the dynamical models of activated sludge system

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International Journal of Biomathematics Vol. 5, No. 5 (September 2012) 1250048 (15 pages) c World Scientific Publishing Company DOI: 10.1142/S1793524512500489

THE DYNAMICAL MODELS OF ACTIVATED SLUDGE SYSTEM: STOCHASTIC CELLULAR AUTOMATON AND DIFFERENTIAL EQUATIONS

Int. J. Biomath. 2012.05. Downloaded from www.worldscientific.com by UNIVERSITY OF ANTWERP on 10/08/12. For personal use only.

JINBAO LIAO∗,†,§, , HUI LU∗,† , ZHENQING LI∗,∗∗ , ¨ ENGLUND¶ XINZHU MENG§ and GORAN ∗State

Key Laboratory of Vegetation and Environmental Change Institute of Botany, Chinese Academy of Sciences Beijing, 100093, P. R. China

†Graduate

University of Chinese of Academy Sciences Beijing, 100049, P. R. China

‡Research

Group Plant and Vegetation Ecology Department of Biology, University of Antwerp (Campus Drie Eiken), Universiteitsplein 1 B-2610 Wilrijk, Belgium

§College

of Information Science and Engineering Shandong University of Science and Technology Qingdao 266510, P. R. China

¶Department

of Ecology and Environmental Science & Ume˚ a Marine Sciences Centre Ume˚ a University, SE-901 87 Ume˚ a, Swede [email protected] ∗∗[email protected]

Received 19 May 2011 Accepted 10 August 2011 Published 9 May 2012 A stochastic cellular automaton (CA) model for activated sludge system (ASS) is formulated by a series of transition functions upon realistic treatment processes, and it is tested by comparing with ordinary differential equations (ODEs) of ASS. CA system performed by empirical parameters can reflect the characteristics of fluctuation, complexity and strong non-linearity of ASS. The results show that the predictions of CA are approximately similar to the dynamical behaviors of ODEs. Based on the extreme experimental system with complete cell recycle in model validation, the dynamics of biomass and substrate are predicted accurately by CA, but the large errors exist in ODEs except for integrating more spatially complicated factors. This is due to that the strong mechanical stress from spatial crowding effect is ignored in ODEs, while CA system as a spatially explicit model takes account of local interactions. Despite its extremely simple structure, CA still can capture the essence of ASS better than ODEs, thus it would be very useful in predicting long-term dynamics in other similar systems. Keywords: Cellular automaton; activated sludge system; ordinary differential equations; dynamic models; wastewater treatment. 1250048-1


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1. Introduction The usefulness of modeling and simulation to gain insight into and evaluate the behavior of wastewater treatment system has been widely acknowledged [20]. During the last decades, the kinetic properties of activated sludge system (ASS) have been studied extensively, and several studies have been presented and documented by ordinary differential equations (ODEs). For instance, the traditional models based on Monod equation have been applied widely [1, 19, 24]. However, some experiments suggested that these models based on equilibrium state lost lots of instantaneous information of treating process and ignored some important dynamical phenomenon, so that building dynamical models is necessary to capture more details of ASS, such as the relatively reliable, simple steady-state models [6, 22, 30] and simulation models [5, 17, 23, 31]. Recently, the International Water Association Task Group provided a standardized set of basic models of ASM1 [11], ASM2 [12] and ASM3 [9, 13], and they have been widely accepted in scientific fields but still have some weaknesses in diagnosing and controlling ASS. For example, these models still cannot reveal some complex and poorly known phenomena in operation, local competition and spatial effect among activated sludge particles, and strongly nonlinear relationship among the important process variables. Besides these mathematical models, some simulations have also attracted considerable attention in modeling in order to better evaluate the performance of wastewater treatment plant (WWTP), like artificial neural networks (ANN) [10, 25], a novel interval arithmetic simulation approach [28], the intelligent modeling approaches [4, 37] and Monte Carlo [16]. However, the available information on the structure of microbial community based on simulations is still fragmentary and unclear due mainly to technical difficulties in enumerating and identifying bacteria present in ASS. To improve monitoring, control, and performance for WWTP, structuring good predictive models is urgent to understand actual behaviors of these complex systems and develop optimal control and management schemes. Based on discrete time and space, a cellular automaton (CA) which consists of some lattices we called cells can simulate the complexity of biological system [7, 14, 33–35]. For example, CA has been used widely to describe the formation of spatial structure of micro-colonies and biofilm [2, 18, 27, 32]. Here, considering three treatment processes of adsorption, degradation and movement, a CA model is structured by using transition rules upon spatial interaction and resource competition. The primary objective is to demonstrate whether the CA predictions on ASS dynamics can be consistent with ODEs. First, we describe the ODEs on ASS based on steady-state. Second, CA is structured by developing a set of adequate transition rules. Then CA model is verified in model validation by comparing with experimental data published by Boudrant et al. [3]. Finally, some advantages and future improvements of CA model are drawn in discussion.

