Real-time monitoring of a microbial electrolysis cell using an electrical equivalent circuit model

Abstract

Efforts in developing microbial electrolysis cells (MECs) resulted in several novel approaches for wastewater treatment and bioelectrosynthesis. Practical implementation of these approaches necessitates the development of an adequate system for real-time (on-line) monitoring and diagnostics of MEC performance. This study describes a simple MEC equivalent electrical circuit (EEC) model and a parameter estimation procedure, which enable such real-time monitoring. The proposed approach involves MEC voltage and current measurements during its operation with periodic power supply connection/disconnection (on/off operation) followed by parameter estimation using either numerical or analytical solution of the model. The proposed monitoring approach is demonstrated using a membraneless MEC with flow-through porous electrodes. Laboratory tests showed that changes in the influent carbon source concentration and composition significantly affect MEC total internal resistance and capacitance estimated by the model. Fast response of these EEC model parameters to changes in operating conditions enables the development of a model-based approach for real-time monitoring and fault detection.

Appendices

Appendix A: equivalent circuit model

Current at the internal capacitors shown in Fig. 4 (equivalent circuit Model #1 with two R/C circuits) can be described as

$${i_1}={C_1}\frac{{{\text{d}}{U_{C1}}}}{{{\text{d}}t}}$$

(15)

$${i_2}={C_2}\frac{{{\text{d}}{U_{C2}}}}{{{\text{d}}t}}$$

(16)

By applying Kirchhoff’s current and voltage laws to the EEC model diagram in Fig. 4 and expressing voltages using Ohm’s law the following model equation can be written:

$${U_{\text{s}}}+{U_{{\text{emf}}}}={i_0}{R_0}+{i_1}{R_1}+{i_2}{R_2}+{i_0}{R_{{\text{ext}}}}$$

(17)

After substituting the currents and rearranging the terms in Eq. 17 voltages across internal capacitors can be expressed as:

$$\frac{{{\text{d}}{U_{C1}}}}{{{\text{d}}t}}=\frac{{{U_{\text{s}}}+{U_{{\text{emf}}}}}}{{{C_1}\left( {{R_0}+{R_{{\text{ext}}}}} \right)}} - \frac{{{U_{C1}}\left( {{R_0}+{R_1}+{R_{{\text{ext}}}}} \right)}}{{{C_1}{R_1}\left( {{R_0}+{R_{{\text{ext}}}}} \right)}} - \frac{{{U_{C2}}}}{{{C_1}\left( {{R_0}+{R_{{\text{ext}}}}} \right)}}$$

(18)

and

$$\frac{{{\text{d}}{U_{C2}}}}{{{\text{d}}t}}=\frac{{{U_{\text{s}}}+{U_{{\text{emf}}}}}}{{{C_2}\left( {{R_0}+{R_{{\text{ext}}}}} \right)}} - \frac{{{U_{C2}}\left( {{R_0}+{R_2}+{R_{{\text{ext}}}}} \right)}}{{{C_1}{R_2}\left( {{R_0}+{R_{{\text{ext}}}}} \right)}} - \frac{{{U_{C1}}}}{{{C_2}\left( {{R_0}+{R_{{\text{ext}}}}} \right)}}$$

(19)

Once again, voltage across MEC can be expressed by applying Kirchhoff’s laws and rearranging terms as

$${U_{{\text{MEC}}}}={U_{\text{s}}} - ({U_{\text{s}}} - {U_{C1}} - {U_{C2}} - {U_{{\text{emf}}}})\frac{{{R_{{\text{ext}}}}}}{{{R_0}+{R_{{\text{ext}}}}}}$$

(20)

Furthermore, a single R/C circuit model (Model #2) can be obtained by assuming R₂ = 0, and U_C2 = 0. In this case the dynamic model equations are simplified to

$$\frac{{{\text{d}}{U_{C1}}}}{{{\text{d}}t}}=\frac{{{U_{\text{s}}}+{U_{{\text{emf}}}}}}{{{C_1}\left( {{R_0}+{R_{ext}}} \right)}} - \frac{{{U_{C1}}\left( {{R_0}+{R_1}+{R_{{\text{ext}}}}} \right)}}{{{C_1}{R_1}\left( {{R_0}+{R_{{\text{ext}}}}} \right)}}$$

