phenol degradation

8/7/2019 phenol degradation

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Volume 3, Issue 1 2008 Article 46

Chemical Product and Process

Modeling

Dynamic Simulation of Phenol Biodegradation

in a Fluidized Bed Bioreactor Using Genetic

Algorithm Trained Neural Network

Venu Vinod Ananthula, National Institute of Technology

Warangal

Venkat Reddy Goli, National Institute of Technology

Warangal

Neelima Murapaka, National Institute of Technology

Warangal

Recommended Citation:

Ananthula, Venu Vinod; Goli, Venkat Reddy; and Murapaka, Neelima (2008) "Dynamic

Simulation of Phenol Biodegradation in a Fluidized Bed Bioreactor Using Genetic Algorithm

Trained Neural Network," Chemical Product and Process Modeling: Vol. 3 : Iss. 1, Article 46.

Available at: http://www.bepress.com/cppm/vol3/iss1/46

DOI: 10.2202/1934-2659.1203

2008 Berkeley Electronic Press. All rights reserved.


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Dynamic Simulation of Phenol Biodegradation

in a Fluidized Bed Bioreactor Using Genetic

Algorithm Trained Neural Network

Venu Vinod Ananthula, Venkat Reddy Goli, and Neelima Murapaka

Abstract

The aim of this work is to simulate the dynamic behavior of a phenol biodegradation process

in a fluidized bed bioreactor (FBR). Pseudomonas putida is used for the biodegradation of phenol.

A mathematical model was developed to describe the dynamic behavior of the biodegradation

process. The model equations describing the process have been solved, and the rate of

biodegradation and the biofilm thickness at different points of time have been determined. The

mathematical model has been directly mapped onto the network architecture. The network is used

to find an error function. Minimization of error function with respect to the network parameters

(weights and biases) has been considered as training of the network. A real-coded genetic

algorithm has been used for training the network in an unsupervised manner. The system is tested

for two different inlet concentrations of feed. The results obtained are then compared with the

experimental results. It is found that there is a good agreement between the experimental results

and the results obtained from the model.

KEYWORDS: biodegradation, phenol, dynamic, neural network, simulation


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1. INTRODUCTION

Fluidized bed bioreactors have been receiving considerable interest in wastewater

treatment. The fluidized bed bioreactor has been shown (Van Den Heuvel andBeeftink, 1988; Denac and Dunn, 1988; Tang et al., 1987; Tang and Fan, 1987;

Holladay et al., 1978) to outperform other types of reactors. The superior

performance of the fluidized-bed bioreactor is due to very high biomassconcentration due to immobilization of cells onto the solid particles; intimate

contact between gas, liquid and solid phases; decoupling of residence times of

liquid and microbial cells due to immobilization. Extensive information isavailable in literature on the biodegradation of phenol and fluidized-bed

bioreactors (Venu Vinod and Venkat Reddy, 2006; Venu Vinod and Venkat

Reddy, 2005; Sa and Boaventura, 2001; Gonzalez et al., 2001; Alvaro et al., 2000;Livingston and Chase, 1991; Fan et al., 1990; Livingston and Chase, 1989;

Worden and Donaldson,1987). Phenolic wastewater treatment is done by usingmethods like freely suspended cell systems, trickling filters, rotating disc,activated sludge, biological fixed film methods and fluidized bed bioreactors.

Tricking filters are more advantageous over freely suspended cell systems (Van

Den Heuvel and Beeftink, 1988). However fluidized bed bioreactors have been

found to be superior to other type of reactors (Denac and Dunn, 1988; Holladay etal., 1978) in relation to volumetric biodegradation capacity. It is highly desirable

to understand the transient behavior of a fluidized bed bioreactor for wastewater

treatment. Fluidized bed bioreactor operation is always in the transient conditionduring start-up and shutdown periods. The transient solution of a set of governing

differential equations provides dynamic information.

