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Application of Monte Carlo Method for Evaluation of Uncertainties of Temperature Measurement by Sprt XXI IMEKO World Congress “Measurement in Research and Industry” August 30 September 4, 2015, Prague, Czech Republic APPLICATION OF MONTE CARLO METHOD FOR EVALUATION OF UNCERTAINTIES OF TEMPERATURE MEASUREMENT BY SPRT Rudolf Palencar1, Stanislav Ďuriš1, Miroslav Dovica2 , Jakub Palenčár1 1Slovak University of Technology in Bratislava, Faculty of Mechanical Engineering, Slovakia rudolf. [email protected], [email protected], [email protected] 2 The Technical University of Košice, Slovakia, [email protected] hydrostatic-head effect (corresponding to location of SPRT Abstract - Procedure for evaluation of uncertainties of the sensing element in DFP), self-heating effect of the SPRT, ITS-90 based on the propagation of distributions using the immersion effect of the SPRT, effect of gas pressure in the Monte Carlo method is presented. It is based on generation of DFPs, choice of fixed point value from plateau isotopic pseudo-random numbers for input variables of the multivariate variations (for triple point of water (TPW) only), residual Gaussian distribution. This allows to take into account the gas pressure in TPW cell, changes of resistances of standard correlations between the resistances at the defining fixed points. resistor caused by changing of its temperature, non-linearity Assumption of Gaussian probability density function is of the resistance bridge, uncertainty of calibration of acceptable with respect to the several sources of uncertainties of resistance standard etc. resistances. The aim of this study is a) presentation of the procedure for MCM method for Keywords: Correlation, International Temperature Scale uncertainty evaluation of the temperature scale by 1990 (ITS-90), Monte Carlo method, Standard Platinum using SPRT calibrated at DFPs; Resistance Thermometer (SPRT), Uncertainty b) validation of the process by using the law of uncertainty propagation according to the GUM for 1. INTRODUCTION specific conditions. The procedure should allow to take into account the Evaluation of the uncertainties of the temperature scale correlation between the SPRT resistances calibrated at the ITS-90 realization is an important task in metrological DFPs and in temperature measurements. practice. Approaches are based on the GUM [1] and Influences such as temperature gradients, fluctuations, Supplements 1 and 2 [2, 3]. Widespread practices are based drifts and further, there are not analysed. Also, the on the law of uncertainty propagation. There are also uncertainties due to non-uniqueness and consistency approaches based on orthogonal polynomials. The overview subranges are not covered. These questions are presented is presented in [4-9]. Propagation of distributions using e.g. in [4]. Monte Carlo Methods (MCM) based on Supplement 1 to the The case study concerns the determination of GUM [2] occurs in a few isolated cases [10]. uncertainties of temperature scale realization using SPRT This paper presents a method based on the propagation calibrated in the range from 0 ° C to the freezing point of of distributions by Monte Carlo method. The procedure is aluminium (660.323 ° C). based on the generation of pseudo-random numbers of input variables (SPRT resistances at defining fixed points (DFPs), 2. INTERNATIONAL TEMPERATURE SCALE ITS-90 SPRT resistances in temperature measurements) of multi- – THEORETICAL BASES dimensional distribution. Multi-dimensional distribution is used because it allows to take into account correlation Temperature according to the ITS-90 can be determined between the SPRT resistances from calibration and in from the inverse function temperature measurement. In the case of uncorrelated resistances it is sufficient to generate data from the one- T f Wr (1) dimensional distribution of input variables. For the actual implementation of MCM for propagation For the corresponding subranges of the ITS-90 from 0 °C of distributions is necessary to know the probability the functions (1) are stated in [11]. Function Wr is given by distributions of input quantities and in the case of correlated N W -W a f W (2) input quantities multivariate distribution function. We r ∑ i i assumed normal distribution for all input SPRT resistances i1 and for correlated resistances multivariate normal where distribution. This assumption is eligible regarding to several R sources of uncertainties of SPRT resistances at the DFPs. W (3) E.g. chemical impurities of the substance in the DFPs, RTPW and their uncertainties and covariances. It means while covariance matrix of type 2N 2N . R is the SPRT resistance at temperature T and RTPW is SPRT resistances R corresponding to measured the SPRT resistance at TPW, temperatures T , SPRT resistance at TPW RTPW and their fi(W) are functions of the individual subranges in [11], uncertainties and covariances are determined in phase of ai are the coefficients of deviation function from SPRT temperature measurement. It means the vector calibration at DFPs and T Rmeas R, RTPW and its covariance matrix. ai=g1(RTPW1, …, RTPWN, RDFP1,…, RDFPN) or