Microbial fuel cells: a fast converging dynamic model

for assessing system performance based on bioanode

kinetics

Siddharth Gadkaria,b, Mobolaji Shemfeb, Jhuma Sadhukhana,b

aDepartment of Chemical and Process Engineering, University of Surrey, Guildford GU27XH, United Kingdom

bCentre for Environment and Sustainability, University of Surrey, Guildford, SurreyGU2 7XH, United Kingdom

Abstract

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 be-

tween reduced and oxidized forms. A two-population model is used to de-

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 reac-

tion 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 den-

sity of the MFC system is studied. Even with the simple formulation, model

predictions were found to be in agreement with observed trends in experi-

∗Corresponding authorEmail address: s.gadkari@surrey.ac.uk (Siddharth Gadkari)

1

mental 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.

Keywords: Microbial fuel cell, Bioelectrochemical system, Mathematical

analysis, Dynamic model, Fast convergence

1. Introduction1

Wastewater treatment consumes large amounts of energy and is a major2

source of greenhouse gas emissions. Moreover, wastewater when discharged3

without treatment causes serious damage to human health and the environ-4

ment [1]. Rising urbanization is leading to increased wastewater generation,5

which further aggravates these challenges, while at the same time using more6

of our planet’s diminishing resources [2]. Thus, recovering energy and re-7

sources from wastewater should be no longer considered a choice, but a key8

opportunity that must be seized for a low carbon future. Wastewater treat-9

ment technologies that can provide energy-neutral operation and help to10

mitigate the multiple challenges of energy efficiency, resource scarcity and11

environmental pollution, can make this process sustainable [3]. In this re-12

gard, microbial fuel cells (MFCs) that use microorganisms as a biocatalyst13

and convert chemical energy from organics in wastewater to (bio)electricity,14

offer a sustainable technological solution [1, 4]. MFCs can transform wastew-15

ater treatment, which is traditionally an energy-consuming process, into an16

energy-neutral and potentially an energy-producing process [3, 5]. This real-17

ization has led to an incremental increase in MFC research in the last 10-1518

years [4–7].19

2

Focused research has helped in improving the power densities with several20

MFC systems reaching close to 1 W m−2 [8, 9]. Few optimized designs have21

managed to obtain even higher power densities up to 3-7 W m−2 [8, 10–13].22

Ren et al. [11] developed a miniaturized MFC with 3D graphene macroporous23

scaffold anode and obtained a record high current density of 15.51 A m−224

(31040 A m−3) and a power density of 5.61 W m−2 (11220 W m−3). Chen25

et al. [12] have reported current densities as high as 390 A m−2 using layered26

corrugated carbon (LCC) as electrodes. However despite this progress, the27

power density of MFCs are still orders of magnitude lower than the chemical28

fuel cells (which are typically > 104 W m−2). This highlights the need for29

further optimization.30

As the electrochemical reactions in an MFC involve biochemical path-31

ways, its performance is not only dependent on operational and design param-32

eters but also on several biological factors [14]. The different processes in an33

