# Thermographic Stall Detection by Model-Inspired Evaluation of the Dynamic Temperature Behaviour
## 2 Principle of Measurement  
In case of temperature differences between the rotor blade surface of wind turbines and the incoming fluid, the thermographic measurable surface temperature correlates with the fluid-surface interaction in the boundary layer flow. However, the detected surface temperature is also influenced by other quantities. In order to derive novel measurement concepts with enhanced interpretability and contrast, the current understanding of the classical measurement chain of thermographic flow visualisation, including all influencing quantities is first presented. For a differential volume element on the surface of a rotor blade--see Figure 1--the temperature \(T_{\text{s}}\) can be calculated using a heat balance, as follows  
\[\rho\cdot c_{\text{P}}\cdot\frac{\partial T_{\text{s}}}{\partial t}=-\
abla\cdot \dot{q}. \tag{1}\]  
As a result, spatial differences of the temperature on airfoils are influenced by spatial changes in density \(\rho\) and specific heat capacity \(c_{\text{P}}\), as well as the following contributions to the heat flux density \(\dot{q}\):  
* Convective heat transfer \(\dot{q}_{\text{c}}\);
* Heat conduction \(\dot{q}_{\text{cond}}\);
* Absorbed radiant heat flux \(\dot{q}_{\text{r}}\). For incompressible flows, the convective heat transfer \(\dot{q}_{\text{c}}\) depends on the flow-dependent heat transfer coefficient \(\alpha\) as well as the temperature difference between the fluid temperature \(T_{\text{fl}}\) and the surface temperature \(T_{\text{s}}\), and can be calculated by  
\[\dot{q}_{\text{c}}=\alpha\cdot(T_{\text{s}}-T_{\text{fl}}). \tag{2}\]  
Since the intensity of the heat transfer depends on the fluid-surface interaction, the heat transfer coefficient \(\alpha\) contains information about the amount and the dynamic of the convective heat transfer for each flow region [25]. Thus, according to Equation (2), different flow conditions lead to different surface temperatures. In contrast, heat conduction \(\dot{q}_{\text{cond}}\), non-isotropic absorbed solar radiation \(\dot{q}_{\text{r}}\) and local changes in shape and material \(M\) lead to flow-independent spatial temperature gradients. In addition, the dynamic behaviour of the radiation intensity and the temperature of the incoming flow, as well as the initial surface temperature \(T_{0}\), influence the distribution and dynamics of the measured surface temperature field. Due to these cross-sensitivities, the different flow regions are not always visible, which means that the thermal contrast between the different flow regions is decreased. Regardless of these influences, the classical concept of thermographic flow visualisation is the evaluation of surface temperature fields as a function of the heat transfer coefficient \(\alpha\) as the input signal--see Figure 2a. The new approach aims to use the thermal response of the surface temperature to dynamic changes in the flow temperature or the radiation intensity, respectively--see Figure 2b. As a result, the flow-dependent heat transfer coefficient \(\alpha\) is considered as a parameter that determines the thermal response of the surface. The visualisation of different flow regions (and, in particular, stall condition) has a higher contrast and interpretability, because the influence of the convective heat transfer is considered after reducing the influence of the heat input variations so that the former cross-sensitivity is reduced. Figure 1: Schematic illustration of the different boundary layer flow regions on the suction side of an airfoil with trailing edge turbulent flow separation. A differential volume element with heat input and output normal to the surface by absorbed radiant heat flux \(\dot{q}_{\text{r},y}\), convective heat transfer \(\dot{q}_{\text{c},y}\) and heat conduction \(\dot{q}_{\text{cond},y}\) are also shown. In the direction of flow, the lateral heat conduction \(\dot{q}_{\text{cond},x}\).