## 2 Model Description
The mass balances for the CSTRs are:  
\[\frac{dS_{1}}{dt}=(S_{in}-S_{1})\frac{Q}{v_{1}}-r_{s1}, \tag{1}\]  
\[\frac{dS_{2}}{dt}=(S_{1}-S_{2})\frac{Q}{v_{2}}-r_{s2}+\frac{Q_{D}}{v_{2}}(S_{3 }-S_{2}), \tag{2}\]  
\[\frac{dS_{3}}{dt}=(S_{2}-S_{3})\frac{Q_{D}}{v_{3}}, \tag{3}\]  
\[S_{in}=\overline{S}_{in}+\delta_{sin}, \tag{4}\]  
where \(S_{1}\) is the pollutant concentration in CSTR 1; \(S_{2}\) is the pollutant concentration in CSTR 2; \(S_{3}\) is the pollutant concentration in CSTR 3; \(S_{in}\) is the inlet pollutant concentration; \(Q\) is the wastewater flow entering CSTR 1 and CSTR 2; \(Q_{D}\) is the wastewater flow entering and leaving CSTR 3; \(v_{i}\) is the liquid volume of the ith CSTR; \(r_{s1}\) is the reaction rate of CSTR 1; \(r_{s2}\) is the reaction rate of CSTR 2. In addition, \(S_{1}\) is defined in \(\mathbb{R}^{+}\), \(S_{2}\) is defined in \(\mathbb{R}^{+}\), \(S_{3}\) is defined in \(\mathbb{R}^{+}\). **Assumptions:**  
**Assumption 1**. : \(Q\)_, \(Q_{D}\), \(v_{1}\), \(v_{2}\), and\(v_{3}\) are positive and constant. This implies that \(Q/v_{1}\), \(Q/v_{2}\),\(Q_{D}/v_{2}\) and \(Q_{D}/v_{3}\) are constant and positive._  
**Assumption 2**. : \(\overline{S}_{in}\) _is positive and constant,\(\delta_{sin}\) is varying but bounded, \(\max\{\delta_{sin}\}>0\), \(min\{\delta_{sin}\}<0\)._  
Figure 1: Schematic structure of the series/parallel continuously stirred tank reactor (CSTR)-based model. **Assumption 3**. : _The reaction rate \(r_{s1}\) only depends on \(S_{1}\), it is bounded for \(S_{1}\) bounded, \(\frac{dr_{s1}}{dS_{1}}\geq 0\); \(r_{s2}\) only depends on \(S_{2}\), it is bounded for \(S_{2}\) bounded, and \(\frac{dr_{s2}}{dS_{2}}\geq 0\)._  
In the case of constant \(S_{in}\), \(\delta_{sin}=0\) in Equation (4), \(S_{1}\) converges to \(S_{1}^{eq}\), \(S_{2}\) converges to \(S_{2}^{eq}\) and \(S_{3}\) converges to \(S_{3}^{eq}\), where \(S_{1}^{eq}\), \(S_{2}^{eq}\) and \(S_{3}^{eq}\) are positive constants obtained by equating Equations (1)-(3) to zero:  
\[0=\big{(}\overline{S}_{in}-S_{1}\big{)}\frac{Q}{v_{1}}-r_{s1}, \tag{5}\]  
\[0=(S_{1}-S_{2})\frac{Q}{v_{2}}-r_{s2}+\frac{Q_{D}}{v_{2}}(S_{3}-S_{2}), \tag{6}\]  
\[0=(S_{2}-S_{3})\frac{Q_{D}}{v_{3}}. \tag{7}\]  
Let  
\[x_{1}=S_{1}-S_{1}^{eq}, \tag{8}\]  
\[\frac{dS_{1}}{dt}=(S_{in}-S_{1})\frac{Q}{v_{1}}-r_{s1}, \tag{1}\]0  
\[\frac{dS_{1}}{dt}=(S_{in}-S_{1})\frac{Q}{v_{1}}-r_{s1}, \tag{1}\]1  
The \(x_{1}\) dynamics is obtained by subtracting Equation (5) from Equation (1):  
\[\frac{dS_{1}}{dt}=(S_{in}-S_{1})\frac{Q}{v_{1}}-r_{s1}, \tag{1}\]2  
\[\frac{dS_{1}}{dt}=(S_{in}-S_{1})\frac{Q}{v_{1}}-r_{s1}, \tag{1}\]3  
where \(D=Q/v_{1}>0\), \(r_{s1}^{eq}\) is \(r_{s1}\) evaluated at \(S_{1}=S_{1}^{eq}\). The \(x_{2}\) dynamics is obtained by subtracting Equation (6) from Equation (2):  
\[\frac{dS_{1}}{dt}=(S_{in}-S_{1})\frac{Q}{v_{1}}-r_{s1}, \tag{1}\]4  
\[\frac{dS_{1}}{dt}=(S_{in}-S_{1})\frac{Q}{v_{1}}-r_{s1}, \tag{1}\]5  
The term \(r_{s2}^{eq}\) is \(r_{s2}\) evaluated at \(S_{2}=S_{2}^{eq}\). The \(x_{3}\) dynamics is obtained by subtracting Equation (7) from Equation (3):  
\[\frac{dS_{1}}{dt}=(S_{in}-S_{1})\frac{Q}{v_{1}}-r_{s1}, \tag{1}\]6  
In addition, \(x_{1}\) is defined in \([-S_{1}^{eq},\ \infty),\ x_{2}\) is defined in \([-S_{2}^{eq},\ \infty)\), and \(x_{3}\) is defined in \([-S_{3}^{eq},\ \infty)\). The model is fitted to current (no tracer) data in Section 3.