Full Length Article

Mathematical modeling of gas phase and biofilmphase biofilter performance

V. Meena a, L. Rajendran b,*, Sunil Kumar c, P.G. Jansi Rani b

a Department of Mathematics, Mangayarkarasi College of Engineering, Madurai 625018, Tamil Nadu, Indiab Department of Mathematics, Sethu Institute of Technology, Virudhunagar 626115, Tamil Nadu, Indiac Department of Mathematics, N I T, Jamshedpur 831001, Jharkhand, India

A R T I C L E I N F O

Article history:

Received 24 May 2014

Received in revised form 28

September 2015

Accepted 29 September 2015

Available online 2 November 2015

A B S T R A C T

In this paper, mathematical models of biofilteration of mixtures of hydrophilic (methanol)

and hydrophobic (α-pinene) volatile organic compounds (VOC’s) biofilters were discussed.

The model proposed here is based on the mass transfer in air–biofilm interface and chemi-

cal oxidation in the air stream phase. An approximate analytical expression of concentration

profiles of methanol and α-pinene in air stream and biofilm phase have been derived using

the Adomian decomposition method (ADM) for all possible values of parameters. Further-

more, in this work, the numerical simulation of the problem is also reported using the Matlab

program to investigate the dynamics of the system. Graphical results are presented and dis-

cussed quantitatively to illustrate the solution. Good agreement between the analytical and

numerical data is noted.

© 2015 Mansoura University. Production and hosting by Elsevier B.V. This is an open

access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-

nc-nd/4.0/).

Keywords:

Methanol

Biofilters

VOCs

Mathematical modeling

Non-linear differential equations

Adomian decomposition method

1. Introduction

Different cleaning technologies of gaseous effluents have beendeveloped. Among these technologies, biological methods areincreasingly applied for the treatment of air polluted by a widevariety of pollutants. Biofilteration is certainly the most com-monly used biological gas treatment technology. Biofilterationinvolves naturally occurring microorganisms immobilized inthe form of biofilm on a porous medium such as peat, soil,compost, synthetic substances or their combination.

The medium provides to the microorganisms a hospitableenvironment in terms of oxygen, temperature, moisture, nu-trients and pH. As the polluted airstream passes through thefilter-bed, pollutants are transferred from the vapor phase tothe biofilm developing on the packing particles [1,2].

Recently Li et al. [3] as well as other research groups [4–10]have investigated emissions of VOCs into the atmosphere. Cur-rently, biological control processes have become an establishedtechnology for air pollution control. Biological control pro-cesses have many advantages over traditional methods suchas lower operating fees and less secondary pollution, which

* Corresponding author. Tel.: +91 9442228951.E-mail address: raj_sms@rediffmail.com (L. Rajendran).

http://dx.doi.org/10.1016/j.ejbas.2015.09.0072314-808X/© 2015 Mansoura University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

e g y p t i an j o u rna l o f b a s i c and a p p l i e d s c i e n c e s 3 ( 2 0 1 6 ) 9 4 – 1 0 5

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is rather true for the removal of readily biodegradable VOCsat low concentrations, so these processes are investigated largelyand widely. Bioreactors for VOC removal can be classified asbiofilters, bio scrubbers, biotrickling filters, or rotating drumbiofilters, and choice of reactors should be based on manyfactors including the characteristics of the target VOCs [11–15].

In order to control the emission of volatile organic com-pounds (VOC) like methanol, α-pinene, etc. from industries,biofilters are being used nowadays instead of chemical complexabsorption method [16–20]. Biofilters offer two major advan-tages to an energy-starved country like India. A mathematicalmodel is describing the dynamic physical and biological pro-cesses occurring in a packed trickle-bed air biofilters to analyzethe relationship between biofilter performance and biomassaccumulation in the reactor [4].

For the treatment of mixed VOCs [21–23], the presence ofmethanol and α-pinene in the air stream significantly influ-enced the removal of pollutants. The removal capacity formethanol and α-pinene per unit volume of the bed decreasedlinearly with increasing loading rates of methanol and α-pinene.The presence of this easily biodegradable compound sup-pressed the growth of the methanol and α-pinene degradingmicrobial community, thereby decreasing methanol andα-pinene removal capacity of the biofilters. Some researchershave studied the biofitration of pure methanol [24–26] and pureα-pinene [27,28].

