Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 512404, 10 pages

http://dx.doi.org/10.1155/2015/512404

## Dynamical Analysis of a Continuous Stirred-Tank Reactor with the Formation of Biofilms for Wastewater Treatment

^{1}Departamento de Matemáticas y Estadística, Universidad Nacional de Colombia, Sede Manizales, Manizales, Colombia^{2}Departamento de Matemáticas y Estadística, Universidad del Tolima, Ibagué, Colombia

Received 2 February 2015; Revised 7 April 2015; Accepted 7 April 2015

Academic Editor: Seungik Baek

Copyright © 2015 Karen López Buriticá et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper analyzes the dynamics of a system that models the formation of biofilms in a continuous stirred-tank reactor (CSTR) when it is utilized for wastewater treatment. The growth rate of the microorganisms is modeled using two different kinetics, Monod and Haldane kinetics, with the goal of studying the influence of each in the system. The equilibrium points are identified through a stability analysis, and the bifurcations found are characterized.

#### 1. Introduction

Water is the most important natural resource without which life could not exist. It is the most abundant liquid on Earth, covering 71% of the Earth’s surface, of which only 3% is fresh water. Water use results in a decrease in water quality, and serious environmental deterioration results from directly returning water to the environment; therefore, in recent years, the construction of increasingly more efficient treatment plants has gained vital importance. Consequently, the role of biological processing in wastewater treatment has increased considerably where biofilm reactors are one of the most interesting [1]. Biofilms are thin layers of microorganisms that adhere to a solid surface; they can develop on almost any type of surface exposed to an aqueous medium, and they are utilized in wastewater treatment to eliminate and oxidize organic and inorganic components. The biomass concentration in a biofilm can be ten times larger than its concentration in a liquid culture [2]. This reduces the volume of the equipment for a constant rate of elimination per unit volume.

A theoretical analysis of the formation of biofilms was performed in [2], and in [3, 4] characterization and analysis of biofilm use in wastewater treatment were conducted. Experimental results on a pilot scale have demonstrated the efficiency of biofilm reactors for treating wastewater. For example, (1) that contained molasses as a source of carbon in different conditions of the influent [1]; (2) with a vertically moving biofilm reactor (VMBR) followed by a stratified sand filter [5]; and (3) in combination with treatment by activated sludge [6].

The implementation of a pilot plant to study these procedures can be costly, which makes mathematical models and computer simulations essential to describe, predict, and control the complex interactions present in the reactor [7]. For several years, researchers have focused on developing mathematical methods that more realistically simulate the behavior of a biofilm reactor. For example, in [8] a biofilm reactor model was proposed based on physiological aspects. By time, the models began to involve more variables [9–11]. “However, these often turned out to be unsatisfactory due to certain intrinsic deficiencies. For example, in [8] the model only took account of the accumulation of active biomass and inactive biomass of biofilms and in [9–11] the model was mostly used to study the morphology and structure of biofilms, so mechanisms and kinetics processes of biofilm formation have not yet been well understood” [12].

In [12] a model is proposed “which attempts to summarize the factors influencing biofilm formation and eliminating the deficiency of existing models and to study the kinetic mechanisms from a new perspective.” This model describes the formation of microbial populations in an aqueous medium inside a reactor and the mechanism under which bacteria can be suspended in the water or colonize the surface. The model in [12] considers extremely thin biofilms so it can be considered that all the microorganisms receive the substrate at the same time, therefore the model incorporates this condition by ignoring diffusion reactions. There are many applications where thin biofilms are desirable and detachment procedure is considered to keep such a condition [13–15].

In this paper, the model in [12] is also considered. However, our bifurcational and dynamical analysis included not only the Monod kinetics but also the Haldane kinetics. Some bifurcations are found, including a transcritical bifurcation that does not observe the conditions in Sotomayor theorem.

In [16] the model proposed in [12] is reformulated to study the behavior of the suspended microorganisms.

#### 2. Materials and Methods

##### 2.1. Mathematical Model

The continuous stirred-tank reactor described in [12] was chosen as the model reactor. The system consists of three ordinary nonlinear differential equations and models the formation of heterotrophic aerobic biofilms inside the reactor, taking into account both the microorganisms adhered to the biofilms and the microorganisms that are suspended. In addition to studying the influence of the various parameters in biofilm formation, in [12] the system is established and modeled using Monod kinetics, the equilibrium corresponding to the washing condition. In the present work, the system is also analyzed as modeled by Haldane kinetics and more equilibrium points are found and their stability is studied, in addition to characterizing the bifurcations that they represent. As bifurcation parameter of the system, the dilution coefficient was chosen, due to this parameter allows to control the residence time of wastewater in the reactor.

The following assumptions are made with respect to the dynamics of the formation of biofilms. The three-dimensional structure of the biofilms is ignored because the biofilms are considered to be infinitely thin. There are a finite number of colonization sites available on the support surface, as well as maximum possible surface density of adhered microorganisms. The adhered cells are separated outside the fluid at a rate proportional to its density and the daughter cells of adhered microorganisms compete for space on the support surface—a fraction of the daughter cells finds adhesion sites and a fraction does not. The function decreases with the number of daughter cells (as the number of daughter cells increases, the number of available sites to occupy decreases), where , which is a function of probability [12].

The reactor dynamics are described by the following equations: The following parameters are used for the numerical simulation: These values are taken from [12].

