# State Analysis of the Water Quality in Rivers in Consideration of Diffusion Phenomenon
## 3 Results and Discussion
Further consideration of the effect of diffusion, on the self-cleaning process, is to consider the situation when the inflowing water on the river banks is clean. This causes pollutants to be \"entrained\" by the river and to decompose along its entire length. Taking into account the phenomenon of diffusion at different river speeds, the spread of pollutants along the same stretch of river is different. The results of this process model, shown in Figure 5 and Table 2, summarise the values used in the experiments. At the initial moment, the value of the pollution was relatively high, but as time and length change, the pollution decreases and changes in length and time. Further considerations included ignoring the phenomenon of diffusion, considering different velocities over the same length, and taking into account the self-cleaning process of the river. The simulation results are shown in Figure 6, and the values used for the simulation are summarised in Table 2. A significant effect of the self-cleaning process can be observed, which influences the faster distribution of pollutants over time and length. Figure 4: Distribution of BOD including diffusion at different times \(t\). Figure 5: BOD distribution for \(V=0.005\) (**a**) \(V=0.01\) (**b**) including the self-cleaning process. An interesting situation arises at different river velocities. The velocity of the river causes the emerging pollutant to be \"entrained\" by the river and to decompose along its length (see Figure 7 and Table 3). When diffusion is taken into account, the distribution of BOD is strongly influenced by velocity as shown successively in Figure 8, and the values used for the simulations are summarised in Table 4. A change in velocity results in a change in BOD concentration values, and a higher velocity results in higher BOD concentrations. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Figure** & \(BOD\) [**mg/L**] & \(A\) & \(\mathbf{D}\left[\mathbf{m^{2}/h}\right]\) & \(V\left[\mathbf{m/h}\right]\) & \(dz\left[\mathbf{m}\right]\) & \(dt\left[\mathbf{h}\right]\) \\ \hline Figure 7a & 10 & 0 & 0 & 0.001 & 0.001 & 0.0014 \\ Figure 7b & 10 & 0.2 & 0 & 0.001 & 0.001 & 0.0014 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The initial values adopted in the simulation studies for the cases shown in Figure 7.  
Figure 6: BOD distribution for \(V=0.005\)**(a)** excluding the self-cleaning process and (b) including the self-cleaning process. Figure 7: BOD distribution excluding the self-cleaning process (**a**) and including the self-cleaning process (**b**). As the velocity increases over the same length of time, the value of the BOD concentration increases. Changes in the BOD concentration at different velocities in the initial time moments are shown in Figure 9.  
An interesting phenomenon occurs when diffusion is not considered, as shown in Figure 10 (cf. Figure 8a). Once a pollutant is introduced into the water, the lack of diffusion results in a prolonged period of dangerous conditions in the river, only after the water masses have moved over a certain length and time does the concentration of the pollutant degrade. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Value** & **Figure 8a** & **Figure 8b** & **Figure 8c** \\ \hline \(BOD\left[\text{mg/L}\right]\) & 10 & 10 & 10 \\ \(A\) & 0 & 0 & 0 \\ \(D\left[\text{m}^{2}/\text{h}\right]\) & 3600 \(\times\) 5 \(\times\) 10\({}^{-9}\) & 3600 \(\times\) 5 \(\times\) 10\({}^{-9}\) & 3600 \(\times\) 5 \(\times\) 10\({}^{-9}\) \\ \(V\left[\text{m}/\text{h}\right]\) & 0.001 & 0.005 & 0.01 \\ \(dz\left[\text{m}\right]\) & 0.001 & 0.001 & 0.001 \\ \(dt\left[\text{h}\right]\) & 0.0014 & 0.0014 & 0.0014 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The initial values adopted in the simulation studies for the cases shown in Figure 8.  
Figure 8: BOD distribution including diffusion for \(V=0.001\) (**a**) \(V=0.005\) (**b**) \(V=0.01\) (**c**). Figure 9: BOD distribution including diffusion at different velocities in the same time moments.