bimetallic thermometer, consolidated in Table 1. This
table highlights in red some values considered as possible outliers.
Table 1 Experimental data of the calibration in the
laboratory
Reference temperature in
the Dry Block 
Instrument (I) / 
Temperature Measurements 
Average 
Standard Desviation  
#1 
#2 
#3 
#4 
#5 
#6  
°C 
°C 
°C 
°C 
°C 
°C 
°C 
°C 
°C  
30 
I 
30 
29 
30 
30 
31 
30 
30 
0.6325 
R 
30.6 
30.4 
30.4 
30.3 
30.2 
30.2 
30.4 
0.1517  
40 
I 
39 
40 
40 
39 
39 
41 
40 
0.8165 
R 
39.1 
39.3 
39.3 
39.2 
39.2 
39.3 
39.2 
0.0816  
50 
I 
50 
50 
50 
50 
49 
50 
50 
0.4082 
R 
49.5 
49.5 
49.5 
49.5 
49.4 
49.5 
49.5 
0.0408  
60 
I 
61 
61 
61 
60 
60 
61 
61 
0.5164 
R 
60.2 
60.2 
60.2 
60.4 
60.3 
60.2 
60.3 
0.0837  
70 
I 
70 
70 
70 
69 
69 
70 
70 
0.5164 
R 
69.5 
69.6 
69.6 
69.6 
69.6 
69.5 
69.6 
0.0516  
80 
I 
80 
79 
79 
79 
80 
80 
80 
0.5477 
R 
79.7 
79.7 
79.7 
79.9 
79.9 
79.9 
79.8 
0.1095  
90 
I 
89 
90 
90 
90 
89 
90 
90 
0.5164 
R 
88.6 
88.6 
88.6 
88.6 
88.5 
88.5 
88.6 
0.0516  
100 
I 
100 
99 
99 
99 
100 
100 
100 
0.5477 
R 
99.7 
99.6 
99.6 
99.7 
99.7 
99.6 
99.7 
0.0548 
This hypothesis is based on the
observation of the experimental data collected for each experimental point,
since they are more dispersed when compared to the other 5 data collected.
However, this hypothesis must be confirmed by applying the Chauvenet method, as
evidenced in the results section.
This
section consolidates the main results of the investigation. It was divided into
three large blocks to facilitate its development: (i) Analysis of
outliers. In this stage, the parametric technique proposed by Chauvenet
will be applied to eliminate aberrant values, i.e., outliers. The reason why
these values ??are produced is very varied. Consequently, the uncertainty
associated with the temperature measurement increases and unquestionably, the
impact of a given measurement process will be negative both technically and
economically. Hence, the importance of carrying out this important stage within
the proposed methodology; (ii) Estimation of the adjustment polynomial.
By applying the ordinary least squares method, it is expected to estimate a
polynomial that best represents the physics of the problem studied. This
polynomial is associated with the least adjustment uncertainty, also known as
the quadratic mean deviation. In addition to allowing the obtaining of adjusted
temperature values, the polynomial allows obtaining errors and uncertainties
for any indication in the calibration range, even in those points where it was
not possible to perform the experimental calibration; (iii) Application
of techniques to estimate uncertainty. In this stage, the enshrined
theory was studied in detail for three uncertainty estimation techniques: GUM,
Kragten, and Relative Uncertainty. Subsequently, the experimental data analysis
was performed to establish the expanded uncertainty associated with the
temperature measurement, as well as the associated to systematic errors.