Recently, few researchers have studied the biofitration ofmixtures of pure methanol and pure α-pinene. Also a few re-searchers have tried to examine the treatment of mixtures ofhydrophobic and hydrophilic VOCs and to understand the in-teractions between these compounds despite the fact that thissituation exists in larger amount of air emissions. Mohseni andAllen [16] developed a mathematical model for methanol andα-pinene removal in VOC’s biofilteration. Lim et al. [29] devel-oped the steady state solution of biofilter model only for thelimiting cases (first order and zero order kinetics). Also Lim et al.[30,31] obtained the non-steady solution of biofilter model usingnumerical methods. Recently some authors [32,33] solved thenon-linear problems using fractional reduced differential trans-form method (FRDTM). To the best of our knowledge, to date,a rigorous analytical expression of concentrations of sub-strate in the biofilm phase and air phase has been reported.The purpose of this communication is to derive approximateanalytical expressions for the concentrations in both the phasesusing the Adomian decomposition method [34–40].

2. Mathematical modeling of the boundaryvalue problem

The mathematical model relating the biofiltration of blends ofhydrophilic and hydrophobic VOCs is based on the biophysi-cal model proposed by Mohseni and Allen [16]. It includes twomain processes of diffusion of the compounds methanol andα-pinene through the biofilm and their degradation in thebiofilm. Fig. 1 illustrates a schematic diagram of a single par-ticle, in the biofilter, covered with a uniform layer of biofilmin which the simultaneous biodegradation of methanol andα-pinene takes place. The experimental setup for thebiofilteration of this organic compound is given in Fig. 2.

2.1. Mass balance in the biofilm phase

The removal of methanol and α-pinene in the biofilm at steadystate is described by the following system of non-linear dif-ferential equations (Mohseni and Allen [16]):

Dd Sdx

XY

SK S

emm

m

m m

m m

2

2=

+( )μmax

(1)

Dd Sdx

XY

SK S

epp

p

p p

p p

2

2=

+( )α μmax (2)

where Sm and Sprepresent the concentration of methanol andα-pinene respectively. μmax, , ,K Y D xe and are maximum spe-cific growth rate, half saturation constant, yield coefficient,effective diffusion coefficient and the distance respectively. Sub-scripts m and p represent methanol and α-pinene respectively.The dry cell density in the biofilm X represents the overall popu-lation of microorganisms that consist of methanol and α-pinenedegraders.The coefficient for the effect of methanol on α-pinenebiodegradation is defined as follows:

α = + ( )( )1 1 2C Km i (3)

where Ki and Cm are the inhibition constant and the concen-tration of methanol in the air phase respectively.The boundaryconditions are

SCm

S SCm

S xmm

mim P

P

PiP= = = = =and at 0 (4)

dSdx

dSdx

xm P= = =0 at δ (5)

2.2. Mass balance in gas phase

The concentrations of methanol and α-pinene in the air, alongthe biofilter column, are described by

UdCdh

A DdSdx

gm

s emm

x

= ⎡⎣⎢

⎤⎦⎥ =0

(6)

Fig. 1 – Biophysical model for the biofilm structure on thebiofilters packing materials and the concentration profilesacross the biofilm.

95e g y p t i an j o u rna l o f b a s i c and a p p l i e d s c i e n c e s 3 ( 2 0 1 6 ) 9 4 – 1 0 5

UdCdh

A DdSdx

gp

s epmp

x

= ⎡⎣⎢

⎤⎦⎥ =0

(7)

where Cm and Cp represent the concentration of methanol andα- pinene in the air phase. U A D D hg s em ep, , , and are the super-ficial gas velocity, biofilm surface area, effective diffusivity ofmethanol, effective diffusivity of α-pinene and position alongthe height of the biofilters respectively.The corresponding initialconditions are

C C C C hm mi p pi= = =and at 0 (8)

where the subscript i represents the concentration of the VOCsat the biofilters inlet.