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Dynamical Models of Activated Sludge System

2. Model Descriptions The ASS consisting of aeration tank and secondary settler mainly considers biological, chemical and physical processes. Here the reaction in settler is ignored, and assume settler is mainly used to concentrate the sludge in order to recycle into aeration tank. The mathematical and CA models are outlined as follows.


2.1. The ODEs model A recycle control based on a Monod kinetics process has been derived for ASS dynamics (see Fig. 1). Upon chemostat principles, the equations are listed below:  µXV dS   = FS 0 + rFS r − (1 + r)FS − , V dt YX/S (2.1)   V dX = rFX r − (1 + r)FX + µXV , dt where X is the biomass of activated sludge (kg/m3 ), S is the substrate concentration of wastewater (kg/m3 ) and V is the volume of aeration tank (m3 ). S0 is the initial substrate concentration (kg/m3 ), F denotes the volumetric wastewater flow rate (m3 /h), YX/S is the yield coefficients (kg/kg), r is the sludge recycle ratio, Sr represents the recycling substrate concentration (kg/m3 ), Xr is recycling biomass concentration of sludge (kg/m3 ) and µ denotes the specific growth rate of biomass (h−1 ). Suppose substrate is not separated from the settler, so Sr = S. Define dilution rate D = F/V (h−1 ) and Xr = g · X, where g (g >1) represents concentration coefficient of biomass in secondary settler. Monod equation is modified by considering the inactivation of activated sludge due to self-autolysis µ=

µmax S − Kd , KS + S

(2.2)

Fig. 1. The activated sludge system is equipped with aeration tank and secondary settler in continuous operation. 1250048-3


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where Kd is the death coefficient (h−1 ), KS is the kinetic coefficient (kg/m3 ), and µmax is the maximal specific growth rate of biomass (h−1 ). Therefore, the Eq. (2.1) can be rewritten as  dS µX   ,  dt = D(S0 − S) − Y X/S (2.3)    dX = [(rg − 1 − r)D + µ]X, dt where (1 + r − rg) < 1 for r > 0 and g > 1. Letting dX/dt = 0, we have D = µ/[1 + r · (1 − g)] > µ in the feedback process, which implies r and g are constrained by each other, and holds −1 < r · (1 − g) < 0. The qualitative and stability analysis for ODEs of ASS similar to Zhao and Chen [36] indicate that the ODEs are globally asymptotically stable in positive equilibrium point. 2.2. Cellular Automaton model The CA system is defined as A = {t, cells, cellspace, neighborhoods, rules} in a square lattice with L × L (Fig. 2) [34], where L is the length of the lattice, A represents the CA system, t is simulation time step, cells is the basic element of CA, cellspace is a set of all cells, neighborhoods represents a set of neighboring cells for a target site, and rules is the transition function of cell state. Moore neighborhoods are taken into account that each cell has eight neighbors (see Fig. 2). Further, each cell state can be influenced by both its own state and neighboring states [32]. Here we consider three cell-states Rij (t) ∈ {0, 1, 2}, where Rij (t) represents the cell

Fig. 2.

The schematic illustration on the structure of cellular automaton in views of Wolfram [34]. 1250048-4


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state in the position cell(i, j) at t. cell(i, j) is empty when Rij (t) = 0, cell(i, j) is occupied by small activated sludge particle if Rij (t) = 1, and Rij (t) = 2 represents cell occupied by substrate. The state of a target cell in next time step can be determined by its surrounding neighbors according to transition rules. For simplicity, let nk be the number of neighboring cells whose state is k (k = 0, 1, 2). Activated sludge can randomly choose one of neighboring cells with substrate and replace its position by offspring, and the parental activated sludge still stays in its original site. In CA system, suppose substrate could be degraded immediately if it is absorbed by activated sludge, and such process would be completed within one time step. Based on above design, the following rules are listed below. Rule 1:

If

Rij (t) = 1,

P10 ≤ Pthreshold ,

then Rij (t + 1) = 0,

where Pk1 k2 is the probability of cell state transiting from k1 to k2 (k1 , k2 ∈ {0, 1, 2}). For example, P10 is the probability that a cell occupied by activated sludge becomes empty. Pthreshold is the death probabilistic threshold, determined by neighboring competition. Define Pthreshold = eσ(n1 −8) that the inactive probability of activated sludge can be exponentially increased with increasing n1 (0 ≤ Pthreshold ≤ 1). Given n1 =

