Bifurcation analysis of continuous aerobic nonisothermal bioreactor for wastewater treatment Rigel V. Gomez-Acata * , Pablo A. Lopez-Perez * , Rafael Maya-Yescas ** , Ricardo Aguilar-Lopez * *Department of Biotechnology & Bioengineering, CINVESTAV-IPN, No. 2508 D.F. Mexico (Tel: 55-5747-3800; e-mail: raguilar@cinvestav.mx). ** School of Chemical Engineering, Universidad Michoacana de San Nicolas de Hidalgo Michoacan, Mexico (e-mail: rmayay@umich.mx) Abstract: The petrochemical industry water effluents must be treated before their discharge to lakes, rivers and the ocean so that activated sludge plants are used. It is of great importance to satisfy several environmental regulations in order to achieve the lowest impact in the ecosystem. This paper explore the behaviour of a wastewater treatment plant model of a petrochemical industry in order to define the optimal operation conditions for the discharge of effluent under the safety regulations, furthermore risk operation areas where stable and unstable equilibrium points existed are outlined. Keywords: Water pollution, Dynamic behaviour, Limit cycles, Bifurcation analysis, Reactor modelling, Stability analysis 1. INTRODUCTION The complexity of the dynamics of biological and ecological processes and their interaction with other systems offer many challenges to control engineering. For example, many bioprocesses involved in water treatment are operated continuously; the complexity of these processes is related to variety of microorganisms involved, variability of environmental disturbances, large perturbations to feed flow rate and uncertainties about composition of the incoming wastewater; they are also nonlinear systems, which could exhibit multiplicity of steady states and instabilities areas (Ajbar and Gamal, 1997; Ibarra-Junquera et al., 2004; Ewart et al., 2006; Femat et al., 2004; Gwaltney and Stadtherr, 2007; Hess and Bernard, 2008; Lara-Cisneros and Femat, 2009). Activated sludge treatment is a complex process where many states and nonlinear relationships are involved, in addition relationships among inputs and outputs in a bioreactor are also complex (nonlinear and time varying) (Fujie et al., 1998; Maqueda et al., 2006). Several studies about wastewater treatment models are focused in their stability and dinamic behaviour (by bifurcation analysis) ir order to investigated the conditions for increasing the performance of the biological system, or avoid risk areas during operation of the bioreactors, examples of the above are the work of Volcke et al., 2010, that constructing operating diagrams for the nitrification process (comounly present in wastewater effluents) by two- step biological conversion, they specified unstable and steady state multiplicity zones and pointed out that the choice of a kinetic structure implies a certain steady steate behaviour; Jianqiang & Ray, 2000, studied the dynamic behaviour of two coupled continuous stirred-tank reactors with recycle for biological wastewater treatment when one of the reactors is operated under self-sustained natural oscillations in order to improve the reactor performance; Bertucoo et al., 1990 examined the stability and bifurcation characteristics of the activated sludge reactors with solids recycle and showed the existence of steady-state multiplicity for some range of operating parameters; Ibrahim et al., 2006 described a model bioreactor for an activated sludge process. The stability and bifurcation characteristics of the model were investigated, the bifurcation analysis of the model showed simple and complex dynamic behaviour over a wide range of the model parameters. The model exhibited a new interesting behaviour (in some range of parameters) including four static limit points (turning points) and two Hopf points, that cause different kinds of stability characteristics ranging from asymptotically stable equilibrium and hysteresis phenomen to periodic and complex behaviour. The objetive of bifurcation analysis is to characterize changes in the qualitative behaviour of a mathematical model defined by ODE´s and/or PDE´s by varying key parameters. In the case of bioreactor operation is common to use the dilution rate as a bifurcation parameter. Many important features as well as potential limitations hard to find with simple simulations are uncovered through this analysis (Zhang and Henson, 2001; Garhyan et al., 2003) Furthermore, this analysis can be used to find optimal operating conditions and to avoid hazardous conditions for the process (Namjoshi, Kienle, & Ranmkrishna, 2003). The bifurcation occurs when one of the eigenvalues is close to the axis of imaginary numbers in the complex plane. The simplest bifurcations are associated when one of the 2012 IFAC Conference on Analysis and Control of Chaotic Systems The International Federation of Automatic Control June 20-22, 2012. Cancún, México 978-3-902823-02-1/12/$20.00 © 2012 IFAC 24 10.3182/20120620-3-MX-3012.00054