MFC with a biocatalysed anode and a chemical cathode, involves a complex34

interplay between electrochemistry, microbiology, material science, transport35

phenomena, and environmental biotechnology [15]. Thus, an in-depth under-36

standing of an MFC system requires a multidisciplinary approach. While the37

advances in MFC design, membrane and electrode materials have certainly38

helped to reduce the cost and increase the power outputs of MFC manifolds,39

a number of challenges still limit the applicability when compared to existing40

technologies [7]. Mathematical models along with experimental studies have41

played an important role in improving MFC performance and facilitating the42

design and system optimization [16–19].Different approaches, ranging from43

the simple formulations describing the bulk bioelectrochemical reactions to44

3

2D/3D formulations accounting for lateral biofilm growth and electrode ge-45

ometries, have been used to model MFC [20, 21].46

Zhang and Halme [22] developed one of the first mathematical models for47

MFC. This was a simple model based on ordinary differential equations, and48

used Monod equation, Faradays law and Nernst equation to calculate the49

MFC current. While the simplistic formulation is based on a lot of assump-50

tions, it served as a good starting point for more advanced analysis. Pinto51

et al. [23] also developed a dynamic model based on ODEs for MFC but they52

used a two population model for representing the microbial population at the53

bioanode, and used the multiplicative Monod kinetics to represent the biofilm54

growth dynamics. They also considered the material balance for substrate55

and mediators and derived the expression for MFC current using Ohms law56

and the voltage over-potentials. While this model was easy to implement and57

allowed fast numerical simulations, the internal resistance and open circuit58

voltage were calculated based on concentration of anodophilic microorgan-59

isms, which is something difficult to measure experimentally, which limits60

the applicability of this approach. Other than the dynamic models based on61

ODEs, more comprehensive mathematical models have also been proposed.62

For e.g. Picioreanu et al. [24] developed a multidimensional mathematical63

model focusing on the anode chamber of MFC considering a mixed culture64

of bacteria. They assumed mediated electron transfer and derived the cell65

voltage using Butler-Volmer equation and Ohms law, while accounting for66

the activation, ohmic and concentration over-potentials at the anode. They67

used the comprehensive model to understand the effect of system parameters68

such as initial substrate and mediator concentrations, mass transfer bound-69

4

ary layer, ratio of suspended to biofilm cells, external load resistance, etc.70

on the power generation capacity and the substrate consumption rate. How-71

ever the added physics, also correspondingly increased the computational72

expense of the simulation, with the full 3D model taking close to 14 h for73

solving about 15 days of MFC operation [24]. More recently Esfandyari et al.74

[25] developed a two chamber batch MFC model consisting of three main do-75

mains, bulk liquid in the anode chamber, biofilm attached to anode and76

bulk liquid in the cathode chamber. They assumed a direct electron transfer77

mechanism and used the Nernst-Monod equation to derive the substrate con-78

sumption rate. The model included the material balance equations for the79

active & inactive bacteria, substrate (lactate), carbon dioxide and protons80

in the biofilm, along with the substrate and CO2 mass balance in the anode81

chamber and oxygen in the cathode chamber. The MFC output voltage was82

calculated by accounting the different overpotential losses. The results from83

the model showed good agreement with the voltage and current observed84

experimentally. While this model was based on ODEs it was relatively more85

comprehensive than Pinto et al. [23]. However on account of improving the86

model rigor and including the various phenomena in the anode & cathode87

chambers and the biofilm, the total number of model equations (total 1088

ODEs and 9 algebraic equations) also correspondingly increase the compu-89

tational expense of the simulation. Additionally, because the model is based90

on lumped formulation, it does not capture the spatial biofilm dynamics even91

with the high computational cost [25].92

Thus, while the more detailed models typically based on partial differen-93

tial equations are expensive in terms of time and computational resources,94

5

the formulations based on ordinary differential equations (ODEs) sometimes95

trivialize the complex process dynamics or are based on parameters that96

are difficult to calculate/measure experimentally. [22–24, 26–28]. The goal97

of this study is to present a simple mathematical formulation that can suffi-98

ciently describe the important processes in an MFC with governing equations99

for mass transfer, bio-electrochemical kinetics, and charge transfer, without100

being computationally intensive. One important distinction of the proposed101

model from the previous approaches is the derivation of concentration over-102

potential, internal resistance and open circuit voltage, which is based here103

on an easily measurable quantity such as the substrate concentration. This104

simplifies the calculation of MFC voltage and provides an easier method for105

assessing system performance and process optimization. In this paper, the106

mathematical model is used to understand the effect of operating parame-107

ters on key performance indicators of MFC such as the, Coulombic efficiency,108

substrate (COD) removal rate and power density of the system.109

2. Model formulation and parameter estimation110

The dynamic model developed in this work is based on the batch operated111

single chamber MFC consisting of a biotic anode and an air-cathode, as112

described in Liu and Logan [29]. The microbial population at the anode113

is represented using a simple two population model, clubbing the bacteria114

into two types, one that consume the substrate (COD) and release electrons115

which are then transferred to the anode via intracellular mediators (primary116

microbial population) and second type that decompose the substrate but do117

not contribute to power generation (secondary microbial population).118

6

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.122

Any gases (CO2, H2, etc.) released during substrate oxidation at the123

anode remain dissolved in the bulk solution.124

MFC is operated in fed-batch mode.125

Changes in pH and temperature are negligible.126

Electrons are transferred from the cells to the anode using intracellu-127

lar mediators, as they undergo transformation between reduced and128

oxidized forms.129

Continuous supply of intracellular mediators is maintained.130

2.1. Mass balance131

The primary microbial population (xp) consumes the substrate (S) and,132

in this process, the oxidized form of the intracellular mediator (Mox) is also133