8

XΩ (t)/Xcell,

Ω=1

where XΩ (t) represents the biomass in each neighbor at t, and Xcell denotes the biomass per cell when a target cell is at R = 1. σ is the coefficient closely relevant to the size of the lattice (L2 ) and here σ = 1, which means the activated sludge would become inactive due to self-autolysis if n1 = 8, so the death of activated sludge depends on density dependence for space and resource competition with neighboring activated sludge. Rule 2: If

Rij (t) = 1, 1 ≤ n2 ≤ 8

and Psurvival ≤ (1 − Pthreshold ) ,

then Rij (t + 1) = 1,

where Psurvival is the survival probability of activated sludge, and 1−Pthreshold is the survival threshold that is applicable for the reproductive activated sludge. In such case, if substrate only exists one of neighbors, activated sludge would adsorb and degrade this substrate firstly to survive from the neighboring competition. Simultaneously, if activated sludge is in reproductive state, it may reproduce offspring in the next time step. Especially, suppose the event of offspring being reproductive state needs four time steps upon cell growth cycle. 1250048-5


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Rule 3:

If

Rij (t) = 0, P01 ≤ PX(t) ,

then Rij (t + 1) = 1,

where PX(t) is the probability of recycling activated sludge distributing on empty cell determined by Xr and the number of empty cells Nempty at t, so PX(t) = r · D · Xr /(Nempty · Xcell ). Consider recycling activated sludge randomly distributes on each empty cell with equal probability. Rule 4:


If

Rij (t) = 0, P02 ≤ PS(t) ,

then Rij (t + 1) = 2.

In the continuous operation, each cell can be occupied by inflow substrate with the equal probability. The threshold PS(t) depends on Nempty and inflow S0 , so PS(t) = D · S0 /(Scell · Nempty ), where Scell represents substrate concentration per cell when R = 2. Here Scell and Xcell are determined by the size of cell designed in CA system. In other cases, when activated sludge is in non-reproductive state, activated sludge would choose and degrade one of the neighboring substrates randomly, and replace its position. In addition, activated sludge can immigrate randomly into one of neighboring empty cells in order to search for substrate in next time step if n2 (t) = 0. Initially, each case is simulated with 50 replicates. Substrate and activated sludge are distributed randomly on the square lattice which is a torus edge that cells on the right edge are neighbors of those on the left and cells on the bottom are neighbors of those on the top to avoid any spurious edge effects. 3. Results We suppose an activated sludge system of V = 30 L, F = 9 l/h and S0 = 50 mgCOD/l. The other parameter values are given based on the conventional activated sludge process [8, 26] (see Table 1). Table 1. Parameters for numerical simulation in ASS. Parameter KS Kd µmax r g YX/S Xcell Scell

Value

Unit

30 0.3 0.8525 0.5 2.65 0.67 0.035 0.01

mgCOD/l mgCOD/l h−1 — — gCOD/gCOD mgCOD/l mgCOD/l

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(a)

(b)

(c) Fig. 3. Dynamic responses of substrate and biomass are performed by ODEs and CA model (average on 50 simulation replicates). The dynamics under the different initial concentrations of S(t = 0) and X(t = 0) are shown (other parameters are shown in Table 1).

The result in Fig. 3 shows that the initial concentration cannot affect dynamics in steady-state, because ASS has the stably positive equilibrium point [36]. Additionally, the trend of CA dynamics is in agreement with ODEs, which indicates the transition rules of CA are suitable though some slight fluctuations exist in CA. On the other hand, in order to exam whether the recycling parameters can affect substrate concentration and biomass at equilibrium state, one may observe (Fig. 4) that biomass and substrate in CA are almost equal to ODEs, but different recycling parameters can lead to different equilibrium concentrations. Generally, with increasing r · g, the result of increasing biomass but decreasing substrate suggests that ASS is similar to the prey–predator system. The dilution rate D = F/V strongly influences dynamics at equilibrium state because increasing biomass leads to substrate decline as D increases (Fig. 5). Alternatively, this shows that more substrate entering ASS with an increment of D provides more resource for activated sludge. Moreover, the fact of CA predictions similar to ODEs further demonstrates that CA design plus its concentration fluctuation can capture the actual phenomenon in complicated ASS. In general, applying 1250048-7


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(a)

(b)

(c) Fig. 4. Dynamic responses of substrate and biomass to different recycling parameters are shown by both ODEs and CA models (average on 50 simulation replicates). Parameters: S(0) = 20 mgCOD/l and X(0) = 15 mgCOD/l (others are shown in Table 1).

