Cellular non-nonlinear network model of microbial fuel cell
Michail-Antisthenis Tsompanas, Andrew Adamatzky, Ioannis Ieropoulos, Neil Phillips, Georgios Ch. Sirakoulis, John Greenman
CCellular non-nonlinear network model of microbial fuelcell
Michail-Antisthenis Tsompanas a, ∗ , Andrew Adamatzky a , Ioannis Ieropoulos b ,Neil Phillips a , Georgios Ch. Sirakoulis c , John Greenman b a Unconventional Computing Centre, University of the West of England, Bristol BS16 1QY,UK b Bristol BioEnergy Centre, University of the West of England, Bristol BS16 1QY, UK c Department of Electrical and Computer Engineering, Democritus University of Thrace,Xanthi 67100, Greece
Abstract
A cellular non-linear network (CNN) is a uniform regular array of locally con-nected continuous-state machines, or nodes, which update their states simul-taneously in discrete time. A microbial fuel cell (MFC) is an electro-chemicalreactor using the metabolism of bacteria to drive an electrical current. In aCNN model of the MFC, each node takes a vector of states which represent ge-ometrical characteristics of the cell, like the electrodes or impermeable borders,and quantify measurable properties like bacterial population, charges producedand hydrogen ions concentrations. The model allows the study of integral re-action of the MFC, including temporal outputs, to spatial disturbances of thebacterial population and supply of nutrients. The model can also be used toevaluate inhomogeneous configurations of bacterial populations attached on theelectrode biofilms.
Keywords:
Microbial fuel cells, cellular non-linear network, spatial models
1. Introduction
Microbial Fuel Cells are renewable bioelectrochemical transducers that con-vert biochemical energy into electricity. MFCs empower simultaneous treatmentof waste-water and energy extraction from mixed organic media via the usageof microbial consortia as bio-catalysts. In addition to treating anthropogenicwaste and wastewater, whilst producing rather than consuming electrical energy,MFCs have the ability of degrading toxic pollutants and the advantage of notburdening further the carbon cycle in the way fossil fuels do [1]. They bear someresemblance to conventional fuel cells, given that they comprised two compart-ments, the anode and the cathode, divided by a Proton Exchange Membrane ∗ Corresponding author
Email address: [email protected] (Michail-Antisthenis Tsompanas)
September 3, 2018 a r X i v : . [ n li n . C G ] M a r PEM), where oxidation and reduction reactions occur. A major difference isthat MFCs use abundant, renewable fuels, such as organic substrates, that aremetabolised by bacteria, whereas chemical fuel cells are fuelled by pure com-pounds (which could be toxic as methanol or explosive as hydrogen) oxidised byprecious metals. As a result, their inexpensive functionality and maintenancedesignate MFCs as a viable solution for producing energy in isolated areas.Despite the fact that MFCs were proposed more than a century ago [2], it isstill a subject of rigorous research, due to the increased power densities that havebeen achieved in the last decade. In addition, since they operate in ambient con-ditions (ambient environment temperature, atmospheric pressure, neutral pH)and given the aforementioned advantages, they can efficiently support systemsfor applications like remotely deployed sensors and robotics [3]. Nonetheless,their analysis in copious conducted experiments reveal significant limitations ontheir performance due to low microbial activity (low growth rate or metabolicrate, due to non-optimal growth conditions or unsuitable microcosm — insu-ficient anodophiles), ohmic losses, mass transfer limitations on the electrodesurfaces, non-optimised electrode architectures and transfer potential throughthe PEM. The process of defining the factors that limit the performance ofMFCs can lead to more efficient designing methods. Although some techniquesused in the conventional chemical fuel cells could be adopted, they can not beexpected to provide the same results due to the fundamental differences of thesystems particularly due to the biological nature.MFCs are complicated devices that contain bio-electrochemical reactions,mass and charge balance principles, biotic or abiotic transformation processes.As a result, their analysis and design process require a multidisciplinary ap-proach with background in electrochemistry, microbiology, physics and engi-neering. On top of that, there are numerous differentiations in the MFCs stud-ied which range from their configurations (having two chambers separated bya PEM or a membrane-less single-chamber MFCs) to the type of their incor-porated mechanism of donating electrons to the anode electrode (mediated ormediator-less MFCs).Given the complexity of these systems, the number of parameters that affecttheir outputs and the costs in time and money needed to perform laboratoryexperiments, the development of computerised mathematical models simulatingthese systems is of great importance. The implementation of modelling tech-niques can contribute to the investigation of the principles covering their oper-ation and affecting their performance, producing better arrangement designs ofMFCs and working circumstances.To address spatial dynamics of biophysical processes in a MFC we designed acellular non-linear network (CNN) model. A CNN is a uniform regular array oflocally connected continuous-state