## 4 Conclusions  
The model proposed in this study was based on the assumption that the anode attached to the bacterial monolayer served as biocatalysts for MFC exoelectrogenesis. By modifying the Freter model and combining it with the Butler-Volmer equation, this model could adequately describe the processes of electricity generation, substrate utilization, and suspended and attached biomass growth, in both batch and continuous operating mode. MFC performances were impacted by various operating variables such as initial substrate concentration, external resistor, influent substrate concentration, and dilution rate, and their interactions, which were revealed by data simulation to be complicated. The simulation results explained that in batch mode, an intermediate initial substrate concentration (\(S_{0}\) about 100 mg L\({}^{-1}\) at this reactor configuration) was appropriate to achieve maximum overall energy efficiency. With the \(S_{0}\) of 100 mg L\({}^{-1}\), an external resistor (about 300 \(\Omega\)) with the value of around the internal resistance (\(R_{in}\) of about 471 \(\Omega\)) could boost the power generation, and a resistor of about twice (about 1000 \(\Omega\)) that value achieved better overall energy efficiency. However, a small external resistor should be adopted for MFC reactors aiming to rapidly remove organic substrates. In continuous mode and at \(R_{ext}\) = 2200 \(\Omega\), the anode-attached biomass could fully grow when the influent substrate concentration was equal to or higher than 100 mg L\({}^{-1}\) at any dilution rates within the tested range. The maximum external power of 0.106 mW could be achieved at \(R_{ext}\) = 352 \(\Omega\) when there was enough substrate to sustain the power generation. An influent substrate concentration of 100 mg L\({}^{-1}\) achieved the overall energy efficiency greater than 10%. The high substrate concentration, e.g., 2400 mg L\({}^{-1}\)  
Figure 7: Simulated MFC performance at steady state under different electrical currents (or external resistors), dilution rates and influent substrate concentrations. (**A**) steady state power density; (**B**) MFC cell energy efficiency; (**C**) overall energy efficiency at \(S_{in}\) = 100 mg L\({}^{-1}\); and (**D**) \(S_{in}\) = 2400 mg L\({}^{-1}\); (**E**) effluent substrate concentration at \(S_{in}\) = 100 mg L\({}^{-1}\); and (**F**) \(S_{in}\) = 2400 mg L\({}^{-1}\).  
encouraged substantial biomass growth and thus reduced the overall energy efficiency to less than 2%. For the low influent substrate concentration (100 mg L\({}^{-1}\)), the maximum power generation and overall energy efficiency could not be concurrently achieved, and operating MFC at an external resistor \(R_{ext}\) close to the internal resistance (\(R_{in}\)) or two of \(R_{in}\) was recommended to obtain optimal values for both indices. Overall, this relatively simple model provided a convenient way for evaluating and optimizing performance of MFC reactors by regulating operating parameters.  
All authors were involved in conceiving and designing the experiments. [PERSON] and [PERSON] conducted the experiments and analyzed the data. [PERSON] wrote the manuscript with [PERSON] and [PERSON] reviewing the draft and providing edits and corrections. [PERSON] finalized the final version and acted as the subnitting and corresponding author for the manuscript.  
This research received no external funding.  
The authors declare no conflicts of interest.