2.3. Dimensionless mass balance equation in the biofilmphase

The non-linear differential Eqs. (1) and (2) are made dimen-sionless by defining the following dimensionless parameters:

β ϕ μ δ= = =( )SK

XY D K

SSS

im

m

m

m em mm

m

im

, ,max1

2

* (9)

β ϕ μ δ1 1

2

= = =( )SK

XY D K

SSS

ip

p

p

p ep pp

p

ip

, ,max * (10)

Using the above dimensionless variables, Eqs. (1) and (2)reduce to the following dimensionless form:

d SdX

SS

m m

m

2

2 1* *

**=

+

⎛

⎝⎜⎜

⎞

⎠⎟⎟ϕ

β (11)

d SdX

SS

p p

p

2

2 111

* ***

=+

⎛

⎝⎜⎜

⎞

⎠⎟⎟αϕ

β (12)

The corresponding boundary conditions for the above Eqs.(11) and (12) can be expressed as

S S Xm p* * at *= = =1 1 0, (13)

dSdX

dSdX

Xm p**

**

at *= = =0 1 (14)

2.4. Dimensionless mass balance in the gas phase

The differential Eqs. (6) and (7) are made dimensionless by de-fining the following parameters:

AHA D S

U CA

HA D SU C

hhH

CCC

CCs em im

g mi

s ep ip

g pim

m

mip= = = = =

δ δ, , , * , *1 * pp

pmiC(15)

Using Eq. (15), Eqs. (6) and (7) can be expressed in the di-mensionless form as follows:

dCdh

AdSdX

m m

X

* ** *

*

= −⎛

⎝⎜⎜

⎞

⎠⎟⎟

=0

(16)

dCdh

AdSdX

mp p

X

* ** *

*

= −⎛

⎝⎜⎜

⎞

⎠⎟⎟

=

α 1

0

(17)

The corresponding initial conditions for the above Eqs. (16)and (17) can be expressed as

CCC

CCC

hmm

mip

mp

pi

* *= = =and at * 0 (18)

3. Analytical expression for the concentrationof methanol and α-pinene using the Adomiandecomposition method (ADM)

In recent years, many authors have applied the ADM [35–40]to various problems and demonstrated the efficiency of theADM for handling non-linear and solving various chemistry

Fig. 2 – Experimental setup for the bio filtration of methanol and α-pinene.

96 e g y p t i an j o u rna l o f b a s i c and a p p l i e d s c i e n c e s 3 ( 2 0 1 6 ) 9 4 – 1 0 5

and engineering problems. Using ADM (refer to Appendix A),we can obtain the concentration of methanol and α-pinene inthe biofilm phase (see Appendix B) as follows:

S XS x

SX

Y D K Sx x

mm

im

m

m em m im

* max*( ) = ( ) = ++( )

−⎛⎝⎜

⎞⎠⎟

=

( )12

1

2 2

2

μ δδ δ

+++( )

−⎛⎝⎜

⎞⎠⎟

ϕβ1 2

2XX

** (19)

S XS x

SX

Y D K Sx x

pp

ip

p

p ep p ip

* max*( ) =( )

= ++( )

−⎛⎝⎜

⎞⎠⎟

=

( )12

2 2

2

μ αδδ δ

111 2

1

1

2

++( )

−⎛⎝⎜

⎞⎠⎟

αϕβ

XX

** (20)

Also solving Eqs. (6–7) and (16–17) using the analyticalmethod, we can obtain the concentration of methanol andα-pinene in the air phase.

C hC h

CA X

Y U K S Kh

Am

m

mi

s m

m g m im m

* max*( ) = ( ) −+ ( )( )

= −+( )

= ( )11

11

μ ϕβ

δ hh* (21)

C hC h

CA X

Y U K S Kh

Ap

p

pi

s p

p g p ip p

* max*( ) = ( ) = −+ ( )( )

= −( )11

11

1 1α μ α ϕδ

++( )β1

h* (22)

4. Analytical expression for theconcentrations of methanol and α-pinene forunsaturated (first order) kinetics

Now we consider the limiting case where the substrate con-centrations of methanol and α-pinene in biofilm phase arerelatively low. In this case S K S Km m p p≤ ≤and . Eqs. (1) and (2)now reduce to the following form.

