## 3 Results  
### Longitudinal Dispersion  
During the tracer test, the analysis of the specific conductivity \(S\) in the filter effluent enabled us to determine three parameters of the model (1):  
- The pre-exponential factor, as a constant value for a given experiment \(\frac{M}{C}\frac{L}{2\sqrt{mD_{x}^{*}}}\);  
- The actual filtration velocity (capillary velocity) \(V_{x}^{*}\);  
- The actual coefficient of longitudinal turbulent diffusion (coefficient of longitudinal capillary turbulent diffusion) \(D_{x}^{*}\).  
The tests were carried out in columns with internal diameters of \(D\) = {15, 3} cm. The results are summarized in Table 1.  
Both columns were filled with the same material. The characteristic \(D\)-values for anthracite particles {\(d_{10}\), \(d_{50}\), \(d_{90}\)} were {0.73, 0.95, 1.2} mm, respectively, while for sand, these values were {0.60, 0.77, 0.86} mm, respectively. The average diameter of the particle weighted by the height of the layers was \(d_{z,50}\) = (0.5 m\(\cdot\)0.95 mm + 0.5 m\(\cdot\)0.77 mm)/(0.5 m +  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**No.** & \begin{tabular}{c} \(D\) \\ **(cm)** \\ \end{tabular} & \begin{tabular}{c} \(Q\) \\ **(mL/s)** \\ \end{tabular} & \begin{tabular}{c} \(p_{p,sorb}\) \\ **(g/m\({}^{3}\) of Bed)** \\ \end{tabular} & \begin{tabular}{c} \(V_{x}^{*}\) \\ **(10\({}^{-2}\) m/s)** \\ \end{tabular} & \begin{tabular}{c} \(D_{x}^{*}\) \\ **(10\({}^{-4}\) m\({}^{2}\)/s)** \\ \end{tabular} & \
\begin{tabular}{c} \(Pe\) \\ **(10\({}^{-4}\) m\({}^{2}\)/s)** \\ \end{tabular} &
\begin{tabular}{c} \(Re\) \\ **0.5 m) = 0.86 mm. The relationship between the Peclet number (_Pe_), the Reynolds number (Re) and the apparent sorbent density (\(\rho_{p,sorb}\)) was established as:  
\[Pe=\left(0.007975+0.058937\cdot\text{Re}^{-0.886653}\right)\cdot\left(1+\frac{ \rho_{p,sorb}}{\rho_{0,p,sorb}}\right)^{-0.249469} \tag{8}\]  
\[Pe=\frac{V_{x}^{*}\cdot 4\cdot R_{h}}{D_{x}^{*}} \tag{9}\]  
\[\text{Re}=\frac{V_{x}^{*}\cdot 4\cdot R_{h}}{v_{w}} \tag{10}\]  
\[N=\frac{\frac{\pi D^{2}}{h}}{\frac{4}{3}\pi\left(\frac{d_{z,50}}{2}\right)^{3}} \tag{11}\]  
\[R_{h}=\frac{V_{z}}{A_{z}}=\frac{\frac{\pi D^{2}}{4}h}{4\pi\left(\frac{d_{z,50 }}{2}\right)^{2}N}=\frac{d_{z,50}}{6} \tag{12}\]  
where  
\(R_{h}\)--hydraulic radius (m);  
\(N\)--numbers of particles in a filter bed;  
\(D\)--internal diameter of the filter column (m);  
\(h\)--height of the filter bed (m);  
\(V_{z}\)--volume of the filter bed (m\({}^{3}\));  
\(A_{z}\)--wetted surface area of bed particles (m\({}^{2}\));  
\(d_{z,50}\)--average diameter of the particle (m), \(d_{z,50}\) = 0.00086 m;  
\(\rho_{p,sorb}\)--apparent sorbent density (g PAC/m\({}^{3}\) of bed);  
\(\rho_{0,p,sorb}\)--standard apparent sorbent density (g PAC/m\({}^{3}\) of bed), assumed \(\rho_{0,p,sorb}\) = 1000 g PAC/m\({}^{3}\) of the bed;  
\(\
u_{w}\)--kinematic water viscosity (m\({}^{2}\)/s), assumed as \(\
u_{w}\) = 10\({}^{-6}\) m\({}^{2}\)/s.  
Equation (8) is valid within the following range: Re \(\in\)\(\langle\)0.75; 4.15\(\rangle\) and \(\rho_{p,sorb}\)\(\in\)\(\langle\)0; 12,575.2\(\rangle\) g PAC/m\({}^{3}\) of the bed.  
Due to the multidimensional nature of Equation (8), the symbols \(\Circuit\) and \(\blacksquare\) in Figures 4 and 5 represent a pair of points corresponding to each other at a given Re or apparent density \(\rho_{p,sorb}\). If Equation (8) was a perfect description of the measurements and such data were accurate, the signs would coincide. Equation (8) provides a very good description of the measured values with a high correlation coefficient R = 0.9892 (R\({}^{2}\) = 0.9786) (Figure 6).  
Figure 4: Model (9) fit—theoretical _Pe_ vs. measured _Pe_ as an Re function.