machines, or nodes, which update their statessimultaneously in discrete time [4, 5]. Essentially, CNN is a finite-differencescheme with time step 1. A CNN is a subset of cellular automata (CA). A CAis the same as CNN but states of nodes are discrete. CA and CNN are oftenmixed, many researchers do not differentiate these two types of machines. Whatis imitated in CA can be imitated in CNN and both offer powerful modelling2apability as well. There are several studies published on CNN/CA or hybridmodels of reaction-diffusion [6, 7, 8, 9, 10, 11, 12, 13, 14, 15] and prototypingof chemical computers [16, 17, 18, 19, 20, 21] and molecular computers [22],spatial dynamics of bacterial colonies [23, 24, 25, 26, 27]. Other approachesrelevant to CNN modelling of MFCs are CA models of fuel loading patternsin nuclear reactors [28], dynamics of nuclear reactors [29, 30, 31, 32], neutrontransport [33], waste water treatment by aerobic granules [34], sequencing batchreactor [35], fluid flow in a porous medium [36].The proposed CNN-based model simulates biochemical and electrochemicalreactions in a MFC based on synthetic redox mediators. To the best of the au-thors’ knowledge this is the first attempt to simulate the outputs of a MFC witha CNN model. Despite the fact that the application of CNNs have been widelyused to simulate several biological, chemical and physical processes as indictedpreviously, the novelty of this study can be pinpointed in the fact that all theprocesses and, thus, the behavior of a batch-fed MFC has not been previouslypresented in a single CNN lattice with a local state comprised of all the criticalquantities (several chemical species concentrations, biomass concentrations andcurrent produced). Such a model will allow for a detailed analysis of integral out-comes of spatial processes inside MFCs, including a possible uneven distributionof nutrients in the MFC chamber, patterns of bacterial population in biofilmscovering electrodes and distributions of diffusing metabolites. Nonetheless, theuse of CNN as the mathematical basis for the model allows the employment ofthe inherent fully parallel nature of synchronised locally interconnected simpleunities. The subject device is a two chamber MFC with the presence of elec-troactive microbes in suspension in the bulk liquid and forming biofilms on thesurface of a planar anode electrode and assuming electron transfer from the mi-crobes to the electrode with the use of an externally added diffusible chemicalmediator. That mechanism can be differentiated to emulate various types ofMFCs, that will be the aim of future works. Nonetheless, the present study canlead towards exploiting the parallelism of the simulating tools and, as a result,intensively accelerating the simulation of MFCs, by the implementation of theCNN-based algorithm on parallel hardware, as illustrated in [52, 53].
2. Previous work
Despite the intense investigation in the laboratory experimental field to op-timise the performance of MFCs, results from computational models are notderived with the same rate. Moreover, the few models developed are targetingspecific MFC configurations each and are so strictly specified that they becomeimpractical for implementation on different configurations. The first model pre-sented [37], investigated a single population using an external mediator as re-ceptor of electrons. That model analysed the correlation of the concentrationof the external mediator with the higher possible power output.The authors of [38] introduced the simulation of a MFC with an added medi-ator and several populations of suspended and attached biofilm microorganisms.The model was developed on two or three dimensions providing the resultant3urrent produced by homogeneous or not biofilms. The results were derived bytaking into account several parameters, like the content of different microbialspecies, the amount of suspended microbes compared with ones attached, thepotential of the mediator, the initial concentrations of the mediator and thesubstrate and many more. The results provided were compared with experi-mental data from a batch MFC fed with acetate and inoculated with Geobacterand found in a good agreement.The model presented in [38] was updated in [39] with the incorporationof International Water Association (IWA’s) anaerobic digestion model (ADM1)[40]. The coexistence of several types of methanogenic and electroactive bacteriais simulated, taking into account whether they are suspended or attached tothe anode electrode. A batch MFC was simulated to test the effects of theelectrical circuit on the population of the microorganisms and the results werecompared with laboratory data. The model is also based on a one, two orthree dimensions partial differential equations system to represent the spatialdistribution of solutes in the biofilm.In [41], a model simulating a MFC with only suspended microorganismsand externally inserted mediator was studied. In this model the conservationof mass for the dissolved ingredients has a basic role. The biomass growth isnot studied in that model while a batch mode MFC which was periodically fedwas simulated. The results were compared with laboratory data from MFCsinhabited with suspended
