Combined bioelectrochemical–electrical model of a microbial fuel cell

Abstract

Several recent studies demonstrated significant charge storage in electrochemical biofilms. Aiming to evaluate the impact of charge storage on microbial fuel cell (MFC) performance, this work presents a combined bioelectrochemical–electrical (CBE) model of an MFC. In addition to charge storage, the CBE model is able to describe fast (ms) and slow (days) nonlinear dynamics of MFCs by merging mass and electron balances with equations describing an equivalent electrical circuit. Parameter estimation was performed using results of MFC operation with intermittent (pulse-width modulated) connection of the external resistance. The model was used to compare different methods of selecting external resistance during MFC operation under varying operating conditions. Owing to the relatively simple structure and fast numerical solution of the model, its application for both reactor design and real-time model-based process control applications are envisioned.

Appendix: On-line parameter estimation procedure

The voltage dynamics at the capacitor (V _c) of the equivalent circuit shown in Fig. 1 can be described by the following first order differential equation:

$$\frac{{{\text{d}}V_{c} }}{{{\text{d}}t}} = \frac{{E_{\text{oc}} }}{{C\left( {R_{1} + R_{\text{ext}} } \right)}} - \frac{{R_{1} + R_{2} + R_{\text{ext}} }}{{R_{2} C\left( {R_{1} + R_{\text{ext}} } \right)}}V_{c} .$$

(28)

By applying Kirchhoff’s law and solving Eq. (28), the analytical solution of MFC output voltage (V _cell) is obtained in the following form:

$$V_{\text{cell}} = E_{\text{oc}} - \left[ {U_{\text{c,final}} + \left( {U_{\text{c,initial}} - U_{\text{c,final}} } \right)e^{{{{ - t} \mathord{\left/ {\vphantom {{ - t} \tau }} \right. \kern-0pt} \tau }}} } \right]\frac{{R_{\text{ext}} }}{{R_{\text{int1}} + R_{\text{ext}} }} ,$$

(29)

where U _c,initial and U _c,final are the initial and final voltage values shown in Fig. 8.

Fig. 8

U _MFC profiles at low and high operating frequencies used for on-line parameter estimation. Operating modes used to estimate R ₁, E _OC, R ₂ and C, are shown as steps 1, 2 and 3, respectively

First, R ₁ estimation is obtained during MFC operation at a high frequency (e.g. 100 Hz) from the following equation:

$$R_{1} = R_{\text{ext}} \frac{{U_{\text{high}} - U_{\text{low}} }}{{U{}_{\text{low}}}}$$

(30)

where U _high and U _low are the high and low output voltage levels measured in the experiment (Fig. 8).

Subsequently, E _oc estimation is obtained by operating the MFC at a low frequency (e.g. 1.0 Hz) and low duty cycle. It is assumed that U _oc is equal to the voltage at the end of the open circuit part of the cycle:

$$U_{\text{oc}} = { \hbox{max} }\left( {U_{\text{MFC}} } \right)$$

(31)

Finally, R ₂ and C estimations are obtained using voltage measurements at a low operating frequency. It is assumed that V _cell reaches a steady-state value at the end of the closed circuit part of each cycle. In Fig. 8 this value is denoted as U _final.

The value of R ₂ is estimated as:

$$R_{2} = U_{\text{final}} \frac{{R_{1} + R_{\text{ext}} }}{{U{}_{\text{oc}} - U_{\text{final}} }}$$

(32)

The value of C is calculated as

$$C = \frac{\tau }{{R_{2} }}\frac{{R_{1} + R_{2} + R_{\text{ext}} }}{{R{}_{1} + R_{\text{ext}} }}$$

(33)

where $\tau$ is the time constant shown in Fig. 8.