Figure 1. A schematic diagram of differential fluidized bed biofilm reactor (DFBBR) system [14].

the biofilm is [13]

(1)

where S is the phenol concentration, is the maximum specific growth rate of substrate, is the average effective diffusion coefficient of the limiting sub strate, is the average biofilm density, is the yield coefficient for phenol, and is the half rate kinetic constant for phenol. The equation can be solved subject to the following boundary conditions [13]:

(2)

(3)

where 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

(4)

Normalized Form

The above differential equation (**Eq.1**) for the model can be simplified by defining the following normalized variables,

(5)

where represent normalized concentration, distance and radius parameters, respectively. α denotes a saturation parameter and f 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. Then **Eq.1** reduces to the following normalized form

(6)

The boundary conditions reduce to

(7)

(8)

The effectiveness factor in normalized form is as follows:

(9)

3. ANALYTICAL EXPRESSION OF CONCENTRATION OF PHENOL USING MODIFIED ADOMIAN DECOMPOSITION METHOD (MADM)

MADM [15-17] 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 [18-22]. Using MADM method, we can obtain the concentration of phenol (see Appendices A & B) as follows:

(10)

provided. Using **Eq.9**, we can obtain the simple approximate expression of effectiveness factor as follows:

(11)

From **Eq.11**, we see that the effectiveness factor is a function of the Thiele modulus f, the saturation parameter α and the radius parameter λ. This **Eq.11** is valid only when

.

4. 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.

5. 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 f. Figures 2(a)-(d) represent a series of normalized phenol concentration for the different values of the Thiele modulus. In this Figure 2, the concentration of phenol decreases with the increasing values of the Thiele modulus f. Moreover, the phenol concentration tends to one as the Thiele modulus f ≤ 0.1. Upon careful evaluation of these figures, it is evident that there is a simultaneous increase in the values of concentration of phenol u when f decreases. Furthermore, the phenol concentration increases slowly and rises suddenly when the normalized radial distance. Figure 3 represents the effecttiveness factor η versus normalized Thiele modulus f for different values of normalized saturation parameter α. From this figure, it is inferred that, a constant value of normalized saturation parameter α, the effectiveness factor decreases quite rapidly as the Thiele modulus f increases. Moreover, it is also well known that, a constant value of normalized Thiele modulus f, the effectiveness factor increases with increasing values of α.

The normalized effectiveness factor η versus normalized saturation parameter α is plotted in Figure 4. The effectiveness factor η is equal to one (steady state value)

(a)(b)

Figure 2. Plot of normalized phenol concentration u as a function of in fluidized bed biofilm reactor. The concentration were computed for various values of the Thiele modulus f and the radius parameter λ = 0.01 using **Eq.10** when the normalized saturation parameter (a) α = 0.1; (b) α = 1; (c) α = 10; and (d) α = 100.

Figure 3. Plot of the normalized effectiveness factor η versus the Thiele modulus f. The effectiveness factor η were computed using **Eq.11** for various values of the normalized saturation parameter α when the normalized radius parameter λ = 0.01.

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 f when the normalized radius parameter λ = 0.01.

when and all values of f. Also the effectiveness factor η is uniform when and for all values of α. From this figure, it is concluded that the effectiveness factor decreases when f increases at x = 0. A three dimensional effectiveness factor η computed using **Eq.11** for as shown in Figure 5. In this Figure 5, we notice that the effectiveness factor tends to one as the Thiele modulus decreases.

6. 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 using the modified Adomian decomposition method. The approximate solution is used

Figure 5. Plot of the three-dimensional effectiveness factor η against f and λ, calculated using **Eq.11** for α = 100.

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.

7. ACKNOWLEDGEMENTS

This work was supported by the Council of Scientific and Industrial Research (CSIR No.: 01(2442)/10/EMR-II), Government of India. The authors also thank the Secretary, The Madura College Board, and the Principal, The Madura College, Madurai, Tamilnadu, India for their constant encouragement.

