WM08
Proceedings of the 36th Conference on Decision & Control San Diego, California USA * December 1997
1:50
ptimal time control of Sequencing Batch Reactors for industrial wastewaters treatment Jaime Moreno Instituto de Ingenieria, UNAM. Ciudad Universitaria, Apdo. Postal 70-472, Coyoach, 04510 Mkxico, D.F., Mkxico Email:
[email protected]. unam,mx Abstract
to the tank, Y the Yield coefficient (constant), and S,, the Substrat concentration in the input flow (constant).
In this work the optimal time control problem is solved for a nonlinear plant with one control input. The model describes a biological sequencing batch reactor, used in the treatment of industrial wastewater. The operation strategy for these plants, derived from the solution of the time-optimal control problem, reduces at a minimum the fill and reaction phases increasing greatly the efficiency of the plant.
Fan, the control variable, can only take values in a positive and closed interval Fan E [0, Fmax],Fmax > 0, and all three state variables ( X , S , V) have to be positive, i.e. X 2 0, S 2 0 and V 2 0. It will be assumed that p ( S ) is defined for S 2 0, p(0) = 0, is positive (i.e. p ( S ) > 0 for S > 0), bounded (i.e. p ( S ) 0 and for some positive constant M ) , and is once continuously differentiable.
The objective of such a reactor is to bring the concentration of the substrate in the tank S under a specified level Smin,while the volume is brought from VOto Vf (Vo5 Vf5 V,,,,). It is of course of interest to optimize the efficiency of the process, defined as the quantity of substrate treated per unit of time. This is equivalent to the minimization of T,, the cycle time.
1 Introduction and problem formulation
Activated sludge is an aerobic biological process in which wastewater is mixed with a suspension of microorganisms to assimilate pollutants and is then settled to separate the treated effluent. In the treatment of industrial wastewaters by the activated sludge process is common the use of Sequencing Batch Reactors (SBR), in which all treatment takes place in a single reactor with different phases separated in time. The cycle in a typical SBR is divided into five discrete time periods: Fill, React, Settle, Draw, and Idle.
Therefore it is sought a state feedback control law for the input variable F,, that brings the system from a given initial state zo to a final state zf in minimal time.
A controllability analysis of the plant shows that the plant (1) is not controllable, i.e. not every pair of points can be connected through a path. For the optimixation So, Vo] and problem to be well posed initial zo = [Xo, Sf,Vf]have to lie on a well definal states zf = [Xf, fined surface. Therefore the controllable part of the plant can be modeled by a bidimensional system.
A reduction of the cycle time of the SBR increases the quantity of water that can be treated by the process. As the settle, draw and idle phases are usually of fixed time or not controllable by the operator, the cycle time of the SBR can be only reduced if the fill and reaction times can be decreased. A simplified model of these phases (1) can be easily obtained using mass balance equations [I]:
s V
=
- - p1( S ) X
Y = Fin ,
Fin (San - S ) +V
This is a typical time optimal control, which can be solved recalling the Maximum Principle of Pontryagin [4] or, as done in [3] and due to the reducibility of the model to a plane system, with a method based on Green’s theorem originally proposed by Miele [2].
2 Time optimal strategy for the biological
(1)
reactor The solution to the time optimal problem for the biological reactor (1) depends on the form of the specific grow rate p ( S ) . Two important and usual descriptions of this function and the corresponding solutions of the
where X is the Biomass concentration, S the Substrate concentration, and V the Volume of water in the tank; p ( S ) is the Specific grow rate, Fin the Input water flow 0-7803-3970-8197 $10.00 0 1997 IEEE
826
optimal problem will be considered. A typical expression for both is given by the equation
and the reaction will be finished (S I S m i n ) .
The control law (3) is Bang-Bang, as is typical for time optimal problems, while (4) is not, because of the singular arc.
where K , [mgll] is the Affinity constant, K i [mgll] is the Inhibition constant and P O [ h - l ] is the maximum specific grow rate.
3 Example of application to an industrial
wastewater treatment plant
2.1 Monotonic ,u (S)(Monod law) In this case p (S) is characterized by the fact that $$ > 0 for every S 2 0. A typical expression is the Monod law given by (2), with Ki = 00. This model is appropriate when the substrate doesn’t inhibit the activity of the biomass.
Some simulations with parameters of a realistic wastewater treatment plant to degrade phenol [3] show that the total duration of the fill and reaction phases with the optimal strategy proposed in this paper is around half of the time needed with a Batch reactor. In a Batch reactor the tank is filled as fast as possible (with Fmax) and during the reaction the volume is constant.
Theorem 1 L e t ,u (S)be strictly m o n o t o n e increasing, i.e. $$$ > 0 f o r e v e r y S 2 0 . I f SO 5 S,, t h e n the tame optimal problem f o r t h e s y s t e m (1) will be solved by the feedback control law
Obviously there is an advantage in using the optimal strategy to run the plant. The complexity in the measurement of the state variables and in the equipment to implant the feedback control law is the pay off for the better performance of the plant. This is really a challenge because the difficulty in the on line measurement of the substrate concentration S and of the biomass concentration X . Somle good results have been obtained using a non linear observer to estimate these variables. This will be rteported elsewhere.
(3) ana! t h e reaction will be finished i f (V = V f ) A (S 5 Smin),
2.2 Haldane-type u , (S) The prototype of this class is the Haldane law, which is described by the equation (2). The maximum value of the specific grow rate p * for the substrate concentration S * is characteristic for the Haldane law.
Acknowledgment: Work financed by DGAPA, UNAM under the Project IN500396.
This type of specific grow rates are typical for processes where the substrate is a toxic substance and, for big concentrations, inhibits the activity of the biomass. This is the case in the treatment of industrial waste water.
References
Theorem 2 L e t ,u ( S ) be a Haldane-type f u n c t i o n , i.e. it i s m o n o t o n i c a l l y increasing (*> 0) u p t o the p o i n t S * , where the m a x i m u m value:* is reached, and f o r S > S* the f u n c t i o n is monotonically decreasing < 0). If So 5 Si, and Smin5 S * < Si, t h e n the t i m e optimal control problem f o r the s y s t e m ( 1 ) will be solved by the feedback control law
(g
Fmax), Fmax
7
if (V = v,,az)v ( S > S * ) if ( S = S * ) A (V < V f ) Z.f(S