ai=g2( WDFP1, WDFP2,…, WDFPN). Furthermore, it is necessary to consider the covariances For the calculation of the coefficients of deviation between SPRT resistances from calibration and from function can be used matrix notation. If the relationship (2) temperature measurement. is applied to N fixed points, than Then we can write the covariance matrix of the vector (8) in the form W f W f W a DFP1 1 DFP1 N DFP1 1 (4) VR VR ,R V meas meas cal (9) R WDFPN f1 WDFPN f N WDFPN aN VR ,R VR meas cal cal where WDFPi Wr,DFPi -WDFPi , WDFPi are resistance ratios for In equation (9) covariance matrix V expresses the Rmeas corresponding DFPi and Wr, DFPi are from the document ITS- uncertainties of SPRT resistances and covariances between them for temperature measurements. Covariance matrix V 90 [11]. Rcal Equation (4) is written in the form expresses uncertainties of SPRT resistances and covariances between them for calibration of SPRT at DFPs. Matrix W M a (5) DFP DFP V expresses the covariances between the SPRT Rmeas ,Rcal Coefficients of deviation function are then (because there resistances from measurement and resistances from -1 exists MDFP ) calibration. Matrix has a nonzero values in-house temperature realization. I.e. the case when SPRT resistance a M -1 W (6) DFP DFP at TPW is used from calibration. If we assume that the Then the equation (2) can be rewritten measurements of SPRT resistances for temperature measurement are under the same conditions as it was in T calibration process then there could exist also covariances Wr W a f W (7) between SPRT resistances for temperature measurement and SPRT resistances from calibration. 3. APPLICATION OF MONTE CARLO METHOD Normally in practice the covariances between resistances are FOR EVALUATIONM UNCERTAINTY OF THE not considered, except for SPRT resistances at TPW. Two TEMPERATURE SCALE cases are considered here, 1) the use of SPRT in the laboratory (in-house), 2) the use of SPRT outside the 3.1 The input data laboratory. Covariance matrix VR has in both cases zero Input data for MCM are the SPRT resistances at DFPs from meas calibration and SPRT resistances from temperature elements. The covariance matrix for the use of SPRT measurement, so input quantities vector of dimension 2N +2 in-house has in the second line the first N elements different values from zero and other elements are zero. For the use of R R, R , R ,, R , R , R T (8) TPW TPW1 TPWN DFP1 DFPN the SPRT outside the calibration laboratory covariance and covariance matrix and vector of input quantities of matrix is zero. The total covariance matrix of vector type (2 2N)(2 2N) . of input resistances is for these cases given below. The SPRT resistances are determined on the basis 3.2 Procedure of calculation of SPRT calibration at DFPs, i.e. vector Procedure for calculating the temperature and its uncertainty R R ,, R , R ,R T cal TPW1 TPWN DFP1 DFPN by propagation of distributions by using Monte Carlo method can be expressed in the Fig. 1. 4. EVALUATION OF UNCERTAINTY AND EXPERIMENTAL DATA We calculated the uncertainties of temperature for specific data from the Slovak Institute of Metrology (SMU) for the sub-range from triple point of water up to freezing point of aluminium. 4.1 Input data The input data are given in Table 1. We measured temperature in the calibration laboratory (in-house) and also outside the calibration laboratory. We used one TPW cell for SPRT calibration. When we considered the SPRT resistances at TPW after tin, zinc and aluminium correlated (for simplicity we considered the correlation coefficient r=1), SPRT resistance of temperature measurement was considered uncorrelated with the other SPRT resistances. Fig.1. Computing phase of uncertainty evaluation by using Monte The SPRT resistances at DFPs are considered uncorrelated Carlo simulation for calculation the temperature and its standard (cases a)-in house, b)-outside laboratory), or correlated uncertainty (cases c)- )-in house, d)-outside laboratory). The correlations between resistances at TPW and at DFPs are considered 3.3 Generating pseudo-random numbers from probability uniformly r=0,4 and between the resistances at DFPs distributions uniformly r=0,3. The results of MCM simulation are To calculate the uncertainties using MCM and the presented in table 2 and table3. creation of applications with graphical interface support was used software Matlab (version 7.14, the 64-bit version), Table 1. Measured values of SPRT resistances at DFPs which uses a pseudo-random numbers generator Marsen Twister, which have undergone by a large number of tests Standard uncertainty of for testing pseudo-random numbers. This program allows Defining fixed Resistance resistance point () () you to generate pseudo-random numbers with the required -5 TPWSn 24,8002001 1,27.10 probability distributions of input quantities and rectangular, -5 TPWZn 24,8001927 1,27.10 triangular and Gaussian distributions. In our case is assumed -5 TPWAl 24,8001872 1,27.10 that all input quantities have a Gaussian distribution and in Sn 46,9397533 3,85.10-5 general they are correlated.
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