Approximate Analytical Expressions for the Concentrations of Acetate and Methane in the Microbial Electrochemical Cell

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.

• Mathematical Modeling
• Microbial Fuel and Electrolysis Cells
• Waste Water Treatment
• Boundary Value Problems
• Non Linear Equations

1. Introduction

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 [1] . The applications of MFC are widespread in different fields including waste water remediation, toxic pollutants/xenobiotics removal, recovery of commercially viable products, sequestration of CO₂ harvesting the energy stored in marine sediments, desalination, etc. [1] . Microbial electrochemical cells are recognized as a modern technology to directly utilize bioenergy stored in organic substances, especially in wastewater [2] . Several experiments have been conducted to evaluate MXCs main performance as a current or hydrogen generator fed with different organic matters [3] and [4] . A simple mediator based model with suspended cells was investigated [5] . A simple model with rapid implementation and computations is used to describe the effect of some operational conditions such as temperature and substrate concentration on in the MFC performance [6] . Pinto developed a time-dependent mathematical model with the uniform distribution of bacteria in the anode chamber. Although a number of MFC mathematical models have been developed and discussed. To the best of the knowledge, only one MEC model has been proposed [7] . Yahya modified this model for a fed-batch reactor. It is a multi population mediator-based model developed based on the Bernard’s anaerobic digestion kinetics model [8] . Alavijeh [2] used a variety of approaches to develop the first generalized conduction-based model for MXCs including both MFCs and MECs. It is a one-dimensional spatial distribution and time-dependent model using Bernard’s anaerobic digestion kinetics model and both biofilm and liquid bulk simulation. The purpose of this communication is to derive the analytical expression for acetate and methane concentration using the Adomain decomposition method. We also provide the tabular complication of concentration of acetate with limiting case results (first order and zero order kinetics).

2. Mathematical Formulation of the Problem

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 [2] . The systematic diagram of the model is represented in Figure 1.

The acetate and methane mass transfer equations through the biofilm are described as follows [2] :

Figure 1

Schematic representation of the model [ 1 ]

where and the diffusion coefficient of acetate and methane in the bioflim. and are the concentration of acetate on the biofilm and methane concentration on the biofilm., and are the density of biomass yield coefficient. The acetate consumption rate by electromagnetic microorganism in the biofuel is represented by the Nenst-Monoid equation [2]

where is the maximum uptake, are the volume fraction, and Half saturated constant. are the Faraday constant, universal gas constant, temperature and local electrical potential of the biofilm respectively. The acetate consumption rate by acetoclastic methangen microorganism in the bioflim is

where and are the volume fraction and Half saturated constant. At the anode surface, there is no substrate flux and at the surface of the biofilm, there is an interface transfer. The boundary conditions for the above equations are given by [2]

where represents the space coordinate in the biofilm (m), thickness of the biofilm (m) and boundary layer thickness (m) respectively. and are the diffusion coefficient of acetate and methane in the liquid. is the acetate concentration in the liquid bulk and in the biofilm interface

and is the methane concentration in the liquid bulk and in the biofilm interface. We introduce the following set of dimensionless variables:

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:

3. Approximate Analytical Expression of Concentration of Acetate and Methane

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 [9] - [13] . The modified Adomain decomposition method [11] is used to give the approximate solutions of the non-linear Equations ((7) and (8)). Many researchers find that the ADM requires less computational work than traditional approaches [11] - [13] . Other advantages include the ability to solve nonlinear problems without linearization, the wide applicability to several types of problems and scientific fields, and the development of a reliable, analytic solution. Many researchers find that the ADM requires less computational work than traditional approaches [11] - [13] . Other advantages include the ability to solve nonlinear problems without linearization, the wide applicability to several types of problems and scientific fields, and the development of a reliable, analytic solution. Using this method (refer Appendix A), we can obtain the concentrations acetate and methane as follows:

where the constants

Equations ((13) and (14)) are valid provided and. This is the only limitations in this method.

4. Limiting Case

4.1. Unsaturated (First Order) Catalysis

We initially consider the situation where the concentration of acetate and methane is less than the half saturation constants and. In this case Hence, Equations ((7) and (8)) reduces to

Hence, the non-linear Equations ((7) and (8)) have been reduces to linear equations. Now, the concentration of acetate and methane for corresponding boundary conditions (9a) to (9c) becomes as follows:

where and. Equations ((18) and (19)) are the exact solution of Equations ((16) and (17)).

4.2. Saturated (Zero-Order) Catalysis

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 and. In this case,. Hence, the non-linear Equations ((7) and (8)) have been reduces to

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)).

4.3. Saturated Electrogenic Microorganism and Acetoclastic Methanogens Are Equal (α = β)

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 to and k are given in Equation (12), when replacing. Equations ((26) and (27)) are the approximate analytical expression of concentration of acetate and methane.