4.1. Analysis
of Outliers
It is
emphasized that the value of
Table 2 Calculation of parameter and the
Instrument (I) / 
Calculation of ? from the Temperature Measurements 
Critical value of _{c}  
_{1}  _{2}  _{3}  _{4}  _{5}  _{6}  
I 
0.0 
1.6 
0.0 
0.0 
1.6 
0.0 
1.73  
R 
1.6 
0.3 
0.3 
0.3 
1.0 
1.0  
I 
0.8 
0.4 
0.4 
0.8 
0.8 
1.6  
R 
1.6 
0.8 
0.8 
0.4 
0.4 
0.8  
I 
0.4 
0.4 
0.4 
0.4 
2.0 
0.4  
R 
0.4 
0.4 
0.4 
0.4 
2.0 
0.4  
I 
0.6 
0.6 
0.6 
1.3 
1.3 
0.6  
R 
0.6 
0.6 
0.6 
1.8 
0.6 
0.6  
I 
0.6 
0.6 
0.6 
1.3 
1.3 
0.6  
R 
1.3 
0.6 
0.6 
0.6 
0.6 
1.3  
I 
0.9 
0.9 
0.9 
0.9 
0.9 
0.9  
R 
0.9 
0.9 
0.9 
0.9 
0.9 
0.9  
I 
1.3 
0.6 
0.6 
0.6 
1.3 
0.6  
R 
0.6 
0.6 
0.6 
0.6 
1.3 
1.3  
I 
0.9 
0.9 
0.9 
0.9 
0.9 
0.9  
R 
0.9 
0.9 
0.9 
0.9 
0.9 
0.9 
Table 2 highlights in red the
These values () were highlighted in red color (Table 2), and the correspondent temperature values were highlighted in red color (Table 1). These values correspond to measures 5 of the instrument (49^{o}C) and reference instrument (49.4^{o}C), whose reference value of the dry block is 50^{o}C, as well as measure 4 of the reference instrument (60.4^{o}C) whose value of the dry block reference is 60^{o}C. Once the mean and the standard deviation were recalculated for those temperature values where the outliers were eliminated, it was found that the standard deviation decreased when compared with the standard deviation of the original data.
4.2. Estimation of the Adjustment Polynomial
In the sequence of the
analysis and using the experimental data, without considering the outliers, it
was possible to establish a mean value that represents the total of the
measurements obtained for each experimental point. Thus, Table 3 consolidates
the average values for the bimetallic thermometer (instrument) and the
reference instrument. Likewise, the value of the systematic error is shown.
This value is obtained by the difference between the temperature indicated by
the instrument and the temperature of the reference instrument, as established
by the International Vocabulary of Metrology (VIM) (JCGM, 2012). According to the International Vocabulary of
Metrology (VIM) (JCGM, 2012), the value that represents the discrepancy between
the temperature measured by an instrument and the temperature measured by a
reference instrument is defined as the difference between the two readings.
Table 3 Average temperature and calculation of systematic
error
Average temperature of
the instrument (T_{i}) 
Average temperature of
the reference instrument (T_{p}) 
Systematic error (E) 
°C 
°C 
°C 
30 
30.4 
0.4 
40 
39.2 
0.8 
50 
49.5 
0.5 
61 
60.2 
0.8 
70 
69.6 
0.4 
80 
79.8 
0.2 
90 
88.6 
1.4 
100 
99.7 
0.3 
Using
the values of Ti and Tp, it was possible to determine the interpolating
polynomial that best represents the physical nature of the experiment by
applying the ordinary least squares method. The objective is to identify the
best polynomial and the uncertainty of adjustment associated with each
polynomial (u_{s}) must be determined. The polynomial with the
smallest (u_{s}) is considered to be the best representation of
the experimental physical model. Regarding the number of degrees of freedom,
this parameter is calculated by the difference between the number of
possibilities (number of experimental points n) and the number of restrictions
imposed (in the case of using fit polynomials, the number of restrictions is
given by a number of coefficients of the polynomial). Table 4 consolidates the
results obtained for the adjusted temperature, applying each polynomial, and
the table 5 shows the final value for the adjustment uncertainty associated
with each polynomial.