Proteus cells and incorporating thionine as a media-tor, proving accurate representation of the system. The model introduces theideas of endogenous metabolism or intracellular substrate storage to justify asmall amount of current present between the feeding pulses. As the conditionsand concentrations in the anode were considered uniform, a one dimensionalsolution of the algorithm’s equations was presented. Some key parameters ofthe equations used were extracted from fitting the outputs to the experimentaldata, while some others were estimated.Another study [42] proposing a mathematical model for MFCs, was basedon two dimensional macroscale mass balance equations and microscale biofilmevolution. The model contains hydrodynamic calculations and mass and chargebalances through diffusion, convection and electromigration to simulate the cur-rent output, species concentrations and pH distributions throughout the anode.Nonetheless, the possibility of depicting on two or three dimensions irregularbiofilm and electrodes configurations and simulate the effects on the MFC oper-ation and outputs was provided. The model was used to reproduce the system’soutputs such as its pH distributions, the effect of multiple communities of elec-troactive, methanogenic and fermentative bacteria existing in the anode biofilmand the effect of the flow over or through complex electrodes.The authors of [43] presented a model simulating a MFC with inspirationof models simulating chemical fuel cells. A mediator-less two-chamber con-figuration was studied in steady and dynamic states, with the combination ofbiochemical reactions, ButlerVolmer expressions, mass and charge balance equa-tions. Also, given the assumption that the anode is under anaerobic conditionsthe ADM1 was used to simulate the biochemical reactions. The authors argue4hat the effects imposed by the reactions in the cathode on the performance ofthe MFC are noteworthy. There were some laboratory experiments conductedin the context of this study to determine some of the parameters required by themodel. Despite that, it was not possible to extract some parameters that wereestimated by a mathematical method of best fitting the results of the modelwith the experimental data.A one dimensional model of a steady state MFC incorporating charge, heatand mass transfer and biofilm formation was suggested [44]. The model sim-ulates the electrical performance of the MFC and the evolution of the biofilmformation that are affected by the inputs, like temperature and substrate con-centration. The configurations simulated were the same as in [43], however aminor improvement in the results’ agreement was realised mainly because of theaddition of the effects that heat transfer has on the anode and the cathode.The authors of [45] presented a model that concentrates on the effects thatthe operational parameters of a MFC, such as the external ohmic resistanceand organic loads, have on communities of anodophilic and methanogenic mi-crobes. The ordinary differential equations that are included by the modelwere subjected to fast numerical solution techniques in order for the model tobe more efficient than the one presented in [38] but not oversimplified as theone presented in [37]. Moreover, the model was calibrated with the analysis oftwo single-chamber membrane-less aircathode MFCs and then validated on twoother MFCs.Another type of MFCs, namely the membrane-less single-chamber MFC,with molasses as fuel was modelled in [46]. The model consisted of a systemof differential equations depicting the diffusion and concentration profiles of themolecules, which are solved with a numerical approximation technique, namelythe implicit finite difference method. Despite the fact that the authors find theresults of the model in accordance with the theoretical principles, they suggestthat more detailed evaluation of the model is required in order to serve theoptimisation of the performance of MFCs.The novel model based on CNN simulating the performance of MFCs pro-posed here, was motivated by the limitations imposed by the previously re-ported models. Some of these limiting factors are the representation of theMFC compartments in one dimension disregarding possible inhomogeneities inthe electrodes or the biofilms and the implementation of complicated differentialequations that make models time consuming and compute-intensive. Moreover,the use of CNN is advantageous compared with past models, as complicatedcomputations are emerging from synchronised local interactions of basic sim-plistic entities. The given homogeneity of CNNs and the local interconnectionsare essential features for mapping great amounts of cells in digital circuit sys-tems, taking full advantage of a parallel implementation of the model.
3. The proposed model
The model proposed here is simulating the performance of a two-chamberacetate-fed MFC with a biofilm attached on the anode electrode in batch mode.5he following assumptions are made:1. The electrode is considered to be a 2D solid, which although is in contrastwith best practices [47], it is a necessary simplification for the model.Important characteristics such as low biofilm populations — due to thelimited available outer area for colonisation — and biofilm erosion will beconsidered in future studies.2. The microbial growth is proportional to the spatial concentrations ofgrowth-limiting factors like acetate and oxidised mediator; however themetabolic products formed are not considered as primary factors effectingthe performance of the MFC and, thus, neglected by the model;3. An ideal PEM is considered, allowing