Mathematical Models of Single Stage Monod Model of Land ll Degradation using Laplace Transform and Homotopy Perturbation Method

A mathematical model to predict waste degradation and land ll gas production is discussed. The model is based on two reaction-diffusion equations containing a nonlinear term related to Michaelis–Menten kinetics of the enzymatic reaction. In this paper, we present an approximate analytical solution of the non-linear differential equations that describe the diffusion coupled with a Michaelis–Menten kinetics law. Approximate analytical expressions for substrate and biomass concentrations have been derived for all values of parameter using Laplace transform method and homotopy analysis method. These results are compared with the numerical result and satisfactory is noted. The obtained results are valid for the whole solution domain.


Introduction
Landfi ll gas is a complex mix of different gases created by the action of microorganisms within a landfi ll. A landfi ll gas generation model is a tool that simulates in simple terms the complex changes that occur during decomposition of waste in a landfi ll (Lamborn 2012). Mathematical modeling of the biochemical, physical and chemical processes inside landfi lls has been widely reported (Young, 1989;El Fadel et al., 1996;Haarstrick et al., 2001;White, 2004

Mathematical Formulation
In examining the accuracy of model predictions and looking at the effects of scale, it is worthwhile having several models to compare. This single-stage Monod model has the following form and this model is based on the work by Monod (1942), El-Fadel et al. (1996) and White et al. (2004). The non-linear differential equations presented below represent the rate of change of biomass and substrate for a single-stage model.
where C b is the biomass carbon concentration (kg/m 3 ), C s is the substratecarbon concentration (kg/ m 3 ), K b the half saturation constant for biomass (kg/m 3 ), and K b is the biomass death rate constant (1/day) respectively. Y s is the mass of biomass formed per mass of substrate carbon utilized (kg/kg) and μ b is the mass specifi c growth rate constant(1/day).The initial conditions are

Method of Solution: Homotopy Perturbation Method and Laplace Transform Method
According to the homotopy perturbation method form of the Eqns. (1) and (2) is constructed as follows The approximate solution of the equations (9) and (10) are C b = C b0 p 0 + C b1 p 1 + C b2 p 2 + ....

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Adding the coeffi cients of we get

Numerical Simulation
The non-linear differential equations (1) and (2) are also solved using numerical methods. The function pdex4 in Matlab software which is the function of solving the initial value problems for ordinary differential is used to solve this equation. The Matlab program is also given in Appendix B. The numerical results are also compared with our analytical results in Figs.   (2c), it is observed that the concentration of biomass carbon decreases when K db (biomass death rate constant) and μ b (mass-specifi c growth rate) increases.
Figs.3 (a)-(c) represent the of substrate carbon concentration C s for all values of rate constant. From the fi gure, it is observed that the substrate carbon concentration is decreases when half saturation constant for biomass (K b ) . From the Figs. (2a) and (2c),it is inferred that the substrate carbon concentration is increases when mass of substrate concentration (Y s ) and the mass specifi c growth rate constant (μ b ) increases.

Conclusion
The mathematical model is developed for waste degradation and landfi ll gas production. A non-linear time-dependent reaction-diffusion equations containing a non-linear term related to Michaelis-Menten kinetics of the enzymatic reaction has been solved analytically using laplace transform method and homotopy perturbation method. The analytical expressions are compared with numerical results using Matlab software. This analytical method is used for other non-linear problems. Good agreement is noted.