Dd Sdx

XY K

Semm m

m mm

2

2= ( )μmax (23)

Dd Sdx

XY K

Sepp p

p pp

2

2= ( )α μmax (24)

The analytical expression for concentrations of methanoland α-pinene in the biofilm phase becomes

S XS x

S

XY D K

x

Xm

m

im

m

m em m

m

*cosh

cosh

max

max

*( ) = ( ) =−( )⎛

⎝⎜

⎞

⎠⎟

( )

(

μ δ

μ ))⎛

⎝⎜

⎞

⎠⎟

=−( )

Y D K

X

m em m

δ

ϕϕ

coshcosh

* 1(25)

S XS x

S

XY D K

x

Xp

p

ip

p

p ep p*

cosh

cosh

max

max

*( ) =( )

=−( )⎛

⎝⎜

⎞

⎠⎟

( )α μ δ

α μ pp

p ep pY D K

X

( )⎛

⎝⎜

⎞

⎠⎟

=−( )

δ

αϕαϕ

coshcosh

1

1

1*

(26)

Using Eqs. (6–7) and (25–26) we obtain the analytical ex-pression of the concentrations of methanol and α-pinene inthe air phase.

C hC h

CA D S

UXY D K

XY

mm

mi

s em im

g

m

m em m

m

m

* tanhmax max*( ) = ( ) = − ( ) ( )1μ μ

DD K

A h

em m

δ

ϕ ϕ= − ( )1 tanh * (27)

C hCp h

CA D S

UX

Y D KX

ppi

s ep ip

g

p

p ep p

p* tanhmax max*( ) = ( ) = − ( ) ( )1α μ α μ

YY D K

A h

p ep p

δ

αϕ αϕ= − ( )1 1 1 1tanh * (28)

5. Analytical solutions for the concentrationsof methanol and α-pinene for saturated (zeroorder) kinetics

Next we consider the limiting case where the substrate con-centrations of methanol and α-pinene in biofilm phase isrelatively high. In this case S K S Km m p p≥ ≥and and Eqs. (1) and(2) reduce to the following form.

Dd Sdx

XY

emm

mm

2

2= ( )μmax (29)

Dd Sdx

XY

epp

pp

2

2= ( )α μmax (30)

Then the analytical expressions for concentration of metha-nol and α-pinene in the biofilm phase are as follows:

S XS x

SX

Y D Sx

XY D S

xmm

im

m

m em im

m

m em im

* max max*( ) = ( ) = − + =( ) ( )12

12μ μ++ −( )αϕ

βX X* *2 2

(31)

S XS x

SXY D S

xXY D S

xpp

ip

p

p ep ip

p

p ep ip

* max max*( ) =( )

= − + =( ) ( )12

12μ μ++ −( )αϕ

β1

1

2 2X X* *

(32)

Using Eqs. (6–7) and (31–32), we obtain the analytical ex-pression of the concentrations of methanol and α-pinene inthe air phase.

C hC h

CX A h

Y UA

hmm

mi

m s

m g

* max* *( ) = ( ) = − = −( )12

12μ δ ϕ

β(33)

C hC h

CX A h

Y UA

hpp

pi

p s

p g

* max* *( ) = ( ) = − = −( )12

12 1 1

1

α μ δ α ϕβ

(34)

6. Removal ratio of methanol and α-pinene

The percentage of the methanol removal ratio is

methanolRmi mf

mi

C C

C=

−×

* *

*100 (35)

where C Cmi mf* *and are the initial (before treatment) and the final

(after treatment) concentrations of methanol in the air phase,respectively. The percentage of the α-pinene removal ratio is

97e g y p t i an j o u rna l o f b a s i c and a p p l i e d s c i e n c e s 3 ( 2 0 1 6 ) 9 4 – 1 0 5

α − =−

×pineneRpi pf

pi

C C

C

* *

*100 (36)

where Cpi* and Cpf* are the initial (before treatment) and the final

(after treatment) concentrations of α-pinene in the air phaserespectively.

