Application of Microbial Fuel Cell technology for a Waste ... · PDF fileApplication of Microbial Fuel Cell technology for a Waste Water Treatment Alternative Microbial fuel cells

Application of Microbial Fuel Cell technology for aWaste Water Treatment Alternative

Eric A. Zielke

February 15, 2006

i

Application of Microbial Fuel Cell technology for a

Waste Water Treatment Alternative

Microbial fuel cells (MFCs) are devices that use bacteria to generate electricity fromorganic matter. Most of the current research performed on MFCs is concerned with increasingthe power density of the system with respect to the peripheral anode surface area; littleresearch has been done on large scale applications. This study analyzed a Monod-typeequation relating the empirical data of voltage output to the substrate concentration of theinoculum. The unknowns of maximum voltage and the half-saturation constant within theMonod-type equation were determined by using the Newton-Raphson numerical method fornon-linear equations in a Fortran 90 computer program. Results demonstrated that voltageoutput follows saturation kinetics as a function of substrate concentration. The size of thesystem was based on the size of one primary clarifier tank from the Arcata Waste WaterTreatment Facility. A capital cost was determined to be $31,250,000 with projections of 10,20 and 30 years in the future.

Microbial Fuel Cell technology Zielke ii

Contents

1 Introduction 1

2 Problem Formulation 1

3 Literature Review 2

3.1 Biological Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3.2 Design Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.3 The Monod Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.3.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.4 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.5 Root-Finding Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Methodology 8

4.1 Least-Squares Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.2 Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Application 10

5.1 Numerical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.2 Assumptions in Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6 Results and Discussion 12

6.1 Numerical Method Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6.1.1 Sensitivity Analysis Results . . . . . . . . . . . . . . . . . . . . . . . 12

6.2 Design Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Microbial Fuel Cell technology Zielke iii

6.2.1 Time Value of Money . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7 Conclusion 15

8 References 16

9 Appendix 18

Microbial Fuel Cell technology Zielke iv

List of Tables

1 Table of Initial Conditions and Parameters . . . . . . . . . . . . . . . . . . . 11

2 Table Parameter Estimations . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Sensitivity Analysis Specific to the Newton-Raphson Computer Program . . 13

Microbial Fuel Cell technology Zielke v

List of Figures

1 Representation of Anaerobic (Anode portion) and Aerobic (Cathode portion)Biological Degradation Simultaneous to Electricity Generation in a SingleChamber Microbial Fuel Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Primary Clarifier Tank Dimensions . . . . . . . . . . . . . . . . . . . . . . . 12

3 Power Density (mW/m2) as a function of Acetate substrate concentrationrepresenting a variance in resistance . . . . . . . . . . . . . . . . . . . . . . 14

Microbial Fuel Cell technology Zielke vi

Table of Nomenclature

c = mg

e = mg

E = mg

E − S = mg

Ks = mg/L

Km = mg/L

k+1 = /s

k−1 = /s

k+2 = /s

P = mW/m2

Pmax = mW/m2

S = mg/L

v = mg/s

µ = mg/s

µmax = mg/s

Microbial Fuel Cell technology Zielke 1

1 Introduction

Renewable energy is an increasing need in our society. Microbial fuel cell (MFC) technology

represents a new form of renewable energy by generating electricity from what would other-

wise be considered waste. This technology can use bacterium already present in wastewater

as catalysts to generating electricity while simultaneously treating wastewater (Lui et al.,

2004; Min and Logan, 2004). Although MFCs generate a lower amount of energy than hydro-

gen fuel cells, a combination of both electricity production and wastewater treatment would

reduce the cost of treating primary effluent wastewater. Currently, most of the research per-

formed on MFCs is concerned with increasing the power density of the system with respect

to the peripheral anode surface area, while little research has been done on determining the

size of a microbial fuel cell needed for a typical waste water treatment facility.

The power density produced in a single chamber MFC can be modeled as a function

of substrate concentration using an empirical Monod-type equation (Lui et al. 2005). This

equation exhibits power density as a state variable, the substrate concentration as the in-

dependent variable and maximum power density and the half-saturation constant as two

parameters (Lui et al. 2005). This particular equation uses empirical data of the power

density and substrate concentration needed to solve for the parameters of maximum power

density and the half-saturation constant. These parameters are then used to model a func-

tion of power density, which is the state variable, versus substrate concentration, which is

the independent variable.

