# Method for Volume of Irregular Shape Pellets Estimation Using 2D Imaging Measurement
## 3 Results
### Estimation of Pellet Volume by Dividing Them into Elliptical Layers  
For compensation of pellet volume calculation when the pellet is irregularly shaped, convexity of the area limited by the pellet was evaluated. After filling the notch in the pellet with its convex shape, its depth can be rationally evaluated. The ellipse geometric shape fills the cross-section area of a pellet more precisely. The three-dimensional model of irregular pellet presented in Figure 13.  
Figure 11: Analysis of example of pellets sample: (**a**) distribution of pellets of first sample according to roundness; (**b**) distribution of pellets of second sample according to roundness. Figure 12: Largest possible ellipse, whose volume is limited by region of pellet, is drawn inside of a pellet. Based on the pellet model presented in Figure 13, an assumption can be made that the cross-section area of the pellet limited by the envelope line is better overlapped by an ellipse than a circle. The volume is calculated by using both contours of the pellet--the real and notched area limited by the envelope line. The pellets volume is divided into cross-section areas, whose height is equal to one point of the image and which are perpendicular to longitudinal axis; then the pellets volume is calculated by integrating (according to Figure 13) Equations (7) and (8):  
\[\text{S(h)}=\pi\text{r}\frac{\text{d}_{2}}{2}, \tag{7}\]  
\[\text{V}=\int\limits_{0}^{\text{h}}\text{S(h)}\text{dh}, \tag{8}\]  
where S(h)--area of ellipse limiting the layer; r--half of cross-section of real area limited by pellet; d\({}_{2}\)--diameter of cross-section limited by pellets envelope line; h--length of a pellet, perpendicular to planes of layers that limit it; V--volume limited by pellet. For testing the reliability of the measuring method evaluating irregularly shaped pellets, models of plasticine were made (Figure 14a). Formed models of the pellets were scanned in two directions, using video camera, with the pellets position covering the largest possible area in relation to position of the video camera and the pellet rotated by 90\({}^{\circ}\) from the largest area position. This allowed us to evaluate the reliability of the measuring method by analyzing irregular geometric shapes from different positions in space. Calibrated experimental equipment was used for measuring. After carrying out primary morphological operations in the scanned images, the contour limited by the model was identified (Figure 14b). One of more important objectives of the primary processing is elimination of very small pellets that are in the background of the image and are not affected by gravity. This task is carried out by applying evaluation of the background [39], which allows us to filter out unneeded objects [40]. Figure 13: Three-dimensional (3D) model of irregular pellet: (**a**) the pellet’s volume consists of the sum of layer circles; (**b**) shape of the regular pellet is limited by pellet envelope line; (**c**) discrepancy between irregular volume and volume limited by envelope line; (**d**) cross-section of pellet limited by its envelope line. The general volume of the pellet is calculated by the previously used method of the sum of the separate volumes of layers, whose height is equal to one point of the image. The improved method is based on the calculation of the ellipse-like areas (based on Figures 8 and 13). The received results are presented in Table 2. The measuring data was compared to the method of the layer sum of circle-like pellets. Data received using the water displacement method is considered benchmark. Measuring results are presented in graphs of Figure 15.  
According to the calculations results, for the evaluation of volume of the irregularly shaped pellet, the evaluation method of pellet layer area approximation using the ellipse can be selected as a rational choice.