Study of methodological and instrumental components of the error on the accuracy of the reconstruction of temperature field on a wall surface

In the paper the reconstruction of temperature distribution based on resistance measurements of linear sensing elements using tomography method are considered. The methodical and instrumental errors of temperature distribution are investigated and analyzed. In particular the first component depends on number of sensors and degree of used approximation of temperature distribution and the second component depends on the level of random and systematic additive and multiplicative components in measurements. Two schemes of placing of the linear temperature resistivity sensors on the investigation object are researched in the paper (Fig. 1). Also, three approximation models of the temperature distribution in the form of two-dimensional cosine, asymmetrical cosine and Gaussian with initial temperature Θ0 = 100 ºС and different maximal change temperature Θm = 25; 10 and 5 ºС are investigated. The spatial resistivity distribution can be approximated by known two-dimensional basic functions are presented by formula (3). The resistances of linear resistive temperature sensors depend on resistivity are represented by formula (5). Coefficients' vector of the basic functions was calculated using the method of least squares with regularization (formula (22)). Then approximated spatial temperature distribution can be calculated on the basis of approximation model of the spatial distribution of resistivity (formula (14)). In the article proposed method is investigated for sensitive elements with the following parameters: resistivity ρ0 = 0.01724 μΩ m, temperature coefficient of resistance α = 4.3∙10-3 1/ºС, diameter of sensitive element d = 0.2 mm is simulated. The temperature distribution on the wall size of 2×2 m×m is investigated. The normalized to the maximum temperature error of reconstructed temperature distribution and root mean square error are calculated (formula (15), (16)). By using Monte-Carlo method (number of simulations M = 104) was performed simulation and in each simulation the surface average value, its standard deviations, minimum and maximum errors were determined by formulas (18) and (19). The characteristics of methodical error of reconstruction of temperature distribution for connection points on the side k = 6; 8; 12, algebraic polynomial of order p = 2 and different schemes (Fig. 1(a) and (b)) are presented in Fig. 2 and 3 respectively. The characteristics of methodical error of reconstruction of temperature distribution for scheme (Fig. 1(b)), approximation model 1(b), connection points on the side k = 6; 8; 12 and algebraic polynomial of order p = 2; 3 is presented in Fig. 4. The characteristics of instrumental error of reconstruction of temperature distribution for scheme (Fig. 1(b)), approximation model 3 (a) and (b), connection points on the side k = 8 and algebraic polynomial of order p = 2 are presented in Fig. 5. The results of this investigation showed that methodical component the most depends on approximation model and order of algebraic polynomial. The influence of additive systematic component of approximation is twice smaller than the influence of random. The influence of multiplicative systematic component of approximation is close to the influence of additive systematic component. The influence of additive random is amplified in 5–10 times.