converted into its reduced form (Mred). This reduced intracellular mediator134

transfers the electron to the anode and also releases a proton as it regains its135

oxidized form. A conceptual schematic of this process is described in figure136

1. Meanwhile, the secondary microbial population (xs) also consumes the137

substrate but does not release any free electrons which can be intercepted by138

the intracellular mediators.139

7

Figure 1: Conceptual schematic showing substrate (S) degradation by a primary bacterial

cell (xp) and electron (e−1) transfer to the anode via reduced (Mred) and oxidized (Mox)

intracellular mediators.

Considering a fed-batch operation, the rate of change of substrate and

biomass concentrations can be expressed as follows:

dS

dt= −qpxp − qsxs (1)

dxpdt

= µpxp −Kdpxp (2)

dxsdt

= µsxs −Kdsxs (3)

where S represents the substrate concentration (g-S L−1), x is the microbial140

concentration (g-x L−1), q is the substrate consumption rate (g-S g-x−1 d−1),141

µ is the microbial growth rate (d−1) and Kd is the microbial decay rate (d−1).142

The subscripts ‘p’ and ‘s’ represent the primary and secondary microbial143

populations respectively.144

The intracellular mediator exists in either oxidized (Mox) or reduced form

(Mred), however the total mediator concentration Mtotal remains constant,

8

and can thus be expressed as:

Mtotal = Mox +Mred (4)

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:

dMox

dt= −Y qp +

γ

V xp

(IMFC

mF

)(5)

where, Y is the dimensionless mediator yield, IMFC is the MFC current (A),145

m is the number of electrons transferred per mol of mediator, F is the Faraday146

constant (A s mol−1), V is the working volume of the anode chamber, and γ147

is the molar mass of the mediator (g mol−1).148

The substrate consumption rate by the primary microbial population and

the corresponding microbial growth rate are dependent on both the oxidized

mediator concentration and the substrate concentration, and are thus de-

scribed using the multiplicative Monod kinetics. On the other hand, the

substrate consumption rate by the secondary bacteria and the corresponding

microbial growth rate are only limited by the substrate concentration, and

are represented using standard Monod kinetics. The substrate consumption

and growth rates are described as follows:

qp = qmaxp

(S

KSp + S

)(Mox

KM +Mox

)(6)

µp = µmax,p

(S

KSp + S

)(Mox

KM +Mox

)(7)

qs = qmaxs

(S

KSs + S

)(8)

µs = µmax,s

(S

KSs + S

)(9)

9

where, KS and KM are the Monod half saturation coefficients for the sub-149

strate and mediator respectively (g L−1), qmax and µmax represent the max-150

imum substrate consumption rate and maximum growth rate respectively151

(d−1).152

2.2. Ohm’s Law and voltage losses153

The standard electrode potentials determine the maximum theoretical

voltage that can be generated from the MFC. However there are activation,

concentration and ohmic voltage losses in the system. Accounting for these

overpotential losses and the internal & external resistance, the current gen-

erated in the MFC can be expressed using Ohm’s Law as follows [23]:

IMFC =Eocv − ηact − ηconc

Rext +Rint

(10)

where Eocv is the open circuit voltage (V), Rext and Rint are the external154

and internal resistance in the electrochemical cell, ηact and ηconc represent155

the activation and concentration over-potentials (V).156

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 (Sin) and the dynamic substrate concentration (S),

as follows:

ηconc =RT

mFln

(SinS

)(11)

10

Activation overpotential represents losses due to the slow electrochemical

kinetics, and can be expressed by the following equation:

ηact =RT

βmFsinh−1

(IMFC

Ani0

)(12)

where T is the system temperature (K), R is the universal gas constant (J157

mol−1 K−1), β is the charge transfer coefficient, An is surface area of anode158

(m2), and i0 is the exchange current density (A m2−1).159

It has been shown that increase in organic substrate concentration in-

creases the ionic strength of the anolyte, which helps in improving the open

circuit voltage and reducing the internal resistance of the cell [30–32]. These

correlations can be used to further improve the model efficacy by incorporat-

ing dynamic expressions for open circuit voltage and internal resistance as a

function of substrate concentration:

