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Mathematical models of steady-state biofilteration are discussed. The theoretical results are much useful for the design of biofilters. This model is based on the system of non-linear reaction/diffusion equations contains a non-linear term related to Monod kinetics, Andrews kinetics, interactive model from Monod kinetics and Andrews kinetics. Analytical expression of concentration of VOC (Volatile organic compounds) and oxygen are derived by solving the system of non-linear equations using Adomian decomposition method (ADM) method. Our analytical results are also compared with the simulation results. Satisfactory agreement is noted.

Biological system for elimination of volatile organics have been explored on experimental studies [1-3]. However, researches into the theoretical studies regarding biofilter models is rather limited. The pioneering contribution of Ottengraf and co-workers [1,2] of this model is based on some rather simplistic assumptions. The closed analytical expression have been used in validating-scale experimental data and actual design of pilot-scale biofilter units. Recently Zarook et al. [

Recently Zarook et al. [

A steady-state biofilteration model (

with the boundary conditions

where and are the concentration of VOC and oxygen at a position x in the biofilm, and are the effective diffusion coefficient of VOC and oxygen in the biofilm. denotes biofilm density, and is the amount of biomass produced per amount of VOC consumed and amount of biomass produced per amount of oxygen consumed. For biological systems, The growth rate, for various reaction kinetics are given as follows:

Monod kinetics:

Andrews kinetics:

When oxygen limits the biodegradation rate, the growth rate is given by interactive model. The above Equations (5) and (6) are written as follows:

Interactive model from Monod kinetics:

Interactive model from Andrews kinetics:

In order to obtain numerical solution of model these equations are brought in dimensionless form through the dimensionless variables and groups. We make the above non-linear partial differential Equations (1) and (2) in dimensionless form by defining the following dimensionless parameters:

where and denotes the dimensionless concentration of VOC and oxygen, X is the dimensionless position in the biolayer. represents the Thiele modulus, , M, L and N are dimensionless constants. By substituting the Equation (9) in Equations (1) and (2), we can obtain the following dimensionless non-linear equation for Monod kinetics:

Using Equation (9), in the non-linear Equations (1) and (2), we can obtain the following dimensionless nonlinear equation for Andrews kinetics:

Using Equation (9), the dimensionless non-linear equation for Interactive model of Monod kinetics becomes

The dimensionless non-linear equation for Interactive model of Andrews kinetics of the Equations (1) and (2) is

Now the boundary condition in dimensionless form may be represented as follows:

For all the above cases, we can obtain the relation between and as follows:

Integrating the above equation twice and using the boundary condition (18) and (19) we get

When, the concentration of VOC and oxygen becomes equal. When, the concentration of oxygen. When, the concentration of VOC becomes one.

Nonlinear phenomena play a crucial role in physical chemistry and biology (heat and mass transfer, filtration of liquids, diffusion in chemical reactions, etc.). Constructing a particular, exact solution for these equations remains an important problem. Finding an exact solution that has a physicochemical or biological interpretation is of fundamental importance. This model is based on a non-stationary system of diffusion equations containing a nonlinear reaction term. It is not possible to solve these equations using standard analytical techniques. The investigation of an exact solution of nonlinear equations is interesting and important. In the past several decades, many authors mainly paid attention to studying the solution of nonlinear equations by using various methods, such as the Backlund and the Darboux transformation [15,16], the inverse scattering method [

By solving the Equation (12), we can obtain the concentration of VOC for Andrews kinetics as follows,

Solving the dimensionless form of Interactive model from Monod kinetics Equation (14), we get the concentration as,

Solving the dimensionless form of Interactive model from Andrews kinetics Equation (16), we get the concentration as,

where

,

,

, and

.

The effectiveness factor is defined as the ratio of actual rate of reaction to the rate of reaction that would result if the entire biofilm was exposed to the concentration at the gas/biofilm interface. The effectiveness factor of various kinetics are as follows:

Monod kinetics

Andrews kinetics

Interactive model from Monod kinetics

Interactive model from Andrews kinetics

The dimensionless form of Equations (10)-(17) corresponding to the boundary conditions (18) and (19) were solved by numerical methods. We have used pdex4 to solve these equations (Pdex4 in MATLAB is a function to solve the initial-boundary value problems of differential equations. Matlab program to find the numerical solution of Equations (10) and (11) is given in the Appendix B. The numerical solution is compared with our analytical results and is shown in Figures 2-9. A satisfactory agreement is noticed for various values of the Thiele modulus and possible small values of reaction/ diffusion parameters.

