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In this paper, mathematical models of biofilm mixtures of n-butanol biofilters were discussed. The model is based on the mass transfer in the biofilm interface and chemical oxidation in the biofilm phase and gas phase. An approximate analytical expression of concentration profiles of n-butanol in the biofilm phase and gas phase has been derived using the homotopy perturbation method and hyperbolic function method for all possible values of parameters. Furthermore, in this work, the numerical simulation of the problem is also reported using the Matlab program. Good agreement between the analytical and numerical results is noted. Graphical results are presented and discussed quantitatively to illustrate the solution. The analytical results will be useful in finding the yields of biomass and oxygen consumption, the specific biomass surface area, activation energy and saturation constant for the Michaelis-Menten kinetics.

Several volatile organic compounds such as n-butanol, acetone, styrene, toluene, and DMDS are emanated from industrial sources that are unhealthy to humanistic well-being and may motive vomiting, irritability and upset the nervous and respiratory systems [

To the best of our knowledge, there is no rigorous analytical expression available to date for the steady-state concentration. As a result, in this work, we focus on obtaining a feeling for the steady-state concentration of n-butanol in the biofilm phase and gas phase. Further, the expression helps us to analyze the physical response related to the parameters in the biofilter model.

The modeling was developed by using the following assumption [

· The biofilm is formed on the outside surface of the packing materials, and there is no reaction in the pores and the biofilm completely covers the surface of the packaging materials.

· Compared to the size of solid particles, the biofilm is very thin; hence planar geometry is used.

· n-butanol is the sole reactant that influences the biodegradation rate, and oxygen does not limit the reaction.

· Arrhenius equation is used for the temperature dependence of the biodegradation rate constant.

· The plug flow model is applied to the gas phase.

· The air/biofilm interface concentration of n-butanol meets Henry's rule by assuming the same air/water partition coefficients.

· There is no boundary layer at the air/biofilm interface. Thus, gas-phase resistance is assumed negligible.

· The biofilm properties ( δ , A s and density) are constant all over the bed.

· The temperature gradient inside the biofilm is negligible.

1) Mass Balance in the Biofilm Phase

The steady-state mass balance equation for Michaelis-Menten kinetics in the biofilm may be written as follows [

D d 2 S d x 2 = e − E R ( 1 T − 1 T r e f ) E C max S K m + S (1)

where S is the concentration of n-butanol, E C max is the maximum of elimination capacity K m is the Michaelis-Menten constant, D is the diffusion coefficient, E is the activation energy, R is the ideal gas constant and T is the kelvin temperature. The boundary conditions Eshraghi et al. [

At x = 0 , S = C H (2)

At x = δ , d S d x = 0 (3)

2) Mass Balance in Gas Phase

The concentration profile of n-butanol in the gas phase may be written as follows:

u d C d z = A S D [ d S d x ] x = 0 (4)

where u is the superficial velocity of gas flow, A S is the biofilm specific area, C is the n-butanol concentration in the gas phase and D is diffusion coefficient. The corresponding boundary condition is

At z = 0 , C = C i (5)

3) Dimensionless Mass Balance Equation in the Biofilm Phase

The non-linear differential Equation (1) is made dimensionless form by defining the following dimensionless parameters:

S * = S S i , x ∗ = x δ , ϕ = δ 2 e − E R ( 1 T − 1 T r e f ) E C max D K m , β = S i K m (6)

Using the above dimensionless variables, Equation (1) reduces to the following dimensionless form:

d 2 S * d x * 2 = ϕ ( S * 1 + β S * ) (7)

The corresponding boundary conditions for the above Equation (7) can be expressed as

S * = 1 at x * = 0 (8)

d S * d x * = 0 at x * = 1 (9)

4) Dimensionless Mass Balance in the Gas Phase

By defining the following dimensionless parameters, the differential Equation (4) is made dimensionless form:

C * = C C i , z * = z H , x * = x δ , A = H A s D S i u δ C i (10)

Using the variables, Equation (4) can be expressed in the dimensionless form as follows:

( d C * d z * ) = A ( d S * d x * ) x * = 0 (11)

The respective boundary condition for the above mentioned Equation (11) can be described as

C * = 1 at z * = 0 (12)

HPM couples the homotopy technology and perturbation. The primary deficiencies in applying perturbation methods are that a small parameter is needed in the equations. The HPM was further developed and improved and applied to nonlinear oscillators [

S * ( x * ) = cosh ϕ ( x * − 1 ) cosh ϕ − β cosh 2 ϕ ( x * − 1 ) 6 cosh 2 ϕ + β 2 cosh 2 ϕ + [ β cosh 2 ϕ 6 cosh 2 ϕ − β 2 cosh 2 ϕ ] [ cosh ϕ ( x * − 1 ) cosh ϕ ] (13)