7. Numerical simulation

In order to investigate the accuracy of the ADM solution witha finite number of terms, the system of differential equa-tions was solved numerically. To show the efficiency of thepresent method, our analytical results are compared with nu-merical results graphically. The analytical solution of theconcentrations of methanol and α-pinene in air phase andbiofilm phase are compared with simulation results in Figs. 3–6.Upon comparison, it gives a satisfactory agreement for all values

of the dimensionless parameters S S C Cm p m p* * and, , * *. The de-

tailed Matlab program for numerical simulation is provided inAppendices C and D.

8. Results and discussion

Eqs. (19–22) represent the simple and new analytical expres-sion of the concentrations of methanol and α-pinene in biofilmphase S Sm p* *, and in the air phase C Cm p* *, respectively.The con-centrations of methanol and α-pinene in the biofilm phase andthe air phase depend upon the parameters φ and β. The varia-tion in the dimensionless variable φ can be achieved by varyingeither the thickness or dry cell density of the biofilm. The pa-rameter β depends upon the initial concentration and halfsaturation constant.

Fig. 3 represents the concentration of methanol Sm* in thebiofilm phase versus dimensionless distance X* for differentvalues of φ and β. From Fig. 3a, b, it is inferred that the con-centration of methanol increases when the initial concentrationof methanol β increases for the fixed values of φ. For largevalue of dimensionless parameter β, the concentration ofmethanol remains constant. In Fig. 3c, d, we present the con-centration of methanol in the biofilm phase for various valuesof φ and for some fixed values of β. Maximum specific growthrate of methanol biodegradation φ decreases the concentra-tion of methanol slowly and reaches the constant level. Theminimum value of S X S Xm p* ** and *( ) ( ) are 1 2 1− +( )( )ϕ β and1 2 11 1− +( )( )αϕ β respectively.

Fig. 4 exhibits the concentration of α-pinene Sp* in thebiofilm phase versus dimensionless distance X* for differentvalues of α ϕ β, 1 1and . From Fig. 4a, b, it is inferred that the con-centration of α-pinene increases when the initial concentrationof α-pinene β1( ) increases for the fixed values of dry cell densityφ1. For large value of β1, the concentration of α-pinene isuniform. In Fig. 4c, d, we show that the concentration ofα-pinene in the biofilm phase for various values of cell densityφ1 and for some fixed values of dimensionless parameter β1.From this figure, we conclude that the concentration ofα-pinene Sp* increases when thickness of the film decreases.The concentration of α-pinene is equal to one Sp* =( )1 whenϕ β2 1 2+( )( ) < .

Fig. 5a, b shows the dimensionless concentration of metha-

nol Cm* versus dimensionless height h*. From Fig. 5a, it is

described that the concentration of methanol slowly reachesthe constant when the biofilm thickness or φ increases. InFig. 5b, it is labeled that the concentration of methanol de-creases when half saturation constant of methanol β decreasesfor the fixed value of other parameter.

Fig. 6a, b demonstrates the concentration of α-pinene Cp*

in the air phase versus dimensionless height h*. From Fig. 6a,it is inferred that the concentration of α-pinene slowly reachesa constant level when the diffusion coefficient φ1 increases forthe fixed value of other parameter. In Fig. 6b, it is labeled thatthe concentration of α-pinene attains the steady state valueswhen the initial concentration of α-pinene β1( ) increases.

Fig. 7 shows the profile of dimension concentration ofmethanol and α-pinene in the air phase versus height h forsome fixed value of the parameters. From these figures it isinferred that the concentration is linearly proportional to the

height of the biofilter. And also the concentration of C Cm p* *and

decrease when the height of the biofilter increases. Fig. 8a, billustrates the removal ratio of methanol and α-pinene in theair phase. From this figure it is observed that the removal ratiois directly proportional to the inlet loading. Our analytical resultsare compared with the experimental result and excellent agree-ment is noted.

9. Conclusion

In this paper, the non-linear differential equations inbiofiltration model have been solved analytically. Approxi-mate analytical expressions pertaining to the concentrationsof methanol and α-pinene in the biofilm phase for all the valuesof parameters are obtained using the Adomian decomposi-tion method. This solution of the concentrations of methanoland α-pinene in the biofilm phase and air phase are com-pared with the numerical simulation results.This model is alsovalidated using experimental results. These analytical resultsprovide a good understanding of the system and the optimi-zation of the parameters in biofiltration model.

