## 3 Results and Discussion  
### Parameter Estimation  
Two identically-made MFC reactors with respect to reactor configuration and electrode materials were inoculated with the same anaerobic sludge and operated in parallel. After MFC were fully inoculated and operated for 30 d, polarization curves of MFC was obtained by varying the external resistor from 2200 to 220 \(\Omega\), with the electrode potentials recorded (Figure 2A). The resultant electrochemical parameters estimated by fitting polarization curves of electrodes were listed in Table 2. The simulation results based on the estimated parameters were in good agreement with the experimental data of both reactors, as shown in Figure 2B.C. The anodic exchange current density of 62.5 mA m\({}^{-2}\) was in the same order of a literature value of 29 mA m\({}^{-2}\)[17]. The cathodic exchange current density (0.975 mA m\({}^{-2}\)) observed in this study was higher than those of non-catalyzed and bio-catalyzed cathodes [18; 19] but lower than another Pt-based cathode [20]. To further validate the model, the external voltage (\(U_{ext}\)) during inoculation stage was simulated based on the listed parameters except the unit exchange current, which was fitted from the experimental data at the inoculation stage (0.129 mA mg\({}^{-2}\) biomass) and had a difference by 32% from 0.189 mA mg\({}^{-1}\) which \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter Symbol** & **Description** & **Value** & **Units** \\ \hline Physical & & & \\ F & [PERSON]’s constant & 96,485 & C/mol \\ R & Ideal gas constant & 8.314 & J/mol/K \\ T & Room temperature & 298 & K \\ \(\Delta\)H\({}_{\mathrm{c}}\) & Heat of combustion of acetic acid & \(-\)875,000 & J/mol \\ M & Relative molecular weight of acetic acid & 60 & g/mol \\ Electrochemical & & & \\ E\({}_{A}\)\({}^{O}\) & Formal reduction potential of anode & \(-\)0.335 & V vs. NHE \\ E\({}_{C}\)\({}^{O}\) & Formal reduction potential of cathode & 0.51 & V vs. NHE \\ Reactor configuration & & & \\ A & Projected surface area of electrode & 0.002 & m\({}^{2}\) \\ d & Distance between anode and cathode & 0.075 & m \\ V & Volume of MFC reactor & \(1.55\times 10^{-4}\) & m\({}^{3}\) \\ \(\delta\) & Ratio of anode surface area to MFC volume & 13 & m\({}^{2}\)/m\({}^{3}\) \\ R\({}_{\mathrm{c}}\) & Contact resistance of MFC & 20 & \(\Omega\) \\ Operating & & & \\ D & Dilution rate & 0–2 & d\({}^{-1}\) \\ \(S_{in}\) & Substrate concentration & 10–2400 & g/m\({}^{3}\) \\ Q & Flow rate & 0–7.8 \(\times\) & m\({}^{3}\)/d \\ \(\sigma\) & Conductivity of MFC medium & 1.1 & S/m \\ \(R_{ext}\) & External resistor & 2200 & \(\Omega\) \\ [HCO\({}_{3}\)\({}^{-}\)] & \(\mathrm{Bicarbonate}\) concentration & \(1\times 10^{-4.5}\) & kmol/m\({}^{3}\) \\ P\({}_{02}\) & Partial oxygen pressure at cathode & 0.2 & atm \\ pH & \(-\)log10 of proton concentration & 7 & Dimensionless \\ Microbial & & & \\ \(\gamma\) & Biomass yield from substrate & 0.05 & g/g \\ k\({}_{\mathrm{u}}\) & Death rate of suspended bacteria & 0.02 & d\({}^{-1}\) \\ k\({}_{\mathrm{w}}\) & Death rate of attached bacteria & 0.02 & d\({}^{-1}\) \\ \(\alpha\) & Bacterial attachment rate & 0.05 & d\({}^{-1}\) \\ \(\beta\) & Bacterial detachment rate & 0.05 & d\({}^{-1}\) \\ m\({}_{\mathrm{u}}\) & Maximum specific growth rate of suspended bacteria & 2.4 & d\({}^{-1}\) \\ m\({}_{\mathrm{u}}\) & Maximum specific growth rate of attached bacteria & 2.4 & d\({}^{-1}\) \\ a\({}_{\mathrm{u}}\) & Half saturation coefficient of suspended bacteria & 100 & g/m\({}^{3}\) \\ a\({}_{\mathrm{w}}\) & Half saturation coefficient of attached bacteria & 100 & g/m\({}^{3}\) \\ w\({}_{\mathrm{max}}\) & Maximum bacterial attachment & 0.33 1  & g/m\({}^{2}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Constants, and design and operating parameters of a microbial fuel cell system. NHE: the normal hydrogen electrode. was fitted from polarization curves. This difference could be a result of the change of the activity or catalytic effect of the attached biomass during the inoculation and 30 d operation.