Eocv = Emin + (Emax − Emin) e−1

Kr1S (13)

Rint = Rmin + (Rmax −Rmin) e1

Kr2S (14)

where, Emin and Emax represent the lowest and the highest observed open160

circuit voltage in the system, Rmin and Rmax represent the lowest and the161

highest observed internal resistance in the system. Kr1 and Kr2 are constants162

[L g−1] which determine the slope of the curve. Similar expressions have been163

previously derived by Pinto et al. [23], considering open circuit voltage and164

internal resistance to be a function of anodophilic microorganisms.165

MFC performance is typically assessed based on the maximum voltage

that can be generated and the substrate or COD removal efficiency. In addi-

tion to these two, Coulombic efficiency (εE) is another important performance

11

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 con-

sumed substrate contributed to current generation. In the present analysis,

the MFC is assumed to be operating in fed-batch mode and thus εE can be

expressed as [33]:

εE =Mo

∫ t0IMFCdt

FbV∆S(15)

where Mo is the molecular weight of oxygen, b is number of electrons166

exchanged per mole of oxygen and ∆S is the change in substrate (COD)167

concentration over time t.168

2.3. Parameter estimation169

The governing equations described in sections 2.1 and 2.2, can be solved170

in an ODE solver to determine the influence of different parameters on the171

MFC performance. However most of the biological parameters such as the172

maximum substrate consumption rates, maximum microbial growth rates,173

yield coefficient, half saturation constants, etc., are specific to the microbial174

population and the substrate used in the experiment. Thus these parameters175

are estimated using best-fit regression analysis by comparing the numerical176

results with experimental values obtained from Liu and Logan [29].177

12

Figure 2: Predicted values of MFC voltage (black solid line) obtained from curve fitting

using Nelder-Mead Simplex algorithm compared to the corresponding experimental values

red solid circles for 3 days of MFC operation. The experimental data has been adopted

from Liu and Logan [29].

In numerical curve fitting, the objective function (J), which is defined178

as, J =∑t

(yexp(t) − ysim(t)

)2, is minimized to obtain the parameter values.179

Here yexp(t) and ysim(t) represent the values of y obtained from experiment180

and numerical simulation respectively at a particular time t. In this analy-181

sis, the objective function is defined as the difference between theoretical and182

measured MFC voltage. The simplex search method is used for minimizing183

the objective function [34]. Figure 2 shows the experimental and fitted values184

of MFC voltage as a function of time. As can be seen, there is satisfactory185

agreement between the experimental and fitted values, indicating a reason-186

ably good estimate of the parameters. The final parameter values obtained187

from the curve fitting, and other parameters that were directly obtained from188

13

literature are given in Table 1.189

Table 1: Model parameter values

Parameter Value Unit Notes/Reference

qmaxp 0.9 day−1 Estimated

µmaxp 7.1 day−1 Estimated

qmaxs 0.84 day−1 Estimated

µmaxs 0.6 day−1 Estimated

KSs 425 mg L−1 Estimated

KM 0.11 1 Estimated

Rmin 8 Ω Estimated

Rmax 500 Ω Estimated

Kr1 0.5 L mg−1 Estimated

Kr2 0.004 L mg−1 Estimated

i0 0.048 A m−2 Estimated

Y 30.25 1 Estimated

Emin 0.01 V [23]

Emax 0.66 V [23]

Kdp 0.02*µmaxp day−1 [23]

Kds 0.02*µmaxs day−1 [23]

m 2 1 [23]

γ 663.4 g mol−1 [23]

KSp 103 mg L−1 [29]

An 7 cm2 [29]

V 28 cm3 [29]

b 4 1 [33]