Equations (21)-(24) represents the new analytical expressions of the concentration of VOC for Monoid, Andrews, Interactive Monoid and Interactive Andrews kinetics for all values of the parameter. Using the relation (Equation (20)), we can also obtain the concentration of oxygen for all the kinetics. Zarook et al. [

analytical expressions of concentration of VOC and oxygen only for the limiting cases (Zero kinetics and First-order kinetics). Concentration of VOC and oxygen depends upon the value of parameters, , L, N and.

The Thiele module, essentially compares biodegradation rate with diffusion rate. We observe the rise and downfall of concentration profiles in two cases. 1) If Thiele modulus is small, then enzyme kinetics predominate. The overall kinetics is governed by the total amount of active enzyme; 2) The response is under diffusion control, if the Thiele module is large, which is observed at high catalytic activity and active membrane thickness or at low reaction kinetic constant or diffusion coefficient values.

Equation (21) represents the concentration of VOC for Monod kinetics in the biofilm.

Figures 2(a)-(d) is the plot of dimensionless concentration versus dimensionless distance X for various values of Thiele modulus and the dimensionless quantity M using Equation (21). From this

Figures 3(a)-(d) is the plot of dimensionless concentration for various values of dimensionless quantity, and the Thiele modulus. From this figure it is observed that the concentration of oxygen is decreases when increases.

Equation (22) is the concentration of VOC for Andrews-type kinetics.

Figures 4(a)-(d) is the plot of dimensionless concentration versus dimensionless distance X for various values of dimensionless parameters. From this figure, it is noted that the concentration of VOC decreases when, , increases.

Figures 5(a)-(d) represents the dimensionless concentration for various values of dimensionless quantity, , and.

Equation (23) represents the concentration of VOC of Interactive model from Monod kinetics.

Figures 6(a)-(d) is the dimensionless concentration versus dimensionless distance X for various values

of parameters using Equation (23). From this

Equation (24) is the concentration of VOC of Interactive model from Andrews kinetics.

Figures 8(a)-(d) is the dimensionless concentration versus dimensionless distance X for various values of Thiele modulus, , , and using Equation (24).

Figures 9(a)-(d) is the dimensionless concentration for various values of dimensionless parameters. From this figure it is inferred that the concentration of VOC is constant when bio-filter thickness decreases.

Figures 10(a)-(d) represent the effectiveness factor versus the Thiele modulus using Equations (25)-(28). From this figure it is observed that the effectiveness factor = 1 when for all mechanisms. Also the effectiveness factor decreases when increases and the values of parameters and decreases.

The non-linear differential equations in biofilter models have been solved analytically for various kinetics using the Adomian decomposition method. Analytical expression of concentration of VOC and oxygen and corresponding effectiveness factor have been obtained for Monoid, Andrews, Interactive Monoid and Andrews kinetics and for all values of parameters. These analytical reactions very much useful for designing or scaling-up of biofilters.

This work was supported by the University Grants Commission (F. No. 39-58/2010(SR)), New Delhi, India and Council of Scientific and Industrial Research (CSIR No.: 01(2442)/10/EMR-II), New Delhi, India. The authors are thankful to Dr. R. Murali, The Principal, The Madura College, Madurai and The Secretary, Madura College Board, Madurai for their encouragement.

Analytical Solution of Non-Linear (Equation (16)) Using The Adomian Decomposition Method In the operator form, Equation (16) becomes

where

and

Applying to both sides of (B1) yields

where a and b are constants of integration. To solve (B3) by the Adomian method, we get

In view of the Equations (B4)-(B5), Equation (B3) gives

we identify the zeroth component as

and the remaining components as the recurrence relation,

where are the Adomian polynomials that represent the non-linear term in (B8).

Using (B9) we can find the first few as follows:

The remaining polynomials can be generated easily. The corresponding boundary condition becomes

Substitution of (B10) and (B11) in (B8) and operating with in conjunction with the boundary conditions (B12) in each case separately, we obtain

Substituting the Equations (B13)-(B15) in

we can obtain the Equation (24) in the text. Similarly, applying the above same procedure, we obtain the Equations (21)-(23).

Matlab/Scilab program to find the numerical solution of the Equations (10)-(11).