## 2 Materials and Methods
### Equivalent Diameter of the Unobstructed Cross-Sectional Area  
When designing cooling channel systems, the channel diameter is selected based on the expectation of a given flow through-put, but when considering LPBF channels, the unobstructed cross-sectional area (or equivalent diameter) may be far from the nominal design. Figure 7a shows a 3D visualization of the \(D_{nom}\) = 3 mm channel investigated using the default image stack size of 500 images, as discussed in Section 2.2. Figure 7b shows a view of the cross-sectional area obtained by stacking all 500 images on top of each other and projecting a \"light\" through, corresponding to the yellow area. Both figures have been oriented to correspond to the defined orientation, shown in Figure 6b, with the build direction being towards \(\beta\) = 180\({}^{\circ}\). By comparing Figure 7a,b it can be seen how the contour of the white center in Figure 7a is closely represented by the contour of the yellow area in Figure 7b, except for the perspective view in Figure 7a. The current work suggests a methodology for estimating the equivalent diameter (\(D_{eq}\)) of the unobstructed cross-sectional area as a function of the channel length. The approach was similar to that of fractal dimensions but instead of considering window lengths, in this proposed methodology, batch sizes containing a set number of images were considered. Figure 8 shows an example of one stack of images that was divided into smaller batches. For each batch size, an average value for \(D_{eq}\) could be calculated, giving an estimate of \(D_{eq}\) at the corresponding length. For a batch size of 125 images, representing  
Figure 7: Equivalent diameter of \(D_{nom}\) = 3 mm. (**a**) Through view of 3D visualization; (**b**) through view of stacked cross-sectional images. a length of 125 images \(\times\) 9 \(\upmu\)m/image = 1.13 mm, the estimated equivalent diameter was \(D_{eq}\) = 2.648 mm, using the following equation:  
\[D_{eq}=\frac{\sum_{i=1}^{N}\sqrt{\frac{4A_{i}}{\pi}}}{N} \tag{11}\]  
where \(A_{i}\) is the area calculated from the pixels corresponding to the unobstructed cross-sectional area (the yellow area in Figure 7b) from batch \(i\), and \(N\) is the stack divider. The initial image stack with a batch size equal to 500 was divided into smaller batches using the following eight stack dividers: 1, 2, 4, 8, 16, 32, 64, 128, and 256. Figure 9a shows the nine points plotted with a linear fit, a second-order polynomial fit, a third-order polynomial fit, and a natural logarithmic fit, using increasing weights of [1, 10, 100, 103, 104, 105, 106, 107, 108] with length, for the \(D_{nom}\) = 3 mm channel. The logarithmic fit is shown to be highly representative of the calculated \(D_{eq}\)-values. The shown estimation was based on the default maximum batch size of 500. Moreover, it was investigated if this batch size was adequate in predicting \(D_{eq}\) for longer channel lengths. The investigation was conducted by considering the 500 images as a \"normal\" stack size and comparing it to a larger stack size. The larger stack size consisted of all the good quality images available for the channel. For \(D_{nom}\) = 3 mm a total of 1890 cross-sectional images were used (the remaining 110 images available did not have a high enough quality). The data of the two stacks can be seen in Figure 9b, together with the logarithmic fits. For the fit of the large data set, no weighing was used. A simple quantitative comparison between the two stack sizes was performed by calculating the average difference in the estimated \(D_{eq}\)-value for the two stacks sizes over a length of 50 mm. The resulting deviation was 1.4%, proving a good agreement between the two calculations. Using the default stack size of 500 images can then be considered enough to give a good approximation of the actual \(D_{eq}\). The points representing the longest lengths in Figure 9 were calculated using stack dividers of 1 and therefore have no standard deviations.