Acknowledgements

This work was supported by the Department of Science andTechnology (DST) (No. SB/S1/PC-50/2012).The authors are thank-ful to The Principal, The Madura College, Madurai and TheSecretary, Madura College Board, Madurai for theirencouragement.

Appendix A: Basic concepts of the Adomiandecomposition method (ADM)

Consider the non-linear differential equation

′′ + ( ) = ( )y N y g x (A.1)

98 e g y p t i an j o u rna l o f b a s i c and a p p l i e d s c i e n c e s 3 ( 2 0 1 6 ) 9 4 – 1 0 5

Fig. 3 – Dimensionless methanol concentration Sm* in the biofilm phase versus dimensionless distance X* for the variousvalues of the parameters β and φ. When (a) φ = 10, (b) φ = 100 for various values of the parameter β and (c) β = 10, (b) β = 100for various values of the parameter φ. The key to the graph: solid line represents Eq. (19) and the dotted line represents thenumerical simulation.

99e g y p t i an j o u rna l o f b a s i c and a p p l i e d s c i e n c e s 3 ( 2 0 1 6 ) 9 4 – 1 0 5

Fig. 4 – Dimensionless α-pinene concentration Sp* in the biofilm phase versus dimensionless distance X* for the parameters

α β ϕ, 1 1and . The parameter α = 1 is fixed, when (a) φ1 = 1, (b) φ1 = 100 for various values of the parameter β1 and (c) β1 = 10, (b)β1 = 100 for various values of the parameter φ1.The key to the graph: solid line represents Eq. (20) and the dotted linerepresents the numerical simulation.

100 e g y p t i an j o u rna l o f b a s i c and a p p l i e d s c i e n c e s 3 ( 2 0 1 6 ) 9 4 – 1 0 5

with boundary conditions

y A y b B0( ) = ( ) =, (A.2)

where N(y) is a non-linear function, g(x) is the given functionand A, B and b are given constants. We propose the new dif-ferential operator, as below

Ld

dx=

2

2 (A.3)

So, Eq. (A.1) can be written as

L y g x N y( ) = ( ) − ( ) (A.4)

The inverse operator L−1 is therefore considered as a two-fold integral operator (Duan and Rach [37]), as below

L dxdxb

xx− ( ) = ( )∫∫1

0

. . (A.5)

Applying the inverse operator L−1 on both sides of Eq. (A.4)yields

y x L g x L N y y b x y( ) = ( )( ) − ( )( ) + ′( ) −( ) + ( )− −1 1 0 0 , (A.6)

Fig. 5 – Dimensionless methanol concentration Cm* in the air phase versus dimensionless height h* for some fixed values ofthe parameters A Cmi= =70 1and . When (a) β = 100 for various values of the parameter φ and (b) φ = 1 for various values of theparameter β. The key to the graph: solid line represents Eq. (21) and the dotted line represents the numerical simulation.

Fig. 6 – Dimensionless α-pinene concentration Cp* in the air phase versus dimensionless height h* for some fixed values of

the parameters α = = =1 70 0 271, , .A Cpiand . When (a) β1 = 10 for various values of the parameter φ1 and (b) φ1 = 0.1 for variousvalues of the parameter β1.The key to the graph: solid line represents Eq. (22) and the dotted line represents the numericalsimulation.