only protons to pass through butnothing else (like other cations, carbon dioxide, acetate and oxygen) anddoes not limit the performance of the MFC;4. An added mediator mechanism is used; as a result, electrons are shuttledfrom the microbes to the anode electrode by the reduced form of a chemicalwhich is oxidised at the surface of the electrode;5. The heat generation by electrochemical reactions occurring on both elec-trodes, anode compartment and biofilm and the heat flux is not considered;6. Temperature is considered fully controlled and kept constant;7. Solute materials are transported in space through molecular diffusive forcesand migration of charged molecules, due to the electrical potential field.The latter is not studied in the current version of the model;8. No gravity forces were included for any of the material to keep the modelas straightforward as possible;9. The anode compartment is defined as a continuously stirred tank reactor;10. The reactions in the cathode side are prescribed as constant and are nota limiting factor of the MFC’s performance.Note here, that neglecting the migration of charged molecules in the electricalpotential field generated by the voltage outputs of the MFC can not be treatedas an oversimplification of the model. That assumption stands, given that theeffects of that phenomenon are minor due to the low voltages produced by thesystem and the high conductivity of the anolyte solution. Moreover, previouslypublished models of MFCs [48] suggested that the results with and without theincorporation of the phenomenon of electrical migration provides results withan average difference in power densities of 1.92% throughout the range of MFCvoltage outputs. Nonetheless, most of the previous work done on the field ofMFC modelling neglected that phenomenon [38, 43, 44].The grid of the CNN model is n × n (here is consisted of 68 ×
68 cells forillustration reasons), and is representing a cross section of an area in the anodecompartment near the electrode of a given batch-mode MFC. The size of thesimulated area, represented by one CNN cell is set with the actual geometricaldistances in mind. The size of each cell has been chosen to be sufficient toillustrate an area where the abstraction of homogeneous reactions occurring canbe justified and fluxes of soluble chemicals can be depicted. Namely, the area of6 CNN cell is defined as a 1 µm × µm area of the anode compartment, an areacomparable with the typical dimensions of some species of bacteria. As a result,the CNN cell is small enough to accommodate the two dimensional projection ofthe existence of a single bacterium in a the specified area. The size of a CNN cellcan be trivially increased after the appropriate decrease of the dimension-relatedparameters and without changing anything in the algorithmic configurations ofthe model. This procedure would effect negatively the accuracy of the model,as a higher level of abstraction will be assumed, however the execution time willbe decreased as a smaller amount of elements will be calculated.The von Neumann neighbourhood was used, meaning that each cell hasfive neighbours, the four closer adjacent cells to the central one (located in thenorth, south, west and east directions), including the central one. Note here thatwhile the Moore neighbourhood could be used to provide more realistic results(as a more extended neighbourhood would be used including nine cells), the vonNeumann neighbourhood was chosen to maintain an efficient point in the trade-off between complexity and accuracy. While the accuracy of the calculationsis slightly reduced and the complexity of the model is reduced, its speed ofexecution is considerably enhanced. Each cell is defined by its state consisted ofseveral parameters that are simultaneously updated throughout the simulationtime steps, according to the states of its neighbours and by the local rule. Theparameters consisting the state of each cell are the following: S ti,j = [ T ti,j , X ti,j , ( C Ac ) ti,j , ( C Mred ) ti,j , ( C Mox ) ti,j , ( C H ) ti,j , O ti,j , I ti,j ] (1)where i and j are the dimension indexes that establish the location of each cell inthe grid and t is the current time step. T t ( i,j ) is illustrating the way the relativearea, represented by each cell, can be classified depending on its characteristicstructure. This parameter can change through the simulation time steps andcan have the following values: T t ( i,j ) = , for borders (confining the movement of elements),1 , for anode electrode surface,2 , for biofilm attached to the electrode,3 , for bulk liquid of anolyte. (2)Parameters X t ( i,j ) , ( C Ac ) ti,j , ( C Mred ) ti,j , ( C Mox ) ti,j and ( C H ) ti,j indicate the con-centrations of ingredients that participate in the significant reactions in a MFCwhich are investigated for designing the model. Note here, that the units of allthe parameters in the following equations and their values for an example con-figuration presented in the following section can be found in Table 1. Namely, • X t ( i,j ) represents the bacteria populations present in the anode compart-ment either in suspension or constituting a biofilm on the electrode,7 ( C Ac ) ti,j represents the electron donor material for the microbes (the fuelof the MFC), here assigned as acetate, • ( C Mred ) ti,j represents the added mediator chemical in reduced form, while • ( C Mox ) ti,j represents the added