Periodic and nonperiodic oscillatory behavior in a model for activated sludge reactors

Periodic and nonperiodic oscillatory behavior in a model for activated sludge reactors
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  Pergamon Mathl. Comput. Modelling Vol. 25, No. 10, pp. 9-27, 1997 Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved PII: s0895-7177(97)00071-x 0895-7177/97 17.00 + 0.00 Periodic and Nonperiodic Oscillatory Behavior in a Model for ctivated Sludge Reactors A. AJBAR* AND G. IBRAHIM Department of Chemical Engineering, King Saud University P.O. Box 800, Riyedh 11421, Saudi Arabia f45k002QKSU.EDU.SA Received and accepted March 1997) Abstract-A dynamic model for an activated sludge process is proposed to investigate the stability and bifurcation characteristics of thii industrially important unit. The model is structured upon two processes: an intermediate particulate product formation and sctive biomass synthesis processes The growth kinetics expressions are bssed on substrate inhibition and noncompetitive inhibition of the intermediate product. The bifurcation analysis of the process model shows static ss well es periodic behavior over a wide range of model parameters. The model also exhibits other interesting stability characteristics, including bistability and transition from periodic to nonperiodic behavior through period doubling and torus bifurcations. For some range of the reactor residence time the model exhibits chaotic behavior ss well. Practical criteria are also derived for the effects of feed conditions and purge fraction on the dynamic characteristics of the bioreactor model. Keywords-Activated sludge process, Inhibition, Structured models, Bifurcation, Oscillations. NOMENCL TURE intermediate product inhibition constant (mg/l) substrate saturation constant (mg/l) saturation constant for biomass growth rate (mg/l) volumetric flow rate (l/h) substrate concentration (mg/l) reactor volume (1) sludge withdrawal fraction biomass concentration (mg/l) intermediate particulate product concentration (mg/l) GREEK SYMBOL S a inverse of substrate inhibition constant (l/mg) resctor residence time (hr) ~1 specific rate for conversion of substrate to intermediate product (hr-‘) ~2 specific rate for conversion of intermediate product to active biomsss (hr-‘) p,,, maximum specific rate (hr-*) . JBSCRIPTS f feed stream R recycle stream *Author to whom all correspondence should be addressed. 9  10 A AJBAR ND G IBR HIM ABBREVIATIONS HB Hopf bifurcation point SLP static limit point PD period doubling bifurcation TR torus bifurcation INTRODU TION The heterogeneity and the complexity of activated sludge processes pose a continuous challenge to developing models that can incorporate all the necessary levels of information concerning the process and be accurate enough for the adequate control and safe operation of the bioreactor. The complete quantification of the microbiological system on the other hand requires the un- derstanding of the complex biological and physicochemical interactions in the process and the measurement of a large number of reaction rates which is often beyond the scope of reasonable measurement techniques. This task is particularly complicated when dynamic modeling is sought. While simple and often unstructured steady-state models are sufficient for the purpose of plant design these models are generally inadequate for dynamic simulations and control of the process since they often fail to predict accurately the effects of process disturbances [1 2]. Structured models on the other hand with different degrees of complexity can supplement the inadequacies of unstructured models. Structured models take into account the inevitable changes in the cell population composition since the model microbial kinetics are constructed on the basis of at least some of the knowledge accumulated in the vast repository of fundamental biochemistry and microbiology. Structured models however may sometimes suffer from over-detailed information that cannot be verified making them inappropriate for practical use. A good structured model should have a reasonable number of parameters to provide it with some levels of flexibility [3 4]. Besides predicting the effects of external disturbances a good dynamic model should also be able to predict the different dynamic behavior the autonomous process may exhibit. Activated sludge reactors have long been known to exhibit a variety of dynamic behavior depending on the process operating conditions [5-71. DiBiasio et al. [8] for instance examined both theoretically and experimentally the occurrence of steady-state multiplicity and hysteresis in activated sludge reactors confirming