## 3 Results and Discussion  
### Analysis of Sand Particle Size Distribution  
The tests were carried out on the samples of self-compacting concrete (SCC) made with Portland cement (C), natural sand 0/2 mm (S) and gravel 2/8 mm (G) at W/C = 0.30 and (S+G)/C = 2.88. The specimens measuring 150 \(\times\) 100 \(\times\) 40 mm were sawn from 150 mm cubes. One surface of each specimen was ground according to the procedure described in EN 480-11 [17]. The specimen was carefully polished, and the clean test surface was treated with phenolphthalein. After drying, it was gently polished by hand with a felt disc. Since the scanning was carried out a few hours after polishing, the surface was damped with a citric acid solution and after about 1 min, it was wiped with a dry cloth and cleaned with a soft brush before taking the photographs. The treatment aimed at moistening the paste and improving the aggregate-paste contrast.  
The tests were carried out on three concrete specimens. One region of each specimen surface--74.1 \(\times\) 67.9 mm = 50.3 cm\({}^{2}\)--was photographed.  
After proper preparation of the binary image of the aggregate using the automated threshold setting, the elements of the aggregate phase were marked, distinguishing the particles smaller than 2 mm in size (Figure 3). The following parameters were calculated for each particle using the automated measurement method: area, equivalent diameter (EqD), Ferete's diameter (Max_Feret). The measurement results were then exported to Excel.  
The sand gradation in 3D space was constructed using the Saltykov method [27; 28]. The particles were assumed to have a spherical shape. An appropriate program was prepared--a VBA macro in the Excel spreadsheet.  
First, the distribution of the measured particle sizes per 1 cm\({}^{2}\) of concrete surface was determined. The EqD and the maximum value of Max_Feret were the basis for the calculations.  
The distributions were determined with the EqD:D\({}_{\mathrm{max}}\) or Max_Feret:D\({}_{\mathrm{max}}\) ratios, where D\({}_{\mathrm{max}}\) = 1995 \(\upmu\)m. Following the principles of the Saltykov method, the logarithmic scale was adopted in such a way that the ratio of diameters of two neighbouring classes was 10\({}^{-0.1}\) = 0.7943. Eighteen size classes were established. The number of particles N\({}_{\mathrm{a}}\)(i) per 1 mm\({}^{2}\) was calculated by dividing the number of particles in a given class by the measured area of the specimen.  
To determine the size distribution of sand particles related to a 3D volume unit, it was necessary to create a table of factors. The algorithm developed by [PERSON] and [PERSON] [29] was used for that purpose.  
The traditional sieve analysis of sand is based on the determination of the sum of the mass of particles passing through a given sieve. In order to determine the sand sieving curve based on the results of the calculations, the sand particle volume (uniform sand density) should be determined in  
Figure 3: NIS Elements window with marked threshold (**left**); sand particle equivalent diameter histogram (**right**).  
individual classes. The sand particles were assumed to be spherical. The calculations made on the basis of the EqD provided better compliance with the traditional sieve analysis than the calculation based on Max_Feret [16].  
The particle volume in individual classes was calculated by taking the arithmetic average of the recorded diameters EqD. The particle volume in a given class V(i) was calculated as the product of the average particle volume V1 and the number of particles N\({}_{\mathrm{v}}\)(i) [28]. The results of the N\({}_{\mathrm{v}}\)(i) calculations are given in Table 1.  
Figure 4 compares the results of the calculations above with the results from the sand sieve analysis. Good agreement was found between these results.