The growth of the suspended and adhered microorganisms in the reactor is a nonlinear natural phenomenon that can be analyzed using nonlinear dynamics, which provides a global perspective of the types of behavior in the system, such as equilibrium states, stability, and bifurcations.

##### 2.2. Monod Kinetics

The suspended and adhered microorganisms inside the reactor consume the organic matter present in the wastewater to be treated. The specific rate of substrate consumption is the “speed” at which the organism consumes the substrate. Although the growth of microorganisms is a complex phenomenon, there are equations that can model this behavior. Monod’s equation is the simplest and most widely utilized equation for describing the kinetics of microbial growth. It is a function of the limiting substrate and is expressed as where is the maximum growth rate. is the concentration of the substrate corresponding to the half of the maximum growth rate of () and represents the affinity of the microorganisms for the substrate. If the organism has a great affinity for the limiting substrate, the value of is low [17].

###### 2.2.1. Equilibrium Points

A point is an equilibrium point or a critical point of if . In this point, the system remains in a stationary state; that is, if is the flux of the system, then [18, 19].

The system has an equilibrium in the washing condition for any set of values of the parameters, in this equilibrium there aren’t microorganisms adhere or suspended, but there is substrate into the reactor. Exists other equilibrium physically feasible when the microorganisms adhere to the support surface and finally two equilibria in the case where suspended microorganisms do not adhere to the support surface.(i)First equilibrium point, corresponding to the washing condition, is .(ii)If , that is, there adherence of microorganisms to biofilms, then a nontrivial equilibrium point exists: . This equilibrium point cannot be found analytically.(iii)If , there exist two equilibrium points: with . This point is physically possible if and ; that is, , where Substituting and in the following equation, one can solve . Consider

###### 2.2.2. Equilibrium Point Stability

An equilibrium point of a dynamical system is stable in Lyapunov sense if all trajectories with initial conditions near the equilibrium point remain in that vicinity. The equilibrium point is asymptotically stable, that is, the trajectories always tend toward the equilibrium point. Lyapunov’s indirect method is outlined in [20] and demonstrated in [21], and it provides a procedure to determine the local stability of a hyperbolic equilibrium point. Furthermore, the Hartman-Grobman theorem states that, near a hyperbolic equilibrium point, the nonlinear system presents a behavior qualitatively equivalent to that of the corresponding linear system [22, 23], where a hyperbolic equilibrium point is an equilibrium point in which its Jacobian matrix does not have eigenvalues with zero real part.

In the following, the stability of the equilibrium points is analyzed by Lyapunov’s indirect method, linearizing the system and calculating its eigenvalues. To analyze the location in the plane of the eigenvalues, the Routh-Hurwitz criterion was utilized.

At the equilibrium point , the Jacobian matrix of the system is Let be a submatrix, such that The eigenvalues , , and of the matrix are the roots of the equation: It is readily apparent that Substituting the values of the parameters, one obtains that . Therefore, is unstable.

For the equilibrium point , numerically performing the stability analysis, we find that it is always stable.

The algorithm is as follows: for a given value of the equilibrium point is found using the Newton-Raphson method, and the equilibrium curve is constructed using the same method taking the previous equilibrium as the initial condition. Every equilibrium point is evaluated in the Jacobian matrix, and the eigenvalues are calculated. If they are all negative, the equilibrium is stable. If either of them is negative, the equilibrium is unstable and if at least one has a zero real part, it is not possible to decide on the stability of the equilibrium.

At the equilibrium point , the Jacobian matrix of the system is and its characteristic polynomial is We establish the following array: The Routh-Hurwitz stability criterion states that the number of roots of the characteristic polynomial with a positive real part is equal to the number of sign changes of the coefficients of the first column of the array.

is negative. Indeed given that and .

If then and .

If then .

Therefore, for , .

If then .

Therefore, for , .

In summary, we have If , there are no sign changes, and, thus, the real part of all the eigenvalues is negative, and the equilibrium point is stable. If there is always a sign change ( is unknown and it does not matter, because anyway there is a sign change), which means that there is an eigenvalue with a positive real part, and, thus, the equilibrium point is unstable.

For the equilibrium point , the characteristic polynomial is , where By virtue of the Routh-Hurwitz criterion, one has Numerically, it is found that, for values , there is a sign change, and, thus, the equilibrium point is unstable, whereas for values , there is no sign change, and, thus, the equilibrium point is stable.

###### 2.2.3. One-Parameter Bifurcations

The fact that the dynamics of a system can change drastically as one or more of its parameters are varied is well known. This qualitative or structural change is known as a* bifurcation*. In general, bifurcation theory studies the structural changes that a dynamical system can undergo as its parameters are varied; further bifurcations only occur in nonhyperbolic equilibria [19, 24].

In what follows, the transcritical bifurcation described in [25] is presented because it is precisely the one studied in this work.

Let be a nonhyperbolic equilibrium point. The structure of the system is characterized as follows.(i)Two curves of equilibrium points and pass through .(ii)Both curves exist on both sides .(iii)The stability of the curves is interchanged as they pass through .

The system presents a transcritical bifurcation in the equilibrium curves and in fact.

When , for the point , an eigenvalue exists if , given that and ; then must be zero. That is,

Thus The equilibrium point in the bifurcation parameter is as follows: .

The equilibrium curve is stable for and unstable for . For the equilibria are unstable for and stable for . Figures 1 and 2 show how the equilibrium curves of the points and cross each other and interchange stability, such that a transcritical bifurcation appears.