Adsorption on Powdered Activated Carbon (PAC) Dosed into an Anthracite-Sand Filter in Water Treatment—Model and Criterion Equations

Abstract

This paper presents research on the mass dispersion and adsorption of organics present in tap water on powdered activated carbon (PAC) in a two-layer filter column. The adsorption rate depends on the difference between the concentration of organics and the equilibrium concentration. In homogeneous flocculators with simultaneous adsorption on PAC, the concentration difference is lower than in a filter column with PAC. Therefore, the utilization of the PAC’s adsorption capacity in filters is higher than in homogeneous flocculators. PAC is introduced into the upper anthracite layer of a filter bed, while the bottom layer is a sand layer, which protects the underdrain system from becoming clogged with PAC particles. The sorbent wis introduced into the bed in the final phase of filter backwashing. The authors present a model of adsorption on PAC in a filter column. Both experiments and calculations confirmed a better utilization of PAC’s adsorption capacity in the filter column compared to its utilization in a homogeneous flocculator. Three criterion equations were developed using dimensionless numbers, Re, Pe and Nu, as well as two similarity moduli related to a sorbent apparent density and an adsorption coefficient. Additionally, a relationship between the Peclet number (Pe) and the Reynolds number (Re) as well as the similarity modulus for the sorbent apparent density were determined for the mass dispersion process. The relationship between the diffusive Nuselt number (Nu) and the Re number as well as the similarity modulus for the sorbent apparent density were determined for the parameter describing an adsorbate permeation rate across a water–sorbent interface. The impact of the Re number and the similarity modulus for the sorbent apparent density on the Henry constant was also investigated. The criterion equations can be used to determine the adsorption model parameters; they may be helpful in designing a filtration system supplemented with PAC. In the capillary velocity range $V_x^{*}$ ∈ ⟨0.15·10⁻²; 0.72·10⁻²⟩ m/s and with a change in the apparent density of the sorbent ${\rho }_{p,sorb}$ from 3000 to 12,000 g PAC/m³ of the bed, as a result of the experimental tests carried out, it was established that the actual coefficient of longitudinal dispersion $D_x^{*}$ varied in the range of 0.16·10⁻⁴ to 2.03·10⁻⁴ m²/s, the product of the constant mass transfer rate and the specific outer surface of sorbent $k\cdot a_m$ varied in the range of 2.23·10⁻⁷ to 1.70·10⁻⁶ (m/s)·(m²/g PAC), while the Henry constant Γ* varied in the range of 7.24 to 44.20 1/m³ of sorbent and the Henry constant Γ varied in the range of 0.0012 to 0.0019 m³ of water/g PAC.

1. Introduction

Natural waters, used as a source for the public water supply, may contain organic chemicals that deteriorate water quality and may serve as possible precursors of oxidation by-products during the water treatment process.

The conventional water treatment process, comprising coagulation, filtration, disinfection and oxidation, may not be sufficient to satisfactorily remove organic substances. Therefore, adsorption on granulated activated carbon (GAC) or powdered carbon (PAC) is used. Granulated carbon is used as a filling material for adsorption columns, while powdered carbon is mainly added to the treated water during coagulation.

Usually, PAC is dosed into the technological system together with a coagulant [1,2], and there are numerous studies that investigate the efficiency of removal of natural organic substances and THM precursors using PAC [2,3,4,5,6,7,8,9].

PAC doses ranging from several to a dozen grams per cubic meter of water have been used to improve the settling of coagulant flocs, while doses reaching several dozen or even several hundred grams per cubic meter of treated water are required to observe satisfactory sorption effects.

This work looked at adsorption in a filter bed with PAC dosed into its top layer. A mathematical description of the adsorption effects requires the use of appropriate models of mass transport with simultaneous adsorption.

In the case of batch reactors, first- and second-order models have been used in which the rate of change of the adsorbate is proportional to the difference between adsorption at the equilibrium state and the current adsorption [10,11,12,13]. The second-order model provides a weak theoretical justification, while the first-order model is used by many authors as a universal one, regardless of the nature of adsorption (related to the adsorption isotherm). A detailed analysis of this model [14,15,16,17] shows that the first-order model refers to adsorption described by the linear Henry isotherm. Moreover, the rate constant in the first-order model provides a variety of information, such as the specific outer surface of sorbent particles, the sorbent dose, the water/other fluid density, the system density, the mass share of water in a water–sorbent (fluid–sorbent) system and the Henry constant [14,15,16,17]. The adsorption model in a batch reactor [14,15,16,17] can also be used when a fragment of any adsorption isotherm is approximated by a straight line. When a mixture of natural substances is adsorbed on a sorbent with a variety of capillary sizes and properties of active centres, good modelling results are obtained from a linear combination of solutions found for the adsorption of one substance on a sorbent with a small variability of its characteristic features [14].

Adsorption studies on PAC are usually carried out in batch reactors [12,13]. This type of research is difficult to transfer to technological systems operating continuously with variable flow. Kinetic constants determined for models describing adsorption in batch reactors are not useful when describing adsorption with variable flow turbulence through technological systems.

To describe the adsorption effects in homogeneous flow reactors, general mass balance models with an adsorption rate component [17] from the Whitman layer model are used. Advection and diffusion models with the adsorption rate component are used to describe adsorption in dispersive flow reactors [11,15,16,17,18].

With sufficiently slow flows through the filter bed and sufficiently fast adsorption, in some models it is assumed that the adsorbate in the liquid phase is in equilibrium with the adsorbate in the adsorbent [19]. In such a case, in the adsorbate transport equation for the liquid phase, adsorption is represented by the derivative of the adsorption amount with respect to time. The rate of change in the amount of adsorbed substance in the solid phase is also described by the difference between the adsorption rate depending on the current concentration of the adsorbate in the liquid phase and the desorption rate depending on the adsorption amount [19].