mediator chemical in oxidised form, • ( C H ) ti,j represents protons/hydrogen ions that are released from the reac-tions occurring in the anode electrode surface.Parameters O ti,j and I ti,j represent the locally imposed over-potential andproduced current density in each cell ( i, j ). The O ti,j parameter studied is theactivation over-potential, while the effects from the ohmic and concentrationsover-potentials are implemented in the local rule calculations [38].The definition of the anode compartment as a continuously stirred tankreactor is based on the fact that phase mixture is quite faster than the electro-chemical and biochemical reaction rates [43]. As a result, all cells representingan area of anode bulk liquid will obtain the same state at the same time step.Also, the cells representing borders, theoretically undergo no changes through-out time, thus, their states are regulated to a constant set of values, equal tozeros.The local rule used in the model simulates the kinetics and reactions of allmaterials. The calculations depend on the type of the cell represented ( T t ( i,j ) ) atthe given time step and employ the concentrations of chemicals in the predefinedarea (the neighbourhood of each cell) to provide the concentrations throughouttime for each CNN cell. The concentrations within cells simulating the biofilmis given by the following expression:( C Ac ) t +1 i,j =( C Ac ) ti,j + D Ac × (( C Ac ) ti − ,j + ( C Ac ) ti +1 ,j + ( C Ac ) ti,j − + ( C Ac ) ti,j +1 − N i,j × ( C Ac ) ti,j ) − ( rf Ac ) ti,j (3)where D Ac is the diffusion coefficient of acetate, N i,j is the number of avail-able neighbour cells of the cell ( i, j ) (not borders or the electrode) and rf Acti,j represents the rate at which the acetate is consumed by bacteria. This consump-tion rate depends on the local concentrations of acetate, biomass and oxidisedmediator and is defined as a double Monod limitation kinetic equation [49]:( rf Ac ) ti,j = Q Ac × X ti,j × ( C Ac ) ti,j K Ac + ( C Ac ) ti,j × ( C Mox ) ti,j K Mox + ( C Mox ) ti,j (4)where K Ac is the Monod half-saturation coefficient for acetate, K Mox is theMonod half-saturation coefficient for the oxidised mediator and Q Ac is the max-imum specific rate constant for microbial consumption of acetate.Equation 3 is actually the numerical approximation of the implicit finitedifference method (eq. 5) of Fick’s second law in two dimensions [46], with the8ddition of the consumption rate of acetate by bacteria. Fick’s second law (eq.6) is used to simulate the transport of solutions or liquids in other liquids, statesthat the change through time of concentration is depending on the differencesof concentration and is derived from mass balance principle of a species in afluid continuum [50]. d fdx (cid:12)(cid:12)(cid:12)(cid:12) x = f ( x + ∆ x ) − f ( x ) + f ( x − ∆ x )∆ x (5) ∂c∂t = D ∂ c∂x (6)where c is the concentration of a species, D is the diffusion coefficient, t is timeand x is the length.Furthermore, the concentration of acetate in the bulk liquid is equal for allcells, as mentioned previously, due to the assumption of the anode as a con-tinuously stirred tank reactor. Also, biofilm development, along with substrateconsumption rate, and solute mass transport, through diffusion, occur at differ-ent time scales, in the order of hours and seconds, respectively. Consequently,an immediate effect of the substrate consumption in the biofilm on the bulkconcentration will be demonstrated in the mathematical formulas. Taking allthe above into consideration, the concentration of acetate in the bulk liquid isgiven by the following equation:( C Ac ) t +1 i,j = ( C Ac ) ti,j + ( rb Ac ) ti,j + (cid:80) ( rf Ac ) tk,l V ta + (cid:80) ( re Ac ) tm,n V ta (7)where ( rb Ac ) ti,j is the net rate of reactions in the bulk, (cid:80) ( rf Ac ) tk,l is the overallreaction rates in the whole biofilm ( ∀ k, l that T t ( k,l ) = 2), (cid:80) ( re Ac ) tm,n is theoverall reaction rates in the surface of the electrode ( ∀ m, n that T t ( m,n ) = 1) and V a is the volume of the bulk liquid, here measured by cells that have an area of1 µm .The rates of reactions in the bulk liquid ( rb Acti,j ) are calculated the sameway as the rates of reactions in the biofilm ( rf Actk,l ), specifically by eq. 4 withthe use of local concentrations for the bulk liquid.Similar rules apply for the rest of chemical species that are important in thefunctionality of a MFC, namely the mediator in reduced and oxidised form andhydrogen ions. The following equations describe the evolution of the concentra-tions in the biofilm region:( C Mred ) t +1 i,j =( C Mred ) ti,j + D M × (( C Mred ) ti − ,j + ( C Mred ) ti +1 ,j + ( C Mred ) ti,j − +( C Mred ) ti,j +1 − N i,j × ( C Mred ) ti,j ) + Y M × ( rf Ac ) ti,j (8)9 C Mox ) t +1 i,j =( C Mox ) ti,j + D M × (( C Mox ) ti − ,j + ( C Mox ) ti +1 ,j + ( C Mox ) ti,j − +( C Mox ) ti,j +1 − N i,j × ( C Mox ) ti,j ) − Y M × ( rf Ac ) ti,j (9)( C H ) t +1 i,j =( C H ) ti,j + D H × (( C H ) ti − ,j + ( C H ) ti +1 ,j + ( C H ) ti,j − + ( C H ) ti,j +1 − N i,j × ( C H ) ti,j ) + Y H × ( rf Ac ) ti,j (10)where D M and D H are the diffusion coefficients of the mediator in reduced andoxidised form and the hydrogen ions, respectively. Y M and Y H are the yield ofmediator in both forms and protons from the acetate substrate, respectively.The concentration of the aforementioned chemical species in the bulk liquidare calculated