action is described by the Monod relationship, which is a nonlinear expression. The differential equation for diffusion with Monod reaction within
Figure 1. A schematic diagram of differential fluidized bed biofilm reactor (DFBBR) system .
the biofilm is 
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 :
where denotes biofilm surface substrate concentration, r is the radial distance, rp is the radius of clean particle, and rb is the radius of biofilm covered bioparticle. The effectiveness factor for a spherical bioparticle is
The above differential equation (Eq.1) for the model can be simplified by defining the following normalized variables,
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
The boundary conditions reduce to
The effectiveness factor in normalized form is as follows:
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:
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 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)
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.
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.
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.
Basic Concept of the Modified Adomian Decomposition Method (MADM)
Consider the nonlinear differential equation in the form
with initial condition
where is a real function, is the given function and A and B are constants. We propose the new differential operator, as below
So, the problem (A1) can be written as,
The inverse operator is therefore considered a two-fold integral operator, as below.
Applying of (A4) to the first three terms
of Eq.A1, we find
By operating on (A4), we have
The Adomian decomposition method introduce the solution and the nonlinear function by infinity series
where the components of the solution will be determined recurrently and the Adomian polynomials An of are evaluated [23-25] using the formula
By substituting (A7) and (A8) into (A6),
Through using Adomian decomposition method, the components can be determined as
From (A9) and (A10), we can determine the components, and hence the series solution of 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 yields
where A and B are the constants of integration. We let,
Now Eq.B.2 becomes
We identify the zeroth component as
and the remaining components as the recurrence relation
We can find An as follows:
The initial approximations (boundary conditions Eqs.7 and 8 are as follows
Solving the Eq.B.7 and using the boundary conditions Eqs.B.10 and B.11, we get
Now substituting n = 0 in Eqs.B.8 and B.9, we can obtain
By operating on (B.16),
Now Eq.B.15 becomes
Solving the Eq.B.18 and using the boundary conditions Eqs.B.12 and B.13, we get
Similarly we can get
Adding Eqs.B.14, B.19 and B.20, we get Eq.11 in the text.
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);
function [c,f,s] = pdex4pde(x,t,u,DuDx)
c = 1;
f = DuDx;
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;