## Materials and Methods
### Diagnostics
#### 2.2.2 Compensation of Astigmatism  
The raw image shown in Figure 6a features another imaging problem to cope with. The shape of the droplets is frequently distorted in two characteristic directions, which are perpendicular to each other. Strongly distorted droplet images even seem like they occur in pairs of two. Furthermore, no droplet is imaged as sharp as for a hot gas temperature of 293 K and very small droplets in the range of 5 um in diameter are not visible. The described imaging error is caused by an astigmatism phenomenon. It implies that no distinctive focal point is present in the optical path. This phenomenon can be exploited to determine the position of droplets in the direction perpendicular to the image plane. This way, all three components of the velocity field can derived using a single camera [35]. However, one major goal of this study is the detection of the droplet diameter with high accuracy, which is severely compromised by the astigmatism phenomenon.  
One reason for the astigmatism may be a misalignment of the optical access module with respect to the camera due to thermal expansion. However, exact alignment of camera and windows at elevated temperature did not improve the situation. Hence, it was studied whether an inhomogeneous thermal expansion of the inner window may be responsible. A plausible deformation of this window, which may result from the thermal load of the hot gas flow at elevated temperatures, is schematically illustrated in Figure 7. An assumed thermal expansion at the centre of the rectangular glass window will result in a curvature, which is dependent on the spatial direction.  
Hence, the image of a droplet is distorted depending on the distance of the droplet from the theoretical focal plane of the imaging system. This implies that a droplet can be imaged sharp in one direction while being completely out of focus in another direction. In summary, the inner window acts at elevated temperatures like a lens with an aberration, which causes the visible astigmatism phenomenon. Furthermore, the optical axis of the camera is not positioned at the centre of the window, which may enhance this effect. It is important to emphasize that the astigmatism phenomenon only appears at elevated temperature levels and that it is present, no matter whether the water cooling module is used or not.  
Since it is very difficult to eliminate the root cause of the problem, an astigmatism correction plate was placed between the camera and the test section. A rotation of the correction glass plate, as depicted in Figure 5, induced an additional astigmatism, which in turn compensated the unwanted astigmatism caused by the inner window. An adjustment in two angular directions was performed in order to properly implement the correction of the astigmatism. This was required to ensure that  
Figure 7: Schematic shape of the inner glass window of the optical access module at elevated temperature and its astigmatism effect on droplet images at the resulting two focal lines and the so-called circle of least confusion.  
the rig had reached a stable temperature level prior to the fine adjustment of the correction device. An exemplary raw image after the application of the astigmatism correction is shown in Figure 6b. Obviously, no astigmatism is present and the image is generally much sharper, which facilitates the detection of very small droplets. Hence, it was possible to acquire raw images of sufficient quality for post-processing. The influence of a pressure difference across the outer window of up to 0.5 MPa was investigated as well. It was found that it has no direct influence on the quality of recorded images, but leads to an amplification of the astigmatism effect due to an increased heat transfer from the hot gas flow to the window assembly.