Approximate analytical solution of non-linear reaction diffusion equation in fluidized bed biofilm reactor

A mathematical model for the fluidized bed biofilm reactor (FBBR) is discussed. An approximate analytical solution of concentration of phenol is obtained using modified Adomian decomposition method (MADM). The main objective is to propose an analytical method of solution, which do not require small parameters and avoid linearization and physically unrealistic assumptions. Theoretical results obtained can be used to predict the biofilm density of a single bioparticle. Satisfactory agreement is obtained in the comparison of approximate analytical solution and numerical simulation.


INTRODUCTION
There has been much interest in the development of biofilms.Biofilms play significant roles in many natural and engineered systems.The importance of biofilms has steadily emerged since their first scientific description in 1936 [1].Mechanistically based modeling of biofilms began in the 1970s.The early efforts focused mainly on substrate flux from the bulk liquid into the biofilm.Biofilms have been used to treat wastewater since the end of the 19th century.Biofilm reactors with larger specific surface areas were developed starting in the 1980s [2][3][4].Biofilm modeling was advanced by Rittmann and Mc-Carty [5,6], who based their models on diffusion (Fick's law) and biological reaction (Monod kinetics) within the biofilm and liquid-layer mass transfer from the bulk liquid.The mathematical model to describe the oxygen utilization for a TFBBR in wastewater treatment was developed by Choi [7], which was proposed to describe the oxygen concentration distribution.This model consisted of the biofilm model that described the oxygen uptake rate and the hydraulic model that presented characteristics of liquid and gas phase [8].
The fluidized bed reactor (FBB) is the reactor which carries on the mass transfer or heat transfer operation using the fluidization concept.At first it was mainly used in the chemical synthesis and the petrochemistry industry.Because this kind of reactor displayed in many aspects its unique superiority, its application scope was enlarged gradually to metal smelting, air purification and many other fields.Since 1970's, people have successfully applied the fluidization technology to the wastewater biochemical process field.An FBB is capable of achieving treatment in low retention time because of the high biomass concentrations that can be achieved.A bioreactor has been successfully applied to an aerobic biological treatment of industrial and domestic wastewaters.An FBB offers distinct mechanical advantages, which allow small and high surface area media to be used for biomass growth [9][10][11][12].
Fluidization overcomes operating problems such as bed clogging and the high-pressure drop, which would occur if small and high surface area media were employed in packed-bed operation.Rather than clog with new biomass growth, the fluidized bed simply expands.Thus for a comparable treatment efficiency, the required bioreactor volume is greatly reduced.A further advantage is the possible elimination of the secondary clarifier, although this must be weighed against the mediumbiomass separator [10][11][12].
Abdurrahman Tanyolac and Haluk Beyenal proposed to evaluate average biofilm density of a spherical bioparticle in a differential fluidized bed system [13].To our knowledge, no general analytical expressions for the concentration of phenol and effectiveness factor have been reported for all values of the parameters λ, α and .
However, in general, analytical solutions of non-linear differential equations are more interesting and useful their numerical solutions, as they are used to various kinds data analysis.Therefore, herein, we employ analytical method to evaluate the phenol concentration and effectiveness factor for all possible values of parameters.

MATHEMATICAL FORMULATION OF THE PROBLEM
The details of the model adopted have been fully described in Haulk Beyenal and Abdurrahman Tanyolac [14].The biological reaction is described by the Monod relationship, which is a nonlinear expression.The differential equation for diffusion with Monod reaction within the biofilm is [13]   where S is the phenol concentration, max  is the maximum specific growth rate of substrate, f D is the average effective diffusion coefficient of the limiting sub strate, f X is the average biofilm density, X P Y is the yield coefficient for phenol, and o K is the half rate kinetic constant for phenol.The equation can be solved subject to the following boundary conditions [13]: where b denotes biofilm surface substrate concentration, r is the radial distance, r p is the radius of clean particle, and r b is the radius of biofilm covered bioparticle.
The effectiveness factor for a spherical bioparticle is

