One of the major applications of the dielectrophoresis is selective trapping and fractionation in lab-on-a-chip devices. Nevertheless, many-particles effects due to high concentration of biological material around electrodes can cause a rapid decrease in trapping efficiency in dielectrophoretic devices. In this paper we present an approach based on a drift-diffusion dynamics to study the particles behavior near electrodes surroundings. Within this approach, we introduce many-particles effects by invoking the effective medium approximation, electrodes can saturate losing their capability to attract further particles and thus leading to a more realistic scenario never shown in past literature.