CA simulation can reveal the actual properties of ASS, such as slight fluctuation in equilibrium concentration, complexity and non-linearity of ASS.

4. Model Validation In order to test whether CA simulation fits the actual experiment, it is applied on both batch operation and continuous operation based on the former empirical data described by Boudrant et al. [3] (Table 2). Based on the experimental data, reactor worked in the batch operation during 0 < t < 15 h without recycling, so Eq. (2.3) can be rewritten as  X dS µmax S   = − − K  d ,  dt YX/S (KS + S) (4.1)  µmax S dX   = − Kd X.  dt (KS + S) For the second period between 15 h < t < 62 h, reactor operated with complete cell recycling, which means (1 + r) · F · X = r · F · Xr based on g · X = Xr , thus we 1250048-8


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(a)

(b)

(c) Fig. 5. The dynamic responses of substrate and biomass in different dilution rate are shown by ODEs and CA model (average on 50 simulation replicates). Here S(0) = 60 mgCOD/l and X(0) = 10 mgCOD/l (others are shown in Table 1).

obtain 1 + r = g · r. The ODEs can be improved according to Eq. (2.1)  dS µX   = D(S0 − S) − ,  dt Y X/S µmax S dX   = − Kd X.  dt KS + S

(4.2)

As shown in Fig. 6, the spatial patterns of activated sludge and substrate are generated by simulation upon six time intervals, where the equilibrium distributions in CA system are shown at t = 40, 50 h. The substrate is markedly distributed at different time step, which provides useful information for further manipulation. The biomass dynamics, on the other hand, seem to be closely related to the spatial positions and competition with neighbors. The numerical simulation is also conducted by using CA and ODEs (Fig. 7), one can observe that the results of ODEs fit the experimental data accurately at t = 0–20 h, but a great deviation exists when t > 20 h. This is due to that the high biomass concentration can lead to strong mechanical stress for spatial crowding effect in this extreme case. However, since the spatial interaction (e.g. competition) for activated sludge and substrate are considered in CA model, the results shows 1250048-9


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J. Liao et al. Table 2. Parameters for numerical simulation in the actual ASS system.


Parameter V S0 D KS Kd µmax r g YX/S Xcell Scell

Value

Unit

5 100 0.6 0.12 0.12 0.46 0.5 2.65 0.17 0.035 0.01

L g/l h−1 g/l g/l h−1 — — g/g g/l g/l

(a)

(b)

(c)

(d)

Fig. 6. The spatio-temporal patterns of biomass and substrate in CA system are revealed at t = 0, 10, 20, 30, 40, 50 h, respectively. In particular, the distributions at t = 40, 50 h are the steady-state of ASS. The gray color represents substrate, and the green ball is activated sludge particle (others are shown in Table 2). 1250048-10


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(e)

(f) Fig. 6.

(Continued )

Fig. 7. The results of ODEs, CA model (average on 50 simulation replicates) and improved ODEs are shown by comparing with experimental data. The dot line is the numeric results simulated by improved ODEs, the dash line is simulated by ODEs and dark solid line is given by CA model. Experimental data consists of biomass (◦) and substrate () (others are seen in Table 2).

that the simulation is accurate and able to reveal the experimental data though it is not perfect fit in the batch operation. Thus CA model is more accurate and realistic than ODEs in this case. Based on the description in experiment [3], we improve the ODEs by considering high concentration of biomass and mechanical stress of activated sludge particles during cell recycling, excluding the cell energy maintenance µ=