by the following:( C Mred ) t +1 i,j = ( C Mred ) ti,j + Y M × ( rb Ac ) ti,j + (cid:80) [ Y M × ( rf Ac ) tk,l ] V ta + (cid:80) ( re Mred ) tm,n V ta (11)( C Mox ) t +1 i,j = ( C Mox ) ti,j − Y M × ( rb Ac ) ti,j + (cid:80) [ − Y M × ( rf Ac ) tk,l ] V ta + (cid:80) ( re Mox ) tm,n V ta (12)( C H ) t +1 i,j = ( C H ) ti,j + Y H × ( rb Ac ) ti,j + (cid:80) [ Y H × ( rf Ac ) tk,l ] V ta + (cid:80) ( re H ) tm,n V ta (13)It must be mentioned here that the reaction rates on the electrode surface forthe model are designed having in mind the oxidation mechanism of the mediatorand the fact that the electrode stands theoretically as an impermeable borderfor all the other solute materials. As a result the reaction rate for the acetatesubstrate is equal to zero (( re Ac ) ti,j = 0), whilst, for the both mediator forms isdepending on the produced current density and calculated as:( re Mred ) ti,j = − I ti,j F (14)( re Mox ) ti,j = I ti,j F (15)10he reduction reactions that involve the hydrogen ions, are occurring in thecathode electrode, which is not studied in detail in the present study. Thus,in order to simulate the decrease in the concentration of hydrogen ions in theanode compartment, which is due to the flux of the ions towards the cathodeelectrode, an upper limit was imposed in all the anode area. If the concentrationof hydrogen ions exceeds a predefined value (namely C H max ) anywhere in theanode, its value will be fixed to the limit.The over-potential imposed locally on the surface of the electrode is basedon the concentrations of protons, reduced and oxidised mediator and the currentproduced. O ti,j = E c − R tot I ti,j − ( E M + RT F ln ( C Mox ) ti,j (( C H ) ti,j ) ( C Mred ) ti,j ) (16)where E c is the constant value assumed for the cathode potential, R tot is thetotal resistance (internal and externally connected), E M is the standard reduc-tion potential for the mediator, R is the gas constant, T is the temperature and F is the Faraday constant.For the calculation of the current density produced on the anode electrode’ssurface, the widely used Butler-Volmer equation [38, 51] is implemented in themodel: I ti,j = I ref (cid:18) ( C Mred ) ti,j ( C Mred ) ref (cid:19)(cid:18) ( C Mox ) ti,j ( C Mox ) ref (cid:19) − (cid:18) ( C H ) ti,j ( C H ) ref (cid:19) − · (cid:18) exp (2 . · O ti,j /b ) − exp ( − . · O ti,j /b ) (cid:19) (17)where I ref is the exchange current density for mediator oxidation in referenceconditions and b is the Tafel coefficient for mediator oxidation (or referred toas the anodic/cathodic Tafel slope, representing the over-potential increase re-quired for a ×
10 increase in current: b = . RTαF , where α is the anodic transfercoefficient).It is worth-mentioning that the over-potential and current density are mean-ingful only on the cells representing the electrode’s surface. Thus, the afore-mentioned equations provide these parameters for cells that have T t ( i,j ) = 1,otherwise they are equal to zero.As the calculations of the over-potential and the current density are implicit,a random positive value of the total current is assumed and then a numericalmethod is used to approximate the lowest error in the calculation of the totalcurrent, given the concentrations of chemical species on the electrode surface.On the first step, the randomly chosen current value is used (in eq. 16) tocalculate the over-potential, which is then used to calculate the new current11ensity (in eq. 17) and, thus, the total current produced. The current densityerror, which is targeted to be minimised, is given by: e tI = | I tnew − (cid:88) ( I ti,j ) | (18)where I new is the randomly assumed value for the first step or the calculatedvalue by the previous step.The evolution of the biofilm is proportional to the concentrations of chemicalspecies as described by a double Monod limitation kinetic equation (as in eq.4). The calculation of its expansion in the area of the bulk liquid of the anodeis based on a simple algorithm to keep the overall computational model asstraightforward and fast as possible. The concentration of biomass material inthe biofilm CNN cells is calculated by the following equation: X t +1 i,j = X ti,j + ( rf X ) ti,j (19)where ( rf X ) ti,j is the rate of the biomass production in the biofilm and equals:( rf X ) ti,j = Y X × ( rf Ac ) ti,j (20)where Y X is the biomass yield on acetate substrate. The same equations areused for the approximation of the suspended biomass in the bulk liquid, withthe usage of the appropriate local concentrations. X t +1 i,j = X ti,j + ( rb X ) ti,j (21)where ( rb X ) ti,j is the rate of the biomass production in the bulk liquid andequals: ( rb X ) ti,j = Y X × ( rb Ac ) ti,j (22)An upper limit on the concentration of biomass in the bulk liquid ( X (cid:48) max )is imposed to control the biomass growth. Moreover, the following method,which is written in pseudo-code, is performed to simulate the release of pressureproduced by the creation of new biomass inside the region of the biofilm. Thisis to ensure a more realistic expansion of the biofilm attached on the anodeelectrode towards the bulk liquid. The algorithm is performed with initial cells12hat are part of the biofilm (for cells i, j that have parameter T t ( i,j ) = 2). if X t +1 i,j > = X max thenif a random neighbour cell ( k, l ) with T t ( k,l ) = 3 can be found then assign T t +1( k,l ) = 2 and X t +1 k,l = X tk,l + X ti,j · .