2 Problem Formulation

The objective of this study is to determine a size of the microbial fuel cell needed for the

Arcata Waste Water Treatment Facility and its associated production costs. Analysis will

include solving for the two parameters of maximum power density and the half-saturation

constant within the Monod-type equation by use of the Newton-Raphson numerical method

for non-linear equations in a FORTRAN 90 computer program. Assumptions pertaining

to the MFCs characteristics and general design will be incorporated in the design. The

information obtained from this feasibility study will be useful for integrating this alternative

into future design proposals.


3 Literature Review

The purpose of this literature review is to organize relevant information and apply the

principles of a feasibility study to MFC technology. This section contains an overview of

the biological mechanism of a MFC, current design structures, the Monod model and its

development, parameter estimation, and root-finding numerical methods.

3.1 Biological Mechanism

In normal microbial catabolism, a substrate such as a carbohydrate is initially oxidized

anaerobically when its electrons are released by enzymatic reactions (Bennetto 1990). The

electrons are stored as intermediates (e.g., NADH, quinones) which become reduced and are

then used to provide the living cell with energy (Bennetto 1990). The ending location for

the electrons is molecular oxygen or dioxygen at the end of the respiratory chain (Bennetto

1990).

A MFC uses bacteria to catalyze the conversion of organic matter into electricity by

transferring electrons to a developed circuit (Bond et al. 2002). Microorganisms can trans-

fer electrons to the anode electrode in three ways: exogeneous mediators (ones external to

the cell) such as potassium ferricyanide, thionine, or nuetral red; using mediators produced

by the bacteria; or by direct transfer of electrons from the respiratory enzymes (i.e., cy-

tochromes) to the electrode (Bond et al. 2003, Min 2004). These mediators can divert

electrons from the respiratory chain by entering the outer cell membrane, becoming reduced,

and then leaving in a reduced state to shuttle the electron to the electrode (Bennetto 1990).

Shewanella putrefaciens, Geobacter sulfurreducens, Geobacter metallireducens and Rhod-

oferax ferrireducens have been shown to generate electricity in a mediatorless MFC (Bond

et al. 2003). Bacteria present in mediatorless MFCs have electrochemically active redox en-

zymes on their outer membranes that transfer the electrons to external materials and there-

fore, do not require exogeneous chemicals to accomplish electron transfer to the electrode

(Oh et al. 2004). When these bacteria oxidize the organic matter present in the wastewater,

the electron is shuttled to the electrode and the protons produced diffuse through the water

to the counter electrode (cathode) giving this particular electrode a positive characteristic.

Oxygen, the hydrogen protons, and the electron that is connected by a circuit from the anode

to the cathode, are then catalytically combined with a platinum catalyst to form water at

the cathode (Bond and Lovely 2003, Bond et al. 2002, Lui et al. 2004). A representation


of the biological mechanism previously mentioned is shown within a single chamber MFC

(Figure 1).

Figure 1: Representation of Anaerobic (Anode portion) and Aerobic (Cathode portion) Bi-ological Degradation Simultaneous to Electricity Generation in a Single Chamber MicrobialFuel Cell

Note that the mechanism of MFC technology is still in research stages and that many

possible reasons for electricity generation that cannot be answered without a better under-

standing of the characteristics of the electricity generating bacteria in MFCs (Min 2004).

3.2 Design Structures

A typical microbial fuel cell (MFC) consists of two separate chambers which can be inocu-

lated with any type of liquid media. These chambers, an anaerobic anode chamber and an

aerobic cathode chamber, are generally separated by a Proton Exchange Membrane (PEM)

such as Nafion (Oh and Logan 2004). A MFC such as this can be classified into two types.

One type generates electricity from the addition of artificial electron shuttles (mediators) to


accomplish electron transfer to the electrode. The other type does not require these addi-

tions of exogenous chemicals and can be loosely defined as a mediatorless MFC (Bond and

Lovely 2003, Chundhuri and Lovely 2003, Lui et al. 2004).

