Mathematical stimulation for bioelectrochemical behavior of a dual- chambered microbial fuel cell (MFC)

In this study, a steady-state bioelectrochemical model was developed to simulate the correlation between the acting overpotential and the produced current that is accounting for anode polarization. This study aimed to analyze the performance of a dual-chambered microbial fuel cell (MFC) equipped with two bio-anodes and fueled with real refinery oily sludge having a COD concentration of 13890 mg /L. Anode polarization data revealed a maximum current density of 6.07 A/m of the substrate at an overpotential of 1.83 V. In addition, the behavior of the experimental measurements revealed the dominance of the ohmic losses in the overall anode overpotential compared to activation and mass transfer losses, respectively. On the other hand, the suggested mathematical model was verified significantly by the obtained experimental data, achieving a determination coefficient (R) of 0.96. Actual sustainable energy was also obtained using the reductive decrease of anode potential (RDAP) and it was found that the sustainable energy for this corresponding system can be attained when applying 17.6 KΩ as the external resistor.


Introduction:
With the increasing demand of a new sustainable energy source to be an effective alternative for the fossil fuels and to solve the greenhouse emission crises, bioelectrochemical systems especially microbial fuel cell (MFC) have paid more attention in the last few decades as a possible main energy source [1]. MFC systems brought a lot of attention as they have the capability of simultaneously remediate and produce electricity directly from both domestic and industrial wastes. It has been proved by intensive studies that MFC can degrade high organic content wastes, inorganic constituents and even toxic contaminants [2]. This type of fuel cell has some common properties with chemical fuel cells since bothcases (anode and cathode side) are separated by a cation exchange membrane, the fuel is oxidized on the anode side and the oxidant (oxygen) reduced on the cathode side. However, MFC uses organic substrates as fuels to generate power and these organics are catalyzed by microbial communities while in chemical fuel cells, the fuel oxidation is catalyzed by noble metals [3]. Hence, power generation in MFC is conducted by bacterial catabolism, electron transfer from bacteria to the anode, reduction of electron acceptors at the cathode and proton transfer from the anode to the cathode via the separating membrane. The performance of any type of MFC relies on several parameters such as the type of biocatalyst and substrate components, membrane or separator characteristics, mixing and diffusion phenomena, surface area of electrodes [4]. In order to enhance the performance of MFC, it is necessary to optimize the design and operation of such fuel cells by developing mathematical models. Various mathematical models were developed to describe biofilm kinetics, biochemical reactions, and biosensors for toxic constituents, enzymatic reaction, electrochemical limitations, and polarization models. A comprehensive review on the most common mathematical models describing MFC, outlining the advantages and disadvantages of these models was conducted [5]. Moreover, a wide survey on the dynamic models of MFC and along with the general review on bioelectrochemical systems BES optimization and control was shown [6]. They point out that mathematical models that account for the biofilm growth dynamics catalyzed by mixed consortium of microbes can be most useful in BES system optimization. A detailed classification of MFC models into mechanism-based and application-based models was presented [7]. While describing these two categories of models, they presented the underlying approaches and usability of the different types of models.
In this study, a bioelectrochemical mathematical model was suggested through the use of

MFC installation and operation
A cylindrical-shaped, dual-chambered MFC was constructed from the perspex material.
The anodic chamber and cathodic chamber were identical in shape having the same full capacity of 1 liter as presented in Figure (  (2) Electron transfer between the two chambers through the separating membrane were considered to be negligible, hence, all the produced electrons are either transformed directly into electricity or consumed in multiple paths in the anodic-chamber. (4) The microbial kinetics is the limiting factor in the overall anode potential.
(5) Oxygen transport from the cathodic-chamber through the membrane was neglected.
(6) Constant temperature and pH were considered fully controlled and maintained constant.

Model Formulation
The Butler-Volmer-Monod (BVM) is a model established depending on easy mathematical expressions of the basic biochemical reactions and electrochemical reactions. The BVM uses the idea of redox state of microorganisms that controls the anodic processes. As the previously reduced microorganisms by chemical electron donors touch the anode, they transfer to the surface of anode their outer orbit electrons to return to its original oxidized form. Oxidized microorganisms are then free in interacting with another donor molecule [8]. In this work, the anode overpotential is precisely demonstrated by the Bulter-Volmer-Monod model as many researchers confirmed that the Butler-Volmer-Monod model was a better tool to describe experimental data than the Nernst-Monod model. The following three chemical equations are applied at the anode:

No.26-(3) 2020 Journal of Petroleum Research & Studies (JPRS)
E75 dt dt S + X ox X c P + X red (1) where; K 1, K 3 , K 5 and K 2, K 4 , K 6 are heterogeneous forward and backward rate coefficients, respectively. S and P are the substrate and product concentrations.
K 5 and K 6 was formulated as described by [8,9]: Where I is the forward current density, n number of electron moles participated in the reaction and F is the Faraday constant (96485 C/mol).
However, by assuming quasi-steady-state conditions, then X ox and X red can be expressed by the following differential equations depending on the mass balance of the biocatalyst: d X ox = 0 = -k 1 .S . X ox + k 2 . X c + k 5 . Xr edk 6 X ox (4) d X red = 0 = k 3 . X c -k 4 . P . X red -k 5 . X red + k 6 X ox (5) Assuming that X red is at a constant value and quantity X c could be represented by the following equation: Where X T is the total amount of redox. In order to simplfy the eqs. (4) and (5), k 1, k 2 and k 3 were eliminated by introducing the substrate affinity (K m ) which can be defined as the halfsaturation constant analogous to that of Michaelis-Menten enzyme kinetics [10], in addition to the product inhibition constant (Kp) refers to the effect of product concentration (P) at which an inhibition of the turnover rate is occured [11]: Moreover, by assuming microbial activity is limiting in the overall anode performance as mentioned earlier, a maximum current density can be introduced and defined as the produced current when all microorganisms generate reduced intermediate X red , thus: Applying more simplifications and binding all the previously presented definitions with the differential balances in eqs. (4) and (5) yields the used Butler-Volmer-Monod equation: Where, ɳ is the anode overpotential vs. reference electrode. (K 1 ) is a lumped parameter represent the ratio between biochemical and electrochemical reaction rate constants, while, (K 2 ) is a lumped parameter represent the ratio between the forward and backward biochemical rate constants, (α) is the anodic charge transfer coefficient.

Experimental procedure
The MFC system was continuously operated for 60 days of full operation period in a continuous mode. After a stable cell potential was achieved , a reference electrode was connected to the anode in the MFC. The (Ag/AgCl) reference electrode was selected to be utilized as it is commonly used in electrochemical measurements for environmental purposes due to its simplicity, stability and nontoxicity [12]. In a half cell measurement, the reference electrode should be considered as the reduced electrode (i.e. anode) and should be connected to the cell as demonstrated in Figure (2). Anode polarization curve was obtained and by connecting the MFC with various external resistances ranged from 10 Ω to

No.26-(3) 2020 Journal of Petroleum Research & Studies (JPRS)
E77 100 KΩ then recording the measured anode overpotential to be plotted against the produced current density.
In order to be able to perform the BVM model, it was necessary to obtain one essential standard parameter, the substrate affinity constant (K m ), which was specified and outlined [8]. The MFC was fed with several concentrations of acetate (as substrate) and by specifying the acetate concentration at which the current was first produced, this point represents the K m.

Sustainable Energy
A steady-state operating MFC can only be attained, when the overall produced current equals the current consumed for an extended time operating at steady-state conditions and hence, the power production is then said to be sustainable. Many scholars have considered the steady-state operation to be sustainable [13,14]. It is important to identify the steady state potential value to determine the maximum sustainable power that can be obtained from this MFC system. By increasing the external applied resistances within equal time intervals (10 min), anode current and potential was recorded to evaluate the harvested

No.26-(3) 2020 Journal of Petroleum Research & Studies (JPRS)
E78 sustainable energy from the system. Sustainable power calculations were conducted by calculating the considering the relative decrease in anodic potential (RDAP) as described: Where, E OCV is the open circuit potential of the anode (V) and E A is the measured anode potential at a specific external resistance (V).

Results and Discussion
The performance of this MFC system was evaluated according to the anodic electrochemical performance which is likely to govern the overall cell power production.  were first evaluated and adjusted to obtain better fittings. For α, it was assumed to have the value of 0.9. This value was based on the behavior of the experimental curve, where it can be clearly noticed form Figure (6) that the region of the mass transfer loss was very minimal. A for the lumped parameters: K 1 and K 2 , they were obtained by conducting a numerical method of fitting the experimental data to the simulated data using MATLAB software version (R2017b). It was It was found that the optimum K 1 and K 2 were 40 and 26.5, respectively, approaching the values estimated by [15]. The BVM model estimated, measured and assumed parameters and constants used in the simulation are given in Table   (1).

Obtained sustainable energy
An estimation of the relative decrease in the anodic potential (RDAP) was conducted in order to assess the maximum sustainable power. The variation of percent deviation of anodic potential with respect to applied external resistance is shown in Figure (7). At lower resistances, the linear fit represents the region at which energy generation is only limited by kinetics or internal resistance. While, at higher external resistances, the explained linear fit represents the region at which the energy generation is only limited by the external resistance. At certain operating conditions, external resistance (R ext ) equals internal resistance (R int ) limitations; identifying this region is important to determine the external resistor at which the sustainable energy is approached [16]. This specific region always lays between the intersect of the two plotted lines. The dotted horizontal line plotted from the intersect point to meet the curve at a certain point, which represents the desired resistor value. As can be clearly seen in Figure (7), this point was spotted at 17600 Ω for this specific MFC system and the corresponding sustainable power generation was 0.17 W/m 3 .

Activation loss
Ohmic loss

Conclusions:
This study demonstrated the feasibility of the kinetic steady-state, one-dimensional model represented by Butler-Volmer-Monod model for describing the enzymatic reactions and the electrochemical reactions that governs the anode potential and consequently the MFC electrical generation. The simulated data showed a very good and reasonable match to the experimental data verifying the applicability of this model to this MFC system. Moreover, the good performance of the BVM model refers to the efficient adjustment of the assumed, estimated and fitted parameters such as the transfer coefficient α, the lumped parameters K 1 and K 2 and the substrate affinity constant K m . Hence, the suggested model is a useful tool to improve MFC understanding and to optimize MFC design and operation. Nevertheless, this mathematical model is simple to be implemented and suitable for use in real-time MFC operation.