(21)

and

$${U_{{\text{MEC}}}}={U_{\text{s}}} - ({U_{\text{s}}} - {U_{C1}} - {U_{{\text{emf}}}})\frac{{{R_{{\text{ext}}}}}}{{{R_0}+{R_{{\text{ext}}}}}}$$

(22)

Finally, Eqs. 21, 22 can be further simplified by assuming U_emf = 0 (Model #3).

Appendix B. analytical solution of single R/C circuit EEC model #2

Analytical solution of the EEC Model #2 with one R/C circuit described by Eqs. 21, 22 can be obtained by the variable separation method. For simplicity, let us define the following variables:

$$K=\frac{{{U_{\text{s}}}+{U_{{\text{emf}}}}}}{{C({R_0}+{R_{{\text{ext}}}})}},\;L=\frac{{{R_0}+{R_1}+{R_{{\text{ext}}}}}}{{C{R_1}({R_0}+{R_{{\text{ext}}}})}}$$

(23)

Then Eq. 21 can be written as

$$\frac{{{\text{d}}{U_{C1}}}}{{{\text{d}}t}}=K - {U_{C1}}L$$

(24)

Isolating the variables and integrating both sides of the equation we obtain

$${\int\limits_{{{U_0}}}^{U} {\left( {K - {U_{C1}}L} \right)} ^{ - 1}}{\text{d}}{U_{C1}}=\int\limits_{{{t_0}}}^{t} {{\text{d}}t}$$

(25)

Solving Eq. 25 we obtain

$$\frac{{K - {U_{C1}}L}}{{K - {U_0}L}}={e^{\left( {{t_0} - t} \right)L}}$$

(26)

Substituting the values of K and L into Eq. 26 and simplifying the algebraic expression:

$$\left( {{U_s}+{U_{{\text{emf}}}}} \right){R_1} - {U_{C1}}\left( {{R_0}+{R_1}+{R_{{\text{ext}}}}} \right)={e^{\left( {{t_0} - t} \right)\frac{{{R_0}+{R_1}+{R_{{\text{ext}}}}}}{{{C_1}{R_1}\left( {{R_0}+{R_{{\text{ext}}}}} \right)}}}}\left( {\left( {{U_s}+{U_{{\text{emf}}}}} \right){R_1} - {U_0}\left( {{R_0}+{R_1}+{R_{{\text{ext}}}}} \right)} \right)$$

(27)

Isolating U_C1:

$${U_{C1}}=\frac{{({U_{{\text{emf}}}}+{U_{\text{s}}}){R_1}}}{{{R_0}+{R_1}+{R_{{\text{ext}}}}}}+{e^{({t_0} - t)\frac{{R0+{R_1}+{R_{{\text{ext}}}}}}{{C{R_1}({R_0}+{R_{{\text{ext}}}})}}}}\left( {{U_0} - \frac{{({U_{\text{s}}}+{U_{{\text{emf}}}}){R_1}}}{{{R_0}+{R_1}+{R_{{\text{ext}}}}}}} \right)$$

(28)

The above equation can be written as

$${U_{C1}}={U_{{\text{final}}}}+\left( {{U_0} - {U_{{\text{final}}}}} \right){e^{ - \frac{{t - {t_0}}}{\tau }}}$$

(29)

where

$${U_{{\text{final}}}}=\frac{{({U_{{\text{emf}}}}+{U_{\text{s}}}){R_1}}}{{{R_0}+{R_1}+{R_{{\text{ext}}}}}},\;\tau =\frac{{{C_1}{R_1}({R_0}+{R_{{\text{ext}}}})}}{{{R_0}+{R_1}+{R_{{\text{ext}}}}}}$$

(30)

Here, U_final and time constant τ can be determined from an experiment.