Mediatorless MFCs can be considered to have more commercial potential then MFCs

that require mediators because the typical mediators are expensive and toxic to the microor-

ganisms (Bond et al. 2003). However, one major disadvantage of the two chamber system is

that the cathode chamber needs to be filled with a solution and aerated to provide oxygen

to the cathode.

In hydrogen fuel cells, the cathode is bonded directly to a PEM which allows for oxygen

from the air to directly react at the electrode (Gottesfeld 1997). This same principle can

be used to design a single chamber MFC (SCMFC) where the anode chamber is separated

from the air-cathode chamber by a gas diffusion layer (GDL) allowing for a passive oxygen

transfer to the cathode, eliminating the need for energy intensive air sparging of the liquid

(Lui and Logan 2004).

Studies have shown that the MFCs using expensive solid graphite electrodes, but less

expensive graphite felt and carbon cloth, can also be used (Bond and Lovely 2003, Tender

et al. 2002, Chundhuri and Lovely 2003, Lui et al. 2004). PEMs such as Nafion are also

expensive and if removed, can substantially reduce the overall cost of the MFC. Platinum is

a critical catalyst at the cathode and no alternative metal has been proven to catalyze the

combination of oxygen, the hydrogen proton, and the electron in a more efficient manner

(Oh et al. 2004). These design parameters set a limit on the overall cost reduction of the

MFC.

3.3 The Monod Model

The Monod model can describe several important characteristics of microbial growth in a

simple periodic culture of microorganisms (Dette et al., 2003). This model was first proposed

by Nobel Laureate J. Monod more then 50 years ago and is one of the basic models for

quantitative microbiology (Monod, 1949; Dette et al., 2003). This model is typically defined

by a differential equation (Dette et al., 2003). Many limitations of this model, as well as

restrictions of its applications, are well known (Baranyi and Roberts, 1995; Ferenci, 1999;

Dette et al., 2003). With these limitations, many modifications of the model have also been

proposed within specific cases (Ellis et al., 1996; Fu and Mathews, 1999; Schirmer et al.,

1999; Vanrolleghem et al., 1999; Dette et al., 2003). The Monod-type equation used in this

analysis is in an algebraic form under the assumption of a steady state condition.


3.3.1 Model Formulation

The Monod model was developed from the Michaelis-Menton equation. The formulation

of the Michaelis-Menton equation begins with examining the reaction of an enzyme and a

substrate (Aiba et al., 1965)

E + Sk+1→←k−1

E S

E S k+2→ E + P

whereE = enzymeS = substrate

E S = enzyme-substrate complexP = product

k+1 = forward reaction rate constantk−1 = reverse reaction ratek+2 = reaction rate constant

If the variables e, S, and c are denoted as the concentrations of the total enzyme, substrate,

and enzyme-substrate complex, the rate of change of the enzyme-substrate complex, dcdt

, can

be expressed as (Aiba et al., 1965)

dc

dt= k+1(e− c)S − k−1c− k+2c

This equation assumes the value of S is considerably greater then that of e. Assuming a

steady state condition, the left hand side of the equation becomes zero and c can be solved

for and expressed as (Aiba et al., 1965)

c =eS

k−1+k+2

k+1+ S

From the above enzyme reaction equations, the rate of product formation, v, can be expressed

as (Aiba et al., 1965)

v = k+2c =k+2eS

k−1+k+2

k+1+ S

=V S

Ks + k+2

k+1+ S

=V S

Km + S


whereV = ek+2 = maximum rate of production

Ks = k−1

k+1= equilibrium constant

Km = Ks + k+2

k+1

If the value of k+2 is considerably smaller than that of k+1, Km reduces to Ks. When

Km = Ks, the Michaelis-Menton equation becomes analogous to the Monod where the

Monod-type equation replaces V and v with µmax (maximum value of specific growth rate)

and µ (specific growth rate). The Monod-type equation assumes only unicellular growth and

can be expressed in the form (Aiba et al., 1965)

µ = µmax

(S

Ks + S

)The power density produced in a single chamber MFC can be modeled as a function of

substrate concentration, S, using an empirical Monod-type equation (Lui et al., 2005)

