## 6 Results of Uncertainty Simulations for Multiple Cases of Use of Radiation Thermometry
### Case 4: Precise and Traceable Measurement of Real Objects with Evaluated Spectral Emissivity  
Case 4 is an attempt of traceable measurement of a real object, evaluated for emissivity by Fourier transform infrared spectroscopy (FTIRS). In this case, instead of the grey body emissivity, the actual spectral emissivity of non-oxidized copper, brass and stainless steel samples with a thin coating of chromium nitride (CrN) was measured with FTIRS (Figure 24) and used for uncertainty simulation. Effective temperature-dependent emissivity values, calculated using Equations (7)-(10) of the hybrid inverse model in the temperature range from -50 \({}^{\circ}\)C to 500 \({}^{\circ}\)C, ranged from 0.338 to 0.351 for copper, from 0.0325 to 0.0304 for brass and from 0.267 to 0.280 for stainless steel.  
Figure 23: Monte Carlo simulated standard uncertainty of measurement (\(k=1\)) of a grey body temperature measurement using vanadium oxide-based thermal imager.  
Table 13 presents the input uncertainties in this case. Uncertainty due to emissivity was simulated with scalar uncertainty deviations (\(\epsilon(\lambda)\pm u_{\epsilon}\)). [PERSON] et al. [13] calculated the best practically achievable relative spectral emissivity evaluation uncertainty in order of 1% of the evaluated emissivity value; however, as our FTIR spectrometer was not recently calibrated and optimized for low uncertainty, we chose to expand this standard relative uncertainty to a more realistic value of 5%. A fixed value for the general emissivity uncertainty of 0.005 and a uniform distribution was assumed in this simulation. As a general measurement case, the use of a thermal camera with a VOx microbolometer and practical uncertainties from Table 2 was assumed.  
Figure 25 displays the calculated measurement uncertainties of measuring the three metal samples. As high measurement uncertainty was not expected when nearing the environmental temperature, we further analyzed the results. Emissivity compensation operates by subtracting measured radiance and (assumed) reflected environmental radiance, obtaining the emitted radiance of the surface as the difference. At low emissivity values and for temperatures close to the environmental temperature, the emitted radiance of the surface is calculated as very small in magnitude. However, the uncertainty of this near-zero difference is relatively high, as it is inherited from the reflected temperature. During division with very low emissivity, the uncertainty of the emitted radiance of the surface is greatly amplified. The uncertainty result is therefore exaggerated when measuring surface temperatures close to environmental temperature at very low emissivity values.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{\
\begin{tabular}{c} **Value and Standard** \\ **Uncertainty** \\ \end{tabular} } & \multicolumn{2}{c}{
\begin{tabular}{c} **Standard Uncertainty Contributions** \\ **for CrN Coated Metals at 35 \({}^{\circ}\)C** \\ \end{tabular} } \\ \cline{3-4}  & & **Stainless Steel** & **Brass** \\ \hline Reflected temperature & (23 \(\pm\) 1) \({}^{\circ}\)C & 2.4 \({}^{\circ}\)C & 35 \({}^{\circ}\)C \\ Air temperature & (23 \(\pm\) 1) \({}^{\circ}\)C & 5.7 mK & 51 mK \\ Emissivity & \(\epsilon(\lambda)\) \(\pm\) 0.005 & 0.57 mK & 5.5 mK \\ Relative humidity & (40 \(\pm\) 5)\% RH & 0.99 mK & 0.96 mK \\ Transfer path distance & (1 \(\pm\) 0.05) m & 0.85 mK & 2.1 mK \\ Atmospheric pressure & (1013.25 \(\pm\) 10) hPa & 0.26 mK & 0.73 mK \\ \hline \hline \end{tabular}
\end{table}
Table 13: Model parameter values, standard uncertainties (\(k=1\)) of the direct model parameters and standard uncertainty contributions (\(k=1\)) of the simulated temperature measurement in case 4.  
Figure 24: Spectral emissivities of CrN-coated samples of copper, brass and stainless steel, measured by FTIR spectrometer. Areas represent the standard uncertainty interval of spectral emissivity.