There are also known models that result from approximate analytical solutions of adsorption dynamics [20]. The Thomas model was developed for plug flow and the Langmuir isotherm, assuming that external and internal diffusion can be neglected [20]. The second model cited in [20] is the Clark model. It was developed for the Freundlich isotherm and contains parameters correcting the lack of symmetry of the bed breakthrough curve in relation to the point corresponding to half the adsorbate concentration at the entrance to the adsorption bed. Both models cannot be used in unsteady-state conditions. Similar semi-empirical models developed for the Henry, Langmuir and rectangular isotherms are cited in [21]. These are the Zhukhowicki, Zabiezinski and Tikhonov models. The model developed by Yoon and Nelson assumes that the rate of decline in the adsorption probability is proportional to the probability of adsorption of the adsorbate and breakthrough of the adsorbate on the adsorbent [20,22]. A model of a similar class is the Klinkenberg model, enabling the determination of the bed breakthrough curve [23]. This is a model obtained from solving the advective mass transport equation and the Whitman model for the adsorption rate. The adsorption equilibrium is described by Henry’s isotherm.

Analytical models make it possible to determine the breakthrough curve for an adsorption column operating in steady states, i.e., at a constant flow and constant concentration of adsorbate at the entrance to the column.

Universal models, enabling simulation of the operation of an adsorption column in unsteady-state conditions, require the solution of a system of differential equations describing the advective–diffusive transport of adsorbate in a liquid and the adsorption rate on the sorbent at the appropriate adsorption isotherm. This system of equations does not have an exact analytical solution and can only be integrated numerically. Such models enable the determination of adsorbate concentrations in the adsorption reactor [15,16,17], inside the capillaries of the sorbent [24,25] or inside the adsorption layer [26,27].

This article presents a model enabling calculations in transient states for a filter bed with advective–dispersive flow through two layers: anthracite with PAC and sand. This model takes into account the kinetics of adsorption onto PAC. A computer program was written to simulate the course of adsorption in the filter bed and determine the parameters of the adsorption model. The additionally proposed criterion equations enable the prediction of the dispersion coefficient and the product of the mass transfer rate constant and the specific outer surface of sorbent for various technological solutions of the PAC filter bed.

2. Data, Methods and Techniques

2.1. Filter Supplemented with PAC

The filter bed designed for adsorption on powdered activated carbon (PAC) may consist of two or three-layers. PAC, as an aqueous suspension, is dosed by a peristaltic pump into the top layer of a two-layer bed or the top/middle layer of a three-layer bed. PAC is introduced into the filter after backwashing while this layer is still in a fluidized state; the dosing is continued until traces of carbon start to appear above the layer. Once the appropriate dose of PAC suspension has been introduced, the backwash flow is turned off and PAC stays deposited inside the layer. The lower one/two layers of the filter protect the installation against carbon particle wash-out. Once the sorbent adsorption capacity is exhausted, PAC is removed from the filter during backwashing.

In the study, a column with an internal diameter of 3 cm was used (Figure 1). It comprised four segments, each 0.5 m long, joined together with flanges and screws. The filter was a dual media filter (total height of 1.1 m) with an upper anthracite layer 0.5 m in depth (Figure 2) and a lower supporting layer of sand and gravel 0.6 m in depth (Figure 3); the depth of each layer was measured at the end of each experiment. A tube with an external diameter of 0.6 cm was permanently inserted into the bed (Figure 1) to supply the PAC suspension into the fluidized anthracite layer. Similar experiments have been reported using a filter filled with pumice particles [28,29]. The authors used PAC collected at the Dłubnia Water Treatment Plant in Kraków (Poland), a product of AquaSorb C, Jacobi Carboons GmbH, widely used as a sorbent in both industrial and municipal applications.

The PAC’s main parameters are as follows:

• Methylene number: min. 20 mL,
• Iodine number: min. 850 mg/g
• Moisture: max. 8–10%
• Surface area: min. 800 m²/g

In the filter, the sorbent stays in contact with organic substances at concentrations comparable to the feed concentrations for a long time (several or several dozen hours, depending on the filter cycle). In homogeneous flow units, i.e., units with a plug or dispersed flow (e.g., flocculators), concentrations of the adsorbate in contact with PAC are lower than in the filter. Therefore, the adsorption rate in the drive modules for the filter will be higher than that for the other units [14,15,16,17]. A higher adsorption rate in drive modules guarantees better utilization of the PAC’s adsorption capacity. Higher sorbent doses to flocculators would be required to observe filter adsorption effects similar to those of PAC [14,15,16,17].

In additional tests on a semi-technical scale, two columns with a diameter of d = 0.150 m were used. The columns were filled from the bottom with a layer of gravel 0.15 m in depth. The next layer was sand with a diameter of 0.3–1.02 mm, diameter d₁₀ = 0.61 mm, with a coefficient of grain unevenness d₆₀/d₁₀ = 1.3 and a depth of 0.5 m. Above the sand, there was a layer of anthracite with a particle diameter of 0.75–1.5 mm, a diameter of d₁₀ = 0.73 mm, with a coefficient of grain unevenness d₆₀/d₁₀ = 1.37 and a depth of 0.5 m. The filter bed used in the laboratory tests also had the same technical parameters. Water from the settling tank at the Dłubnia Water Treatment Plant in Kraków (Poland) was pumped into the filter beds. Water from the Dłubnia River was initially purified in the coagulation process using the PAX−16 coagulant, a product of Kemipol, Police, Poland.

2.2. Experiments

Tap water, produced by the Cracow Waterworks, with natural organic substances (at a temperature of 293 K) was dosed into the filter at the rates of 0.8, 1.0, 1.5 and 2.2 mL/s; water was stored in the 200 dm³ tank. The organic content in water samples was measured as the absorbance A at a wavelength of 254 nm in a 5 cm long quartz cuvette using a SPECORD 210 PLUS spectrophotometer, Analytik Jena, Jena, Germany.

Ultraviolet absorbance is proportional to the concentration of organic substances C in water [14,15,16,17]. Since the concentration C can be linked to absorbance A, the units g/m³ can be replaced with 1/m in the authors’ mathematical models. Depending on the units adopted, model parameters and changes in concentration C or absorbance A over time may have different dimensions. The transition from one dimension to another requires substituting g/m³ for 1/m, or vice versa. UV₂₅₄ absorbance characterizes the content of natural matter containing mainly aromatic groups [30]. These are substances that are precursors of trihalomethanes (THM). Removing these substances from water is important in limiting the formation of THMs that are hazardous to health in the water purification process.