P =PmaxS

KS + S

whereP = power density

Pmax = maximum power densityS = substrate concentration

Ks = half-saturation constant

3.4 Parameter Estimation

Linear regression is a parameter estimation optimization problem where Pmax and Ks are

the decision variables of the Monod model . The main objective can take one of several

forms, all of which attempt to reduce the residuals, thereby improving the model. A residual

is the difference between the observed emperical data to which the model is being fit, and

the predicted value output by the model:

ei = yi − yi


whereei = residualyi = observed valueyi = predicted value

The most common objectives for this parameter estimation include minimizing the sum

of absolute deviations, minimizing the maximum absolute deviation of the residuals, and

minimizing the sum of the squared residuals. This study is concerned with minimizing the

sum of the squared residuals because it makes very efficient use of a small amount of data,

and it is built into Solver available on Microsoft Excel to use as a reference when writing the

FORTRAN 90 source code.

3.5 Root-Finding Numerical Methods

All numerical root-finding methods utilyze an iteration process producing a sequence of

numbers converging towards a limit which is a root. A numerical method is needed to solve

for the Monod-type equation since there are two unknown parameters to solve for. Some of

the more common root-finding algorithms include the Bisection method, the Secant method

and the Newton-Raphson method.

The Bisection is the most simple root-finding numerical method, but is generally used

to converge to the answer instead obtaining the answer. This method requires f to be a

continuous function and requires previous knowledge of two initial guesses, a and b, such

that f(a) and f(b) have opposite signs to bisect the root.

The Secant method requires any two points for the two initial guesses and will converge

faster then the Bisection Method. However, the method will only solve an equation of one

unknown and not two such as in the Monod-type equation.

The Newton-Raphson method will solve a non-linear equation of two or more unknowns

and is one of the most common multivariate iterative numerical methods. Since the Monod-

type equation is non-linear and requires two unkowns to be solved, the Newton-Raphson

method with be the method of choice. The Newton-Raphson method can also be coupled

with Guass elimination, LU decomposition or Cramer’s rule.


4 Methodology

This section details the numerical approach applied to this project. The methodology in-

cludes a section explaining the least squares method and another for explanation of the

Newton-Raphson method.

4.1 Least-Squares Approach

The Least-Squares approach can be expressed as,

min z = Σni=1(yi − yi)

2

whereyi = observed value

yi = predicted value

now applied to the Monod-type equation,

min z = Σni=1(Pabs(i)− PmaxS(i)

(Ks+S(i)))2

For this unconstrained optimization problem, the optimal values for Pmax and KS can be

found by taking partial derivatives of the objective function with respect to each parameter

being solved for and then setting the equations equal to zero,

∂z

∂Pmax

= f1(Pmax, Ks) = 2[Σni=1

(PmaxS(i)2

(Ks + S(i))2− Pabs(i)S(i)

Ks + S(i)

)= 0

∂z

∂Pmax

= f2(Pmax, Ks) = 2[Σni=1

(Pabs(i)PmaxS(i)2

(Ks + S(i))2− P 2

maxS(i)2

(Ks + S(i))3

)= 0

The result is a system of non-linear equations which can now be solved for iteratively in a

computer program utilizing the Newton-Raphson methodology.


4.2 Newton-Raphson Method

The numerical approach of the Newton-Raphson method is widely used for solving non-

linear equations of two or more unknowns. This model uses the derivative of the function

to determine the root (Chapra and Canale, 2002). The root is defined as the intercept of

derivative line and axis of the independent variable. In the case of only two variables, a first

order Taylor series can be expressed as (Chapra and Canale, 2002),

ui+1 = ui + (xi+1 − xi)∂ui

∂x+ (yi+1 − yi)

∂ui

∂y

The root estimate corresponds to values of x, y, and ui+1 equaling zero where the previous

equation can be rearranged and expressed in its general form as (Chapra and Canale, 2002),