Table 4
Adjusted
Temperature (T_{a}) for each polynomial
Polynomial of Degree 1 
Polynomial of Degree 2 
Polynomial of Degree 3 
°C 
°C 
°C 
29.8 
30.1 
30.2 
39.7 
39.8 
39.8 
49.6 
49.5 
49.7 
60.5 
60.4 
60.9 
69.5 
69.3 
70.2 
79.4 
79.3 
80.8 
89.3 
89.4 
91.4 
99.2 
99.6 
102.1 
Table 5 Calculation
of the uncertainty associated with the polynomial adjustment (u_{s})
Polynomial of Degree 1 
Polynomial of Degree 2 
Polynomial of Degree 3 
°C 
°C 
°C 
0.1568 
0.1522 
0.2826 
The result above shows as the lower value of uncertainty is 0.1522^{o}C and corresponds to the polynomial of degree two. Thus, this result confirmed that, contrary to the expected, a polynomial of degree two models the physical nature of the problem with an uncertainty less than that associated with a polynomial of the first degree. Thus, in this procedure, it was evident that the best interpolating polynomial of adjustment corresponds to a seconddegree polynomial under the following equation:

Therefore, it is good calibration practice to
test at least three degrees of a polynomial to define the one that offers the
lowest adjustment uncertainty and, therefore, the lowest uncertainty associated
with the measurement. The evaluation of a polynomial of the fourth degree or
higher, although it offers a better result from a mathematical point of view
(i.e.: smaller fit errors), does not, however, represent the physical nature of
the phenomenon studied.
The application of a seconddegree
interpolating polynomial allows for correcting the experimental results
measured by the measuring instrument, thus eliminating the systematic error
inherent to the measurement process, and facilitating the process of
incorporating calibration in the remote data processing.
4.3. Application of Techniques to Estimate
Uncertainty
Once
the uncertainty of the adjustment is determined, we estimate the expanded
uncertainty associated with the measurement by the three methods under study:
GUM, Kragten, and Relative Uncertainty.
4.3.1.
Uncertainty Analysis: GUM Method
In the table above it is possible to identify that the greatest source of uncertainty is associated with the resolution of the measurement instrument, while repeatability is the source of the lowest contribution.
Table 6 Uncertainty
analysis: GUM method
Uncertainty associated to instrument
resolution 
Uncertainty associated to
reference instrument 
Uncertainty associated to
repeatability 
Uncertainty associated to
Polynomial Adjustment 
Combined Uncertainty (u_{c}) 
Coverage Factor (k) 
Expanded Uncertainty (U_{E}) 
Confidence Level  
°C 
°C 
°C 
°C 
°C 
 
°C 
%  
0.5774 
0.0475 
0.0619 
0.1522 
0.6022 
2.0 
1.22 
95.45%  
0.5774 
0.0475 
0.0333 
0.1522 
0.5999 
2.0 
1.22  
0.5774 
0.0475 
0.0000 
0.1522 
0.5990 
2.0 
1.21  
0.5774 
0.0475 
0.0183 
0.1522 
0.5992 
2.0 
1.22  
0.5774 
0.0475 
0.0211 
0.1522 
0.5993 
2.0 
1.22  
0.5774 
0.0475 
0.0447 
0.1522 
0.6006 
2.0 
1.22  
0.5774 
0.0475 
0.0211 
0.1522 
0.5993 
2.0 
1.22  
0.5774 
0.0475 
0.0224 
0.1522 
0.5994 
2.0 
1.22  
Table 7 consolidates the final results of the method. It shows the temperature set by the polynomial, as well as the corrected systematic error module. This parameter was determined by calculating the difference between the temperature indicated by the instrument and the temperature set by the interpolating polynomial. In addition, the expanded uncertainty associated with the temperature measurement is shown, and finally, the calculation of the total error of the measurement: Total Error =Systematic error+Expanded uncertainty (Process tolerance).
Table 7 Uncertainty
analysis: GUM method
Adjusted Temperature 
Correction of Systematic Error Module 
Expanded Uncertainty 
Error Total 
°C 
°C 
°C 
°C 
30.0 
0.1 
1.22 
1.3 
40.0 
0.2 
1.22 
1.5 
50.0 
0.5 
1.21 
1.7 
61.0 
0.6 
1.22 
1.8 
70.0 
0.7 
1.22 
1.9 
80.0 
0.7 
1.22 
1.9 
90.0 
0.6 
1.22 
1.8 
100.0 
0.4 
1.22 
1.6 
From the Table 7 it is observed that the expanded
uncertainty associated with the temperature measurement applying the GUM method
is equal to 1.22^{o}C in all the calibration points except for point
50^{o}C, where an expanded uncertainty equal to 1.21 was estimated.
4.3.2.
