## 2 Materials and Methods
### Heat Conduction Model Used to Identify the Boundary Conditions of the Heat Transfer
The minimization of the objective function made it possible to determine the values of the unknown vector parameters, \(p_{i}\). This vector was composed of the HTC value in the nodes of the approximation of HTC in time. The set of \(p_{i}\) parameters determined the change in HTC on the cooled surface as a function of time and the sensor radius. The variable metric method [40], based on the BGFS algorithm [41; 42; 43; 44], was used to minimize the objective function. The derivative of the objective function with respect to the unknown \(p_{i}\) parameters was estimated numerically. The distribution of the HTC along the radius of the sensor was determined using Equation (18):  
\[\alpha(r,\tau)=\Big{(}1-7\theta^{2}+6\theta^{3}\Big{)}P_{1}(\tau)+\Big{(}8 \theta^{2}-8\theta^{3}\Big{)}P_{2}(\tau)+\Big{(}2\theta^{3}-\theta^{2}\Big{)} P_{3}(\tau) \tag{18}\]  
\[\vartheta=\frac{r}{R_{max}} \tag{19}\]  
where  
\(\theta\)--dimensionless sensor radius;  
\(R_{max}\)--maximum sensor radius, m.  
The \(P_{1}(\tau)\) parameter determined the HTC values ON the sensor axis (\(r=0\)), the \(P_{2}(\tau)\) parameter determined the values of this coefficient for \(r=1/2 Rmax\) and for the \(P_{3}(\tau)\) parameter, \(r=R_{max}\). The variability of parameters \(P_{1}(\tau)\), \(P_{2}(\tau)\) and \(P_{3}(\tau)\) (20)-(22) over time was approximated by third-degree polynomials [39]. \[P_{1}(\tau)=\sum_{i=1}^{4}H_{i}P_{1i}(\tau_{i}) \tag{20}\]  
\[P_{2}(\tau)=\sum_{i=1}^{4}H_{i}P_{2i}(\tau_{i}) \tag{21}\]  
\[P_{3}(\tau)=\sum_{i=1}^{4}H_{i}P_{3i}(\tau_{i}) \tag{22}\]  
The cooling time was divided into periods in which \(\tau\in(\tau_{1}\), \(\tau_{2})\). The third-degree Hermite polynomial was used to approximate the HTC distribution (23)-(26) [37]:  
\[H_{1}=\frac{(1-\eta)(9\eta^{2}-1)}{16} \tag{23}\]  
\[H_{2}=\frac{9(1-3\eta)(1-\eta^{2})}{16} \tag{24}\]  
\[H_{3}=\frac{9(1+3\eta)(1-\eta^{2})}{16} \tag{25}\]  
\[H_{4}=\frac{(1+\eta)(9\eta^{2}-1)}{16} \tag{26}\]  
where  
\(\eta\)--dimensionless time. Dimensionless time was determined using Formula (27):  
\[\eta=\frac{2\tau-\tau_{1}-\tau_{2}}{\tau_{2}-\tau_{1}} \tag{27}\]  
On the rest of the surfaces, a second boundary condition was assumed in the form of HF. The amount of heat transferred in the air gap, between the sensor surface and the inside surface of the casing, was determined using Equation (28) [5]:  
\[\vartheta=\frac{r}{R_{max}} \tag{19}\]0where  
\(\sigma\)--Stefan-Boltzmann radiation constant, W/(m\({}^{2}\cdot\)K\({}^{4}\));  
\(T_{1}\)--temperature of the inner surface of the casing, K;  
\(T_{2}\)--temperature of the sensor side, K;  
\(\varepsilon_{1}\)--emissivity of the sensor surface;  
\(\varepsilon_{2}\)--emissivity of the casing surface;  
\(\lambda_{p}\)--thermal conductivity of air, W/(m\(\cdot\)K);  
\(L_{co}\)--distance between sensor and casing, m;  
\(Pr\)--Prandtl number;  
\(Ra\)--Rayleigh number. The emissivity of both the cylinder surface and its casing was \(\varepsilon_{1}=\varepsilon_{2}=0.2\). The thermophysical properties of air, necessary to determine the value of Rayleigh and Prandtl numbers, were determined for the average water temperature. The boundary condition on the lower outer surface of the casing considered heat losses due to natural convection and heat radiation to the environment (29) [5]:  
\[\vartheta=\frac{r}{R_{max}} \tag{19}\]1  
where  
\(T_{ot}\)--ambient temperature, K;  
\(L_{o}\)--diameter of the outer casing, m.  
On the outer side surface of the casing, due to the presence of gaps in the insulating material surrounding the sensor (Figure 5), which allowed water to flow off the sensor surface, the boundary condition was defined by Equation (30) [45]:  
\[\vartheta=\frac{r}{R_{max}} \tag{19}\]2 \[\vartheta=\frac{r}{R_{max}} \tag{19}\]3 \[\vartheta=\frac{r}{R_{max}} \tag{19}\]4  
where  
\(\alpha_{b}\)--heat transfer coefficient on the side surface of the casing, W/(m\({}^{2}\cdot\)K);  
\(t_{b}\)--temperature of the outer surface of the side casing.