∂ui

∂xxi+1 +

∂ui

∂yyi+1 = −ui + xi

∂ui

∂x+ yi

∂ui

∂y

Similar the the single equation case of Newton’s method, the root estimate corresponds to

the values of x and y, where ui+1 and vi+1 equal zero. The equations can then be rearranged

and expressed as,∂ui

∂xxi+1 +

∂ui

∂yyi+1 = −ui + xi

∂ui

∂x+ yi

∂ui

∂y

∂vi

∂xxi+1 +

∂vi

∂yyi+1 = −yi + xi

∂vi

∂x+ yi

∂vi

∂y

Now the only unknowns are xi+1 and yi+1 and the Newton Raphson Method utilizing

Cramer’s Rule can be expressed in its general form,

xi+1 = xi −ui

∂vi

∂y− vi

∂ui

∂y

∂ui

∂x∂vi

∂y− ∂ui

∂y∂vi

∂x

yi+1 = yi −vi

∂ui

∂x− ui

∂vi

∂x∂ui

∂x∂vi

∂y− ∂ui

∂y∂vi

∂x

The denominator portion of these equations is known as the determinant of the Jacobian

matrix system.


5 Application

This section details the application specific to the numerical method as well as the design.

The numerical application describes the computer algorithm and its associated parameters.

The design application details attributes specific to the Arcata Wastewater Treatment Fa-

cility as well as the components and assumptions used in the the design.

5.1 Numerical Application

The subroutine exited do loop iterations by use of stopping criteria. The stopping criteria

tested for the maximum number of iterations, maxit, and the difference between the first

two partial derivatives, f1(Pmax, Ks) and f2(Pmax, Ks), from the unconstrained optimization

problem. The difference value was denoted as ε. The do loop of the Newton-Raphson method

evaluated both roots, replacing f1 with f1old and f2 with f2old, and tested the stopping

criteria by use of if commands. The main program asked the user for the first two parameter

estimates of Pmax and Ks and two separate data files containing the 13 data points of power

density and substrate concentrations from a case study on single chamber MFC research.

Each stopping criteria case was written in the main program to address the user of the

stopping criterion encountered.

The initial conditions were set to evaluate the value of Pmax and Ks corresponding to

resistance loads of 218 ohms, 1000 ohms and 5000 ohms denoted by runs 1, 2 and 3 (Table

1). A set of ten runs for a sensitivity analysis on the Newton-Raphson method computer

program to ensure accurate results. This was accomplished by increasing the initial values

of Pmax and Ks until the program diverged (Table 2). The computer program Solver built

into Microsoft Excel was also used to verify results (Appendix).

5.2 Assumptions in Design

Some of the assumptions pertaining to the theoretical design and cost of a MFC for a wastew-

ater treatment facility include the characteristics of the MFC. These assumptions include,

1. Single chamber MFCs treat roughly 50 to 70% of the chemical oxygen demand (COD)

present in wastewater in a 12 hour time span (Lui, 2004).


Table 1: Table of Initial Conditions and Parameters

Variables ValuesPmax (run 1) 600 (mW/m2)Pmax (run 2) 300 (mW/m2)Pmax (run 3) 100 (mW/m2)KS (run 1) 100 (mg/L)KS (run 2) 35 (mg/L)KS (run 3) 1 (mg/L)maxit* 1000ε* 0.0001*Note: These values were repeated for every run

2. Gas Diffusion Electrode (carbon paper with Pt) 5 cm by 5 cm; price = $12.50 (Fuel,

2005) Since the platinum catalyzed GDE is the most expensive component of the fuel cell,

this project will only consider the GDE’s cost in the final evaluation.

3. The Monod-type equation used in this particular study only accounts for a single sub-

strate type (Acetate was investigated in this study).

4. Voltage and Power is not dependent on volume (Appendix).

5. Power generation is highly dependent on peripheral surface area.

6. A typical trickling filter has 100m2/m2 of surface area to volume and sand has 1200m2/m3

of surface area to volume (Logan, 2005).

7. The price of fuel cells has decreased at a rate equivalent to the increase of the inflation rate.

Other assumptions pertain to the wastewater characteristics, treatment facility and

the current discount rate. Included are,

1. Domestic wastewater contains roughly 220mg/L of COD (Lui, 2004).

2. Acetate make up for 2-10% of the constituents associated with domestic wastewater

(Naidoo, 1999)

3. Power generation increases linearly to COD concentration suggesting that COD is the

culporate in power generation (Lui, 2004).

