# State Analysis of the Water Quality in Rivers in Consideration of Diffusion Phenomenon
## 3 Results and Discussion  
The results of simulation studies for models that do not take into account the diffusion phenomenon are described in [5; 6; 7]. The incentive for the experimental part of this paper is the attempt to test, in the form of simulation studies, the extended mathematical model which has been significantly extended in relation to the previously used simplified mathematical models of the river. The adopted experimental strategy includes sets of changing hydrological and biological parameters of the river, such as the velocity of the flow of water masses, the intensity of diffusion phenomena and the biological oxygen demand correlated with them. In this way, a comprehensive overview of the variability of the individual coordinates of the state of the process occurring in the river was created, in order to obtain an objective assessment of the accuracy of the monitoring of the ecological state of the river, which changes under their influence. These results are the starting point for the construction of an intelligent control module for the treatment process, taking diffusion under consideration. In view of the assumptions thus made based on the mathematical model of a polluted river described by Equations (10) and (13), a number of simulation studies were conducted, taking into account that measurements are made along the characteristics of free-flowing water. The mathematical model was verified for a legitimate subject, the Wislok river, covering the area of southeastern Poland. Field research was conducted on the local Wislok river, which is a hydrological object of significant importance for the Podkarpackie Province. The pollution load of the Wislok river depends on the level of development and industrialisation of individual fragments of its riverbed. Four measurement points were selected to reflect the nature of its flow in a specific way. The places where measurements were carried out are: Zarnowa (21.818889 E, 49.875556 N), Lutoryz (21.9093209 E, 49.9666064 N), Rzeszow (22.0115789 E, 50.0086266 N), and Smolarzyny (22.2924051 E, 50.1256537 N). In addition, selected measurement points are associated with the physical realisation of an intelligent monitoring and control system, using data transmission telemetry. Figure 2 illustrates the cross-sections of the riverbed, including the velocities that occur at given river latitudes. The research presented in this paper, on the mathematical model of the river taking into account the diffusion phenomenon, is based on Equation (22) and is carried out only for the BOD coefficient. Consideration of the diffusion phenomenon is addressed, encompassing two spatial coordinates, which are time and the length of the river section. The simulation results are described in a two-dimensional area \(z\), \(t\) for different values of \(D\), \(V\), and \(A\). A first interpretation of the problem is to consider the diffusion coefficient, assuming that the values \(V=0\) and \(A=0\). The results obtained are shown in Figure 3 and Table 1 summarises the initial values used in the experiments. Figure 2: Cross–section of the riverbed and current velocities—Lutoryz (**a**) and Smolarzyny (**b**). Analysing Figure 3, we can observe the appearance of pollution at the beginning of the river section under consideration. The BOD concentration decreases in the long term, which is caused by the spreading of the pollution centre. The consequence is the distribution of the concentration of pollution over a certain time and space. Here, we can see the influence of diffusion, where, after a certain period of observation, a decrease in its concentration and distribution can be seen. At different lengths, at different moments in time, the BOD concentration takes on a correspondingly different value, as demonstrated in Figure 4. Depending on the time moment, at a given river length position, BOD concentration values vary. Respectively, at the moment of, e.g., \(t=t_{1}\), \(t=t_{2}\), \(t=t_{3}\) the BOD value is within the limits of 5, 3.5 and 2.7 [mg/L].