## 3 Present Method for Determining the Heat Transfer Coefficient
### Measurement Principle  
The dependencies of free and forced convective heat transfer can be found in Section 2. It is important to note that the heat transfer coefficient cannot be measured directly. In the present work, the heat transfer coefficients are determined with energy balances. The IR thermal camera is used to measure the temperature \(T_{xy}\) for each pixel of the heated strip in steady state. The supplied electrical energy \(\dot{Q}_{gen}\) corresponds to the dissipated heat losses \(\dot{Q}_{losses}\) of the sample, as shown in Equation (11):  
\[\dot{Q}_{gen}=\dot{Q}_{losses} \tag{11}\]  
The basic prerequisites for the validity of this energy balance are the properties of the heated strip. The strip made of CuNi44 has an almost constant electrical resistance in the temperature range of \(T\) = 0-600 \({}^{\circ}\)C. This allows easy calculations of the heat generated in the strip without having to consider changes in electrical resistance. Furthermore, CuNi44 has a constant thermal conductivity of \(\lambda\) = 21.5 W/(mK) in the temperature range of \(T\) = 0-300 \({}^{\circ}\)C. However, because of the strip's thickness of \(t_{strip}\) = 0.1 mm, it can be considered as thermally thin (\(Bi=\alpha L/\lambda\ll 1\) with \(L=t_{strip}\) = 0.1 mm). As a result, thermal conduction in the thickness direction is negligible. Accordingly, the heat losses consist of the heat conduction in the longitudinal and in the transversal direction as well as forced convection and radiation on the nozzle-facing side. On the nozzle averted side, heat losses occur due to free convection and radiation towards the surroundings. A schematic representation of the heat flow for one pixel is shown in Figure 4; Equation (12) shows the energy balance of all heat losses as stated in Equation (11). \[\dot{Q}_{gen}=\dot{Q}_{conv,force}+\dot{Q}_{conv,frac}+\sum_{n=1}^{2}\dot{Q}_{ n,rad}+\sum_{k=1}^{4}\dot{Q}_{k,cond} \tag{12}\]  
with  
\(\dot{Q}_{conv,force}\) & as the forced convection heat flow in W;  
\(\dot{Q}_{conv,free}\) & as the free convection heat flow in W;  
\(\dot{Q}_{n,rad}\) & as the radiation heat flow in W;  
\(\dot{Q}_{k,cond}\) & as the conduction heat flow in W.  
The heat generated by the electrical energy in each pixel can be calculated based on [PERSON]'s first law, Equation (13). Since the electrical resistance of the strip material is independent of temperature, the same power is generated in each pixel area. Therefore, the heat calculation can be simplified by dividing the total power by the number of pixels \(n_{pr}\)  
Figure 4: Schematic representation of the heat flow of one pixel. \[\dot{Q}_{gm,yx}=\Delta U_{px}\cdot I\approx\frac{U\cdot I}{n_{px}} \tag{13}\]  
with  
\(\Delta U_{px}\) as the voltage drop per pixel in V;  
\(I\) as the current in A;  
\(U\) as the voltage drop over the strip in V.  
With the introduction of the pixel length \(\Delta x\) and pixel width \(\Delta y\), the following expressions for the convective heat flows result according to Equation (1):  
\[\dot{Q}_{com,force}=h_{force}\cdot\left(T_{x,y}-T_{fluid}\right)\cdot\Delta x \Delta y \tag{14}\]  
\[\dot{Q}_{com,frac}=h_{free}\cdot\left(T_{x,y}-T_{Fluid}\right)\cdot\Delta x \Delta y \tag{15}\]  
The fluid temperature \(T_{fluid}\) is the reference temperature for the convective heat flow, while the ambient temperature \(T_{\infty}\) is used for the heat flow by radiation. The free convection heat transfer coefficient \(h_{free}\) is determined separately using the equation of the Nusselt number of an isothermal horizontal wall. An experimental check analogous to the energy balance is also conducted. To simplify the determination of the radiative heat flow, the strip is coated with a black, temperature-resistant paint. The emissivity of \(\varepsilon\) = 0.9 is included in the balance according to [PERSON]'s results [19].