1 µmax S · − Kd , 2 KS + S + (S /KIS ) 1 + (P/KIP )n 1250048-11

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where KIS and KIP (g/L) are the kinetic coefficients, and n is an inhibition coefficient (dimensionless), P is the products concentration of activated sludge (g/l), and that dP/dt = YP/S · µ · X − D · P , where YP/S is the product yield coefficients (g/g). The new parameter values are defined as n = 3.83, KIS = 904 g/l, KIP = 65 g/l and YP/S = 0.89 g/g. The improved ODEs are slight better but more complicated than before in the continuous operation (see Fig. 7). Furthermore, due to its average on its fluctuation dynamics, CA model still matches the experimental data better than improved ODEs. In fact, both of CA and improved ODEs have minor errors which may largely attribute to system uncertainties in this study, especially for the kinetic data. 5. Discussion and Conclusion Stochastic CA model consisting of so many cells is considered as a very powerful technique [14, 33–35], and the cells in a two-dimensional computational lattice can represent different state, such as empty, and occupied by substrate or activated sludge. These cell states can transit from one state to another in CA system. If one cell interacts with neighbors, an appropriate transition function takes place with a specified probability. The key feature of CA model is that the neighboring cells can influence the target site-state, which plays a significant role in evolution rules. Here we only consider the local interaction and spatial competition of activated sludge, which allow the entire activated sludge population to choose an optimal way to react with surroundings. Compared to ODEs, the advantage of CA approach is that very detailed and complex spatial structure of ASS can be performed upon its simplicity, as well as its ability to allow for more granularity of the simulation. This advantage is hard to be realized by ODEs owing to its mean-field assumption that each site is equally likely to contact any other individuals without local interaction and spatial effect. Simultaneously, since CA highlights the complexity of interactions and competition, some local details and position of substrate and activated sludge can obtain, as well as the over-all information of ASS. Actually, define the inactive probability of activated sludge exponentially increases with an increment of neighboring biomass, CA model integrates the influence of density dependence that is also considered in the improved ODEs. Especially, the results in CA can reflect the characteristic of fluctuation, which may not exhibit strictly periodicity and nonlinearity with several stochastic variables. This fluctuation is mainly shaped by system stochasticity, neighboring interaction and activated sludge random movement, together with the concentration fluctuation of recycling biomass. Contrarily, the mathematical complexity of ODEs is often a drawback for the application and even understanding of detailed models [15], because it is difficult to identify the different states variables, such as those proposed by Wanner and Gujer [29] and Picioreanu et al. [27]. In addition, the mean-field ODEs ignoring the local interaction cannot reflect the fluctuations and strong non-linearity in actual ASS. In 1250048-12


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general, the advantages of CA can be attributed to the adequate transition rules, such as activated sludge particles mobility toward resource and cell-to-cell interaction. Thus CA can provide different spatio-temporal evolutive scales for dynamics. However, the deep understanding on ODEs is also necessary to structure the correct transition rules, and otherwise CA might be meanless if without qualitative results generated from the mathematical model [21]. CA model still has a long way to be applied in actual ASS, because too many factors are not taken into account, including temperature, pH, the size of activated sludge particles, partial pressures of oxygen and carbon dioxide, and various stresses (physical, environmental and chemical, etc.). To continue this effort, the more detailed factors and improvements should be considered into CA model to make it better describe and predict the treatment progressing. Once CA is tested with more actual experiments, it might be applied to evaluate different treatment systems. However, the CA leads to a new and flexible way of modeling ASS, thus it should have the potential to be applied to other similar systems. Acknowledgments This research was supported by the National Natural Science Foundation of China (No. 30870397) and the State Key Laboratory of Vegetation and Environmental Change. References [1] C. E. Adams, W. W. Eckenfelder and C. H. Joseph, A kinetic model for design of completely-mixed activated sludge treating variable-strength industrial wastewaters, Water Res. 9 (1975) 37–42. [2] E. Ben-Jacob, O. Schochet, A. Tenenbaum, I. Cohen, A. Czirok and V. Tamas, Generic modeling of cooperative growth patterns in bacterial colonies, Nature 368 (1994) 46–49. [3] J. Boudrant, N. V. Menshutina, A. V. Skorohodov, E. V. Guseva and M. Fick, Mathematical modelling of cell suspension in high cell density conditions Application to L-lactic acid fermentation using Lactobacillus casei in membrane bioreactor, Process Biochem. 40 (2005) 1641–1647. [4] M. Cote, B. P. A. Grandjean, P. Lessard and J. Thibault, Dynamic modeling of the activated sludge process: Improving prediction using neural networks, Water Res. 29 (1995) 995–1004. [5] P. L. Dold, G. A. Ekama and G. V. R. Marais, A general model for the activate sludge process, Progr. Wat. Tech. 12 (1980) 47–77. [6] G. A. Ekama, J. L. Barnard, F. W. G¨ unthert, P. Krebs, J. A. McCorquodale, D. S. Parker and E. J. Wahlberg, Secondary settling tanks: Theory, modeling, design and operation, IWA Scientific and Technical Report No. 6, IWA, London (1997). [7] G. B. Ermentrout, CA approaches to biological modeling, J. Theor. Biol. 160 (1993) 97–133. [8] C. P. L. Grady and H. C. Lim, Biological Wastewater Treatment: Theory and Application (Marcel Dekker, New York, 1980). [9] W. Gujer, M. Henze, T. Mino and M. C. M. Van Loosdrecht, Activated sludge model No. 3, Water Sci. Technol. 39 (1999) 183–193. 1250048-13


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