## 2 Materials and Methods
### Numerical Calculations  
Numerical calculations that allowed for identifying the HTC were performed for the axially symmetric sensor. Inverse calculations were performed based on measurements of the temperature changes inside the sensor obtained from the experimental cooling tests. Calculations were performed assuming that the thermal conductivity and the specific heat of the sensor could change with temperature. The data given in the data sheet for Inconel 600 [46] were approximated by polynomials (31) and (32):  
\[\lambda=2\cdot 10^{-8}t^{3}-3\cdot 10^{-5}t^{2}+0.0285t+15.124 \tag{31}\]  
\[c_{p}=3\cdot 10^{-5}t^{2}+0.1818t+442.85 \tag{32}\]  
For the calculations, it was assumed that the density of the Inconel did not change with temperature and was \(\rho=8470\) kg/m\({}^{3}\)[46]. A constant sensor emissivity value of 0.2 was assumed. The solution to the heat conduction equation was accomplished with the finite element method. Thirty-three elements were used in the direction of the sensor height y and 22 elements in the direction of its radius \(r\) (Figure 11). The number of intervals of the boundary condition approximation as a function of cooling time length varied from 7 to 10 intervals. Uncertainty of the Inverse Solution  
The inverse solutions to the boundary condition in space and time at the boundary surface depend on the differences between measured and computed temperatures due to a natural consequence of the objective function formulation, and it may show a significant deviation from the real boundary condition. These differences may result, among others, from the temperature measurement errors. For those reasons, the uncertainty test was performed and the boundary condition resulting from Equation (33) was searched:  
\[\alpha(t,\ r)=\alpha(t)\Big{[}\Big{(}\Big{(}1-3\theta^{2}+2\theta^{3}\Big{)}F_ {1}\Big{)}+\Big{(}\Big{(}3\theta^{2}-2\theta^{3}\Big{)}F_{2}\Big{)}\Big{]} \tag{33}\]  
where  
\(F_{1}\)--scaling factor for the heat transfer coefficient in the sensor axis (r = 0);  
\(F_{2}\)--scaling factor for the heat transfer coefficient in the sensor edge. \[\alpha(t)=-1.6544+0.1175t-0.0005t^{2}+6.9\cdot 10^{-7}t^{3}-3.33\cdot 10^{-10}t^{4} \tag{34}\]  
The Inconel cylindrical sensor temperature was simulated using the FEM model with the linear shape functions. Thirty-three elements in the sensor high and 22 elements in the direction of its radius were used. The boundary conditions were defined by heat fluxes: \(\hat{q}_{gz}\), \(\hat{q}_{p}\) and \(\hat{q}_{b}\) were specified at the sensor surfaces. The sensor temperatures at four points were recorded as the simulated temperature sensors indications. Then, these temperatures were disturbed by the random errors that simulated the maximum error of the measured temperature. These errors were estimated to have a value of \(\pm\)5.4 K. The simulated temperature measurement errors assigned to the thermocouple placed at point P2 are shown in Figure 12.  
Figure 11: Finite element mesh with the location of thermocouples. For the inverse solution, the FEM model with 33 elements in the direction of the sensor height and 22 elements in the direction of its radius was employed. The inverse solution was compared to the specific boundary condition obtained from Equation (33). For the inverse solution, the time of cooling was divided into 8 periods. It had given 66 coefficients to be determined from the minimum condition of the objective function (16). The inverse solution converged with the average temperature difference of 2.66 K, with the maximum distance between temperature curves 0.09 K. In Figure 13, a comparison between the measured and calculated temperatures at points P1, P2, P3 and P4 are presented. For all considered points, the results of the calculation stay with good agreement with the measured data. The mean square error of the temperature calculation was 0.61 K. In Figure 14, a comparison of the HTC calculated due to the inverse solution (Inv) with the reference HTC (Ref) obtained from the solution of Equation (33) is shown.