the experimental findings reported in earlier works [6 7]. Bertucoo et al. [9] on the other hand 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. In this paper the dynamic characteristics of a flexible model of activated sludge process with solids recycle are studied. The proposed model is a simplified version of a structured model used by Andrews and his colleagues [HI-121 for the dynamic simulation and control of activated sludge processes. The model is structured upon substrate and intermediate component growth depending processes. The model kinetics are based on the other hand on substrate and inter- mediate product inhibitory effects. A static and dynamic bifurcation analysis is carried out for the evidently nonlinear model using continuation methods. It is known that a nonlinear system can exhibit complex dynamic behavior depending on the values of the model parameters. A non- linear system has often point attractors. It can also exhibit periodic quasiperiodic and even a chaotic behavior. The use of principles of bifurcation theory coupled with continuation methods allows in general an in-depth investigation of the dynamic characteristics of the model including eventual periodic and nonperiodic behavior. The present work has then two objectives. The first objective is to carry out a static and dynamic bifurcation analysis of the model and to show how principles of bifurcation theory can be used to classify different branching phenomena in a realistic model of activated sludge process. It is shown that the proposed model can predict within a wide range of physically realistic param- eters phenomena ranging from hysteresis periodic behavior to bistability and chaotic behavior.  Activated Sludge Reactora The second objective of this work is to analyze the performances of the reactor when operated in different regions of static, periodic, and nonperiodic behavior. Practical strategies are suggested for obviating oscillatory behavior or to determine if such oscillations can be of beneficial use. PRO ESS MODEL The proposed model Substrate (S) 4 Particulate product (Xs) 4 Biomass (Xa) is structured upon two processes: (1) formation of an intermediate particulate product (X,) depending on substrate; (2) active biomass (Xa) synthesis. The model is a simplified version of the srcinal Andrews model [lo-121 for activated sludge process where the decay rate is being assumed negligible. This assumption is acceptable when operating at low cell residence time. The substrate S is converted to a slowly biodegradable particulate product following the rate expression: p1 = (S+$CrS2)Y (1) where ~1 is the specific rate. This kinetic expression accounts for the well-known phenomenon of substrate inhibition. The Haldane expression has been used for this purpose. The equation requires only three parameters: the specific growth rate pm, the substrate saturation constant K,, and the substrate inhibition constant l/a. The Haldane equation is accurate enough to be favored over more complicated inhibitory kinetics models [13]. Substrate inhibition has been extensively studied both in batch reactors and in chemostats, and it was shown that substrate inhibition models are fundamental in predicting the stability characteristics in activated sludge reactors (with or without solids recycle), such as the occurrence of steady-state multiplicity and the hysteresis phenomenon [14,15]. Besides the direct inhibitory effects due to the substrate, a delayed inhibitory effect caused by the intermediate product is also assumed. This inhibition effect is assumed to be noncompetitive, i.e., the growth of the intermediate product affects negatively the maximum growth rate pL,, given by where Ki s the inhibition constant. The biomass growth rate on the other hand, depends on the intermediate product following the common Monod behavior: J77bX, “= K,+X, (3) where ~2 is the specific growth rate and K, is the half saturation constant for biomass synthesis. The intermediate product inhibition affects then both the substrate uptake rate and the biomass growth rate through the term CL,,,. This is in agreement with the observations made in [16] on the influence of particulate intermediate products on the performances of microbial cultures. Equations [l-3] form then the model kinetics. In the following section, the unsteady state component balance equations around the reactor-settler, shown in Figure 1, are written for the different species. Substrate S. The substrate is consumed to produce the uct X#. The unsteady state component balance yields intermediate particulate prod- W)S + v$$ (4)  12 A. AJBAR AND G IBRAHIM Q R) Figure 1. Schematic diagram of the bioreactor with