Uncertainty analysis: Kragten's method
Table 8 consolidates the results of the uncertainty analysis by Kragten's method. It can be observed in this table that, unlike the GUM method, the uncertainty is not constant throughout the instrument's measurement range, but varies from 1.21^{o}C to 1.39^{o}C.
Table 8 Consolidated
results: Kragten's method
(E_{c}) 
[E_{c} (T_{i} )] 
[E_{c} (T_{a} )] 
(u_{Ti}) 
(u_{Ta}) 
(u_{c}) 
(k) 
(U_{E}) 

°C 
°C 
°C 
°C 
°C 
°C 
 
°C 
% 
0.1 
0.6 
0.2 
0.6 
0.2 
0.6523 
2.0 
1.32 
95.45% 
0.2 
0.9 
0.1 
0.7 
0.2 
0.6855 
2.0 
1.39  
0.5 
1.0 
0.3 
0.6 
0.2 
0.5990 
2.0 
1.21  
0.6 
1.2 
0.5 
0.6 
0.2 
0.6350 
2.0 
1.29  
0.7 
1.3 
0.5 
0.6 
0.2 
0.6350 
2.0 
1.29  
0.7 
1.3 
0.5 
0.6 
0.2 
0.6393 
2.0 
1.30  
0.6 
1.2 
0.4 
0.6 
0.2 
0.6350 
2.0 
1.29  
0.4 
1.0 
0.2 
0.6 
0.2 
0.6393 
2.0 
1.30 
(E_{c}): Correction of Systematic Error; [E_{c}
(T_{i} )]: Correction of Systematic Error associated to Indicated
temperature; [E_{c} (T_{a} )]: Correction of Systematic Error
associated to Adjusted temperature; (u_{Ti}): Uncertainty associated to
Indicated temperature; (u_{Ta}): Uncertainty associated to Adjusted
temperature; (u_{c}): Combined uncertainty; (k): Coverage
factor; (U_{E}): Expanded Uncertainty; : Confident level.
4.3.3.
Uncertainty Analysis: Uncertainty Relative Method
Table 9 consolidates the results of the uncertainty
analysis by the Relative Uncertainty method. It can be observed that, unlike
the GUM and Kragten's method, the uncertainty is much lower when compared to
that obtained by the previous methods. This uncertainty varies from 0.0025^{o}C
to 0.012^{o}C.
Table 9 Consolidated
results: Uncertainty relative method
(E_{c}) 
(T_{i }) 
(T_{a} ) 
(u_{Ti}) 
(u_{Ta}) 
(u_{c}) 
(k) 
(U_{E}) 

°C 
°C 
°C 
°C 
°C 
°C 
 
°C 
% 
0.1 
30.0 
30.1 
0.6 
0.2 
0.0012 
2.0 
0.0025 
95.45% 
0.2 
40.0 
39.8 
0.6 
0.2 
0.0037 
2.0 
0.0074  
0.5 
50.0 
49.5 
0.6 
0.2 
0.0056 
2.0 
0.011  
0.6 
61.0 
60.4 
0.6 
0.2 
0.0061 
2.0 
0.012  
0.7 
70.0 
69.3 
0.6 
0.2 
0.0058 
2.0 
0.012  
0.7 
80.0 
79.3 
0.6 
0.2 
0.0050 
2.0 
0.010  
0.6 
90.0 
89.4 
0.6 
0.2 
0.0038 
2.0 
0.0077  
0.4 
100.0 
99.6 
0.6 
0.2 
0.0024 
2.0 
0.0049 
(E_{c}): Correction of Systematic Error; (T_{i}
): Indicated temperature by the instrument; _{ }(T_{a} ): Adjusted temperature
by polynomial; (u_{Ti}): Uncertainty associated to Indicated
temperature; (u_{Ta}): Uncertainty associated to Adjusted temperature;
(u_{c}): Combined uncertainty; (k): Coverage factor; (U_{E}):
Expanded Uncertainty; :
Confident level.