The PAC doses into the anthracite layer were as follows: 1, 2, 3 and 4 g. The apparent PAC density per unit volume of the anthracite layer changed following the current depth of the anthracite layer; also, the depth changed slightly due to the amount of dosed sorbent and arrangement of anthracite particles.

Once the adsorption experiments were completed, a tracer (1 M NaCl) was pulsed into the filter column. The average longitudinal dispersion coefficient for the entire column was calculated from a specific conductivity S in the filter effluent. At low electrolyte concentrations, conductivity is proportional to electrolyte concentrations.

Finally, the filter bed was backwashed (fluidized) to remove the sorbent as well as other undesirable substances and the unit was ready for the next experiment.

2.3. Parameters of the Advection and Diffusion Model

To determine the actual filtration velocity $V_x^{*}$ (capillary filtration velocity) and the actual (capillary) longitudinal dispersion coefficient $D_x^{*}$, a tracer experiment was performed. During the experiment, a tracer (sodium chloride solution) was introduced in a pulse-mode into the filter column. The electrolyte content in the filter effluent was determined through conductivity measurements. The specific conductivity S is proportional to the electrolyte concentration C_e within the analyzed electrolyte concentration range.

For a pulsed tracer mode, there is an analytical solution of the advection–diffusion equation [19]:

$$\frac{S(x,t)}{{\mu }_S}=C_e(x,t)=\frac{M}{Q}\frac{x}{2\sqrt{\pi D_x^{*}{t}^3}}\exp \left(-\frac{{(x-V_x^{*}t)}^2}{4D_{x}t}\right)$$

(1)

where

• $C_e$—electrolyte concentration (val/cm³);
• ${\mu }_S$—equivalent conductivity (μS·cm²/val);
• $S$—specific conductivity (μS/cm);
• $M$—mass of a tracer introduced (val);
• $Q$—flow rate (m³/s);
• $V_x^{*}$—actual filtration velocity (capillary velocity) (m/s), $V_x^{*}=V_x/\epsilon$;
• $V_x$—apparent filtration velocity (empty bed velocity) (m/s);
• $\epsilon$—average porosity of the filter bed;
• $D_x^{*}$—actual coefficient of longitudinal turbulent diffusion (coefficient of longitudinal capillary turbulent diffusion) (m²/s), $D_x^{*}=D_x/\epsilon$;
• $D_x$—longitudinal turbulent diffusion coefficient in an empty bed (m²/s);
• $x$—linear coordinate (height of the bed including the supporting layer and the drain tube x = L = 3.1 m);
• $t$—time (s).

Formula (1) allows us to determine $V_x^{*}$ and $D_x^{*}$ using the least squares method, based on the specific conductivities in the filter effluent.

The parameters $V_x^{*}$ and $D_x^{*}$ were used to calculate adsorption in the filter column.

2.4. 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:

$$\frac{\partial C}{\partial t}\epsilon +V_x\frac{\partial C}{\partial x}=D_x\frac{{\partial }^{2}C}{\partial x^2}-k\cdot a_m\cdot {\rho }_{p,sorb}\cdot \left(C-C_i\right)$$

(2)

where

• $C$—adsorbate concentration (g/m³ of water) for absorbance (1/m);
• $C_i$—equilibrium concentration (g/m³ 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 (m²/g of PAC);
• $V_x$—empty bed (column) velocity (m/s);
• $D_x$—longitudinal dispersion coefficient in an empty bed (m²/s);
• $\epsilon$—filter bed porosity (m³ of pores/m³ of bed);
• ρ_p,sorb—apparent density of sorbent fed to a column (g PAC/m³ 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) (m³ of bed), sum of the volume of anthracite with PAC (m³ of sorbent) and the volume of pores (m³ of pores) is the volume of the bed (m³ of bed); note: the volume of pores (m³ of pores) is equivalent to volume of water (m³ of water); it means that: 1 m³ of pores = 1 m³ of water;
• $t$—time (s);
• $x$—linear coordinate (m).

Adsorption capacity a* in a fixed adsorption layer is described by the equation:

$$\frac{\partial a^{*}}{\partial t}\left(1-\epsilon \right)=k\cdot a_m\cdot {\rho }_{p,sorb}\cdot \left(C-C_i\right)$$

(3)

where

• a*—adsorption capacity (g/m³ of sorbent) for absorbance ((1/m)/m³ of sorbent), $a^{*}=a\cdot {\rho }_{p,sorb}/\left(1-\epsilon \right)$;
• a—adsorption capacity (g/g of sorbent) for absorbance ((1/m)/g of sorbent);
• $\left(1-\epsilon \right)$—share of a anthracite + sorbent volume (i.e., sorbent) in the bed volume (m³ of sorbent/m³ 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:

$$a^{*}={\mathsf{\Gamma }}^{*}{C}_i$$

(4)

where

• Γ*—Henry’s constant (m³ of water/m³ of sorbent) for absorbance (1/m³ of sorbent)

and

$${\mathsf{\Gamma }}^{*}=\mathsf{\Gamma }\cdot {\rho }_{p,sorb}/\left(1-\epsilon \right)$$

(5)

• Γ—Henry’s constant (m³ 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

$$a^{*}=\frac{a_{\max }^{*}{C}_i}{b+C_i}$$

(6)

where

• $a_{\max }^{*}$—maximum adsorption capacity (asymptote of the Langmuir isotherm) (g/m³ of sorbent) for absorbance ((1/m)/m³ of sorbent);
  b—Langmuir isotherm’s constant (g/m³ of water) for absorbance (1/m) (others as above).

Then, the maximum adsorption capacity can be expressed in other units, using the formula

$$a_{\max }^{*}=a_{\max }\frac{{\rho }_{p,sorb}}{\left(1-\epsilon \right)}$$

(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 ∈ (0, L〉, C(x, t) = 0, a*(x, t) = 0, L—length of the bed—3.1 m

-

Boundary: for t ≥ 0, x = 0, C(0, t) = C₀, a*(0, t) = 0

Velocity V_x changed throughout the column. In both the anthracite and sand layer, the interparticle velocity V_x/ε changed as a result of changes in layer porosity. In the drain tube, the velocity V_x = Q/Ap (Ap—internal cross-sectional area of the drain tube 0.35 cm², internal tube diameter 0.668 cm, tube length 200 cm).