F 96,485 A s mol−1 Constant

R 8.314 J K−1 mol−1 Constant

T 298.15 K Assumed

β 0.5 1 Assumed

3. Results and discussion190

The dynamic model is used to study how the operating parameters, in-191

cluding the initial substrate (COD) concentration and the external resistance,192

influence the MFC performance in terms of maximum voltage, COD removal193

14

rate, Coulombic efficiency and power density. While COD removal rate is im-194

portant when discussing MFC as a wastewater treatment technology, power195

density, maximum voltage and Coulombic efficiency values are useful when196

comparing MFC with standard fuel cells or other power generation technolo-197

gies.198

For this analysis, Sin and Rext are assumed to be 1 gL−1 and 1000 Ω re-199

spectively, when not specified separately. To be consistent while comparing200

different initial COD concentrations, it is assumed that the substrate is re-201

plenished when 95 % of the original substrate has been consumed. It should202

be noted that for whole range of parameter values studied in this work, no203

single simulation took more than 3 seconds on a PC with 2.40 GHz dual core204

processor and 8 GB RAM.205

3.1. Effect of initial substrate (COD) concentration206

Figure 3 shows the change in maximum voltage (V), Coulombic efficiency207

(%) and the COD removal rate (g L−1 day−1) of the MFC system as the208

initial COD concentration (Sin) increases from 0.75 gL−1 to 3 gL−1. As209

can be seen from figure 3a, the maximum voltage generated in the MFC in-210

creases with increase in the initial COD concentration. While this increase211

is sharply linear at lower Sin values (< 1.5 g L−1), it begins to plateau with212

further increase in Sin. Such an increase in voltage can be attributed to the213

increased access to the substrate that the primary microbial population has,214

when the initial COD concentration is increased. This results in enhanced215

metabolic activity, thereby providing more electrons that can be transferred216

to the anode, leading to higher voltage generation [35]. However the amount217

of substrate that can be consumed by the microbial populations is also lim-218

15

ited by the maximum consumption rate, and thus the increase in maximum219

voltage slows down at higher values of Sin. The predicted result (figure 3a)220

from the simulation is also fully consistent with that observed in previous221

experimental studies [29, 36–38].222

Figure 3: Predicted values of (A) maximum voltage, (B) Coulombic efficiency, and (C)

COD removal rate as a function of initial COD concentration.

Figure 4 shows the comparison of predicted and experientially observed223

maximum voltage as a function of initial COD concentration. The exper-224

imental data used is adopted from Liu and Logan [29] and refers to the225

experiments where glucose was used as substrate feed at the anode. As can226

be seen from figure 4, there is fairly good agreement between experimen-227

16

tal data and simulation predictions. The percentage difference of maximum228

voltage between predicted and the three experimental values for initial COD229

concentrations of 0.3, 0.6 and 1.2 g/L, is less than 5%. It should be noted230

that plots shown in figure 3, are based on parameters considering wastew-231

ater as the substrate feed. When wastewater is replaced by glucose (as in232

figure 4), the substrate consumption rates are modified and were evaluated233

following the parameter estimation procedure described in section 2.3.234

Figure 4: Predicted values of maximum voltage (black dashed line) obtained from the

simulation compared to the experimental values (red solid circles) as a function of initial

COD concentration. The experimental data is based on glucose concentration (anode

substrate) and has been adopted from Liu and Logan [29].