Modeling of the transient state operation of a liquid-solid fluidized bedbioreactor has been attempted by several researchers. Worden and Donaldson

(1987) conducted experimental and modeling studies to analyze the dynamicresponse behavior of a phenol oxidizing fixed film using an FBR. Experimental

dynamics were compared with a dynamic mathematical model based on diffusion

and reaction within the biofilm. The dynamic model of the biofilm reactor wasable to predict the trends in the data at constant pH. Tang et al. (1987)

investigated the transient response of a draft tube fluidized bed bioreactor (DTFB)

to a step increase in influent phenol concentration. The draft-tube restored the

stable operation when subjected to a 200% step increase in inlet concentration.Simulation results showed good agreement with experimental data. Nakamura and

Sawada (2000)presented a microbial growth model for estimating the dynamicbehavior of cell growth and substrate consumption in the biodegradation ofphenol containing a heavy metal such as Zn or Cu ions in medium. The transient

solution of the set of differential equations governing a system provides dynamic

information such as the order of the step response, time constant, damping factor

1

Ananthula et al.: Simulation of Phenol Biodegradation in a Fluidized Bed Bioreactor

Published by Berkeley Electronic Press, 2008


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and process gains, which are essential to the design of control devices of the

bioreactor. Tsuneda et al. (2002) have studied the dynamic response of

completely mixed three-phase fluidized bed biofilm reactor treating simulated

domestic wastewater after a step change has been given in inlet concentration.Dynamic studies involving phenol has been carried out in other types of

bioreactors, viz., fixed bed biofilm reactor (Leitao and Rodrigues, 1998), trickle

bed bioreactors (Iliuta and Larachi, 2005), fixed biofilm (Tzu-Yang Hsien andYen-Hui Lin,2005).

The model equations for biodegradation of phenol have been

simulated using conventional numerical methods (Livingston and Chase, 1989;Tang and Fan, 1987; Tang et al., 1987). Solution of the differential equations

(linear and nonlinear) has been obtained previously using neural network

approach (Parisi et al., 2003; Lagaris et al., 1998; Meade and Fernandez, 1994a;Meade and Fernandez, 1994b; Lee and Kang, 1990). Though there have been

some many reports in literature (Balan et al., 1999) of application of NeuralNetwork model to predict the biodegradation of phenol, application of NeuralNetwrok GA method to solve model equations of biochemical processes in

FBR has not been reported. In this work an attempt has been made to solve the

dynamic model equations describing the biodegradation process of phenol in a

fluidized bed bioreactor using a combination of neural network GA. The meansquared error (MSE) has been evaluated using a subroutine incorporating ANN

and optimization of weights for ANN has been carried out in the GA module.

ANN as Differential Equation Solver

In early 90s, it was proved that the approximation capabilities of networks makeArtificial Neural Netwroks (ANN) as numerically accurate and predictable as

conventional computational methods (Lagaris et al., 1998; Meade and Fernandez,1994a; Meade and Fernandez, 1994b; Lee and Kang, 1990). Finding a neural

network that approximates the solution of a given set of differential equations has

many benefits compared with traditional numerical methods viz., obtaining aanalytic continuous solution (compared to numerical methods), good

generalization capabilities, tackling real time problems reaching the global

minimum of the error surface etc. (Parisi et al., 2003; Lagaris et al., 1998; Meade

and Fernandez, 1994a). In the approach of solving differential equations by neuralnetworks, the mathematical model of a physical process will be incorporated

directly and accurately in to the architecture of the neural network (Parisi et al.,2003).

2

Chemical Product and Process Modeling, Vol. 3 [2008], Iss. 1, Art. 46

http://www.bepress.com/cppm/vol3/iss1/46

DOI: 10.2202/1934-2659.1203


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Genetic Algorithms

In a genetic algorithm, this principle of natural evolution is introduced through its

operators. Several researchers have attempted to apply genetic algorithms toneural network weight optimization problems (Sexton et al.1998;Karunanithi etal., 1992; Schaffer et al., 1992; Dominic et al., 1991).

2. MATHEMATIACAL MODEL DEVELOPMENT

The following assumptions are made to develop the unsteady state isothermal

biofilm model.1. The rate limiting substrate is only phenol. Sufficient amount of oxygen is

available for the phenol degradation.

2. The substrate is transported from bulk liquid to biofilm phase through thestagnant liquid-layer by molecular diffusion.

3. Diffusivity of phenol through biofilm and density of biofilm are constant.4. Production of biomass is proportional to the consumption of phenol.5. The bed particles may be considered spherical in shape, of uniform size,

and inert to the substrates.