The integration results were compared to the actual measurements and the parameters C₀, $k\cdot a_m$ and Γ were determined using the least squares method. The influent concentration C₀ was estimated as an asymptotic value, after a PAC adsorption capacity was exhausted. Such an approach seems more accurate than a single absorbance measurement in a sample taken from the 200 dm³ feed water tank. In the experiments, the initial absorbance A₀ was determined instead of the concentration.

3. Results

3.1. 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 $\frac{M}{Q}\frac{L}{2\sqrt{\pi D_x^{*}}}$;

-

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. Tracer test results for two columns.

No.	D (cm)	Q (mL/s)	ρp,sorb (g/m3 of Bed)	${\mathit{V}}_{\mathit{x}}^{*}$ (10−2 m/s)	${\mathit{D}}_{\mathit{x}}^{*}$ (10−4 m2/s)	Pe	Re
1	15	1.08	0	0.24	0.2574	0.05284	1.36
2	15	1.13	0	0.27	0.3164	0.04878	1.54
3	15	1.35	0	0.32	0.5020	0.03685	1.85
4	15	1.45	0	0.35	0.4928	0.04086	2.01
5	15	1.93	0	0.47	0.9634	0.02794	2.69
6	3	2.20	3143.8	0.69	1.7250	0.02299	3.97
7	3	2.20	6287.6	0.72	2.7859	0.01488	4.15
8	3	2.20	9431.4	0.71	3.3023	0.01227	4.05
9	3	2.20	12,575.2	0.72	2.0315	0.02039	4.14
10	3	1.50	3010.0	0.47	0.5626	0.04797	2.70
11	3	1.50	6020.0	0.50	1.3640	0.02116	2.89
12	3	1.50	9030.1	0.42	0.5394	0.04414	2.38
13	3	1.50	12,040.1	0.46	0.6536	0.04004	2.62
14	3	1.00	3042.4	0.26	0.4032	0.03708	1.49
15	3	1.00	6084.8	0.25	0.4682	0.03093	1.45
16	3	1.00	8841.9	0.23	0.6161	0.02115	1.30
17	3	1.00	12,040.1	0.21	0.3976	0.03013	1.20
18	3	0.80	3109.3	0.15	0.1605	0.05324	0.85
19	3	0.80	6218.5	0.24	0.2574	0.05318	0.98
20	3	0.80	9327.8	0.27	0.3164	0.04157	0.79
21	3	0.80	12,437.0	0.32	0.5020	0.05659	0.75

.

Both columns were filled with the same material. The characteristic D-values for anthracite particles {d₁₀, d₅₀, d₉₀} 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·0.95 mm + 0.5 m·0.77 mm)/(0.5 m + 0.5 m) = 0.86 mm. The relationship between the Peclet number (Pe), the Reynolds number (Re) and the apparent sorbent density (ρ_p,sorb) was established as:

$$Pe=\left(0.007975+0.058937\cdot {\mathrm{Re}}^{-0.886653}\right)\cdot {\left(1+\frac{{\rho }_{p,sorb}}{{\rho }_{0,p,sorb}}\right)}^{-0.249469}$$

(8)

$$Pe=\frac{V_x^{*}\cdot 4\cdot R_h}{D_x^{*}}$$

(9)

$$\mathrm{Re}=\frac{V_x^{*}\cdot 4\cdot R_h}{{\nu }_w}$$

(10)

$$N=\frac{\frac{\pi D^2}{4}h}{\frac{4}{3}\pi {\left(\frac{d_{z,50}}{2}\right)}^3}$$

(11)

$$R_h=\frac{V_z}{A_z}=\frac{\frac{\pi D^2}{4}h}{4\pi {\left(\frac{d_{z,50}}{2}\right)}^{2}N}=\frac{d_{z,50}}{6}$$

(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 (m³);
• A_z—wetted surface area of bed particles (m²);
• d_z,50—average diameter of the particle (m), d_z,50 = 0.00086 m;
• ρ_p,sorb—apparent sorbent density (g PAC/m³ of bed);
• ρ_0,p,sorb—standard apparent sorbent density (g PAC/m³ of bed), assumed ρ_0,p,sorb = 1000 g PAC/m³ of the bed;
• ν_w—kinematic water viscosity (m²/s), assumed as ν_w = 10⁻⁶ m²/s.

Equation (8) is valid within the following range: Re ∈ 〈0.75; 4.15〉 and ρ_p,sorb ∈ 〈0; 12,575.2〉 g PAC/m³ of the bed.

Due to the multidimensional nature of Equation (8), the symbols ◯ and ∎ in Figure 4 and Figure 5 represent a pair of points corresponding to each other at a given Re or apparent density ρ_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 (R² = 0.9786) (Figure 6).

3.2. Adsorption in an Anthracite-Sand Filter with PAC

The adsorption of organic substances present in the water fed into the filter with PAC proceeded at different rates, depending on sorbent doses and flow rates. Changes in absorbance A over time for PAC doses of 1, 2, 3 and 4 g and flows of 0.8, 1.0, 1.5 and 2.2 mL/s are shown in Figure 7, Figure 8, Figure 9 and Figure 10. The solid line in these Figures represents the calculation results obtained using model (2), (3), (4). The adsorption capacity of PAC, at a dose of 1 g, becomes exhausted much faster than at higher doses. If the absorbance A₀ in the feed is higher at a higher PAC dose than at a lower one, then the curves intersect (Figure 8, doses: 3, 4 g) or the curve for a 4 g dose lies above the curve for a 3 g dose (Figure 9).

Data for the adsorption models (2), (3) and (4) and the calculated model parameters are summarized in

Table 2. Data for the adsorption model (2), (3), (4) and the calculated model parameters Γ*, k·a_m (numerical values refer to the anthracite layer).