As can be seen from figure 3b, the initial COD concentration has an235

inverse relation with the Coulombic efficiency. With increase in initial COD236

concentration, Coulombic efficiency decreases in almost a linear fashion. The237

εE values observed here (10-13 %) are in the similar range as that described238

17

for the air-cathode single chamber microbial fuel cell in Liu and Logan [29].239

As the substrate (COD) at the anode is consumed by both primary and240

secondary microbial populations, an increase in substrate concentration also241

leads to increased competition between the two populations. Thus while242

the primary microbial population consumes more substrate and results in243

higher voltage generation as Sin is increased, the corresponding increment in244

substrate consumption by secondary microbial population leads to decrease245

in the Coulombic efficiency of the cell. Several experimental studies that246

have studied the influence of initial COD concentration have also observed a247

similar decreasing trend of Coulombic efficiency with increase in initial COD248

concentration [32, 39, 40].249

However while increase in Sin results in reduced Coulombic efficiency of250

the system, it helps in improving the COD removal rate, as can be seen in251

figure 3c. Similar to the maximum voltage generation, the increase in COD252

removal rate is sharply linear at low Sin values, but slows down and reaches253

a plateau at higher values. In experimental studies, this improvement in254

COD removal rate at higher Sin values is ascribed to the enhanced hydroxyl255

radical (OH) formation, which helps in further oxidative degradation of the256

COD adsorbed on the surface of the electrode [41]. The observed trend from257

numerical results is consistent with that reported in previous experimental258

studies [32, 40, 42].259

As has been shown, while increasing Sin helps in increasing voltage gen-260

eration and COD removal rate, it also reduces the electron recovery of the261

system. In terms of percentage, increasing the initial COD concentration262

from 0.75 to 3 g L−1, results in 10% increase in maximum voltage, 12.5% de-263

18

crease in Coulombic efficiency, and 75% increase in COD removal rate, for the264

MFC system studied in this analysis. Thus depending on the focus of MFC265

end-use, optimum initial COD concentration can be selected by performing266

a parametric analysis using the current model.267

3.2. Effect of external resistance268

Figure 5 shows the influence of external resistance on the MFC perfor-269

mance. As can be seen, the increase in applied external resistance results in270

decrease in maximum current density (mA cm−2), Coulombic efficiency (%)271

and the COD removal rate (g L−1 day−1) of the system.272

Figure 5: Predicted values of (A) maximum current density, (B) Coulombic efficiency, and

(C) COD removal rate as a function of external resistance.

19

Several studies have suggested the strong influence of external resistance273

on biofilm formation, bacterial diversity, intermediate metabolism, internal274

resistance, and anode potential, which thereby affects the electricity gener-275

ation characteristics and COD removal performance [43–45]. Based on the276

current model, current density is expected to decrease with increase in ex-277

ternal resistance as per Ohm’s law. Higher current densities at low external278

resistance drive more effective utilization of the COD, which subsequently279

subsides with increasing Rext. Also, as the current density decreases (with280

increasing external resistance), more substrate becomes available for sec-281

ondary microbial population for purposes other than electricity generation,282

which would lead to decrease in Coulombic efficiency, as observed in the re-283

sults. Overall, increasing Rext from 100 to 2000 Ω, decreases the maximum284

current density by 70 %, Coulombic efficiency by 55 % and the COD re-285

moval rate by 18.5 %. These trends are in agreement with results reported286

in previous experimental studies [29, 32, 39, 40, 46].287

Li and Hua [46] studied the effect of external resistance on a denitrifying288

microbial fuel cell and found that as the external resistance was decreased, it289

increased the COD removal rate, current density and the Coulombic efficiency290

of the system. Similarly Jadhav and Ghangrekar [47] observed that exter-291

nal resistance is inversely proportional to MFC current and COD removal292

rate, however the decrease in COD removal efficiency was only marginal293

with increasing Rext. Katuri et al. [45] also observed that the extent of COD294

removal in MFC only decreased by 6% even after increasing external resis-295

tance manifolds. The results from the current analysis also show a relatively296

small decrease (18.5 %) in COD removal rate even when Rext is increased 20297

20

times. Buitrn and Moreno-Andrade [48] studied the performance of a single-298

chamber microbial fuel cell degrading phenol and found that while increasing299

external resistance resulted in lower current densities, it did not adversely300

affect the substrate degradation rate.301

3.3. Influence of operating parameters on power density302

Figure 6: Predicted values of maximum power density as a function of initial COD con-

centration at different values of external resistance.

Figure 6 shows the effect of initial substrate concentration on the maxi-303

mum power density (mW m−2) of the MFC system for four different external304

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.306

The increase in power density is consistent with the increase in maximum307

voltage with increasing Sin as observed in figure 3A. For each resistance, the308

21

power density reaches a maximum value and plateaus after showing a linear309

increasing trend at lower Sin values. Also, high current densities and high310

Coulombic efficiencies at lower external resistance promotes more effective311

utilization of substrate and results in higher power density. Similar rela-312

tionship between power density and initial COD concentration & external313

resistance has been reported in previous works [32, 39, 43, 49, 50].314

For the range of parameter values studied here, maximum power density315

of 730 mW m−2 is obtained for Sin=3 gL−1 and Rext=100 Ω. Experimental316

studies have also pointed out that increased utilization of anode for electrode317

respiration by mixed microbial population at low start-up external resistance318

[43] and increase in ionic strengths due to improved solutions conductivities319

at higher COD considerations [32], could be some of the other possible ra-320

tionales for increase in power densities at lower Rext and higher Sin.321

The current model is applied to MFC studied by Zhang et al. [50] to322

compare the effect of external resistance on power density, as obtained from323

the experiments to that predicted from the simulations. Model parameters324

are first estimated by fitting the current density profile from Zhang et al.325

[50] at external resistance of Rext = 250 Ω. Following this, the model is326

used to calculate power densities at two other external resistances, 50 Ω and327

1000 Ω, which are compared with the experimental values as shown in figure328

7. As can be seen from the figure, there is close agreement between power329

density values obtained from model predictions and those measured in the330

experiment.331

22

Figure 7: Predicted values of power density (black dashed line) obtained from the sim-

ulation compared to the experimental values (red solid circles) as a function of external

resistance. The experimental data has been adopted from Zhang et al. [50].

