Microbial fuel cells: A fast converging dynamic model for assessing system performance based on bioanode kinetics

In this work, a dynamic computational model is developed for a single 
chamber microbial fuel cell (MFC), consisting of a bio-catalyzed anode and 
an air-cathode. Electron transfer from the biomass to the anode is assumed 
to take place via intracellular mediators as they undergo transformation between 
reduced and oxidized forms. A two-population model is used to describe 
the biofilm at the anode and the MFC current is calculated based on 
charge transfer and Ohm's law, while assuming a non-limiting cathode reaction 
rate. The open circuit voltage and the internal resistance of the cell are 
expressed as a function of substrate concentration. The effect of operating 
parameters such as the initial substrate (COD) concentration and external 
resistance, on the Coulombic efficiency, COD removal rate and power density 
of the MFC system is studied. Even with the simple formulation, model 
predictions were found to be in agreement with observed trends in experimental studies. This model can be used as a convenient tool for performing 
detailed parametric analysis of a range of parameters and assist in process 
optimization.

scribe the biofilm at the anode and the MFC current is calculated based on charge transfer and Ohm's law, while assuming a non-limiting cathode reaction rate. The open circuit voltage and the internal resistance of the cell are expressed as a function of substrate concentration. The effect of operating parameters such as the initial substrate (COD) concentration and external resistance, on the Coulombic efficiency, COD removal rate and power density of the MFC system is studied. Even with the simple formulation, model predictions were found to be in agreement with observed trends in experi-2D/3D formulations accounting for lateral biofilm growth and electrode ge- 45 ometries, have been used to model MFC [20,21].
MFC. This was a simple model based on ordinary differential equations, and isms, which is something difficult to measure experimentally, which limits 60 the applicability of this approach. Other than the dynamic models based on 61 ODEs, more comprehensive mathematical models have also been proposed.    The important assumptions made in the model are as follows: 119 Substrate in the MFC chamber is perfectly mixed.

120
Microbial population in the biofilm of the anode is uniformly dis-121 tributed.

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Any gases (CO 2 , H 2 , etc.) released during substrate oxidation at the 123 anode remain dissolved in the bulk solution.

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MFC is operated in fed-batch mode.

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Changes in pH and temperature are negligible.

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Electrons are transferred from the cells to the anode using intracellu-127 lar mediators, as they undergo transformation between reduced and 128 oxidized forms.

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Continuous supply of intracellular mediators is maintained. The primary microbial population (x p ) consumes the substrate (S) and, 132 in this process, the oxidized form of the intracellular mediator (M ox ) is also 133 converted into its reduced form (M red ). This reduced intracellular mediator 134 transfers the electron to the anode and also releases a proton as it regains its 135 oxidized form. A conceptual schematic of this process is described in figure   136 1. Meanwhile, the secondary microbial population (x s ) also consumes the  Considering a fed-batch operation, the rate of change of substrate and biomass concentrations can be expressed as follows: where S represents the substrate concentration (g-S L −1 ), x is the microbial µ is the microbial growth rate (d −1 ) and K d is the microbial decay rate (d −1 ).

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The subscripts 'p' and 's' represent the primary and secondary microbial 143 populations respectively.

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The intracellular mediator exists in either oxidized (M ox ) or reduced form (M red ), however the total mediator concentration M total remains constant, and can thus be expressed as: The transfer of electrons from intracellular mediator to anode and then further to cathode, results in current generation. The rate of change of the oxidized mediator concentration is described as follows: where, Y is the dimensionless mediator yield, I MFC is the MFC current (A), where, K S and K M are the Monod half saturation coefficients for the sub-149 strate and mediator respectively (g L −1 ), q max and µ max represent the max- where E ocv is the open circuit voltage (V), R ext and R int are the external 154 and internal resistance in the electrochemical cell, η act and η conc represent 155 the activation and concentration over-potentials (V).

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A constant supply of intracellular electron transfer mediators is assumed, which undergo transformation between reduced and oxidized forms as they transfer electrons to the anode. Thus the major limiting factor influencing the voltage losses at the anode is the substrate concentration. The concentration overpotential at the anode can therefore be expressed as a function of initial substrate concentration (S in ) and the dynamic substrate concentration (S), as follows: Activation overpotential represents losses due to the slow electrochemical kinetics, and can be expressed by the following equation: where T is the system temperature (K), R is the universal gas constant (J where, E min and E max represent the lowest and the highest observed open 160 circuit voltage in the system, R min and R max represent the lowest and the 161 highest observed internal resistance in the system. K r 1 and K r 2 are constants

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MFC performance is typically assessed based on the maximum voltage that can be generated and the substrate or COD removal efficiency. In addition to these two, Coulombic efficiency ( E ) is another important performance indicator and determines the electron recovery of the system. It represents the ratio of the total number of electrons recovered at the anode and the maximum number of electrons that could have been recovered if all the consumed substrate contributed to current generation. In the present analysis, the MFC is assumed to be operating in fed-batch mode and thus E can be expressed as [33]: where In numerical curve fitting, the objective function (J), which is defined   Table 1.
189       resistors. As can be seen, the power density increases with increase in ini-305 tial COD concentration, but decreases as the external resistance is increased.

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The increase in power density is consistent with the increase in maximum