No.	Q (mL/s)	Γ* (1/m3 Sorbent)	PAC Dose (g)	ρp,sorb (g PAC/m3 of Bed)	ε (m3 of Pores/m3 of Bed)	Vx (m/s)	k·am (m/s)∗(m2/g PAC)	Nu	Re
1	2.2	5.58	1	3143.80	0.47	3.251·10−3	4.123·10−7	2.489	4.380
2	2.2	12.00	2	6287.60	0.465	3.362·10−3	6.754·10−7	8.243	4.579
3	2.2	43.78	3	9431.40	0.46	3.251·10−3	9.591·10−7	17.747	4.476
4	2.2	44.20	4	12,575.21	0.45	3.251·10−3	1.703·10−6	42.938	4.575
5	1.5	13.05	1	3010.02	0.47	2.212·10−3	2.597·10−7	1.501	2.981
6	1.5	8.65	2	6020.05	0.465	2.341·10−3	7.457·10−7	8.713	3.188
7	1.5	27.92	3	9030.07	0.46	1.910·10−3	1.092·10−6	19.341	2.630
8	1.5	33.15	4	12,040.09	0.45	2.054·10−3	6.598·10−7	15.932	2.891
9	1	9.03	1	3042.39	0.47	1.226·10−3	6.107·10−7	3.567	1.651
10	1	16.24	2	6084.78	0.465	1.175·10−3	5.848·10−7	6.907	1.600
11	1	34.95	3	8841.94	0.46	1.045·10−3	1.713·10−6	29.722	1.439
12	1	36.94	4	12,040.09	0.45	9.403·10−4	7.569·10−7	18.277	1.323
13	0.8	7.24	1	3109.25	0.47	7.004·10−4	2.229·10−7	1.331	0.944
14	0.8	18.55	2	6218.51	0.465	7.959·10−4	6.234·10−7	7.524	1.084
15	0.8	25.47	3	9327.76	0.46	6.310·10−4	4.770·10−7	8.730	0.869
16	0.8	32.89	4	12,437.02	0.45	5.876·10−4	5.774·10−7	14.402	0.827

.

The experimental results made it possible to establish a relationship between the diffusive Nusselt number (Nu) (equivalent to the Sherwood number Sh), the Re number and the apparent sorbent density (ρ_p,sorb) as:

$$Nu=0.1587\cdot {\mathrm{Re}}^{0.6244}\cdot {\left(\frac{{\rho }_{p,sorb}}{{\rho }_{0,p,sorb}}\right)}^{1.8370}$$

(13)

$$Nu=\frac{\frac{k\cdot a_m}{\epsilon }\cdot {\rho }_{p,sorb}\cdot {\left(d_{50}\right)}^2}{D_m}$$

(14)

$$\mathrm{Re}=\frac{\frac{V_x}{\epsilon }\cdot 4\cdot R_h}{{\nu }_w}=\frac{V_x^{*}\cdot 4\cdot R_h}{{\nu }_w}$$

(15)

$$N=\frac{\frac{\pi D^2}{4}h-V_p}{\frac{4}{3}\pi {\left(\frac{d_{50}}{2}\right)}^3}$$

(16)

$$R_h=\frac{V_z}{A_z}=\frac{\frac{\pi D^2}{4}h-V_p}{4\pi {\left(\frac{d_{50}}{2}\right)}^{2}N}=\frac{d_{50}}{6}$$

(17)

where:

• k—mass transfer rate constant across a water–sorbent interface (m/s);
• a_m—specific outer surface of sorbent particles (m²/g PAC);
• R_h—hydraulic radius (m);
• N—number of particles in a bed;
• D—outer diameter of the filter column (m);
• D_m—molecular diffusion coefficient (m²/s), assumed D_m = 10⁻⁹ m²/s;
• h—height of the bed (m);
• V_z—volume of the bed (m³);
• V_p—volume of the tube dosing the PAC suspension (m³);
• A_z—wetted surface area of the bed particles (m²);
• d₅₀—average diameter of the bed particle (m), d₅₀ = 0.00095 m;
• ρ_p,sorb—apparent sorbent density (g PAC/m³ of bed);
• ρ_0,p,sorb—standard apparent sorbent density (g PAC/m³ of bed), assumed as ρ_0,p,sorb = 1000 g PAC/m³ of the bed
• ν_w—kinematic water viscosity (m²/s), assumed ν_w = 10⁻⁶ m²/s;
• ε—porosity (m³ of pores/m³ of bed).

Equation (13) is valid within the range Re ∈ 〈0.827; 4.579〉 and ρ_p,sorb ∈ 〈3010.02; 12,575.2〉 g PAC/m³ of the bed.

Due to the multidimensional nature of Equation (13), the symbols ◯ and ∎ in Figure 11 and Figure 12 represent a pair of points corresponding to each other at a given Re or apparent density ρ_p,sorb. If Equation (13) was a perfect description of the measurement data, and such data were accurate, the signs would coincide. Equation (13) provides a very good description of the measured values with a high correlation coefficient R = 0.9740 (R² = 0.9486) (Figure 13).

3.3. 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 (Γ*/Γ₀*) was established, as follows:

$$\left(\frac{{\mathsf{\Gamma }}^{*}}{{\mathsf{\Gamma }}_0^{*}}\right)=\left(0.6619+1.2457\cdot {\mathrm{Re}}^{0.1752}\right)\cdot {\left(\frac{{\rho }_{p,sorb}}{{\rho }_{0,p,sorb}}\right)}^{1.1678}$$

(18)

where

• ${\mathsf{\Gamma }}_0^{*}$—standard Henry’s coefficient (1/m³ of sorbent), assumed ${\mathsf{\Gamma }}_0^{*}$ = 1 (1/m³ of sorbent);
• d₅₀—average diameter of bed particle (m), d₅₀ = 0.00095 m;
• Re—Equation (15);
• R_h—Equation (17);
• Others as above.

Equation (18) remains valid for the sorbent within the range Re ∈ 〈0.827; 4.579〉 and ρ_p,sorb ∈ 〈3010.02; 12,575.2〉 g PAC/m³ of bed.

Due to the multidimensional nature of Equation (18), the symbols ◯ and ∎ in Figure 14 and Figure 15 represent a pair of points corresponding to each other at a given Re or apparent density ρ_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 (R² = 0.9024) (Figure 16).

