^{1}

^{1}

^{2}

Mathematical modeling of microbial electrochemical cells (MXCs) for both microbial fuel cell and microbial electrolysis cell is discussed. The model is based on the system of reaction diffusion of reaction-diffusion equation containing a non-linear term related to substrate consumption rates by electrogeneic and methanogenic microorganism in the bioflim. This paper presents the approximate analytical method to solve the non-linear differential equation that describes the diffusion coupled with acetate (substrate) consumption rates. Simple analytical expressions for the concentrations of acetate and methane have been derived for all experimental values of bulk concentration, distributions of microbial volume fraction, local potential in the biofilm and biofilm thickness. In addition, sensitivity of the parameters on concentrations is also discussed. Our analytical results are also validated with the numerical results and limiting cases results. Further, a graphical procedure for estimating the kinetic parameters is also suggested.

Microbial fuel cells (MFC) can be defined as a microbial catalyzed electrochemical system which can facilitate the direct conversion of substrate to electricity through a cascade of redox reactions, especially in the absence of oxygen [_{2} harvesting the energy stored in marine sediments, desalination, etc. [

Anolyte contains fermentative microorganisms and acetoclastic methanogens. Biofilm contains acetoclastic methanogens and anode respiring bacteria (electrogens). Acetate is produced during fermentation process and then diffuses to the biofilm where electrogens consume it and conduct electrons to the anode surface [

The acetate and methane mass transfer equations through the biofilm are described as follows [

where

where

where

where

Using the above dimensionless variables the non-linear reaction-diffusion Equations ((1) and (2)) are expressed in the following dimensionless form:

The boundary conditions can be written as follows:

Recently, many authors have been applied the Adomain decomposition method (ADM) to various problems and demonstrated the efficiency of the ADM for handling non-linear problem in physics and engineering sciences [

where the constants

Equations ((13) and (14)) are valid provided

We initially consider the situation where the concentration of acetate

Hence, the non-linear Equations ((7) and (8)) have been reduces to linear equations. Now, the concentration of acetate

where

We now consider that the second major limiting situation found in practice, when the concentration of acetate and methane is very much greater than the half saturation constants

The above Equations ((17) and (18)) are linear reaction-diffusion equations which are exactly solvable. By solving the above Equations ((17) and (18)), we can obtain the concentration of Acetate (16), and Methane (17).

Equations ((22) and (23)) are the exact solution of Equations ((20) and (21)).

In this case, Equations ((7) and (8)) become as follows:

In this case, the above non-linear equation can be solved using Adomain decomposition method. Now, the concentrations become

where the constants

Equations ((7) and (8)) represent the general closed-form of analytical expression for the concentrations of acetate and methane for non steady state condition and for various system parameters (potential, saturation parameter of electrogenic microorganism and acetoelastic methanogenes, the diffusion coefficient of acetate, ratio of the thickness of the biofilm and boundary layer). It is of interest to compare the influence of each parameter on the concentration of acetate and methane for various realistic experimental parameters.

Influence of Potential on the Concentration of Acetate. The influence of dimensionless potential on the concentration of the acetate for some experimental values of parameters is shown in

Influence of Saturation Parameter of Electrogenic Microorganism

acetate increases when saturation parameter of electrogenic microrganism

Influnce of the Ratio of Thickness of the Biofilm and the Boundary Layer.

Influence of Other Parameters of the Concentration of Methane and Acetate. The concentration of methane versus dimensionless distance x for various experimental values of parameters is plotted in

The non-linear differential Equations ((9) and (10)) for the given initial-boundary conditions are being solved numerically. The function pdex, in Matlab software which is a function of solving the initial-boundary value problems for non-linear ordinary differential equations is used to solve this equation. Its numerical solution is compared with analytical results in

The acetate consumption rate by electromagnetic microorganism in the microbial fuel cell (Equation (3)) can be written as follows:

x | |||||||||
---|---|---|---|---|---|---|---|---|---|

Analytical Equation (10) | Numerical | % of derivation | Analytical Equation (10) | Numerical | % of derivation | Analytical Equation (10) | Numerical | % of derivation | |

0 | 0.418633 | 0.42057 | 0.460515 | 0.81091 | 0.80775 | 0.39996 | 0.9881 | 0.99000 | 0.19199 |

0.2 | 0.438648 | 0.440447 | 0.408479 | 0.81742 | 0.81435 | 0.38407 | 0.98851 | 0.99034 | 0.18524 |

0.4 | 0.498707 | 0.500134 | 0.285452 | 0.83709 | 0.83424 | 0.33744 | 0.98974 | 0.99137 | 0.16517 |

0.6 | 0.59885 | 0.599779 | 0.154921 | 0.86961 | 0.86739 | 0.26308 | 0.99171 | 0.99309 | 0.13176 |

0.8 | 0.739148 | 0.739587 | 0.059418 | 0.91529 | 0.91379 | 0.16528 | 0.99463 | 0.99553 | 0.08536 |