005 ;Run
Algorithm 1 for i = k and j = l ;(recursive execution until X t +1 r,p < = X max ) X t +1 i,j = X ti,j · . endelse X t +1 i,j is calculated as in eq. 19 ; end Algorithm 1: Conditions in the evolution of the biomassDespite the fact that the model is specifically designed to simulate a MFCin batch mode, a system with continuous flow can be studied by changing theboundary conditions. That approach is the subject of a future study.
4. Experiments
The configuration of parameters used by the model to produce the resultspresented in this section, are depicted in Table 1. Each time step representsa time period of one tenth of day. The time period of a time step was chosenbased on the trade-off between accuracy of the results and execution time. Asfor smaller time periods the results were not significantly different the executionspeed was not burdened with a smaller period of time steps.The results derived by the model are illustrated in Figs. 1 and 2. Fig-ure 1 depicts how the average — of the columns of cells along the length ofthe electrode — concentrations of materials are changing in one dimension —illustrated in the x -axis of the graphs —, namely from the electrode moving out-wards to the bulk liquid, throughout the time steps of the model that representthe days of the real experiment. Note here that the thickness of the biofilm canbe identified as the distance in x axis from the electrode surface position to thepoint where the stable values of concentrations start. These stable values depictthe existence of the continuously stirred tank reactor, namely the bulk liquid.Moreover, Fig. 2 describes the evolution of the concentrations of every materialin the bulk liquid, the current produced and the expansion of the biofilm.The results depicted in Fig. 2 are in good agreement with the theoreticalprinciples covering the functionality of MFCs in batch mode with added media-tors which are also depicted in previously proposed models investigating similarsystems [38]. It must mentioned here that, despite the biomass initial concen-trations, all the other parameters used in previous models [38] to provide resultsare the same inserted in the model presented here and illustrated in Table 1.Also, the biomass simulation and the limitation of hydrogen ions are simplifiedcompared to the previous models. 13 a) Day 1(b) Day 5(c) Day 10 Figure 1: The concentrations – averaged over the columns of cells along the length of theelectrode – of materials along the distance away from the anode electrode during differentdays (dashed line represents the oxidized form of the mediator). a)(b)(c) Figure 2: Model’s results: (a) the current output, (b) the evolution of biomass and (c) theevolution of concentrations in the bulk liquid. I ref × − A/m X i,j (in bulk liquid) 0.2 gCOD/m b V X i,j (in the biofilm) 0.8 gCOD/m E c V ( C Ac ) i,j gCOD/m R tot
100 Ω ( C Mred ) i,j mME M V ( C Mox ) i,j mMR J/ ( mol × K ) ( C H ) i,j mMT K D Ac . × − m /dayF C/mol D
Mred × − m /daye I A D
Mox . × − m /dayK Ac gCOD/m D H . × − m /dayK Mox mM C H max mMX (cid:48) max gCOD/m X max gCOD/m Y M ( molmediator )( gCODacetate ) Y H ( molH +)( gCODacetate ) Y X ( gCODbiomass )( gCODacetate ) Q Ac ( gCODacetate )( gCODbiomass ) × day Table 1: Parameters used in the execution of the model.
The following statements which are easily observed in the results of the modelpresented in Fig. 2 are following the general theoretical performance of batchmode MFCs with added mediators. The current output reaches its maximumvalue within three to four days that retains for a short period (less than a day)and then its value is asymptotically reduced. The concentration of acetate isreduced to zero in the first ten days, while the biomass concentration in thebiofilm increases and reaches a plateau when the acetate concentration is lowenough. The mediator is a constant quantity that can be found in either oxidisedor reduced form, thus adding the concentrations in any time step equals to 1 mM which is the initially defined concentration. The oxidised mediator follows thereduction of the acetate substrate, until the current produced is high enough tocause the oxidation reaction in the electrode to overcome the reduction reactionsoccurring in the biomass. Then, as there is no more acetate to fed reductionreactions, the concentration of oxidised mediator is asymptotically reaching itsinitial concentration as a product of the oxidation reaction in the electrode.That is also the reason for the asymptotic reduction of the current, namely thelower ratio between reduced and oxidised mediator. Similar explanation of theprogress of the oxidised mediator stands for the reduced mediator.