For the coefficient $\mathsf{\Gamma }$ calculated based on ${\mathsf{\Gamma }}_0^{*}$ from Equation (5), the following relation was observed:

$$\left(\frac{\mathsf{\Gamma }}{{\mathsf{\Gamma }}_0}\right)=\left(0.000270+0.000662\cdot {\mathrm{Re}}^{0.141957}\right)\cdot {\left(\frac{{\rho }_{p,sorb}}{{\rho }_{0,p,sorb}}\right)}^{0.146392}$$

(19)

where ${\mathsf{\Gamma }}_0$ is the standard Henry coefficient (m³ of water/g PAC), assumed as ${\mathsf{\Gamma }}_0$ = 1 (m³ of water/g PAC) (others as above).

Equation (19) remains valid for the sorbent within the range Re ∈ 〈0.827; 4.579〉 and ρ_p,sorb ∈ 〈3010.02; 12,575.2〉 g PAC/m³ 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 ${\mathsf{\Gamma }}^{*}$ and $\mathsf{\Gamma }$ 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.

4. Discussion

4.1. Longitudinal Turbulent Diffusion

The Peclet number is inversely proportional to the Re number (r. (8)). For Re numbers in the range Re ∈ 〈0.75; 4.15〉, a higher Re number results in a lower Pe number. It is possible to determine the impact of velocity $V_x^{*}$ on a turbulent diffusion coefficient $D_x^{*}$ at different densities ${\rho }_{p,sorb}$ using Equation (8) (Figure 17). At low velocities, a doubled $V_x^{*}$ increases $D_x^{*}$ by 3.5 times; in a medium velocity range, a doubled $V_x^{*}$ increases $D_x^{*}$ by 3.1 times; and in the high velocity range, a doubled $V_x^{*}$ increases $D_x^{*}$ 3.0 times.

In the velocity range $V_x^{*}$ ∈ 〈0.1·10⁻²; 1.06·10⁻²〉 m/s and when the density changes from 3000 to 12,000 g PAC/m³ of the bed, the coefficient $D_x^{*}$ increases 1.34 times (Figure 17).

4.2. Mass Transfer across the Water–Sorbent Interface

The expression $k\cdot a_m$ stands for a proportionality factor in the model of the mass transfer rate. A doubled velocity $V_x^{*}$, and thus a higher Re number, increases this factor by approximately 1.5 times (Equation (13)); at a four-fold increase in $V_x^{*}$, the factor increases by approximately 2.4 times (Figure 18).

An increase in the apparent sorbent density ${\rho }_{p,sorb}$ from 3000 to 12,000 g PAC/m³ of the bed increases $k\cdot a_m$ by as much as 3.1 times due to the larger available permeation surface as ${\rho }_{p,sorb}$ increases. Some PAC particles are trapped in the interparticle cavities of the filter material (anthracite) and the surface of a water–sorbent interface across which contaminants may penetrate is partially covered by anthracite. At higher PAC doses, the sorbent particles may also cover each other, which reduces the effective water–sorbent interface area available for contaminants.

4.3. Adsorption Equilibrium

In Equation (4), Henry’s coefficient Γ* changed mainly due to changes in the amount of PAC introduced into the anthracite layer (Figure 19). Therefore, changes in the sorbent apparent density ${\rho }_{p,sorb}$ from 3000 to 12,000 g PAC/m³ of the bed may result in a significant increase in Γ* from about 7.8 up to about 39.5 1/m³ of sorbent (about 5.1 times, Equation (18)), while the coefficient Γ, based on Equation (19), changes from about 0.00123 to about 0.00150 m³ water/g PAC (about 1.2 times, Equation (19)). The effect of the Re number (ranging from 0.8 to 4.6) on Γ* is shown in Figure 19. The Re number influences the equilibrium constant and can change Γ* by about 1.23 times (Equation (18)) for ${\rho }_{p,sorb}$ ∈ 〈3000; 12,000〉 g PAC/m³ of the bed and Γ by approx. 1.2 times (Equation (19)). This means that a turbulent diffusion affects the adsorption isotherm coefficient of the sorbent, the particles of which may partially overlap. A turbulent motion may enhance transport of adsorbate into the aggregated particles, increasing the degree of adsorption.

The adhesion of PAC particles to anthracite, the distribution of PAC in anthracite cavities, the mutual overlapping of PAC particles, and the expansion of the effective area of adsorbate permeation to the sorbent all accompany higher PAC doses, changing the sorbent adsorption capacity. Therefore, both the Γ* and Γ coefficients change.

4.4. Influence of Mass Transfer Rate on Adsorption Effects

At the same sorbent apparent density ${\rho }_{p,sorb}$ = 9030.07 g PAC/m³ of the bed but with different values of $k\cdot a_m$ {1.092·10⁻⁶ (Table 2, experiment No. 7), 2.729·10⁻⁰⁵} (m/s)^.(m²/g PAC)}, the shapes of the absorbance distribution curves change over time (Figure 20 and Figure 21). At slower adsorption rates, the change in the rate of absorbance has two strong maxima over time (two inflection points in Figure 20), while in fast processes, this effect is hardly noticeable (Figure 21). The nature of changes observed over time for slow and fast adsorption processes is the same. (Figure 22 and Figure 23); if the Γ* value is 100 times lower, the inflection points disappear. Therefore, the shape of the absorbance distribution curve over time depends on $k\cdot a_m$ and the adsorption coefficient Γ* (Figure 24).

4.5. Efficiency of PAC Utilization

It is worth comparing the adsorption efficiency for the filter with PAC and a homogeneous system, i.e., a flocculation chamber treating water with both coagulant and PAC. The effect of adsorption on PAC in homogenous system can be determined with a known adsorption isotherm and may exceed real-life effects, because flocculation time does not exceed 1 h, while adsorption isotherms are determined for equilibrium states (e.g., after 24 h) or from equations describing the adsorption kinetics in a homogeneous reactor in steady states [15,16,17,19].

The adsorption isotherm (Figure 25) of the analyzed PAC is

$$a=\mathsf{\Gamma }\cdot A=0.00167\frac{m^3}{\mathrm{g}\hspace{0.33em}\mathrm{PAC}}\cdot A$$

(20)

where

a—adsorption capacity for absorbance (1/m)·m³/g PAC:

A—absorbance (1/m);

Γ—Henry’s isotherm coefficient (m³/g PAC).