6. The biofilm is uniform in thickness, composition and reactivity.

Under these assumptions, a mass balance for phenol over the fluidized bed reactor

can be written as

)(, binobsS SSQR = (1)

Since phenol is the only growth-limiting substrate and all other nutrients are

present in excess amounts and oxygen is sufficiently supplied, the phenolutilization rate in the biofilm based on diffusion (Ficks law) and biological

inhibition reaction (Haldane kinetics) can be expressed as:

Rpr

S

rr

SD

t

SSf

+

= 2

2

2

(2)

where

I

SK

SSK

SRp 2max

++=

To solve this equation two boundary conditions and one initial condition are

required. Since no substrate can diffuse through the support medium,

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at Prr= 0=

r

S

(3)

At the biofilm and liquid-film interface, mass transfer of phenol into the liquid-

film is equal to the phenol diffused into the biofilm.

at fP Lrr += )( ibSSf SSkr

SD =

(4)

Since, at time t= 0 the concentration of substrate in the biofilm is zero we havethe initial condition:

at t= 0 00 == SS (5)

2.1 Growth of biofilm:

As the phenol diffuses into and through the biofilm during biodegradation, the

biofilm utilizes phenol as carbon source for biosynthesis. Since the density ofbiofilm is assumed constant, the volume of biofilm and thus the thickness of

biofilm must increase with time as the biofilm grows. Therefore, the phenol

diffuses through a boundary, which can be moving with time. The boundary isliquid / biofilm interface. Since the biofilm grows upon the utilization of phenol,

the growth rate of biofilm can be expressed by the following equation (Tzu and

Yen, 2005):

+

++=

fP

P

Lr

r

I

S

sxf

dr

K

SSK

SY

dt

dL2

max/

(6)

Equations (2) (5) can be rewritten in terms of nondimensional variables asfollows:

*1

2

*22

*

*

Rpx

S

L

rx

x

S

L

r

t

S

f

Pf

P

++

=

(7)

at x = 0 0

*

=

x

S

(8)

at x = 1 )1( **

i

Sf

fSS

D

Lk

x

S=

(9)

at t= 0 *0* SS = (10)

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where

bS

SS =* ,

i

i

rr

rrx

=

0

,

)(

2

*

tL

tDt

f

sf= ,

++

=2*

*

**

*2

max*

I

S

Sf

i

K

SSK

S

DS

rRp

b

The phenol degradation rate can be found after equations (6) - (10) have been

solved by integrating the reaction rate over the biofilm for a single bioparticle and

multiplying by the number of particles and reactor as shown in equation (11)

(11)

Experimental (observed) phenol degradation rate Rsobs is calculated from equation(1) and then compared with the simulated degradation from equation (11).

Thickness of the biofilm is calculated from equation (6).

3. EXPERIMENTAL

3.1 The reactor set-up

The schematic diagram of the draft tube fluidized bed bioreactor used in the

present work is shown in the Fig. 1.

3.2 Reactor and the draft tube

The fluidized bed bioreactor and the draft-tube are made up of glass. A sparger

made up of same material has been provided at the bottom of the reactor through

which air can be sparged into the reactor. The total volume of the reactor is about

2.67 10-3 m3 (2.67 liters). The top of the glass reactor is closed with a platethrough which all the probes and sensors are inserted into the reactor. An

overflow line has been provided near the top so that, the reaction medium flowsout of the reactor in continuous operation.

+=

= ++=

fP

P

Lrr

rr

I

SSx

VPcalcS drr

K

SSK

S

Y

NR 2

2

max

/

, 4

5




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Plastic beads with a density of 1005 kg/m3

have been used for

immobilization of the microorganism. The average diameter of the beads is 3.895

mm. Two peristaltic pumps one each for media and feed into the reactor have

been provided. The flow rate of these pumps can be set at the required value using

a flow controller. The capacity of the pumps is 0.11 10-7 9.7 10-7 m3/s (40 3500 ml/h).

Figure 1. Experimental Setup for the Biodegradation of Phenol in a Fluidized Bed

Bioreactor

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The reactor is provided with a glass jacket to maintain the temperature of the

reactor system. Depending on the temperature set for the reactor operation,

controller switches on either the heating or refrigeration circuit. Separate tanks

made of stainless steel have been provided for supplying the feed, medium, acidand base solutions for pH control.

3.3 Reactor instrumentation

To maintain the pH of the system a pH meter and a controller have been provided.

pH has been maintained by addition of acid or base from the tanks provided at thetop. Oxygen will be consumed in the degradation of phenol by microorganism.

Oxygen required for the process has been supplied in the form of air from a

compressor. The oxygen content in the reaction medium can be measured using aDO meter. The flow rate of air can be measured using rotameter, with a range of

0.167 10-4 1.67 10-4 m3/s (110 lpm).