1 | 0.919696 | 0.919789 | 0.010118 | 0.97394 | 0.97347 | 0.04891 | 0.99837 | 0.99862 | 0.02622 |

Average % of deviation | 0.157141 | Average % of deviation | 0.31975 | Average % of deviation | 0.275781 |

As shown in

From Equation (10), we can obtain the concentration of acetate at bioflim and anode interface as

Now the plot of

Symbols | Définitions | Units |
---|---|---|

Acetate concentration in the biofilm | ||

Methane concentration in the biofilm | ||

Acetate concentration on the biofilm surface | ||

Methane concentration on the biofilm surface | ||

Diffusion coefficient of acetate | ||

Diffusion coefficient of methane | ||

Density of biomass | ||

Density of biomass | ||

Half saturated constant of acetate consumed by acetoclastic methanogenic bacteria | ||

Half saturated constant of acetate consumed by electrogenic bacteria | ||

Maximum acetate consumption rate | ||

Thickness of the biofilm | m | |

Liquid concentration boundary layer thickness | m | |

Volume fraction of active electrogenic microorganism | None | |

Volume fraction of active acetoclastic methanogenic microorganism | None | |

Yield coefficient | None | |

Local electrical potential | v | |

Dimensionless potential | None | |

_{ } | Dimensionless parameter | None |

Dimensionless parameter | None | |

Dimensionless parameter | None | |

Dimensionless parameter | None | |

Dimensionless parameter | None |

Dimensionless parameter | None | |
---|---|---|

Dimensionless parameter | None | |

Dimensionless parameter | None | |

Dimensionless concentration of acetate in the biofilm | None | |

Dimensionless concentration of methane in the biofilm | None | |

x | Dimensionless space coordinate in the bioflim | None |

k | Dimensionless parameters | None |

A theoretical model describing the bio energy production using microbial electrochemical cell via Nernst-Mo- noid kinetics is analyzed. The time independent non-linear partial differential equations have been solved analytically using the Adomain decomposition method. The primary result of this work is the approximate analytical expression of concentration of acetate and methane for all values of parameters. The influence of potential, ratio of thickness of biofilm and boundary layer, etc. on the concentration of acetate and methane is discussed. Our results are in excellent agreement with stimulation and limiting case results. Also two graphical procedures are suggested for estimating the kinetic parameters.

This work was supported by the DST SB/SI/PC-50/2012, New Delhi, India. The authors are thankful to Mr. S. Mohamed Jaleel, The Chairman, Dr. A. Senthilkumar, The Principal, Dr. P. G. Jansi Rani, Head of the Department of Mathematics, SethuInistitute of Technology, Kariapatti-626115, Tamilnadu, India for their encouragement.

Sivasamy Pavithra,Lakshmanan Rajendran,Raghavan Ashokan, (2016) Approximate Analytical Expressions for the Concentrations of Acetate and Methane in the Microbial Electrochemical Cell. Natural Science,08,196-210. doi: 10.4236/ns.2016.84023

This is given in the supplementary material of the manuscript.

In this appendix, we indicate how Equation (6) in this paper is derived. Furthermore, an ADM is constructed to determine the solution of Equation (4) in the operator form,

where

where A and B are the constants of integration. We let,

where

From Equations ((B.3) to (B.5)), Equation (B.2) becomes

We identify the zeroth component as

and the remaining components as the recurrence relation

where

Adding (B.9) and (B.10), we can obtain the concentration of acetate as described in Equation (10) in the text. By substituting the values of

This is given in the supplementary material of the manuscript.

Consider the singular boundary value problem of

where N is a non-linear differential operator of order less than n,

where

The inverse operator

By applying

Such that

The Adomian decomposition method introduce the solution

and

where the components

can be used constant Adomian polynomials, when

Through using modified Adomian decomposition method, the components

which gives

From (A.8) and (A.11), we can determine the components

can be used to approximate the exact solution. The approach presented above can be validated by testing it on a variety of several linear and nonlinear initial value problems.

Appendix CScilab/MatlabProgram for the Numerical Solution of Equation (4)function pdex4

m = 0;

x = linspace (0, 1);

t = linspace (0, 100000);

sol = pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t);

u1 = sol(:,:,1);

%――――――――――――――――――――?

Figure

plot(x,u1(end,:))

title('u1(x,t)')

xlabel('Distance x')

ylabel('u1(x,1)')

function [c,f,s] = pdex4pde(x,t,u,DuDx)

c =1;

f =1.* DuDx;

e=0.3;alpha=2;

F =-(e*u(1))/((1+(alpha*u(1))));

s =F;

% ――――――――――――――――――――?

function u0 = pdex4ic(x);

u0 = [

% ――――――――――――――――――――?

function [pl,ql,pr,qr] = pdex4bc(xl,ul,xr,ur,t)

j=10;

pl = [

ql = [

pr = [-j*(1-ur(1))];

qr = [