Analysis of Biodegradation and Microbial Growth in Groundwater System Using New the Homotopy Perturbation Method

In this paper, we drive the concentration of microbial growth in the groundwater system. This model is based on the system of non-linear differential equations. The system of equations is solved by using the new homotopy perturbation method. We followed toluene degradation and bacterial growth by measuring toluene and oxygen concentrations and by direct cell counts. And the total amount of toluene degraded by Pseudomonal putida F1 in the sediment columns increased with rising concentration of the source and flow rate. In contrast, the efficiency of toluene removal slowly decreases. The approximate analytical expression of this model, the concentration of toluene and bacteria also consideration of a metabolite concentration, the microbial growth of attached and suspended bacteria, depending on the simultaneous presence of toluene. Finally, oxygen and dual Monod kinetics are discussed. The analytical solutions are also compared with simulation results and satisfactory the agreement is noted.


INTRODUCTION
The reliability of the design and cost-efficient bioremediation methods, the controls and limitations of the biodegradation potential of natural microbial communities in aquifers [1]. Petroleum hydrocarbons belongs to the most abundant contaminants in aquifers [2]. Among them, the nonaromatic compounds benzene, toluene, ethyl benzene, and xylene (BTEX) are of major concern due to their toxicity [3]. Recent research on the biodegradation of aromatic hydrocarbons in flow through laboratory and field experiments have shed some light on the limitation of biodegradation by transverse dispersive mixing [4]. While, data from natural aquatic systems hint at considerably lower values [5,6]. The media in the sediment column experiments, one containing the electron donor (toluene), and the other the electron acceptor (oxygen or nitrate). These were combined directly at the inlet of the column to prevent bacteria creeping back into the water reservoirs [7]. The new-growing cells are released into the mobile aqueous process in the model, and eventually flushed out of the base. This release of new-grown cells from the sediment surface to the mobile aqueous phase has already been observed under growth conditions in earlier studies on microbial transport [8,9]. MFC methods are innovative modes of restoring / reducing nutrients through waste water. The MFC can also be used to produce a wide range of organic problems, such as: wastewater from of the agro-industry, digested sludge, domestic wastewater, food wastewater, and marine sediments [10,11]. Biosensors are analytical devices interpreting a biochemical recognition reaction into an observable effect [12]. In addition, the attachment to suspended cells ratio was the highest at the lowest concentrations of substrates, and vice versa. It is well trend from marine sediments like aquifers [13 -16]. In additional factors such as food web interactions, grazing, or resource rivalry as well as multiple constraints play an important role within natural microbial communities [17]. In order to clean up hydro carbonates from soil and water, the latest development research is used such as Alberta techniques. This is an all-new approach to clean up polluted sites using bacteria to eat offending water particles while leaving the nice. Recently, Kirthiga and Rajendran [18] have obtained analytical expression on the concentrations of the output of biomass and ethanol from industrial wastewater. Presented a steady-state analysis of the MFC and the analytical expression of a substrate, anodophilic, methanogenic, and oxidized mediators obtained in all parameters using Homotopy perturbation method [19].
In CONCENTRATION OF BIODEGRADATION Biodegradation is the normal deterioration by microorganisms such as bacteria and fungi or other biological activity of the products. Composting is a mechanism powered by humans in which biodegradation occurs under a specified set of circumstances. The Mathematical modeling and solution of biodegradation reactions are as follows.

A. Direct utilization of toluene for growth
In the standard model, we assume that the bacteria directly grow on the degradation of toluene. The electron acceptor is considered available in excess, and biomass decay is neglected. Then the standard Monod equations read as follows [20]: The initial conditions are [µM] are the half saturation concentration of yield coefficient and the toluene.

B. Analytical solutions of the direct utilization of toluene for growth
The analytical expression of concentration of bacteria ) (t X and toluene ) (t c tol are obtained as follows:

Fig. 2.Comparison of analytical expression of the concentration with simulation results and initial condition with various values of parameters
when __ is analytical and, *** is numerical when __ is analytical and, *** is numerical.