3.4 Microbial culture

A strain of microorganism Pseudomonas sp. reported to be capable of utilizing

phenol as the sole carbon and energy source was obtained from Regional

Research Laboratory, Jammu, India.

3.5 Subculture

The bacterium is subcultured once in a month by preparing slants using nutrient

agar. For every 100ml of nutrient broth/agar the growth medium of thecomposition mentioned in Table 1 was added. To each of the test tubes 15ml ofthis nutrient agar solution is added in a tilted position around 30

0to the horizontal.

After the solidification of the nutrient agar in the test tubes, colonies of bacteria is

introduced on it, and is incubated for 24 hours at 30oC and then it is stored at 4

oC

in a refrigerator.

Table 1

For 100ml of Nutrient broth/agar

Compound Concentration, gBeef Extract 0.3

NaCl 0.5

Peptone 0.5

Agar-agar 1.2

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3.6 Culture preparation

The culture was maintained by periodic subculture on nutrient agar and stored in a

refrigerator. The reaction medium was prepared from this strain by growing the

bacteria on 2.6 10-3 m3 (2.6 litres) of 0.05 kg/m3 (50 ppm) of phenol solutioncontaining growth medium of the composition mentioned in Table 2. Before

inoculation of the organism sterilization of the phenol solution was done in

autoclave at a gage pressure of 1.034 105 N/m2 (15 psi) for 20 minutes. This hasbeen done to selectively grow the Pseudomonas species.

Table 2

Composition of growth medium

Compound Concentration, ppm

KH2PO4 420

K2HPO4 375

(NH4)2SO4 240NaCl 15

CaCl2 15

MgSO4.H2O 30

3.7 Growth medium

The growth medium was made up using tap water. Sterile conditions were not

maintained during the continuous operation of the reactor, to simulate treatment

of actual plant wastewater as the latter would contain different contaminants.

3.8 Biomass

25 ml of the reactor medium was taken in every run and filtered through 0.7 mfilter paper to separate the biomass produced. The filter paper was dried at 105

0C

and weighed again after drying to obtain the weight of the biomass produced.

3.9 Measurement of biofilm thickness

At the beginning and end of the experimental run (indicated by the steady state

reached), some bioparticles were taken into a dish and weighed. They were dried

and again weighed. The difference between these two weights is equal to the

amount of water evaporated. Volume of water evaporated (equal to the weight

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evaporated) divided by the number of particles and the average surface area of

each particle gives the thickness of the biofilm.

3.10 Start-up of the equipment

2.6 liters of reaction medium was transferred to fluidized bed bioreactor and theorganism was allowed to grow in batch mode for 36 hours for immobilization of

microorganism on to the solid particles. In the first run thereafter was put in

continuous operation with a feed flow rate of 510 mlhr-1

(corresponding to the

dilution rate of 0.196 h-1

) of inlet phenol concentration of about 64 ppm. Thedissolved oxygen (DO) concentration in the reactor was maintained at 2 ppm.

The pH in all the runs was maintained at 7.0 using 0.1 N HCl and 0.1 N NaOH.

The reaction temperature was maintained at 300C in all the runs using a

temperature controller provided with a heating/cooling circuit. The concentration

of phenol in the overflow from the reactor was analyzed for every 1hriodometrically (Furman, 1959).

4. SIMULATION OF FBR BY FFANN

4.1 Construction of FFANN and evaluation of error function:

A simple multilayer FFANN consisting of one input layer with a single neuron,

one hidden layer with five hidden neurons and an output layer with a single

neuron has been chosen for the solution of differential equations. Each neuron in

the hidden layer uses sigmoidfunction as its activation function, and each neuronin the output layers uses purelinfunction as its activation function.

The two inputs to the networks are the dimensionless time, t* anddimensionless biofilm thickness, x. The output of the network is dimensionless

phenol concentration within the biofilm, S*. The network architecture is shown in

Fig. 2.The input to both networks is the dimensionless biofilm thickness, x

and t. The output from the network is the dimensionless phenol concentrations

within the biofilm S*. S

*can be written as (detailed procedure in Appendix)

outii

i uzvS += = )(*5

1 (12)

hiiii utwxwz ++= 21 )2/tanh()( xx =

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The dimensionless concentration derivative can be written as

(13)

S*3

4

2

5

1

*

t*

Figure 2. Network Architecture

(14)

(15)

Once the values ofS* and its derivatives with respect to x and t* are evaluated,the error function can be calculated as shown below.