Table 3. Summary of adsorption parameters in the filter and the homogeneous reactor (D_h—PAC dose in a homogenous reactor; V—volume of treated water).

No.	Q (mL/s)	Γ* (1/m3 of Sorbent)	Γ (m3 of Water/g PAC) Equation (5)	PAC Dose D (g)	ρp,sorb (g PAC/m3 of Bed)	ε (m3 of Pores/m3 of Bed)	Filter Cycle Time t for 0.9·A0 (s)	A0 (1/m)	Aa Average (1/m)	V = Q·t (m3)	PAC Dose ${\mathit{D}}_{\mathit{h}}=\frac{\left({\mathit{A}}_0-{\mathit{A}}_{\mathit{a}}\right)\cdot \mathit{V}}{\mathbf{\Gamma }\cdot {\mathit{A}}_{\mathit{a}}}$ (g)	$\frac{{\mathit{D}}_{\mathit{h}}}{\mathit{D}}$
1	2.2	5.58	0.00094	1	3143.8	0.47	1250	3.47	2.93	0.00275	0.3035	0.3035
2	2.2	12	0.00102	2	6287.6	0.465	2250	3.41	2.7	0.00495	0.7794	0.3897
3	2.2	43.78	0.00251	3	9431.4	0.46	7250	3.36	2.51	0.01595	3.2344	1.0781
4	2.2	44.2	0.00193	4	12,575.21	0.45	7250	3.06	1.98	0.01595	5.2096	1.3024
5	1.5	13.05	0.00230	1	3010.02	0.47	4000	4.6	3.94	0.006	0.6018	0.6018
6	1.5	8.65	0.00077	2	6020.05	0.465	5500	3.79	2.72	0.00825	1.9434	0.9717
7	1.5	27.92	0.00167	3	9030.07	0.46	9000	2.95	1.86	0.0135	4.7373	1.5791
8	1.5	33.15	0.00151	4	12,040.09	0.45	10,500	3.77	2.49	0.01575	4.8481	1.2120
9	1	9.03	0.00157	1	3042.39	0.47	3500	3.73	2.54	0.0035	0.9819	0.9819
10	1	16.24	0.00143	2	6084.78	0.465	9500	3.54	2.47	0.0095	2.4643	1.2322
11	1	34.95	0.00213	3	8841.94	0.46	14,500	2.86	1.54	0.0145	7.4423	2.4808
12	1	36.94	0.00169	4	12,040.09	0.45	20,500	3.13	1.85	0.0205	8.4933	2.1233
13	0.8	7.24	0.00123	1	3109.25	0.47	9500	3.29	2.55	0.0076	1.3207	1.3207
14	0.8	18.55	0.00160	2	6218.51	0.465	15,000	2.93	1.86	0.012	4.1337	2.0668
15	0.8	25.47	0.00147	3	9327.76	0.46	23,000	2.93	1.78	0.0184	7.1183	2.3728
16	0.8	32.89	0.00145	4	12,437.02	0.45	29,500	2.83	1.53	0.0236	12.0074	3.0018

shows the D_h/D ratio, i.e., the ratio of the sorbent doses to the homogeneous reactor D_h vs. the dose D to the filter. For suitable doses D and flows Q, the D_h/D ratios are greater than 1 (gray areas). This means that in the filter, the same effects can be achieved at D < D_h and the sorbent efficiency in the filter is higher than in the homogeneous reactor. In practice, lower doses of sorbent would reduce water treatment costs. In experiment No. 16 (Table 3), the ratio D_h/D was 3.0018, so to obtain the same filtration effect, the dose per 1 m³ of water for the filter was D/V = 169.5 g PAC/m³, while in a homogeneous reactor (flocculator), it was D_h/V = 508.8 g PAC/m³.

It should be noted that even if the effective coefficient Γ for the adsorption column (Table 3, No.: 7, 8, 10 ÷ 16) was smaller or equal to Γ (0.00167 m³/g PAC) for the adsorption isotherm in bulk adsorption, the ratios D_h/D could exceed 1. Due to the mutual packing of the powdered sorbent in the filter column, the average Henry’s coefficient was 0.00158 m³/g PAC and was slightly lower than Γ for bulk adsorption (0.00167 m³/g PAC).

5. Analysis of Flow Resistance during Tests on a Semi-Technical Scale

In additional tests, carried out using filters installed at the Dłubnia Water Treatment Station in Kraków, the resistance of water flow through the filter bed with and without PAC was determined in each filtration cycle at a constant filtration speed of 5.0 m/h in relation to an empty bed. The obtained results are presented in

Table 4. Results of pressure loss measurements in semi-technical scale tests on a filter without PAC and with PAC.

Average Filter Cycle Time (h)	PAC Dose (g PAC/m3 of the Bed)	Filtration Begins		End of Filtration		Increase in Pressure Losses after Filtration		Increase in Pressure Losses after PAC Dosing, Filtration Begins	Increase in Pressure Losses after PAC Dosing, End of Filtration
		Pressure Loss		Pressure Loss		(cm H2O)		(cm H2O)	(cm H2O)
		(cm H2O)		(cm H2O)
		Filter with PAC	Filter without PAC	Filter with PAC	Filter without PAC	Filter with PAC	Filter without PAC
1	2	3	4	5	6	7	8	9	10
21.25	4750	23.7	21.4	38.8	35.3	15.1	13.9	2.3	3.5
	4750	23.7	20.9	41.5	38.3	17.8	17.4	2.8	3.2
	4750	21	22	45.3	44	24.3	22	1	1.3
	4750	22	24.2	48	45.5	26	21.3	2.2	2.5
22	2380	21.8	20.5	44.1	40.9	22.3	20.2	1.3	3.2
	2380	22.2	21.2	45.2	42	23	20.8	1	3.2
	2380	22.4	20.1	49	45	26.6	21.9	2.3	4
	2380	21.8	19.9	43.7	40	21.8	20.1	1.9	3.7
18.5	7130	22.4	19.6	40	37.5	17.6	17.9	2.8	2.5
	7130	21.8	19.5	45.2	40.2	23.3	20.7	2.3	5
	7130	23.9	21.7	33.9	31.9	15	12.2	2.2	2

. The table determines the increase in pressure losses during water flow through the bed after dosing with PAC and without PAC in the initial phase of the filter cycle and after its completion.