REFERENCES

- Zobell, C.E. and Anderson, D.Q. (1936) Observationson the multiplication of bacteria in different volumes of stored sea water and the influence of oxygen tension and solid surfaces. Biological Bulletin, 71, 324-342. doi:10.2307/1537438
- Williamson, K. and McCarty, P.L. (1976) A model of substrate utilization by bacterial films. Journal of the Water Pollution Control Federation, 48, 9-24.
- Harremoës, P. (1976) The significance of pore diffusion to filter denitrification. Journal of the Water Pollution Control Federation, 48, 377-388.
- Rittmann, B.E. and McCarty, P.L. (1980) Model of steadystate-biofilm kinetics. Biotechnology and Bioengineering, 22, 2343-2357. doi:10.1002/bit.260221110
- Rittmann, B.E. and McCarty, P.L. (1981) Substrate flux into biofilms of any thickness. Journal of Environmental Engineering, 107, 831-849.
- Rittman, B.E. and McCarty, P.L. (1978) Variable-order model of bacterial-film kinetics. American Society of Civil Engineers. Environmental Engineering Division, 104, 889-900.
- Choi, J.W., Min, J., Lee, W.H. and Lee, S.B. (1999) Mathematical model of a three-phase fluidized bed biofilm reactor in wastewater treatment. Biotechnology and Bioprocess Engineering, 4, 51-58. doi:10.1007/BF02931914
- Meikap, B.C. and Roy, G.K. (1995) Recent advances in biochemical reactors for treatment of wastewater. International Journal of Environmental Protection, 15, 44-49.
- Vinod, A.V. and Reddy, G.V. (2003) Dynamic behaviour of a fluidised bed bioreactor treating waste water. Indian Chemical Engineer Section A, 45, 20-27.
- Sokol, W. (2003) Treatment of refinery wastewater in a three-phase fluidized bed bioreactor with a low-density biomass support. Biochemical Engineering Journal, 15, 1-10. doi:10.1016/S1369-703X(02)00174-2
- Gonzalez, G., Herrera, M.G., Garcia, M.T. and Pena, M.M. (2001) Biodegradation of phenol in a continuous process: Comparative study of stirred tank and fluidized-bed bioreactors. Bioresource Technology, 76, 245-251. doi:10.1016/S0960-8524(00)00092-4
- Sokol, W. and Korpal, W. (2004) Determination of the optimal operational parameters for a three-phase fluidised bed bioreactor with a light biomass support when used intreatment of phenolic wastewaters. Biochemical Engineering Journal, 20, 49-56. doi:10.1016/j.bej.2004.02.009
- Tanyolac, A. and Beyenal, H. (1996) Predicting average biofilm density of a fully active spherical bioparticle. Journal of Biotechnology, 52, 39-49. doi:10.1016/S0168-1656(96)01624-0
- Beyenal, H. and Tanyolac, A. (1998) The effects of biofilm characteristics on the external mass transfercoefficient in a fluidized bed biofilm reactor. Biochemical Engineering Journal, 1, 53-61. doi:10.1016/S1369-703X(97)00010-7
- Adomian, G. (1976) Nonlinear stochastic differential equations. Journal of Mathematical Analysis and Applications, 55, 441-452. doi:10.1016/0022-247X(76)90174-8
- Adomian, G. and Adomian, G.E. (1984) A global method for solution of complex systems. Mathematical Model, 5, 521-568. doi:10.1016/0270-0255(84)90004-6
- Adomian, G. (1994) Solving frontier problems of physics: The decomposition method. Kluwer Academic Publishers, Boston, 1994.
- Hasan, Y.Q. and Zhu, L.M. (2008) Modified adomian decomposition method for singular initial value problems in the second-order ordinary differential equations. Surveys in Mathematics and Its Applications, 3, 183-193.
- Hosseini, M.M. (2006) Adomian decomposition method with Chebyshev polynomials. Applied Mathematics and Computation, 175, 1685-1693. doi:10.1016/j.amc.2005.09.014
- Wazwaz, A.M. (1999) A reliable modifications of Adomian decomposition method. Applied Mathematics and Computation, 102, 77-86. doi:10.1016/S0096-3003(98)10024-3
- Wazwaz, A.M. (1999) Analytical approximations and Pade approximants for Volterra’s population model. Applied Mathematics and Computation, 100, 13-25. doi:10.1016/S0096-3003(98)00018-6
- Wazwaz, A.M. (2002) A new method for solving singular initial value problems in the second-order ordinary differential equations. Applied Mathematics and Computation, 128, 45-57. doi:10.1016/S0096-3003(01)00021-2

APPENDIX A

Basic Concept of the Modified Adomian Decomposition Method (MADM)

Consider the nonlinear differential equation in the form

(A1)

with initial condition

(A2)

where is a real function, is the given function and A and B are constants. We propose the new differential operator, as below

(A3)

So, the problem (A1) can be written as,

(A4)

The inverse operator is therefore considered a two-fold integral operator, as below.

(A5)

Applying of (A4) to the first three terms

of **Eq.A1**, we find

By operating on (A4), we have

(A6)

The Adomian decomposition method introduce the solution and the nonlinear function by infinity series

(A7)

and

(A8)

where the components of the solution will be determined recurrently and the Adomian polynomials A_{n} of are evaluated [23-25] using the formula

(A9)

By substituting (A7) and (A8) into (A6),

(A10)

Through using Adomian decomposition method, the components can be determined as

(A11)

which gives

(A12)

From (A9) and (A10), we can determine the components, and hence the series solution of in (A7) can be immediately obtained.

APPENDIX B

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,

(B1)

where .

Applying the inverse operator on both sides of **Eq.B.1** yields

(B2)

where A and B are the constants of integration. We let,

(B3)

(B.4)

where

(B.5)

Now **Eq.B.2** becomes

(B.6)

We identify the zeroth component as

(B.7)

and the remaining components as the recurrence relation

(B.8)

We can find A_{n} as follows:

(B.9)

The initial approximations (boundary conditions **Eqs.7** and **8** are as follows

(B.10)

(B.11)

and

(B.12)

(B.13)

Solving the **Eq.B.7** and using the boundary conditions **Eqs.B.10** and **B.11**, we get

(B.14)

Now substituting n = 0 in **Eqs.B.8** and **B.9**, we can obtain

(B.15)

and (B.16)

By operating on (B.16),

(B.17)

Now **Eq.B.15** becomes

(B.18)

Solving the **Eq.B.18** and using the boundary conditions **Eqs.B.12** and **B.13**, we get

(B.19)

Similarly we can get

(B.20)

Adding **Eqs.B.14, B.19** and **B.20**, we get **Eq.11** in the text.

APPENDIX C

Scilab/Matlab Program to Find the Numerical Solution of Eq.8 Is as Follows

function pdex1 m = 2;

x = linspace(0.01,1);

t = linspace(0,1000);

sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);

u = sol(:,:,1);

figure plot(x,u(end,:))

title(‘u(x,t)’)

% -------------------------------------------------------------

function [c,f,s] = pdex4pde(x,t,u,DuDx)

c = 1;

f = DuDx;

phi=24.5;

alpha=100;

s =-(phi^2*u)/(1+alpha*u);

% -------------------------------------------------------------

function u0 = pdex1ic(x)

u0 = 1;

% -------------------------------------------------------------

function [pl,ql,pr,qr] = pdex4bc(xl,ul,xr,ur,t)

pl = 0;

ql = 1;

pr = ur-1;

qr = 0;