C. Consideration of a metabolite
In this model, we assume that the bacteria first transform toluene to a metabolite without growth, and then grow on the degradation of the metabolite. A suitable candidate metabolite is methyl-catechol. The revised equations read as follows [20]: The initial conditions are

III.
MICROBIAL GROWTH Microbial growth is increased in cell size and frequent cell division. Many microbes have the enzymes and biochemical pathways required for all cellular synthesis Components are made from minerals and energy sources, biomass, nitrogen, phosphorus, and sulfur. Usually, the growth temperature ranges for a particular organism's spans from 30 to 40°C. Some relations are delivered microbial growth in the groundwater system. The concentration of growth is below.
E. Reactive-transport modelling of Governing equations We simulate microbial growth in the column systems coupled to one-dimensional reactive-transport with a numerical model that considers three mobile components, namely toluene oxygen, and suspended bacteria as well as the attached bacteria as immobile component. We model microbial growth of attached and suspended bacteria, depending on the simultaneous presence of toluene and oxygen, by dual Monod kinetics [20]: and attachment of suspended bacteria to the sediment surface is described by the modified first-order attachment rate attach r .
One-dimensional transport of toluene and oxygen in the column system and their consumption due to growth of attached bacteria can be described by a system of coupled advection-dispersion-reaction equations [

Fig. 7. Comparison of analytical expression of the concentration with simulation results and initial condition with various values of parameters
when __ is analytical and, ooo is numerical.
when __ is analytical and, ooo is numerical.

IV.
NUMERICAL SIMULATION A convenient way to introduce variable microbial kinetics in a numerical model is to describe growth dynamics with the help of Monod-type kinetics [21 -23] Growth rates are then governed by the time-varying local concentrations of one or more reactive substances. The homotopy perturbation was first proposed by He et al. [24]. This method is used to find an approximate analytical solution of nonlinear problems. The non-linear differential Eqns. (1) -(3) have been solved numerically using MATLAB software. A respective script pdex4 is provided in Appendix-B. The analytical expressions of concentrations of are obtained from new homotopy perturbation method is compared with simulation results in graphs (2 -10). Satisfactory agreement is noted. V.
RESULTS AND DISCUSSION The above solutions represent the new approximate analytical solutions for the concentration of bacteria and toluene, and metabolite for all values of parameters and Reactive-transport modelling is discussed. As time increases the parameter values coincide with the x-axis. So the value of 't' is stopped after a particular period. Figure 2(a) shows the maximum specific growth rate constant on the concentration of the bacteria. From this figure, we observed that the concentration increases as the increasing value of a particular growth rate. Figure 2(b) illustrates different values of the parameter tol K observed that the concentration increases with the increasing of the half-saturation concentration. Figure 3, represents the value of the 't' values is five since the limit increases the error is also increased, so take lower values and higher values to apply the parameters and found the graph. Figure 3(a) illustrates the effect of maximum specific growth rate constant max  on concentration profile. It represents the concentration profile is decreasing with increasing values of max  . As a result, its toluene coefficient decreases, and boundary decreases. Figures. 3(b) and 3(c), represents the concentration of toluene for different values of half-saturation concentration of the toluene and yield coefficient. It is observed that an increase in the parameters leads to an increase in concentration. This is the direct utilization of toluene for growth concentration. The gap between toluene inlet and outlet concentrations is initially increased, and then a steady value was reached, which we denote as the maximum efficiency of degradation.
In figures 4 to 6, it is observed that there will not be any changes in the graph as t increases; hence the value t is restricted. In these figures represented the consideration of metabolite concentrations. Figures 4(b), 5(a), 6(a), and 6(b) shows the effect of VI.
CONCLUSION The system of differential equations has been formulated and solved analytically using the new Homotopy perturbation method for various values. This work is mainly derived from metabolite concentration, the microbial growth of attached and suspended bacteria, depending on the simultaneous presence of toluene and oxygen, and dual Monod kinetics system. The attached bacteria are responsible for the majority of the observed biodegradation. While attached cells were mainly responsible for toluene degradation, the release of cells into the pure water causes permanent inoculation of the aquifer downstream. The effects of various parameters on concentration profiles are discussed. The obtained results have a satisfactory agreement.