## 3 Results and Discussion
The size and the shape of powder particles (including internal porosity) and the size of void spaces between particles significantly affect wettability and solubility. The measured differences in the microstructure of the analyzed samples can be reflected in various functional properties, such as powder solubility. Therefore, a direct relationship can be observed between the microstructure and functional properties of the tested powder samples. The presence of fine particles and low powder porosity make it difficult for water to penetrate their interior [10]. Agglomerated powders contain larger particles with irregular shapes and a porous surface. According to [PERSON] [28], the flowability of powders is determined mainly by the size of agglomerates, whereas changes in powder composition have no significant effect on flow characteristics. However, the results of this study indicate that the reconstitution properties of high-protein milk powders depend not only on the powder's microstructure, but also (to a large extent) on the concentration and type of protein in the concentrate. In the present study, a moderate correlation (r = 0.4-0.7) was observed between the size of powder particles described by D32, D43, and the insolubility index (positive values of r) vs. density, the Carr index, and the Hausner ratio (negative values of r) (Table 4). The weak correlations between particle size and the analyzed properties of milk powders are explained by the relationship between the mass and size of fractal objects. The fractal dimension, calculated as the relationship between the perimeter and the surface area of a particle, describes surface roughness and supports precise and unbiased comparisons of particle microstructure (Figure 3). \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Diameter**} & **Insolubility** & **Loose Bulk** & **Tapped Bulk** & \multirow{2}{*}{**Carr Index**} & **Hausner Ratio** \\  & **Index** & **Density** & **Density** & & \\ \hline D32 & 0.62 & \(-\)0.58 & \(-\)0.65 & \(-\)0.44 & \(-\)0.47 \\ \hline D43 & 0.41 & \(-\)0.66 & \(-\)0.79 & \(-\)0.69 & \(-\)0.72 \\ \hline \end{tabular}
\end{table} 
Table 4: Correlation coefficients (\(r\)) denoting the strength of the linear relationships between particle size and powder properties. Figure 2: Micrographs of high-protein milk powders at 200\(\times\) (**upper row**) and 800\(\times\) (**bottom row**) magnification. The results of the image analysis are presented in Table 5, and an example of the logarithmic relationship between the perimeter and surface area of the analyzed particles is shown in Figure 4. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Product** & \(\mathbf{D_{L}}\) & \(\mathbf{R^{2}}\) \\ \hline MCC & 1.29 & 0.92 \\ \hline SPC & 1.40 & 0.87 \\ \hline WPC & 1.18 & 0.91 \\ \hline CH & 1.28 & 0.93 \\ \hline WPH & 1.25 & 0.93 \\ \hline WPI & 1.24 & 0.93 \\ \hline \hline \end{tabular}  
\(\mathbf{D_{L}}\)—fractal dimension, \(\mathbf{R^{2}}\)—coefficient of determination.  
\end{table} 
Table 5: Fractal dimensions and determination coefficients for high-protein milk powders. Figure 4: Logarithmic plot for image analysis of whey protein hydrolysate. Figure 3: Examples of micrographs for fractal dimension calculations. The calculated fractal dimensions point to minor differences in the microstructure of milk powders. The calculated fractal dimension was highest in SPC and lowest in WPC, which implies that WPC particles had smoother and less ragged contours (Table 5, Figure 3). The observed differences in the microstructure of the analyzed powders may affect their functional properties, including solubility. For this reason, the relationships between the microstructure and the properties of the tested powders were analyzed. An analysis of the linear relationship between the fractal dimension and powder properties revealed that the fractal dimension was correlated with bulk density (r = 0.73). The fractal dimension was not correlated with the remaining properties of milk powders.