101e g y p t i an j o u rna l o f b a s i c and a p p l i e d s c i e n c e s 3 ( 2 0 1 6 ) 9 4 – 1 0 5

Using the boundary conditions Eq. (A.2), Eq. (A.6) becomes

y x L g x L N y Bx A( ) = ( )( ) − ( )( ) + +− −1 1 (A.7)

The Adomian decomposition method introduces the solu-tion y(x) and the non-linear function N(y) by infinite series

y x y xnn

( ) = ( )=

∞

∑0

(A.8)

and

N y Ann

( ) ==

∞

∑0

(A.9)

where the components yn(x) of the solution y(x) will be deter-mined recurrently and the Adomian polynomials An of N(y) areevaluated using the formula

A xn

dd

N yn

n

nn

nn

( ) = ⎛⎝⎜

⎞⎠⎟=

∞

=∑1

0 0! λλ

λ(A.10)

which gives

A N yA N y y

A N y y N y y

A N y y

0 0

1 0 1

2 0 2 0 12

3 0 3

12

= ( )= ′( )

= ′( ) + ′′( )

= ′( ) +

,,

,

′′′( ) + ′′′( )N y y y N y y0 1 2 0 131

3!, .

�

(A.11)

By substituting Eqs. (A.8) and (A.9) in Eq. (A.7) gives

y L g x L A Ax Bnn

nn=

∞− −

=

∞

∑ ∑= ( )( ) − ⎛⎝⎜

⎞⎠⎟

+ +0

1 1

0(A.12)

Then equating the terms in the linear system of Eq. (A.11)gives the recurrent relation

y L g x Bx A y L A nn n01

11 0= ( )( ) + + = − ( ) ≥−

+−, , (A.13)

which gives

y L g x Ax B

y L A

y L A

y L A

01

11

0

21

1

31

2

= ( )( ) + += − ( )= − ( )= − ( )

−

−

−

−

,

,

,

, .�

(A.14)

From Eqs. (A.11) and (A.14), we can determine the compo-nents yn(x), and hence the series solution of yn(x) in Eq. (A.7)can be immediately obtained.

Fig. 7 – Dimension methanol and α-pinene concentrationsCm and Cp in the air phase versus the height h using thevalues of the parameters S S D D K Kim ip em ep m p, , , . , , ,= = =1 0 004 10

respectively. The solid line represents Eqs. (21–22).

Fig. 8 – (a) The methanol removal ratio methanolR versus inlet load Cmi for some fixed values of the parametersA D S H U hs em im g= = = = = = = = =70 0 04 1 0 1 1 10 1 1 1, . , , . , , , , , ,ϕ β δ . The graph is plotted using Eq. (35). (b) The α-pinene removalratio α − pineneR versus inlet load Cpi for some fixed values of the parametersA D S H U hs em im g= = = = = = = = =70 0 04 1 0 1 1 10 1 1 11 1, . , , . , , , , , ,ϕ β δ . The graph is plotted using Eqn. (36).

102 e g y p t i an j o u rna l o f b a s i c and a p p l i e d s c i e n c e s 3 ( 2 0 1 6 ) 9 4 – 1 0 5

Appendix B: Analytical solution of Eqs. (11) and(12)

In this appendix, we have derived the solution of Eqs. (11) and(12) using the Adomian decomposition method. Eq. (11) can bewritten with the operator form

L S N Sm m( * ) ( * )= ϕ (B.1)

wherethe differential operator*

andLd

dXN S

SS

mm

m

= =+

2

2 1( * )

**β(B.2)

Applying the inverse operator L dX dXxx

− ( ) = ( )∫∫1

10

. .**

* * on both

sides of Eq. (B.1) yields

S X AX B LS

Sm

m

m

**

** *( ) = + +

+

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−ϕβ

1

1(B.3)

Where A Sm= ( )* ′ 1 and B Sm= ( )* 0 . We let,

S X S Xm mn

n* ** *( ) = ( )=

∞

∑0

(B.4)

N S XS

SAm

m

mn

n

[ * ]*

**( ) =

+=

=

∞

∑1 0β

(B.5)

In view of Eqs. (B.4) and (B.5), Eq. (B.3) gives

S X AX B L Amn

nn

n* * *( ) = + +=

∞−

=

∞

∑ ∑0

1

0

ϕ (B.6)

We identify the zeroth component as

S x AX Bm0* * *( ) = + (B.7)

and the remaining components as the recurrence relation

S X L A nm nn+ ( ) = ≥−1

1 0* ,* ϕ (B.8)

where An are the Adomian polynomials of S S Sm m mn1 2* , * *� . We

can find the first few An as follows:

A N SS

Sm

m

m0 0

0

01= =

+( * )

**β

(B.9)

Ad

dN S S

Sm m

m1 0 1

0

121

= + =+( )=λ

λβλ

[ ( * * )]* (B.10)

The remaining polynomials can be generated easily, and so,

S Xm0 1* *( ) = (B.11)

S xX

Xm1

2

1 2* *

**( ) =

+( )−⎛

⎝⎜⎞⎠⎟

ϕβ (B.12)

Adding (B.11) and (B.12), we get Eq. (19) in the text. Simi-larly, we can apply the above method to find the solution ofEq. (12). Higher order iteration will be considered to improvethe accuracy of the results.