4.4. Metrological Comparison of
the GUM, Kragten, and Relative Method
This section consolidates the
uncertainty analysis results by the three evaluation methods: GUM, Kragten, and
Relative. Figure 1 illustrates the different uncertainty values found
by GUM and Kragten methods. The results confirmed that there is a considerable
difference in estimating the measurement uncertainty according to the method
that is applied. In addition, by the method of Relative Uncertainty, it was
amazing to find a much lower value than in the other two cases. This is because
the relative uncertainty method requires a linear independence between the
variables without considering the measurand's mathematical model. Thus, it is
confirmed that this method is unsuitable when the measurand corresponds to
addition or subtraction parameters since it does not represent the physical
nature of the problem.
In relation to the results obtained by the Kragten method and the GUM method, the consolidated results showed that the calculation of uncertainty by the Kragten method is superior to the results obtained by the GUM method. These results can be explained through the sources of uncertainty for each situation. Equation (13) showed that for the Kragten method, there are two main components and, as shown in Table 8, both predominate with a significant contribution: (u_{Ti}): Uncertainty associated to Indicated temperature; (u_{Ta}): Uncertainty associated with Adjusted temperature. On the contrary, for the GUM method, Equation (4) shows that despite having four contributing components of uncertainty (i.e., Instrument resolution (u_{inst}), Reference instrument (u_{p}), Repeatability (u_{r}), and Polynomial adjustment (u_{s})), the consolidated results in Table 6 confirm that two of these components do not contribute significantly to the calculation of uncertainty. Those sources correspond to Reference instrument (u_{p}) and Repeatability (u_{r}). In this way, only two sources contribute directly to calculating uncertainty by the GUM method. When comparing the order of magnitude of the contributing sources of uncertainty by the GUM method (u_{s} and u_{inst}) the values are lower than the contributions to the Kragten method (u_{Ta} and u_{Ti}). Thus, the GUM method offers a lower value than the calculation of uncertainty due to its mathematical conception from the conceptual formulation of the method.
Figure 1 Uncertainty of Measurement by methods: GUM
and Kragten
In the industry, the importance of finding
increasingly more minor uncertainties for the control of their measurement
processes, specifically in the use of bimetallic thermometers for temperature
control, is an activity that directly impacts the industry's economy. The
consolidated results in this work allowed us to compare, from the metrological
rigor, three methods for the evaluation of measurement uncertainty. The
Relative Uncertainty method showed the lowest measurement uncertainty. However,
it is not able to faithfully represent the physical nature of the problem. The
explanation of this behavior that associates that the combination of relative
uncertainties does not consider the mathematical calculation model of the
measurand and should only be applied when it is defined by the multiplication
and division of linear and independent terms. On the other hand, although
Kragten's method is used to estimate uncertainty due to its mathematical
robustness, it offers greater uncertainties when compared to the GUM method.
Thus, the GUM method was more appropriate for estimating the measurement
uncertainty associated with the temperature magnitude. Furthermore,
it is noteworthy that the ordinary least squares method proved to be a suitable
approach for: (i) deriving a second degree interpolating polynomial, which
exhibits reduced uncertainty in comparison to the firstorder polynomial
typically utilized in industrial settings, and (ii) determining the temperature
adjustment for any instrument reading within the calibration range. The methods
applied in the investigation are valid for the physical principle of
measurement associated with the thermal expansion between two metals. However,
depending on the robustness of the methods developed, this study can be replicated
for the evaluation of instruments that control other variables physical in the
industry: pressure, dimensional and electrical measurements, among others.
To the
Vicerrectoría de Ciencia, Tecnología e Innovación (VCTI) of the Universidad
Antonio Nariño for the financing of Project No. 2022008 (February/2022 to
December/2023) entitled: "Development of statistical techniques for the
analysis of uncertainty and metrological performance in industrial
equipment". In addition, this work
was partially supported by the Ministerio de Ciencia, Tecnología e Innovación 
Minciencias, in Colombia by the project
entitled “Gestión de energía basada en computación en la nube para la
interoperabilidad entre un grupo de microrredes aisladas" (70172) , which is part of “Programa de Investigación en Tecnologías
Emergentes para Microrredes Eléctricas Inteligentes con Alta Penetración de
Energías Renovables" (10185270169).
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