## 2 Materials and Methods
### Heat Conduction Model Used to Identify the Boundary Conditions of the Heat Transfer  
The identification of the boundary conditions during cooling with the use of three systems requires the calculation of the temperature field of the cooled cylinder using the heat conduction Equation (10) [4]:  
\[\rho c_{p}(t)\frac{\partial t(r,y,\tau)}{\partial\tau}=\frac{1}{r}\frac{ \partial}{\partial\tau}\bigg{[}\lambda(t)r\frac{\partial t(r,y,\tau)}{ \partial r}\bigg{]}+\frac{\partial}{\partial y}\bigg{[}\lambda(t)\frac{ \partial t(r,y,\tau)}{\partial y}\bigg{]} \tag{10}\]  
assuming that  
\[(\tau>0;0<r<R;0<y<h)\]  
where  
\(\rho\)--density of the sensor material, kg/m\({}^{3}\);  
\(c_{p}\)--specific heat, J/(kg\(\cdot\)K);  
\(t\)--sensor temperature, \({}^{\circ}\)C;  
\(r\), \(y\)--cylindrical coordinates;  
\(\tau\)--time, s;  
\(\lambda\)--thermal conductivity, W/(m\(\cdot\)K);  
\(h\)--sensor height, m.  
The solution of Equation (10) was obtained using the finite element method, which transforms Equation (10) into a system of algebraic linear Equation (11) [37]:  
\[K_{ij}T_{j}+C_{ij}\frac{\partial T_{j}}{\partial\tau}=B_{i} \tag{11}\]  
where  
\(K_{ij}\)--heat conduction matrix;  
\(C_{ij}\)--heat capacity matrix;  
\(B_{i}\)--boundary condition vector. For one element, matrices \(K_{ij}\) and \(C_{ij}\), and vector \(B_{i}\) take forms ((12)-(14)):  
\[K_{ij}=\sum\int_{\Omega_{e}}\lambda\left(\frac{\partial H_{i}}{\partial r}\frac{ \partial H_{j}}{\partial r}+\frac{\partial H_{i}}{\partial z}\frac{\partial H _{j}}{\partial z}\right)rd\Omega_{e}+\sum\int_{S_{e}}rH_{i}H_{j}adS_{e} \tag{12}\]  
\[C_{ij}=\sum\int_{\Omega_{e}}\rho cH_{i}H_{j}rd\Omega_{e} \tag{13}\]  
\[B_{i}=\sum\int_{\Omega_{e}}q_{p}rH_{i}d\Omega_{e}+\sum\int_{S_{e}}H_{i}( \alpha T_{a})rdS_{e} \tag{14}\]  
where  
\(\alpha\)--heat transfer coefficient, W/(m\({}^{2}\cdot\)K);  
\(T_{a}\)--water temperature, \({}^{\circ}\)C.  
The finite element method was used to determine the temperature field from the heat conduction equation. The description of the finite element method used to solve Equation (10) is presented in the paper [38]. The accuracy of the FEM solution was used for modeling the temperature field of the axially symmetrical sensor [5]. The boundary condition at the water-cooled surface was searched in the form of a function depending on time and location (15). Thus, it was possible to determine the local values of the boundary conditions on the cooled surface:  
\[\dot{q}=\alpha(r,\tau)(t_{s}-t_{n}) \tag{15}\]  
where  
\(t_{s}\)--temperature of the cooled surface, \({}^{\circ}\)C;  
\(t_{n}\)--coolant temperature, \({}^{\circ}\)C.  
To solve the inverse problem, the objective function was adopted (16):  
\[E(p_{i})=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{m}\left\{\left[\frac{te^{ii}-t(p_ {i})^{ii}}{\sqrt{1+\left(\frac{\Delta t^{ei}}{\Delta\tau}\right)^{2}}}\right] ^{2}+\left(\frac{\frac{\Delta t^{ei}}{\Delta\tau}-\frac{\Delta t(p_{i})^{ii} }{\Delta\tau}}{Fs^{j\;i}}\right)^{2}\right\} \tag{16}\]  
where  
\(p_{i}\)--vector of unknown parameters;  
\(m\)--number of temperature measurements performed by one sensor;  
\(n\)--number of temperature sensors;  
\(te^{ii}\)--sample temperature measured by sensor \(j\) at time \(\tau_{i}\);  
\(t^{ii}\)--computed sample temperature at the location of sensor \(j\) at time \(\tau_{i}\). The term \(Fs^{j\;i}\) was specified as follow (17):  
\[ \begin{array}{l}Fs^{j\;i}=\frac{\Delta te^{ij}}{\Delta\tau}\;\;\mbox{for} \;\left|\frac{\Delta te^{ij}}{\Delta\tau}\right|>1\\ Fs^{ii}=1\;\;\mbox{for}\;\left|\frac{\Delta te^{ij}}{\Delta\tau}\right|\leq 1 \end{array}  \tag{17}\]  
The objective function (16) defines the temperature difference between the measured and computed temperatures. The temperature difference was extended with the norm of the temperature gradients. In this paper [39], it is shown that the objective function (16) ensures acceptable accuracy of the inverse solution for high temperatures as well as for variable temperature gradients in space and time.