4. This particular study will assume all other constituents in wastewater act as Acetate.

5. The primary clarifier tank at the Arcata Waste Water Facility is 337.8m3 (Figure 2).

6. Typical power demand at a wastewater facility is 1500W at any given time.

7. The current discount rate is 5.0% (Federal, 2005).


Figure 2: Primary Clarifier Tank Dimensions

6 Results and Discussion

This section details the results from the Numerical Method FORTRAN 90 computer pro-

gram, the sensitivity analysis, and the theoretical design. Discussion of the results are

integrated within each subsection.

6.1 Numerical Method Results

Three runs were performed on the data obtained from one case study using Acetate as the

substrate within a batch mode single chamber MFC (Table 3). The values compared to that

obtained in Solver intrinsic to Microsoft Excel showed an insignificant difference verifying

accurate results (Appendix).

6.1.1 Sensitivity Analysis Results

The sensitivity analysis showed great robustness of the FORTRAN 90 computer program.

The program diverged to the incorrect answer when a deviation from the pattern of increasing


Table 2: Table Parameter Estimations

Pmax (mW/m2) Ks (mg/L)Run 1 74.99 0Run 2 333.2 36.48Run 3 593.2 103.9

initial values was performed (Table 4).

Table 3: Sensitivity Analysis Specific to the Newton-Raphson Computer Program

Initial Value of Pmax Initial Value of Ks Iterations ResultRun 1 340 40 62 passRun 2 350 50 67 passRun 3 360 60 71 passRun 4 370 70 72 passRun 5 380 80 72 passRun 6 390 90 72 passRun 7 400 100 73 passRun 8 500 200 74 passRun 9 600 300 75 passRun 10 700 100 8 fail

6.2 Design Results

The values of parameters obtained from the computer program were ran through the Monod-

type equation to develop a theoretical function relating acetate substrate concentration to

power density (Figure 3). The assumption of 70% acetate bio-degredation of COD corre-

sponds to a range of power density at any given instant. The range of interest was chosen as

the function relating to 1000 ohms resistance due to the functions placement on the graph. A

linear average was determined as 240 mW/m2. Assuming half of the volume of the primary

clarifier tank as volume for the platinum catalyzed cathode material, an approximation of

power generation can be calculated as follows,

337.8m3

2 ∗ 100m2/m3(240mW/m2) = 4.053kW


Figure 3: Power Density (mW/m2) as a function of Acetate substrate concentration repre-senting a variance in resistance

Since fuel cell components are very expensive (namely the platinum), a more optimal ap-

proach would determine the exact amount of surface area needed to supply the demand of

power. This calculation is expressed as,

(1500W )(0.240W/m2) = 6250m2

An estimation of the cost of a system such be computed as,

6250m2

0.0025m2($12.50) = $31, 250, 000


6.2.1 Time Value of Money

The time value of money is one of the basic concepts used in financing a project. In the

case of fuel cell technology, a value such as the one previously obtained can be evaluated as

a cost in the future under the valid assumption of fuel cell components decreasing in price

with an increase in research and development. Assuming $31,250,000 as a future cost, the

Single Payment Present Worth factor can be used to determine the present value given a

discount rate. Using 5% as the discount rate this value can be expressed for 10 years in the

future,

$31, 250, 000[SPPW (5%, 10)] = $19, 180, 000

for 20 years in the future,

$31, 250, 000[SPPW (5%, 20)] = $11, 780, 000

for 30 years in the future,

$31, 250, 000[SPPW (5%, 30)] = $7, 230, 000

7 Conclusion

The results of this project demonstrate that MFC technology may someday become another

viable means of treating wastewater. With increasing gas and oil prices, costs to manufacture

the chemicals needed in conventional wastewater treatment facilities will increase greatly

while research continues decreasing the overall costs of fuel cell components associated with

fuel cells, thus opening the doors for applications of MFCs to emerge. Projects such as this

one will further the knowledge of existing technologies in hopes of improving and optimizing

global energy usage.