5. Discussion

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 Figure 2(a). The microbial activity is strongly dependent on the redox potential of the anode. From this figure it is observed that the concentration of acetate decreases when increases or potential decreases.

Influence of Saturation Parameter of Electrogenic Microorganism and Acetoelastic-Methhano- genes on Concentration of Acetate.-As shown in Figure 2(b) and Figure 2(c), the concentration of

Figure 2

Plot of dimensionless concentration versus dimensionless distance x using Equation (10) for various values of the parameter (a) , (b) , (c) and , (d) and for some fixed experimental values of other parameters

acetate increases when saturation parameter of electrogenic microrganism or bulk concentration of the acetate increases and saturation parameter of acetoelastic methanogenes decreases. From Figure 2(c) it is also observed that the concentration of acetate is inversely proportional to diffusion coefficient of acetate.

Influnce of the Ratio of Thickness of the Biofilm and the Boundary Layer. Figure 2(d) represents the concentration verses distance from the anode surface for various values of or ratio of biofilm thickness and boundary layer thickness. From this figure, it is inferred that the concentration of acetate increases when the ratio of thickness increases.

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 Figure 3. From these figure, it is interfered that the concentration of methane increases when increases or decreases.

Figure 4 represents the dimensionless concentrations of acetate versus potential for various values of and. From these figures, it is observed that the concentration of acetate increases when thickness and decreases. From Figure 4(d), it is observed that the concentration of acetate does not differ significantly about the parameter.

Figure 3

Plot of dimensionless concentration of versus dimensionless distance x using Equation (11). (a) For various values of the parameter (a) , (b) , (c) and , (d) and for some fixed experimental values of other parameters

Figure 4

Plot of dimensionless concentration versus dimensionless potential using Equation (10) for various values of the parameter (a) , (b) , (c) , (d) and for some fixed experimental values of other parameters

6. Comparison with Numerical Data and Limiting Case Results

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 Table 1. The maximum relative error between our analytical result and the numerical result is 0.32%. The Matlab program is also given in Appendix C. The concentration of acetate and methane are also obtained for the following limiting cases, that is zero order kinetics, first order kinetics and saturated electrogenic microorganism and saturated acetoclastic methanogens are equal. Also our analytical results are compared with limiting case results in Figure 5 and it gives a satisfactory agreement.

7. Determination of Kinetic Parameters K AC ,E , q AC ,AM,max , φ E ,a , α and β

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

Table 1

Comparison of the acetate concentration calculated using Equation (10) with the numerical simulation for various experimental values of when

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

Figure 5

Plot of two dimensional comparative diagram of general solution of dimensionless Acetate concentration and methane concentration with the limiting cases (Zero order and in (a) and (b) respectively) for the experimental values Comparison of dimensionless concentration of and using Equations ((10), (11)), Equations ((22), (23)), Equations ((26), (27))

As shown in Figure 6, is plotted against to obtain the straight line with the slope and intercept. The slope and intercept yields the value of the parameters and.

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

Now the plot of versus gives the slope and intercept. From these plot, we can obtain the kinetic constant and (Table 2).

Figure 6

Estimation of kinetic parameters and from Equation (28)

Table 2

Nomenclature

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

8. Conclusion

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.

Acknowledgements

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.

Appendix A: Basic Concept of the Adomain Decomposition Method

This is given in the supplementary material of the manuscript.

Appendix B: Approximate Analytical Solution of Non Linear Equation (7) Using ADM

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. Applying the inverse operator on both sides of Equation (B.1) yields

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 are the Adomain polynomials of.We can find the first two as follows:

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 in Equation (8), we get the concentration of the methane (Equation (11)) in the text.

Appendix C: Scilab/Matlab Program for the Numerical Solution of Equation

This is given in the supplementary material of the manuscript.

Supplementary Material of the Manuscript

Appendix A: Basic Concept of the Modified Adomain Decomposition Method

Consider the singular boundary value problem of order nonlinear differential equation in the form

where N is a non-linear differential operator of order less than n, is given function and are given constants. We propose the new differential operator, as below

where, so, the problem can be written as

The inverse operator is therefore considered a fold integral operator, as below [11] [12]

By applying on (A.3), we have

Such that

The Adomian decomposition method introduce the solution and the nonlinear function by infinite series

and

where the components of the solution will be determined recurrently. Specific algorithms were seen in [11] [12] to formulate Adomian polynomials. The following algorithm:

can be used constant Adomian polynomials, when is a nonlinear function. By substituting (A.6) and (A.7) in to (A.5)

Through using modified Adomian decomposition method, the components can be determined as

which gives

From (A.8) and (A.11), we can determine the components, and hence the series solution of in (A.6) can be immediately obtained. For numerical purposes, the n-term approximate

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 = [0];

% ――――――――――――――――――――?

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

j=10;

pl = [0];

ql = [1];

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

qr = [1];

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