solids recycle. with Yzls is the yield coefficient assumed constant. Equation (4) is also equivalent to s,--s-epx,=e X/S (5) where 0 = V/Q is the reactor residence time. Particulate intermediate product X,. The particulate product X, is consumed to produce the biomass. A component balance yields, then Q-&f + QR-&R + V(p2 - pi)-& = QW& + Q 1 + R - W)& + Vf . (6) The assumed ideal conditions in the settler allows the following simple relation between the recycle XsR and the effluent X, concentrations xR=x  l+R-W) 8 R ’ 7) Equation (6) is then equivalent to xsj - wx, + e p2 - pl)xa = edt. 8) Active biomass X,. The component balance equation for biomass is QXaj + QRX,R + Vp2Xa = QWX, + Q l + R - W)Xa + V . (9) Similarly to the intermediate product (equation (7)), Xa~ and the effluent X, biomass concentrations X,,=X (l+R-W) 0 R * Equation (9) is equivalent then to a simple relation links the recycle (19) d a a j WXa + @2X, = 8~. 01) Equations [5,8,11] form the autonomous model of the bioreactor. The three-dimensional model includes a large number (nine) of parameters. Besides the reactor operating parameters, i.e., feed conditions and purge fraction, the nominal values of the other model parameters are given in Table 1. These nominal values (as shown in the table) are taken from realistic ranges given in literature. The bifurcation analysis consists in studying the branching phenomena in the model when its parameters (or a subset of them) are varied around the nominal values. In the first part  Activated Sludge Reactors 13 of this investigation, the effects of the substrate fee concentrations are analyzed. The effects of the purge fraction and other parameters are discussed in later sections. Among all the model parameters the residence time 0 is the easiest parameter to vary and is chosen as the bifurcation parameter. Table 1. Nominal values of model parameters. Parameter Mm0 KS mg/l) Kc ma sj mg/U W X& /l) X.f(w/l) Y x/s 4ed Pm hrsl) Value 10 10 500 500 0.1 80 20 0.50 0.02 3.0 Reference 25) 24) 26) 24) 25) 24) NUMERIC L TOOLS ND PRESENT TION TECHNIQUES The methodology for static and dynamic bifurcation consists in the numerical continuation techniques, coupled with the principles of bifurcation theory. The bifurcation diagrams are ob- tained using the software AUTO of Doedel and Kernevez [17]. This package is able to perform both steady state static) and dynamic bifurcation analysis, including the determination of the entire periodic solution branches. AUTO also computes the Floquet multipliers along periodic solution branches, and therefore, determines the stability of the periodic orbits. A periodic orbit loses its stability by a number of mechanisms. The most common of them are period doubling bifurcation and saddle node tangent bifurcation. Because of periodicity there is always a Floquet multiplier equal to +l. When Floquet multipliers lie inside the unit circle, the periodic solution is asymptotically stable. A Floquet multiplier leaving the unit circle through -1 indicates period doubling bifurcation. Passage of complex Floquet multipliers out of the unit circle indicates that the periodic orbit bifurcates to an invariant torus. Periodic solutions can also lose their stability to chaotic attractors. Chaotic attractors show extreme sensitivity to initial conditions and nearby trajectories diverge. A chaotic attractor is best characterized by its Lyapunov exponents. A chaotic attractor has at least one positive Lyapunov exponent. The technique and the algorithm of Wolf et al. [18] are used to efficiently compute these exponents. The DGEAR subroutine [19] with automatic step-size to ensure accuracy for stiff differential equations is used for numerical simulation of periodic as well ss chaotic attractors. RESULTS ND DISCUSSION The continuity diagram substrate concentration S) vs. residence time 0)) is shown in Fig- ure 2a for the feed condition Sf = 500.0 mg/l and the rest of model parameters shown in Table 1. The continuation diagram shows two stable branches solid lines) connected to a unstable branch dash lines) in the middle. The enlargement of Figure 2a around the unstable region Figure 2b) shows the existence of two static limit points SLP) turning points) occurring at residence times, respectively, of 1.663 hr and 2.354 hr. The diagram is also characterized by the occurrence of two Hopf points HB) at residence times of 1.659 hr and 2.338 hr, respectively. The existence of Hopf points indicate necessarily the occurrence of an oscillatory behavior in the system. A Hopf point arises when a pair of complex eigenvalues of the Jacobean of the model crosses the imaginary axis
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