Solving Equation (11) using boundary condition (12) the concentrations of n-butanol in gas phase can be obtained as follows:

C * ( z * ) = 1 + z * A tanh ϕ [ 2 β ϕ 3 − 1 − ϕ ( β cosh 2 ϕ 6 cosh 2 ϕ − β 2 cosh 2 ϕ ) ] (14)

In order to use the new analytical method, the trail solution for Equation (7) is given below:

S * ( x * ) = A cosh ( m x * ) + B sinh ( m x * ) (15)

where A , B , m are constants. Using the boundary conditions (8) and (9), we get the constant

A = 1 , B = − sinh ( m ) cosh ( m ) (16)

Now Equation (15) reduces to

S * ( x * ) = cosh ( m x * ) − sinh ( m ) cosh ( m ) sinh ( m x * ) (17)

where m is constant. This constant can be obtained as follows:

Equation (7) can be rewritten as

d 2 S * ( x * ) d x * 2 = m 2 cosh ( m x * ) − ( sinh ( m ) cosh ( m ) ) m 2 sinh ( m x * ) (18)

Substituting Equation (18) in Equation (7), we get the following result.

m 2 cosh ( m x * ) − ( sinh ( m ) cosh ( m ) ) m 2 sinh ( m x * ) = ϕ ( cosh ( m x * ) − ( sinh ( m ) cosh ( m ) ) sinh ( m x * ) ) 1 + β ( cosh ( m x * ) − ( sinh ( m ) cosh ( m ) ) sinh ( m x * ) ) (19)

When x = 0, the above results becomes

m 2 = ϕ 1 + β (20)

Using Equation (20), the value of m becomes as follows:.

m = ϕ 1 + β (21)

The concentration of n-butanol can be obtained in the biofilm process by inserting Equation (21) in Equation (17), as follows:

S * ( x * ) = cosh ( ϕ 1 + β x ) − sinh ( ϕ 1 + β ) cosh ( ϕ 1 + β ) sinh ( ϕ 1 + β x ) = cosh ( ϕ 1 + β ( x − 1 ) ) cosh ( ϕ 1 + β ) (22)

Hyperbolic function method is a special case of exponential function method [

Equations (13) & (14) represent the simple and new analytical expressions of the concentration of n-butanol in biofilm-phase ( S * ) and in the gas-phase ( C * ) respectively. The concentration of n-butanol in the biofilm-phase and the gas-phase depends upon the parameters ϕ and β . The variation in the dimensionless variable ϕ can be achieved by varying either the thickness ( δ ) or diffusivity of the biofilm (D). The parameter β depends upon the initial concentration ( S i ) and half-saturation constant ( K m ).

The experimental setup for the biofiltration of this organic compound is given in

represents n-butanol concentration for different values of z * and compared with analytical method numerical simulation and experimental results (Eshraghi et al. 2016).

The sensitivity analysis of the parameter is given in

In order to investigate the accuracy of the HPM solution with a finite number of terms, the nonlinear differential equation is solved numerically. To show the efficiency of the present method, the analytical expressions of the concentration of n-butanol in biofilm-phase and gas-phase are compared with simulation results in Tables 1-3 for the experimental values of parameters. A satisfactory agreement is

x * | Concentration S * | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

when ϕ = 1 | when ϕ = 10 | when ϕ = 100 | |||||||||||||

Simulation | (HPM)Equation (13) | hyperbolic function method Equation (22) | % of error deviation (HPM) Equation (13) | % of error deviation hyperbolic function method Equation (22) | Simulation | (HPM) Equation (13) | hyperbolic function method Equation (22) | % of error deviation (HPM) Equation (13) | % of error deviation hyperbolic function method Equation (22) | Simulation | Equation (13) | hyperbolic function method Equation (22) | % of Error deviation (HPM) Equation (13) | % of error deviation hyperbolic function method Equation (22) | |

0 | 1.0000 | 1.0000 | 1.0000 | 0.00 | 0.00 | 1.0000 | 1.0000 | 1.0000 | 0.00 | 0.00 | 1.0000 | 1.0000 | 1.0000 | 0.00 | 0.00 |

0.2 | 0.8998 | 0.9425 | 0.9236 | 4.53 | 2.64 | 0.6117 | 0.6188 | 0.6498 | 1.14 | 5.86 | 0.1700 | 0.1743 | 0.1810 | 2.46 | 6.07 |

0.4 | 0.8486 | 0.8924 | 0.8657 | 4.90 | 2.01 | 0.3535 | 0.3598 | 0.3701 | 1.75 | 4.48 | 0.0235 | 0.0243 | 0.0249 | 3.40 | 5.62 |