Normalized Form
The above differential equation (Eq.1) for the model can be simplified by defining the following normalized variables, max ; ; ; ; where , and u   represent normalized concentration, distance and radius parameters, respectively.α denotes a saturation parameter and  is the Thiele modulus.
Furthermore, the saturation parameter α describes the ratio of the phenol concentration within the biofilm to the rate kinetic constant for phenol The boundary conditions reduce to The effectiveness factor in normalized form is as follows:

ANALYTICAL EXPRESSION OF CONCENTRATION OF PHENOL USING MODIFIED ADOMIAN DECOMPOSITION METHOD (MADM)
MADM [15][16][17] is a powerful analytic technique for solving the strongly nonlinear problems.This MADM yields, without linearization, perturbation, transformation or discretisation, an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms.The decomposition method is simple and easy to use and produces reliable results with few iteration used.The results show that the rate of convergence of Modified Adomian decomposition method is higher than standard Adomian decomposition method [18][19][20][21][22].
Using MADM method, we can obtain the concentration of phenol (see Appendices A & B) as follows: provided Using Eq.9, we can obtain the simple approximate expression of effectiveness factor as follows: From Eq.11, we see that the effectiveness factor is a function of the Thiele modulus , the saturation parameter α and the radius parameter λ.This Eq.11 is valid only when

NUMERICAL SIMULATION
The non-linear equation [Eq.1] for the boundary conditions [Eqs.7 and 8] are solved by numerically.The function pdex4 in Scilab/Matlab software is used to solve the initial-boundary value problems for parabolic-elliptic partial differential equations numerically.The Scilab/ Matlab program is also given in Appendix C. Its numerical solution is compared with the analytical results obtained using MADM method.

RESULTS AND DISCUSSION
Eq.10 is the new, simple and approximate analytical expression of the concentration of phenol.Concentration of phenol depends upon the following three parameters  , α and .Figures 2(a)-(d

CONCLUSIONS
We have developed a comprehensive analytical formalism to understand and predict the behavior of fluidized bed biofilm reactor.We have presented analytical expression corresponding to the concentration of phenol in terms of , , , and     using the modified Adomian decomposition method.The approximate solution is used to estimate the effectiveness factor of this kind of systems.The analytical results will be useful for the determination of the biofilm density in this differential fluidized bed biofilm reactor.The theoretical results obtained can be used for the optimization of the performance of the differential fluidized bed biofilm reactor.Also the theoretical model described here can be used to obtain the parameters required to improve the design of the differential fluidized bed biofilm reactor. 1 Applying of (A4) to the first three terms The Adomian decomposition method introduce the solution   By substituting (A7) and (A8) into (A6), Through using Adomian decomposition method, the components   n y x can be determined as From (A9) and (A10), we can determine the components   n y x , and hence the series solution of   y x in (A7) can be immediately obtained.

Analytical Expression of Concentration of Phenol Using the Modified Adomian Decomposition Method
In this appendix, we derive the general solution of nonlinear Eq.7 by using Adomian decomposition method.We write the Eq.7 in the operator form,   Applying the inverse operator on both sides of Eq.B.1 where A and B are the constants of integration.We let, We identify the zeroth component as and the remaining components as the recurrence relation We can find A n as follows:   0 Solving the Eq.B. 18 and using the boundary conditions Eqs.B.12 and B.13, we get Similarly we can get

Figure 4 .
Figure 4. Plot of the normalized effectiveness factor η versus normalized saturation parameter α.The effectiveness factor η were computed using Eq.11 for different values of the Thiele modulus  when the normalized radius parameter λ = 0.01.
values of α.From this figure, it is concluded that the effectiveness factor decreases when  increases at x = 0.A three dimensional effectiveness factor η computed using Eq.11 for 100   as shown in Figure 5.In this Figure 5, we notice that the effectiveness factor tends to one as the Thiele modulus decreases.
the solution   y x will be determined recurrently and the Adomian polynomials A n of   , F x y are evaluated [23-25] using the formula Solving the Eq.B.7 and using the boundary conditions Eqs.B.10 and B.11,