# State Analysis of the Water Quality in Rivers in Consideration of Diffusion Phenomenon
## 2 Materials and Methods  
### Water Quality Indicators  
Water is a fundamental element of any national economy. The analyses of changes in water pollution indicators, namely Biochemical Oxygen Demand (BOD) and Dissolved Oxygen (DO) levels, are interesting and complex scientific problems [17; 18]. Mathematical modelling plays a major role here. Organic and inorganic substances contribute to surface water pollution. The most important components affecting water quality status are Total Organic Carbon (TOC) [19], Chemical Oxygen Demand (COD) and BOD. BOD\({}_{5}\) is a standard five-day test at 20 \({}^{\circ}\)C to determine the assimilable fraction of a mixture of organic substances. Complete oxidation of organic compounds, under the given conditions, occurs within 28 days. The BOD\({}_{k}\) or BOD\({}_{c}\) indicators correspond to the final and total oxygen consumption. Biochemical oxygen demand BOD\({}_{5}\) is a conventional indicator of organic pollution, while BOD is a representation of the decomposition processes of organic compounds in water. The process of BOD changes are assessed by measuring dissolved oxygen consumption. This process is very complex and depends on many factors including: oxygen content, type of organic compounds existing in the water, microorganisms and environmental factors [20; 21; 22; 23]. The above processes can be described by a first-order differential equation. It is assumed that the rate of decomposition of organic substances is proportional to the concentration of the remaining nonoxidised substances. The BOD changes by a well-defined value over a specific time interval:  
\[\frac{dx_{1}}{dt}=-k_{1}x_{1} \tag{1}\]  
where \(x_{1}\)--BOD value [mg/L] over time t and \(k_{1}\)--first order reaction rate constant. At the same time, as the substrate concentration decreases further, the decomposition rate changes. Assuming that the initial value of BOD at time \(t_{0}\) is \(x_{0}\) and decreases to \(x_{1}\) after time \(t\), the analytical solution of Equation (1) takes the form:  
\[\ln\frac{x_{1}}{x_{0}}=-k_{1}t,x_{1}=x_{0}e^{-k_{1}t} \tag{2}\]  
Because the rate of the BOD process is strongly influenced by temperature, the BOD process is accelerated with increasing temperature. If the optimum value is exceeded, the process is inhibited or even disappears. The relationships shown above are valid only for values obtained at the same temperatures:  
\[k_{1}=k_{20}\theta^{(t-20)} \tag{3}\]where the \(\theta\)--temperature-dependent thermal coefficient. At temperatures \(\langle 20,30\rangle^{\circ}\)C, the thermal coefficient is 1.056, while at interval \(\langle 4,20\rangle^{\circ}\)C the coefficient takes on a value of 1.135. Theoretically, the \(k_{1}\)-value should be constant and is dependent on the concentration and type of organic matter, the type and activity of the microorganisms, and the temperature. Two phases can be distinguished in the BOD test: fast and slow. The first phase is characterised by a high \(k_{1}\)-value, where fast decomposition and synthesis of biomass occurs, with the rapid consumption of dissolved oxygen in the water. In the second phase, the coefficient \(k_{1}\) takes on small values. In this phase, slow endogenous respiration processes take place. The values of \(k_{1}\) can take values between approximately \(\langle 0.01,0.30\rangle\)d\({}^{-1}\)(d\({}^{-1}\) --per day). At temperature 20 \({}^{\circ}\)C, the average \(k_{1}\) values are: \(\langle 0.04,0.08\rangle\)d\({}^{-1}\) for low-pollution rivers, \(\langle 0.06,0.10\rangle\)d\({}^{-1}\) for low-pollution rivers, \(\langle 0.12,0.22\rangle\)d\({}^{-1}\) for partially treated wastewater, \(\langle 0.15,0.30\rangle\)d\({}^{-1}\) for raw wastewater. The \(k_{1}\) value depends on the activity of the microorganisms and decreases as the activity decreases. The increase in \(k_{1}\) with temperature is represented by relation (3). Dissolved oxygen is one of the most important indicators of water quality: it influences the chemical and biochemical processes that occur in water.