Dosing PAC onto the filter bed causes an initial increase in flow resistance at the level of 1.0–2.8 cm H₂O (Table 4, col. 9). After the end of the filter cycle, there is a greater increase in flow resistance on the filter with PAC than on the filter without PAC (Table 4, col. 10). The difference in resistance values ranges from 1.3 to 5.0 cm H₂O (Table 4, col. 10). Based on the results obtained, it can be concluded that dosing PAC onto the filter bed at an amount of 2380 to 7130 g PAC/m³ does not significantly affect the resistance of water flow through the filter bed and the length of the filter cycle.

6. Conclusions

Powdered activated carbon can be used in water treatment during both coagulation and filtration processes. If a homogeneous or batch reactor is used, lower adsorption effects are observed than in a filter bed to which PAC is introduced. The filter with PAC works as an advection–dispersion reactor in which adsorption process proceeds faster than in other types of reactors. In such a reactor, the average difference between the concentration of contaminants in water and the equilibrium concentration is larger than in other types of reactors.

In particular:

• At a low velocity, doubling $V_x^{*}$ increases the capillary turbulent diffusion coefficient $D_x^{*}$ by 3.5 times; in the medium range, doubling $V_x^{*}$ increases $D_x^{*}$ by 3.1 times; and in the high velocity range, such an increase results in an increase in $D_x^{*}$ by 3.0 times.
• In the velocity range $V_x^{*}$ ∈ 〈0.1·10⁻²; 1.06·10⁻²〉 m/s and with the density ${\rho }_{p,sorb}$ changing from 3000 to 12,000 g PAC/m³ of bed, the coefficient $D_x^{*}$ changes by 1.34 times.
• In the capillary velocity range $V_x^{*}$ ∈ 〈0.15·10⁻²; 0.72·10⁻²〉 m/s and with a change in the apparent density of the sorbent ${\rho }_{p,sorb}$ from 3000 to 12,000 g PAC/m³ of the bed, it was established that the actual coefficient of longitudinal dispersion $D_x^{*}$ varies in the range of 0.16·10⁻⁴ to 2.03·10⁻⁴ m²/s.
• When changing the apparent density of the sorbent from 3000 to 12,000 g PAC/m³ of the bed, the $k\cdot a_m$ value changes by 3.2 times; this change is slightly smaller than for the sorbent apparent density.
• In the capillary velocity range $V_x^{*}$ ∈ 〈0.15·10⁻²; 0.72·10⁻²〉 m/s and with a change in the apparent density of the sorbent ${\rho }_{p,sorb}$ from 3000 to 12,000 g PAC/m³ of the bed, the product of the constant mass transfer rate and the specific outer surface of sorbent $k\cdot a_m$ varies in the range of 2.23·10⁻⁷ to 1.70·10⁻⁶ (m/s)·(m²/g PAC).
• Doubling the filtration velocity $V_x^{*}$ increases the $k\cdot a_m$ value by approximately 1.5 times, while quadrupling it increases the $k\cdot a_m$ value approximately by 2.4 times.
• An increase in the apparent density of the sorbent ${\rho }_{p,sorb}$ from 3000 to 12,000 g PAC/m³ of the bed may increase the adsorption coefficient Γ* by about 5.1 times and Γ by about 1.2 times.
• The Re number affects the adsorption equilibrium constant and can cause changes in Γ* by about 1.23 times for ${\rho }_{p,sorb}$ ∈ 〈3000; 12,000〉 g PAC/m³ of bed and changes of Γ by approximately 1.2 times. This means that turbulent diffusion affects the adsorption isotherm coefficient of the powdered sorbent; the particles of sorbent may partially overlap each other. A turbulent motion may promote the transport of adsorbate into the aggregated particles, increasing the adsorption capacity.
• In the capillary velocity range $V_x^{*}$ ∈ 〈0.15·10⁻²; 0.72·10⁻²〉 m/s and with a change in the apparent density of the sorbent ${\rho }_{p,sorb}$ from 3000 to 12,000 g PAC/m³ of the bed, the Henry constant Γ* varies in the range of 7.24 to 44.20 1/m³ of sorbent and the Henry constant Γ vaires in the range of 0.0012 to 0.0019 m³ of water/g PAC.
• Equations comprising dimensionless criterion numbers (Re, Pe, Nu) and similarity moduli can be used to determine the parameters of the adsorption model that facilitate the design of filters with PAC.
• The high organic removal observed in a filter with PAC means that the PAC dose per unit volume of treated water may be several times lower than such a dose in a homogeneous flocculator at the same average effluent adsorbate concentration.
• Microscopic examination showed no presence of the PAC particles in the filtrate coming from this installation.
• The dosing PAC onto the filter bed in the amount of 2380 to 7130 g PAC/m³ does not significantly affect the resistance of water flow through the filter bed and the length of the filter cycle.

The water treatment method, employing adsorption on PAC dosed into the filter column, better utilizes the activated carbon’s sorption capacity. Such a technological solution will reduce treatment costs. Studies carried out on a semi-technical scale [15,17] have shown that the adsorption process in an anthracite bed with PAC is repeatable with sufficient accuracy for technological needs. The relative repeatability error of the product of the constant mass transfer rate and the specific outer surface of sorbent $k\cdot a_m$ is approximately 32.2%, and the Henry constant Γ* is approximately 23.4%. Laboratory and semi-technical tests [15,17] have shown that the same adsorption effects as in a homogeneous flocculator can be achieved in a filtration column with PAC at a two to three times lower sorbent dose. This will help to reduce the costs of removing organic substances from water that worsen the taste and smell of water and are potentially THM precursors. In practice, the doses of PAC used in combination with coagulation are in the order of several dozen to several hundred grams per cubic meter of treated water [1,2,5,9]. Therefore, reducing the PAC dose is economically justified. The described adsorption model and criterion equations can be used when designing technological systems in which, instead of dosing PAC with the coagulant into raw water pre-treatment devices, it can be introduced into the filtration column.