As seen from figures 3-7, the mathematical model accurately captures the332

trends similar to that observed in vast majority of studies, showing influence333

of external resistance and initial COD concentration on important MFC per-334

formance indices, such as the Coulombic efficiency, COD removal rate, and335

the power density. However some experimental studies that have observed336

either no change or a reverse relationship [48, 50–53]. The factors responsi-337

ble for such variations could be very specific to the design and operational338

conditions of the particular MFC and will need to be studied individually.339

The proposed mathematical model can be easily implemented by exper-340

imentalists and used to run parametric analysis of important operating pa-341

rameters using basic computational resources, to determine the optimum342

range of operation of a given MFC system. The optimum range required343

23

depends on the targeted end-use of the MFC, which can be focused on just344

power generation or wastewater treatment alone, or have a dual goal with im-345

proving the efficiency of both outputs. As seen from figures 3 and 5, increas-346

ing initial substrate concentration increases the maximum voltage and power347

density of the MFC as well as the substrate (COD) removal rate. However348

it also decreases the Coulombic efficiency of the system. Thus even though349

the maximum voltage generated increases initially with increase in substrate350

concentration, the total energy that could be extracted from the substrate is351

getting reduced. Thus it is important to choose the initial substrate concen-352

tration judiciously considering all factors such that the MFC system could353

be operated at optimum efficiency. Similarly the optimum range of operation354

of other important system parameters, like the external resistance, electrode355

surface area, volume of the MFC chamber, choice of substrate and micro-356

bial population (which determine the maximum substrate consumption rate,357

maximum growth rate and half saturation coefficient), can also be studied358

using the proposed modeling framework. The model is not only robust but359

also computationally inexpensive, making it a useful tool for MFC system360

optimization.361

The analysis also suggests that the Achilles’ heel of the microbial based362

electrochemical technology is the potential loss that occurs during transfer363

of electrons from bacteria to the electrode, which along with some other364

factors is one of the major reasons for the poor efficiency of these systems.365

Thus in addition to developing electrodes that can provide better prospect366

for biofilm attachment and growth, as well as increased effective surface area,367

equal or even more focus is needed to be directed towards understanding and368

24

improving the extracellular electron transfer.369

In the current form, the model assumes a non-limiting cathode, however370

including the influence of cathode polarization would enhance the model ac-371

curacy. Similarly in the current model, the presence or influence of any372

ion-exchange membrane between the two electrodes, and dependence of in-373

ternal resistance and open circuit voltage on external load (resistance), has374

not been included. Accounting for these additional factors would further375

increase the applicability of the model to not just two chamber MFC but376

also other bioelectrochemical systems such as microbial electrolysis cell, mi-377

crobial electrosynthesis system, etc. Introducing suitable modifications that378

can improve the model predictions without compromising on the speed of379

convergence, are planned for the near future.380

4. Conclusion381

This study presents a simple, fast converging mathematical model to de-382

scribe a microbial fuel cell system. The model includes governing equations383

for mass transfer of substrate and biomass, bio-electrochemical kinetics and384

charge transfer. The model rigor is further improved by using substrate385

concentration, a simple measurable quantity, to calculate the dynamic open386

circuit voltage and internal resistance of the cell. Model analysis predicts387

a positive influence of increasing initial substrate concentration and lower-388

ing external resistance on the power density and substrate removal rate of389

the system. These predictions are in accordance with previously reported390

experimental studies. The mathematical model is easy to implement and is391

computationally inexpensive, and can thus serve as a good starting point to392

25

determine the operating conditions for fairly optimum system performance,393

before more comprehensive experimental/numerical studies are performed.394

Acknowledgements395

The authors would like to acknowledge the financial support by the Natu-396

ral Environment Research Council (NERC) UK project grant: NE/R013306/1.397

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