## 3 Results
### Adsorption Isotherm  
In dynamic conditions, the adsorption isotherm changes due to the adhesion of PAC particles to the filter medium, in this case anthracite. Moreover, the PAC particles trapped in interparticle cavities of anthracite have a slight chance of coming into contact with the adsorbate compared to free PAC particles in bulk adsorption. Both of these factors reduce the effective area through which adsorbate can penetrate into the sorbent. This may reduce the PAC adsorption capacity. However, at a higher PAC dose, a developing PAC distribution structure may offset other effects and the effective penetration area will increase, resulting in a higher adsorption capacity. Also, a turbulent flow, shaping a turbulent diffusion towards PAC particles, may modify sorbent utilization. Fundamentally, the adsorption isotherm should remain the same regardless of the arrangement of PAC particles relative to each other, their location in the anthracite layer and flow turbulence. However, research does not confirm such formalism.  
Based on the data (Table 2), a relationship for the similarity invariant (\(\Gamma^{*}/\Gamma_{0}\)*) was established, as follows:  
\[\left(\frac{\Gamma^{*}}{\Gamma_{0}^{*}}\right)=\left(0.6619+1.2457\cdot\text{Re }^{0.1752}\right)\cdot\left(\frac{\rho_{p,sorb}}{\rho_{0,p,sorb}}\right)^{1.1678} \tag{18}\]  
where  
\(\Gamma_{0}^{*}\)--standard [PERSON]'s coefficient (1/m\({}^{3}\) of sorbent), assumed \(\Gamma_{0}^{*}\) = 1 (1/m\({}^{3}\) of sorbent);  
\(d_{50}\)--average diameter of bed particle (m), \(d_{50}\) = 0.00095 m;  
Re--Equation (15);  
\(R_{in}\)--Equation (17);  
Others as above.  
Equation (18) remains valid for the sorbent within the range Re \(\in\)\(\langle 0.827;4.579\rangle\) and \(\rho_{p,sorb}\)\(\in\)\(\langle 3010.02;\) 12,575.2\(\rangle\) g PAC/m\({}^{3}\) of bed.  
Due to the multidimensional nature of Equation (18), the symbols \(\Circle\) and \(\blacksquare\) in Figures 14 and 15 represent a pair of points corresponding to each other at a given Re or apparent density \(\rho_{p,sorb}\). If Equation (18) was a perfect description of the measurement data, and such data were accurate, the signs would coincide. Equation (18) provides a very good description of the measured values with a high correlation coefficient R = 0.9450 (R\({}^{2}\) = 0.9024) (Figure 16).  
Figure 13: Model (13) fit—theoretical _Nu_ vs. measured _Nu_.  
For the coefficient \(\Gamma\) calculated based on \(\Gamma_{0}^{*}\) from Equation (5), the following relation was observed:  
\[\left(\frac{\Gamma}{\Gamma_{0}}\right)=\left(0.000270+0.000662\cdot\text{Re}^{0.14 1957}\right)\cdot\left(\frac{\rho_{p,sorb}}{\rho_{0,p,sorb}}\right)^{0.146392} \tag{19}\]  
where \(\Gamma_{0}\) is the standard Henry coefficient (m\({}^{3}\) of water/g PAC), assumed as \(\Gamma_{0}=1\) (m\({}^{3}\) of water/g PAC) (others as above).  
Figure 16: Model (18) fit—theoretical \(\Gamma^{*}\) vs. \(\Gamma^{*}\) from the model.  
Figure 14: Model (18) fit—theoretical \(\Gamma^{*}\) vs. \(\Gamma^{*}\) from the model, as a function of Re.  
Equation (19) remains valid for the sorbent within the range \(\text{Re}\in\langle 0.827;4.579\rangle\) and \(\rho_{p,sorb}\in\langle 3010.02;12,575.2\rangle\) g PAC/m\({}^{3}\) of bed.  
Equations (18) and (19) describe how the Re number and the similarity module for sorbent apparent density impact both of the Henry constants \(\Gamma^{*}\) and \(\Gamma\) for a particular sorbent. Equations (18) and (19) are not used to determine Henry's coefficients--the coefficients for different sorbents must be determined in separate tests.