5. Discussion
The advantages of CNN towards other mathematical methods are their sim-plicity that does not have an effect on their robustness and capability to simulatecomplicated phenomena, their inherent parallel nature that make them ideal for16mplementation in contemporary parallel computing devices and their local ac-tivity character that combined with the aforementioned features empower theirexecution on specialised hardware. As a result, the novelty of the model pro-posed here can be found in the method of numerical approximation, namelyCNN, of the equations giving the kinetics of reactions occurring in a MFC andits performance. As mentioned before, CNN is a simple method as it uses agroup of simple cells located on a grid, characterised by a state (a set of pa-rameters) and updated based on the same local rule. It has been proved [4]that complex computations can emerge from local interactions of basic entities.Moreover, the synchronised functionality of the basic entities enables a fully par-allel execution of computations throughout the grid, accelerating significantlythe production of results by the model. In addition to the ability of CNNsto be efficiently executed in parallel computers, their homogeneity, simplicity,synchronised activity and the local characteristic of the interconnections allowsan effortless implementation in hardware, aiming further acceleration of thecomputations [52, 53]. These computing circuits can be cost efficient and pre-manufactured Field Programmable Gate Arrays (FPGAs) and Graphics Pro-cessing Units (GPUs) or fully custom, providing higher performance efficiencyApplication Specific Integrated Circuits (ASICs).The present study is based on a time explicit scheme which, despite thefact that might not be as accurate as implicit scheme in general, and especiallywith large simulation time steps, allows a less complicated implementation andrequires a lower computational effort. A major advantage of the CNN-basedmodel is that the possible inhomogeneities in the progress of the biofilm or inthe structure of the anode electrode can be easily illustrated by the local ruleor the initialisation of the cells’ states.Some models presented previously, simulate the MFC in just one dimension[44], oversimplifying the reactions occurring. On the other hand, in [38, 39, 42,46] two dimensional and three dimensional representations are provided thatcan account for complicated electrode sizes and biofilm formation; however,inhomogeneities would be difficult to implement. The model presented herecan easily be scaled to three dimensions and as CNN are the basis of the model,inhomogeneities in the whole volume of the MFC (PEM, electrodes and biofilms)can easily be studied and efficiently recreate a wide variety of actual systems.Despite the fact that an homogeneous area (in the x dimension) was studiedhere as a proof of concept, the ability of the model to simulate inhomogeneitiesin two dimensional paradigms is trivial.Moreover, in [38] the competition of two different species, fed on the samesubstrate used, was investigated, while in [39] several communities of methanogenicand electroactive bacteria were simulated. The CNN based model is simulatingthe operation of a MFC inoculated with a single species of bacteria. However,the local dynamics of CNN allow the investigation of more complicated bacterialcommunities. That is an aspect of an ongoing study.Nonetheless, some models are designed to calculate significant parametersof a MFC after it has reached its steady state [44]. Here, the evolution of theoutputs of the MFC through time given its inputs is investigated because the17rocess of reaching the steady state is a demanding procedure and should beoptimised.
Conclusions
The notion of modelling key procedures of complicated processes that occurin real life, enables the scientific community to recreate and study time consum-ing and expensive laboratory experiments. As a result, scientists can investigatethe conditions and parameters of a phenomenon that are difficult to measure inreal life. The model proposed here can serve as a virtual lab, which scientistsand engineers can utilise to test and justify their theoretical approach to thefunctionality of MFCs. Moreover, a more efficient designing of these systemscan be based on the successful modelling of the processes occurring in MFCs.The CNN-based model simulating the performance of a two-chamber acetate-fed MFC described here, is designed in two dimensions and studies cross-sectionof an area near the anode electrode. The concentrations of chemicals that areinvolved in the process of producing current and biomass, throughout time, arestudied. The local rule of the CNN structure is designed to reflect the doubleMonod limitation equation, Fick’s second law of diffusion and the Butler-Volmerequation.During the process of designing and evaluating the model the following con-clusion was reached. The biofilm’s distribution, its initial concentration andthe way it evolves and expands through time are greatly affecting the MFC’soutputs. Biofilms are difficult to predict and it is challenging to formulate al-gorithms that simulate their behaviour. That is a major aspect that limits theproduction of a plethora of MFC models, compared with the ones describingconventional fuel cells.The results provided by the model are in good agreement with the theory andthe general concepts described in the literature for the functionality of MFCsas well the results of previously published works on MFC modelling. Note thatthe results produced by the proposed model simulating the performance of theMFC in 15 days is executed in less than a minute in a contemporary computer.Consequently, shifting from two dimensions to three to get more accurate results,given the possible inhomogeneities in MFC compartments or electrodes, will notbe prohibited in terms of execution times.Nonetheless, the fact that the model is based on CNN can be proved furtheradvantageous, as its simplicity, repeatability and local interactions enable itsimplementation on hardware. That will in turn hugely accelerate the executionof the calculations of the described equations.The incorporation of the reduction reactions that occur in the cathode sideof the MFC and the study of their effects on the performance of the systemare aspects of future work. Moreover, multi-species biofilms will be investigatedincluding non- and electroactive bacteria to illustrate the possible competitionover the common substrate. Furthermore, the simulation of a MFC under con-stant flow will be presented with minor alterations in the local rules of the bulkliquid which will be the first step to study systems that are interconnected.18 cknowledgements
This work was funded by the European Union’s Horizon 2020 Research andInnovation Programme under Grant Agreement No. 686585.https://ec.europa.eu/programmes/horizon2020/