Appendix C: Matlab program for the numericalsolution of Eqs. (11) and (12)

Appendix D: Matlab program for the numericalsolution of Eqs. (16) and (17)

103e g y p t i an j o u rna l o f b a s i c and a p p l i e d s c i e n c e s 3 ( 2 0 1 6 ) 9 4 – 1 0 5

Appendix E: Nomenclature

Symbols Definitions Units

As Biofilm surface area per unit volume of the biofilters m m2 3

Cm Concentration of methanol in the air stream g m3

Cmi Concentration of methanol in the inlet air stream g m3

Cp Concentration of α-pinene in the air stream g m3

Cpi Concentration of α-pinene in the inlet air stream g m3

Dem Effective diffusivity of methanol in the biofilm m h2

Dep Effective diffusivity of α-pinene in the biofilm m h2

h Dimension along the height of the biofilters mH Total height of the biofilters mKi Inhibition constant for α-pinene in the presence of methanol g m3

Km Half saturation constant of methanol in Monod kinetics obtainedfrom differential biofilters experiments

g m3

Kp Half saturation constant of α-pinene in Monod kinetics g m3

mm Air/biofilm partition coefficient for methanol, dimensionless –mp Air/biofilm partition coefficient for α-pinene, dimensionless -Sm Concentration of methanol in the biofilm g m3

Sp Concentration of α-pinene in the biofilm g m3

Ug Superficial velocity of air through the biofilters m/sX Dry cell density of the biofilm kg m3

Y Organic carbon content of the biofilm g gYm Biomass yield coefficient for methanol kg cell kg methanolYp Biomass yield coefficient for α-pinene kg/cell/kg α-pinene

SCm

Smm

mim= = Initial concentration of methanol in the biofilm g m3

SCm

Spp

pip= = Initial concentration of α-pinene in the biofilm g m3

L Linear operatorAn Adomian polynomial

Greek letters

α Coefficient for the effect of methanol on α-pinene biodegradation,dimensionless

–

δ Biofilm thickness mμmax m( ) Maximum specific growth rate for methanol biodegradation h−1

μmax p( ) Maximum specific growth rate for α-pinene biodegradation h−1

ρb Density of the biofilm kg/m3

Dimensionless Parameters:

β = SK

im

m

Dimensionless constant of methanol in Monod kinetics obtained fromdifferential biofilters experiments

–

ϕ μ δ= ( )XY D K

m

m em m

max2

Dimensionless parameter –

Xx

* =δ

Dimensionless coordinate in dry cell density of the biofilm –

SSS

mm

im

* = Dimensionless concentration of methanol in the biofilm –

β1 =SK

ip

p

Dimensionless constant of α-pinene in Monod kinetics obtained frombench-scale biofiltration results

–

ϕ α μ δ1

2

= ( )XY D K

m

m em m

maxDimensionless parameter –

SSS

pp

ip

* = Dimensionless concentration of α-pinene in the biofilm –

AHA D S

U Cs em im

g mi

=δ

Dimensionless parameter –

AHA D S

U Cs ep ip

g pi1 =

δDimensionless parameter –

CCC

mm

mi

* = Dimensionless concentration of methanol in the air stream –

CCC

pp

pi

* = Dimensionless concentration of α-pinene in the air stream –Cmi* Initial (before treatment) concentration of methanol –

Cmf* Finial (before treatment) concentration of methanol –

Cpi* Initial (before treatment) concentration of α-pinene –

Cpf* Initial (before treatment) concentration of α-pinene –

hhH

* = Dimensionless along the height of the biofilters –

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