# Method for Volume of Irregular Shape Pellets Estimation Using 2D Imaging Measurement
## 3 Results
### Estimation of Pellet Volume by Dividing Them into Circular Layers  
Based on experiments of pellet analysis, it can be stated that sufficient precision can be achieved when the pellet's volume is evaluated according to area limited by the pellet's silhouette. After evaluating examination results it was determined that only in specific cases, when dealing with a pellet's shape that is very close to ideal, is the ellipsoid volume formula Equation (2) suitable for calculation of pellet volume. Since roundness index of analyzed material is not very high (evidenced by experimental examination, Figure 4), this mathematical expression does not match the real volume limited by pellet. According to the literature [37], the reduction of error of measurements is affected by the division of the analyzed ellipsoid into layers, i.e., dividing its area into narrow regions that are perpendicular to longitudinal axis of ellipse. The area limited by the pellet's silhouette is divided into layers of least height. This height is equal to the height of one point of the image. Using this method, the area of circle, limiting every layer, is calculated [37,38] with Equation (3):  
\[\mathrm{S_{i}}=\pi\mathrm{r}^{2}=\pi\left(\frac{\Delta\mathrm{y}}{2}\right)^{ 2}, \tag{3}\]  
where \(\mathrm{S_{i}}\)--area of circle that limits the layer; \(\Delta\mathrm{y}\)--the widthof the ellipse layer (particle scheme in Figure 8). Figure 8: The layering of an ellipse: (**a**) side view of an ellipse model; (**b**) top view of ellipse model. Figure 7: Average discrepancy of area limited by the pellet’s silhouette when pellets are evaluated according to their size and roundness. The measuring method studied by [PERSON] and [PERSON] [37] is based on model of the ideal ellipse, because the ellipsoid consists of sum of volumes of circles Equations (4) and (5):  
\[\mathrm{V_{i}}=\mathrm{S_{i}}\mathrm{\Delta x}, \tag{4}\]  
\[\mathrm{V}=\sum_{i=1}^{n}\mathrm{V_{i}}, \tag{5}\]  
where \(\mathrm{S_{i}}\)--area of circle limiting the layer; \(\mathrm{V_{i}}\)--volume of circle limiting the layer; \(\mathrm{V}\)--volume limited by ellipsoid. During the examination, the formed models of the regular shaped pellets were scanned using video camera. Pellets of ellipse shape were formed according to measurement results presented in Figure 7--most pellets are elongated. At the beginning of measuring, the calibration of the experimentation equipment was carried out by evaluating the ratio coefficient between points in the image and real size of a pellet--for recalculation of results into SI (metric) system units. In the later stages, after binarization and threshold segmentation operations were carried out in scanned images, the area of the analyzed pellet was distinguished. The general volume of the pellet was calculated by adding up volumes of the separates layers (perpendicular to longitudinal axis of the pellet), whose height is equal to one point of image. The algorithm of the calculation method is presented in Figure 9. Convexity of the was evaluated experimentally. During the examination of pellet models in course of the experiment, it was determined that this threshold value was 0.8, while there were additional calculations carried out for determining the convexity of a pellet. Figure 9: Algorithm for calculation of pellet volume (pellets divided into layers). After evaluation of results of the pellet volume measurements, presented in Figures 1 and 10, it can be stated that approximation using the ellipse limiting the pellet's area is not a reliable volume measuring method for evaluating the volume of the ellipsoid. In case of ellipsoid approximation, the average of absolute error of measurements is up to 1.258 cm\({}^{3}\), while \(\sigma=0.687\). When the volume of pellets model is calculated by dividing its area into layers of one point of the image and the limited area of layers is added up, more reliable measurement results are produced because of the lowest possible distance between the adjacent analyzed areas of cross-section.