=

=

5

1

1

1 )(*

i

iii zwvx

S

=

)(* 112

12

2

iii zwvx

S

=

=

5

1

1

2 )(

*

*

i

iii zwv

t

S

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(16)

2

2

*)(

=x

SxE ifx = 0 (17)

{ }

+

=2**

2

*

2

*)( ii

R

b

Sf

S SSSS

S

D

k

x

SxE

ifx = 1 (18)

{ }2**)( 02 SSxE = ift* = 0 (19)

Then the mean squared error is calculated as follows:

+

=

pn

i

ji

p

xExEn

MSE1

11

21 )(11

1)(

1(20)

5. RESULTS AND DISCUSSION

The model equations describing the dynamic behavior of the phenol

biodegradation process were solved using a FFANN trained with a real-coded GAto find out the bio-degradation rate and the thickness of the biofilm. A MATLAB

script was developed for this purpose. The model parameters used in the study arementioned in Table 3.

Table 3

Model parameters used in the study

Parameter Value Units

max 1.436 * 10-4

s-1

Ks 21.92* 10-3

Kg/m3

KI 522*10-3

Kg/m3

Yx/s 0.6 Kg/kg

ks 0.4*10-4

m/s

Np 6369 --rp 2.155*10

-3m

Lfo 1.1*10-6

m

Dsf 109 0.042 - 0.051 m

2/s

210 Kg/m3

( )

2

0

2

*22

*

*

1

1

+

++

= RpxS

L

rx

x

Sr

t

S

xE

f

P

i

11




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The architecture of the network was selected by a trail and error

procedure starting with a simple structure of one hidden layer with 3 neurons (the

input and output layer has each one neuron which will be fixed by the problem).

The number of neurons in the hidden layer was increased progressively up to 15.The decrease in error with respect to increase in hidden neuron was observed. The

network with five neurons in the hidden layer was found to be sufficient for thepresent problem. No attempt was made to find out the optimum number of hidden

layers, since it would be better to select a network with minimum number of

hidden layers so that the number of parameters to be optimized will be less in thetraining phase. A batch mode of training was adapted, in that the weights are

updated after the presentation of all the training examples that constitute an

epoch.

5.1 Rate of biodegradation

The rate of biodegradation of phenol observed experimentally and that obtainedfrom simulation have been plotted in Figs. 3 and 4 for feed concentrations of 64

ppm and 76 ppm respectively. The figures show the biodegradation rate increases

with time due to the production of biomass in the reactor. Fig. 3 shows that steadystate is reached at 7.75 hr whereas it takes 9.33 hours for steady state

corresponding to a feed concentration of 76 ppm. This is due to the higher

concentration of phenol in feed. In both the figures there is good agreementbetween the observed values and the simulated values. Maximum deviation

between has been found to be 9.6%.

5.2Growth of biofilm

The thickness of the biofilm calculated from experimental data and the simulated

vales has plotted in Fig. 5, for the data corresponding to the inlet concentration of64 ppm and 76 ppm. Initial thickness of the biofilm is 1.10e-6 m. There is a

gradual increase in thickness, and the thickness goes up to 4.2e-6 m (simulated)

corresponding a steady state at t = 7.75 hr. The steady state biofilm thicknessobtained for feed concentration of 76 ppm is 6.38e-6 m (simulated) at t= 9.33 hr.The deviation between the experimental and the simulated values is about 5%.

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0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

2.50E-05

3.00E-05

3.50E-05

4.00E-05

0 2 4 6 8 10

Time, hrs

Rateofphenolbiodegradation

(kg/h)

Obs

Sim

Figure 3. Observed and Simulated rate of biodegradation of phenol as a functionof time (Feed concentration = 64 ppm)

0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

2.50E-05

3.00E-05

3.50E-05

4.00E-05

4.50E-05

0 2 4 6 8 10 12

Time, hrs

Rateofphenolbiodeg

radation

(kg/h)

Obs

Sim

Figure 4. Observed and Simulated rate of biodegradation of phenol as a functionof time (Feed concentration =76 ppm)

