## 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]. Adaption 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.