## 2 Experimental Method for Determining the Heat Transfer Coefficient
### Evaluation Method  
The measured bath and sample temperatures are evaluated after the trials to determine the heat transfer coefficients. The baseline for this calculation is Fourier's Law, which describes the temperature distribution in an object, in this case, given in cylindrical coordinates in Equation (2) [8]. \[\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}\lambda(T)r\frac{\partial T}{ \partial r}\bigg{)}+\frac{1}{r^{2}}\frac{\partial}{\partial\varphi}\bigg{(} \lambda(T)\frac{\partial T}{\partial\varphi}\bigg{)}+\frac{\partial}{\partial z }\bigg{(}\lambda(T)\frac{\partial T}{\partial z}\bigg{)}+\dot{q}^{\prime\prime \prime}=\rho(T)c_{v}(T)\frac{\partial T}{\partial t} \tag{2}\]  
where \begin{tabular}{l l} \(r\) & radial-coordinate in m; \\ \(\lambda\) & thermal conductivity in W/mk; \\ \(T\) & temperature in K; \\ \(\varphi\) & \(\varphi\) (tangential)-coordinate in \({}^{\circ}\); \\ \(z\) & \(z\) (axial)-coordinate in m; \\ \(\dot{q}^{\prime\prime\prime}\) & heat source/sink in W/m\({}^{3}\); \\ \(\rho\) & density in kg/m\({}^{3}\); \\ \(c_{\varphi}(T)\) & heat capacity in J/kgK; \\ \(t\) & time in s. \\ \end{tabular}   
In the first step, [PERSON]'s Equation is simplified and reduced in order to ensure a safe computation time and reduce the complexity. In this case, the circumferential direction is neglected because the conditions on the surface of the wire samples during cooling in the lead bath do not differ with the circumference. Therefore, no heat conduction is expected in this direction. The axial direction is also neglected since the front surface area is small compared to the area of the circumferential surface, resulting in a smaller heat flux in the z-direction. In addition, the sample is moved into the lead bath at a relatively high traversing velocity. Hence, there is no significant difference in cooling time between the tip and the end of the sample. This leads, in turn, to only a small  
an accurate calculation for all measured samples. Since the thermocouple is located in the centre of the wire during the experimental trials, this thermocouple must be taken into account in the model as well. This is realised by different material data, such as the thermal conductivity, along with the radius of the thermocouple and the remaining radius of the wire. Regardless of the material, temperature-dependent material properties were calculated with the Software JMatPro 13.3 and are used for all calculations. The heat transfer between the lead and the wire surface is calculated with Equation (1). Since the heat transfer coefficient (HTC) is unknown and is still subject to the calculations, an initial guess is necessary. This initial estimation of the heat transfer coefficient is used together with the heat conduction in the wire and the thermocouple to calculate the temperature at the centre for each time step of the cooling process. The discretization regarding the time derivation is realised with a one-step forward Euler method. The used solver determines the step size for each time-step. The whole model was setup in MATLAB(r) R2021a. A schematic overview of the model-setup and functionality is given in Figure 4.  
After calculating the temperature in the wire centre for each time step during the cooling process, the calculated results are compared to the actual measured data from the experimental trials. If the deviation between the two cooling curves is above a certain threshold, the heat transfer coefficient is changed within an upper and a lower boundary according to the algorithm \"sqp\" used in MATLAB(r) for finding the minimum of a constrained nonlinear multivariable function. Subsequently, the calculation is repeated. The upper and lower boundaries are set to avoid unrealistic model results. As soon as the temperature difference is below the threshold, the calculation is stopped.