0.6 | 0.8082 | 0.8543 | 0.8252 | 5.39 | 2.06 | 0.2069 | 0.2107 | 0.2196 | 1.83 | 5.78 | 0.0032 | 0.0033 | 0.0034 | 3.12 | 5.88 |

0.8 | 0.7898 | 0.8306 | 0.8012 | 5.15 | 1.42 | 0.1338 | 0.1367 | 0.1389 | 2.16 | 3.67 | 0.0004 | 0.0004 | 0.0004 | 0.00 | 0.00 |

1 | 0.7768 | 0.8226 | 0.7932 | 5.56 | 2.06 | 0.1121 | 0.1147 | 0.1189 | 2.26 | 5.71 | 0.0001 | 0.0001 | 0.0001 | 0.00 | 0.00 |

Average error % | 4.25 | 1.69 | Average error % | 1.52 | 4.25 | Average error % | 1.49 | 2.92 |

z * | Concentration C * | ||||||||
---|---|---|---|---|---|---|---|---|---|

when ϕ = 0.01 | when ϕ = 0.1 | when ϕ = 1 | |||||||

Equation (14) | Simulation | % of Error deviation | Equation (14) | Simulation | % of error deviation | Equation (14) | Simulation | % of Error deviation | |

0 | 1.0000 | 1.0000 | 0.00 | 1.0000 | 1.0000 | 0.00 | 1.0000 | 1.0000 | 0.00 |

0.2 | 0.9801 | 0.9806 | 0.05 | 0.9400 | 0.9414 | 0.14 | 0.8556 | 0.8566 | 0.11 |

0.4 | 0.9601 | 0.9612 | 0.11 | 0.8800 | 0.8828 | 0.31 | 0.7112 | 0.7131 | 0.26 |

0.6 | 0.9402 | 0.9418 | 0.16 | 0.8200 | 0.8242 | 0.50 | 0.5667 | 0.5697 | 0.52 |

0.8 | 0.9203 | 0.9224 | 0.22 | 0.7600 | 0.7657 | 0.74 | 0.4223 | 0.4263 | 0.93 |

1 | 0.9013 | 0.9040 | 0.29 | 0.7030 | 0.7100 | 0.99 | 0.2851 | 0.2900 | 1.68 |

Average error % 0.138 | Average error % 0.446 | Average error % 0.583 |

z * | Concentration C * | ||||||||
---|---|---|---|---|---|---|---|---|---|

when β = 0.5 | when β = 1 | when β = 1.6 | |||||||

Equation (14) | Simulation | % of error deviation | Equation (14) | Simulation | % of error deviation | Equation (14) | Simulation | % of error deviation | |

0 | 1.0000 | 1.0000 | 0.00 | 1.0000 | 1.0000 | 0.00 | 1.0000 | 1.0000 | 0.00 |

0.2 | 0.8933 | 0.8929 | 0.04 | 0.9405 | 0.9411 | 0.06 | 0.9971 | 0.9971 | 0.00 |

0.4 | 0.7866 | 0.7859 | 0.08 | 0.8810 | 0.8822 | 0.13 | 0.9943 | 0.9942 | 0.01 |

0.6 | 0.6800 | 0.6788 | 0.17 | 0.8215 | 0.8233 | 0.21 | 0.9914 | 0.9914 | 0.00 |

0.8 | 0.5733 | 0.5717 | 0.27 | 0.7620 | 0.7644 | 0.31 | 0.9886 | 0.9885 | 0.01 |

1 | 0.472 | 0.4700 | 0.42 | 0.7055 | 0.7055 | 0.00 | 0.9857 | 0.9858 | 0.01 |

Average error % 0.163 | Average error % 0.118 | Average error % 0.005 |

noted. The detailed Matlab program for numerical simulation is provided in Appendix B and Appendix C.

In this paper, the non-linear differential equations in the biofiltration have been solved analytically. Using the homotopy perturbation method and hyperbolic function method, an approximate and closed-form of analytical representation of the concentrations of n-butanol in the biofilm phase is provided. This solution of the concentrations of n-butanol in the biofilm phase and the gas phase is compared with the numerical simulation results. These new analytical results provide a good understanding of the system and the optimization of the parameters in the biofiltration model.

This work was supported by consultancy project, Academy of Maritime Education and Training (AMET), Deemed to be University, Chennai. The Authors are also thankful to Shri J. Ramachandran, Chancellor, Col. Dr. G. Thiruvasagam, Vice-Chancellor, Academy of Maritime Education and Training (AMET), Deemed to be University, Chennai, for their constant encouragement.

The authors declare no conflicts of interest regarding the publication of this paper.