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0.00E+00

1.00E-06

2.00E-06

3.00E-06

4.00E-06

5.00E-06

6.00E-06

7.00E-06

0 5 10 15

Time, hrs

Biofilmthickness,m

64 ppm Sim

76 ppm Sim

64 ppm Exp

76 ppm Exp

Figure 5. Growth of biofilm for feed concentrations of 64 and 76 ppm

6. CONCLUSIONS

The model equations describing the dynamic behavior of the phenol

biodegradation process in a fluidized bed bioreactor have been solved usingfeedforward artificial neural network and the rate of biodegradation and the

biofilm thickness as a function of time has been calculated. Real-coded genetic

algorithm has been used to train the network in an unsupervised manner. Theprogram script has been developed using MATLAB. The calculated degradation

rates are in good agreement with the experimental values, with the maximum

deviation being 9.6%. From the result we can suggest that genetic algorithm

trained feed forward neural network can be used for the dynamic simulation ofbiodegradation process in a fluidized bed bioreactor.

Nomenclature

SfD Diffusivity of phenol in biofilm [m2/s]

E Error measuref*(y) Approximated output from the network

fL Thickness of biofilm [m]

k Liquid solid mass transfer coefficient for phenol [m/s)

IK Inhibition constant for phenol [kg/m3]

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SK Monod constant for phenol [kg/m3]

*

IK Dimensionless Inhibition constant for phenol [kg/m3]

*SK Dimensionless Monod constant for phenol [kg/m

3

]

n Number of inputs to the FFANN

np Number of points in the time domain

nPOP Population size

N Output of the network

pc Crossover probability

P Number of points in the integration domain

PB Total number of boundary points

Q Flow rate of feed [m3/s]

r Radial position in biofilm [m]

rp Radius of biomass free bioparticle [m]RSobs Observed rate of phenol degradation [kg/hr]

RScalc Calculated rate of phenol degradation [kg/hr]

bS Phenol concentration in bulk liquid [kg/m3]

inS Phenol concentration in feed [kg/m3]

0S Phenol concentration at t= 0. [kg/m3]

Si Phenol concentration at the interface between a

bioparticle and bulk liquid [kg/m3]

*S Dimensionless phenol concentration

t Time [s]

t* Dimensionless timeiu , outu Bias of hidden unit i and output unit, respectively

iv Weight from the hidden unit i to the output unit

ijw Weight from the input unit j to the hidden unit i

x Dimensionless biofilm thickness

sxY / Yield coefficient

Density of biofilm [kg/m3]

max Maximum specific growth rate of biomass [h-1

]

Sigmoid transfer function

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APPENDIX

The Method

Let us consider a general differential equation in the form:

D (f (y)) = 0 (A1)

with respect to boundary conditions

B (f (y)) = 0 (A2)

WhereDandB are any non-linear, inhomogeneous differential operators and f (y)is the solution that satisfies equation (A1) and the boundary conditions (A2).

Considering that an FFANN is a universal function approximator, the goal of themethod is to find a neural network f*(y) which approximates f(y) in the finite

domain n[a,b]y .

Consider a multilayer FFANN with n input units, one hidden layerwith H sigmoid units and a linear output unit. For a given input

vector ( )1 nx= x ,...,xr

the output of the networkN, is given by

1

)H

i i out

i=

v (z uN = + (A3)

Where

1

n

i ij j i

j

z w x u=

= + (A4)

Since is a continuous and derivable function ofx, it can be shown that (Lagariset al., 1998):

1

)k H

k k

i ijkij

i

Nv w

x(z

=

=

(A5)

Wherek

denotes the kth

derivative of the sigmoid function.

Hence it is possible to approximate the differential operators DandB

using the network; in other words f(y) can be approximated by a network (N) witha differentiable activation function. In order to find an approximation of f(y),

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equation (A1) along with boundary conditions (A2) can be chosen as the

performance function of the network. The error measure Emust be evaluated in a

finite number of points (P) into the integration domain n[a,b]i

y .

22

* *1 1[ ( ( ))] [ ( ( ))]( )BPP

i j

i jB

D f y B f yP P

E w += (A6)

PB is the total number of boundary points and f*(y) is the approximated output

from the network corresponding to the input points ( ,i j

y y ). As Etends to zero,

f* tends to fand so the approximate solution for the differential equation system isfound. The efficient minimization of equation (A6) can be considered as aprocedure of training the neural network. At this point, the original problem has

been reduced to an unconstrained optimization problem involving theminimization of the errorEwith respect to the network parameters wij and u i.e.,weights and biases. Since the error does not depend on target outputs (the function

fis unknown a priori) the network is said to be trained in an unsupervised manner.

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