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9 Appendix

Ks= 103.898616 593.24849 Pmax--< 9334.65327power density substrate Pcalc leaszt sqr

0 0 0 0225 80 258.076326 1094.04332250 80 258.076326 65.2270352275 80 258.076326 286.410755375 220 402.949137 781.154234400 220 402.949137 8.69740641420 220 402.949137 290.731944475 400 470.926866 16.5904236500 400 470.926866 845.247142530 400 470.926866 3489.6352480 800 525.057549 2030.18272505 800 525.057549 402.30527530 800 525.057549 24.4278223

Ks= 36.4771335 333.157677 Pmax--< 4567.12025power density substrate Pcalc leaszt sqr

0 0 0 0210 80 228.822717 354.294686225 80 228.822717 14.6131675240 80 228.822717 124.931649270 220 285.774751 248.842776295 220 285.774751 85.1052152320 220 285.774751 1171.36765280 400 305.315126 640.855596300 400 305.315126 28.2505626330 400 305.315126 609.343013290 800 318.629322 819.638075315 800 318.629322 13.1719778340 800 318.629322 456.70588

Ks= -0.00012995 74.9999174 Pmax--< 1800power density substrate Pcalc leaszt sqr

0 0 0 060 80 75.0000392 225.00117775 80 75.0000392 1.5388E-0990 80 75.0000392 224.99882360 220 74.9999617 224.99885175 220 74.9999617 1.467E-0990 220 74.9999617 225.00114960 400 74.9999418 224.99825375 400 74.9999418 3.3917E-0990 400 74.9999418 225.00174760 800 74.9999296 224.99788775 800 74.9999296 4.9591E-0990 800 74.9999296 225.002113

487.685439

Acetate - 218 ohm

Acetate - 1000 ohm

Acetate - 5000 ohm


!<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><

program monod implicit none double precision, dimension(:), allocatable::p,s double precision::Pmax, Ks, newPmax, newKs, f1, f2, oldf1, oldf2, old11, old12, old21, old22, det, useless double precision, dimension(2,2)::j integer::numpts, a, i, exitflag, it, maxit, status, b

open(12,file="p.dat") open(13,file="s.dat")

numpts=0 do read(12,*, iostat=status)useless numpts=numpts+1 if(status<0)exit end do numpts=numpts-1 !write(*,*)"the numbpts=", numpts

allocate(p(numpts)) allocate(s(numpts))

rewind 12 do b=1,numpts read(12,*)p(b) read(13,*)s(b) end do

!do a=1,numpts !write(*,*)"this is the power",p(a) !write(*,*)"this is the substrate",s(a) !end do

write(*,*)"what is Pmax" read(*,*)Pmax !Pmax=0.6d0 write(*,*)"what is Ks" read(*,*)Ks !Ks=140d0

!f1=oldf1+(2d0*(Pmax*s(i)-p(i)*(s(i)+Ks))*s(i))/(s(i)+Ks)**2d0 !f2=oldf2+(2d0*(Ks*p(i)+(p(i)-Pmax)*s(i))*s(i)*Pmax)/(Ks+s(i))**3d0 !write(*,*)"f1 and f2 =",f1,", ", f2

maxit=1000d0 it=0d0 do old11=0d0 old12=0d0 old21=0d0 old22=0d0 oldf1=0d0 oldf2=0d0 do i=1,numpts !write(*,*)"this is the power",p(i) !write(*,*)"this is the substrate",s(i) j(1,1)=old11+(2d0*s(i))/(Ks+s(i))**2d0 j(1,2)=old12+(2d0*(Ks*p(i)+p(i)*s(i)-2d0*Pmax)*s(i))/(Ks+s(i))**3d0 j(2,1)=old21+(-2d0*(2d0*Pmax-p(i))*s(i)*s(i))/(Ks+s(i))**3d0 !this was the fun part j(2,2)=old22+(-2d0*(2d0*Ks*p(i)+s(i)*(3d0*(p(i)-Pmax)*Pmax*s(i)-p(i))))/(Ks+s(i))**4d0 f1=oldf1+(2d0*(Pmax*s(i)-p(i)*(s(i)+Ks))*s(i))/(s(i)+Ks)**2d0 f2=oldf2+(2d0*(Ks*p(i)+(p(i)-Pmax)*s(i))*s(i)*Pmax)/(Ks+s(i))**3d0 old11=j(1,1) old12=j(1,2) old21=j(2,1)



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