Saranya, K., Mohan, V. and Rajendran, L. (2020) Analytical Solution of the Non-Linear Equation in Biodegradation of N-Butanol in a Biofilter. American Journal of Analytical Chemistry, 11, 172-186. https://doi.org/10.4236/ajac.2020.114013

The homotopy perturbation method is used to give the approximate solutions of the non-linear Equation (6). We construct the homotopy for Equation (1) as follows:

( 1 − p ) ( d 2 S * d x * 2 − ϕ S * ) + p ( d 2 S * d x * 2 + β S * d 2 S * d x * 2 − ϕ S * ) = 0 (A1)

The analytical solution of Equation (1) is

S * = S 0 * + S 1 * p + S 2 * p 2 + S 3 * p 3 + ⋯ (A2)

Substituting Equation (A2) into Equation (A1) we get

( 1 − p ) [ d 2 d x * 2 ( S 0 * + S 1 * p + S 2 * p 2 + ⋯ ) − ϕ ( S 0 * + S 1 * p + S 2 * p 2 ) ] + p ( d 2 d x * 2 ( S 0 * + S 1 * p + S 2 * p 2 + ⋯ ) + β ( S 0 * + S 1 * p + S 2 * p 2 + ⋯ ) × d 2 d x * 2 ( S 0 * + S 1 * p + S 2 * p 2 + ⋯ ) − ϕ ( S 0 * + S 1 * p + S 2 * p 2 + ⋯ ) ) = 0 (A3)

Comparing the coefficients of like powers of p in Equation (A3) we get

p 0 : ( d 2 S 0 * d x * 2 − ϕ S 0 * ) = 0 (A4)

p 1 : ( d 2 S 1 * d x * 2 − ϕ S 1 * + β S 0 * d 2 S 0 * d x * 2 ) = 0 (A5)

The boundary conditions for Equation (A1) are as follows

S 0 * = 1 at x * = 0 (A6)

d S 0 * d x * = 0 at x * = 1 (A7)

Solving Equation (A4) and using the boundary conditions Equation (A6) and (A7), we obtain the following results:

S 0 * = C 1 cosh ϕ x * + C 2 sinh ϕ x * (A8)

By applying the boundary conditions, we get C 1 and C 2

C 1 = 1 , C 2 = − sinh ϕ cosh ϕ (A9)

Substitute C 1 & C 2 value in (A8) we get,

S 0 * ( x * ) = cosh ϕ ( x * − 1 ) cosh ϕ (A10)

Now Equation (A5) becomes

d 2 S 1 * ∂ x * 2 − ϕ S 1 * + β S 0 * d 2 S 0 * ∂ x * 2 = 0 (A11)

The boundary conditions for the above equation are as follows:

S 1 * = 0 at x * = 0 (A12)

d S 1 * d x * = 0 at x * = 1 (A13)

Solving Equation (A11) and using the boundary conditions Equations (A12) and (A13), we can get the following result:

S 1 * ( x * ) = β 2 cosh 2 ϕ − β 6 cosh 2 ϕ [ cosh 2 ϕ ( x * − 1 ) ] + [ β cosh 2 ϕ 6 cosh 2 ϕ − β 2 cosh 2 ϕ ] [ cosh ϕ ( x * − 1 ) cosh ϕ ] (A14)

Considering the two terms, we get

s * ( x * ) = S 0 * ( x * ) + S 1 * ( x * ) (A15)

which is Equation (15) in text.

Appendix B: Matlab Program for the Numerical Solution of Equation (1)function pdex4

m = 0;

x = linspace(0,1);

t=linspace(0,1000000);

sol = pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t);

u1 = sol(:,:,1);

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

figure

plot(x,u1(end,:))

title('u1(x,t)')

xlabel('Distance x')

ylabel('u1(x,1)')

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

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

c = 1;

f = 1.*DuDx;

k=5; B=0.005;

F1=-((k*u(1))/(1+(B*u(1))));

s=F1;

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

function u0 = pdex4ic(x)

u0 = [

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

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

pl = [ul(1)-1];

ql = [

pr = [ur(1)-0];

qr = [

function mat1

options=odeset('RelTol',1e-6,'stats','on');

%initial conditions

Xo=1;

tspan=[0 1];

tic

[t,X]=ode45(@TestFunction,tspan,Xo,options);

toc

figure

holdon

plot(t,X(:,1),'-');

t=0;

%plot(n,(100-100*X(:,1)),'-');

lgend('x1')

ylable('x')

xlable('t')

return

function [dx_dt]=Test Function(t,x)

A=0.8;m=1;B=0.01;

dx_dt(1)=A*((-tanh(sqrt(m)))+((B*sech(sqrt(m))*sech(sqrt(m))*sqrt(m)*(sinh(2*(sqrt(m)))))/3))+(((B*sech(sqrt(m))*sech(sqrt(m))*cosh(2*(sqrt(m))))/6)-((B*sech(sqrt(m))*sech(sqrt(m)))/2))*(-tanh(sqrt(m)));

return