## 2 Data, Methods and Techniques
### Model of Adsorption in the Filter with PAC  
The adsorbate transport model in the adsorption column can be described by the advection and diffusion equation, taking into account the adsorption rate [15; 16; 17; 20; 21; 22; 23]. The model [15; 16; 17] was used in the following form:  
\[\frac{\partial C}{\partial t}\varepsilon+V_{x}\frac{\partial C}{\partial x}= D_{x}\frac{\partial^{2}C}{\partial x^{2}}-k\cdot a_{m}\cdot\rho_{p,sorb}\cdot(C-C_{i}) \tag{2}\]  
where  
\(C\)--adsorbate concentration (g/m\({}^{3}\) of water) for absorbance (1/m);  
\(C_{i}\)--equilibrium concentration (g/m\({}^{3}\) of water) for absorbance (1/m);  
\(k\)--mass transfer rate constant across a water-sorbent interface (m/s);  
\(a_{m}\)--specific outer surface of sorbent particles (m\({}^{2}\)/g of PAC);  
\(V_{x}\)--empty bed (column) velocity (m/s);  
\(D_{x}\)--longitudinal dispersion coefficient in an empty bed (m\({}^{2}\)/s);  
\(\varepsilon\)--filter bed porosity (m\({}^{3}\) of pores/m\({}^{3}\) of bed);  
\(\rho_{p,sorb}\)--apparent density of sorbent fed to a column (g PAC/m\({}^{3}\) of bed), \(\rho_{p,sorb}=m_{sorb}/V\);  
\(m_{sorb}\)--mass of sorbent fed to a filter column (g of sorbent);  
\(V\)--volume of an adsorption column (anthracite + PAC) (m\({}^{3}\) of bed), sum of the volume of anthracite with PAC (m\({}^{3}\) of sorbent) and the volume of pores (m\({}^{3}\) of pores) is the volume ofthe bed (m\({}^{3}\) of bed); _note: the volume of pores (m\({}^{3}\) of pores) is equivalent to volume of water (m\({}^{3}\) of water); it means that: 1 m\({}^{3}\) of pores = 1 m\({}^{3}\) of water; \(t\)_--time (s); \(x\)_--linear coordinate (m). Adsorption capacity \(a^{*}\) in a fixed adsorption layer is described by the equation:  
\[\frac{\partial a^{*}}{\partial t}(1-\varepsilon)=k\cdot a_{m}\cdot\rho_{p, sorb}\cdot(C-C_{i}) \tag{3}\]  
where  
\(a^{*}\)--adsorption capacity (g/m\({}^{3}\) of sorbent) for absorbance ((1/m)/m\({}^{3}\) of sorbent), \(a^{*}=a\cdot\rho_{p,sorb}/(1-\varepsilon)\); \(a\)--adsorption capacity (g/g of sorbent) for absorbance ((1/m)/g of sorbent); \((1-\varepsilon)\)--share of a anthracite + sorbent volume (i.e., sorbent) in the bed volume (m\({}^{3}\) of sorbent/m\({}^{3}\) of bed) (others as above). To integrate Equations (2) and (3), the adsorption isotherm has to be identified; it may be the Henry isotherm for low adsorbate concentrations:  
\[a^{*}=\Gamma^{*}C_{i} \tag{4}\]  
where  
\(\Gamma^{*}\)--[PERSON]'s constant (m\({}^{3}\) of water/m\({}^{3}\) of sorbent) for absorbance (1/m\({}^{3}\) of sorbent)  
and  
\[\Gamma^{*}=\Gamma\cdot\rho_{p,sorb}/(1-\varepsilon) \tag{5}\]  
\(\Gamma\)--[PERSON]'s constant (m\({}^{3}\) of water/g PAC) for absorbance (1/g PAC) (others as above). Equation (5) enables us to accommodate the Henry coefficient expressed in different units. Different isotherms can be used in the calculations. If the Langmuir isotherm is used as  
\[a^{*}=\frac{a^{*}_{\max}C_{i}}{b+C_{i}} \tag{6}\]  
where  
\(a^{*}_{\max}\)--maximum adsorption capacity (asymptote of the Langmuir isotherm) (g/m\({}^{3}\) of sorbent) for absorbance ((1/m)/m\({}^{3}\) of sorbent);  
\(b\)--Langmuir isotherm's constant (g/m\({}^{3}\) of water) for absorbance (1/m) (others as above). Then, the maximum adsorption capacity can be expressed in other units, using the formula  
\[a^{*}_{\max}=a_{\max}\frac{\rho_{p,sorb}}{(1-\varepsilon)} \tag{7}\]  
where \(a_{\max}\) is the maximum adsorption capacity (asymptote of the Langmuir isotherm) (g/g PAC) for absorbance (1/g PAC). Equations (2) and (3) were solved numerically together with the isotherm (4). The following conditions were assumed:  
* Initial: for \(t=0\), \(x\in(0,\) L), \(C(x,t)=0\), \(a^{*}(x,t)=0\), L--length of the bed--3.1 m
* Boundary: for \(t\geq 0\), \(x=0\), \(C(0,t)=C_{0}\), \(a^{*}(0,t)=0\)  
Velocity \(V_{x}\) changed throughout the column. In both the anthracite and sand layer, the interparticle velocity \(V_{x}/\varepsilon\) changed as a result of changes in layer porosity.