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An Optimal Operating Strategy forFixed-Bed Bioreactors Used inWastewater Treatment

C. Benthack,* B. Srinivasan, D. Bonvin

Institut dAutomatique, Ecole Polytechnique Federale de Lausanne,

Lausanne, Switzerland

Received 3 July 1999; accepted 29 June 2000

Abstract:Optimization of a fixed-bed bioreactor used inwastewater treatment is addressed. The objective of op-timization is to maximize the treatment efficiency of thebiofilter by manipulating the feed flow rate while satis-fying operational constraints. Numerical results indicatethat the optimal input is characterized as being on theboundary of the admissible region. Thus, the character-ized optimal solution is implemented using a simplefeedback control law, which provides the optimal inputprofile despite variations in substrate inlet concentration

and biomass growth rate. 2001 John Wiley & Sons, Inc.Biotechnol Bioeng72: 3440, 2001.

Keywords:fixed-bed bioreactors; wastewater treatment;optimal control; numerical optimization; dynamic optimi-zation

INTRODUCTION

A typical wastewater treatment plant consists of primarytreatment, in which suspended particles are removed fromthe wastewater by mechanical operations such as screeningand sedimentation, and secondary treatment, where, in gen-eral, dissolved carbon- and nitrogen-containing wastewatercomponents are removed by microbial activity. In someplants, a tertiary treatment step is added to achieve betterpurification results (e.g., for the removal of phosphorus-containing components by microorganisms). Thus, biologi-cal processes may be employed in secondary and/or tertiarytreatment.

The two main types of bioreactors utilized in biologicalwastewater treatment are activated sludge processes andfixed-bed bioreactors. While in traditional activated sludgetreatment microorganisms are suspended in the liquid, theyare fixed on a stationary support in fixed-bed bioreactors.The oldest form of fixed-bed bioreactors are the so-calledtrickling filters that have been utilized since the nineteenthcentury (Tchobanoglous and Schroeder, 1985). But, it has

only been during the last few decades that fixed-bed reac-tors have gained interest, compared with the activatedsludge process, due to their smaller reactor size, improvedremoval efficiency, reduced odor annoyance, and robust-ness toward hydrodynamic variations and toxic shocks in

the inlet concentration. Another main feature is the filtrationof suspended particles that enables operation of the unitwithout a downstream clarifier, which constitutes an intrin-sic part of the activated sludge process. As a result, fixed-bed bioreactors have emerged as an alternative to traditionalactivated sludge secondary treatment and as a complemen-tary tertiary treatment step after the activated sludge process(Pujol et al., 1992, 1993).

Numerous approaches exist for modeling biofilm pro-cesses (Chaudry and Beg, 1998; Jacob, 1994). On the otherhand, much work has been dedicated to the modeling andoptimization of activated sludge treatment, resulting in thegeneral and well-accepted IAWRQ model, (Henze et al.,1987). The IAWRQ model is a kinetic model that comprisesaerobic and anoxic growth of heterotrophic and autotrophicbiomass, decay of biomass, ammonification, and hydrolysisof entrapped particulate organic matter and nitrogen. Themodel presented here for fixed-bed bioreactors uses onlyone part of the IAWRQ model (i.e., the aerobic growth ofheterotrophic biomass). In addition, the necessary transportphenomenon of the fixed-bed structure is described by par-

tial differential equations. Such a simple model has beenverified experimentally by Samb et al. (1996a).

The main contribution of this study is with regard tooptimization, where very little work has been done. Fixed-bed bioreactors are often operated with a constant feed flowrate at the available inlet concentration. That is, the operat-ing conditions are not adjusted according to the varyingtreatment potential of the biomass present in the reactor.The operating conditions are often chosen in a very conser-vative manner so as not to violate the quality requirementsat the reactor outlet for the worst-case scenario. In thiswork, the feed flow rate is considered the manipulated vari-able for the purpose of optimizing the treatment efficiency

of a fixed-bed bioreactor.In general, such an optimization problem involves the

implementation of time-varying input profiles, which haveto be calculated using computationally expensive numericalalgorithms (Edgar and Himmelblau, 1988). Also, the inputprofile changes with the growth rate of the biomass and thesubstrate inlet concentration, which can vary considerably.However, in this work, because optimal input is character-ized as being on the boundary of the admissible region, a

Correspondence to:C. Benthack*Present address:BASF AG, DWF/AM-L440, Ludwigshafen D-67056,

Germany. Telephone: +49-621-60-79569; fax: +49-621-60-55647; [email protected]

2001 John Wiley & Sons, Inc.


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simple feedback strategy is used to keep the bioreactor onthe boundary. This, on the one hand, avoids computing theoptimum numerically and, on the other hand, provides theoptimal input profile even in the presence of variations inthe substrate inlet concentration and biomass growth rate.

MODELING

This study considers an aerobic biofilter operated as a co-

current ascending column. The flow direction examined hasthe advantage of reducing possible odor problems. The bio-filter is employed for the removal of carbonaceous substratedue to the action of microorganisms that grow in a biofilmon the support that makes up the fixed bed. This support canbe made of different materials and different forms. Here, agranular support of expanded clay is employed. Figure 1provides a schematic view of the biofilter.

It is not possible to operate this unit continuously, be-cause the biofilm growth reduces the reactor volume avail-able for liquid and air flows and, eventually, clogs the filter.Thus, before clogging occurs, the operation has to bestopped and the reactor backwashed. In practice, it is back-washed regularly (i.e., every 24 or 48 h depending on thesetup). Therefore, this process has no steady state due to theincreasing biofilm thickness that reduces the bed void frac-tion and, thus, influences the flow velocities of both theliquid and gas phases.

A very simplified macroscopic model of the aerobicfixed-bed bioreactor is considered, because, in the contextof characterizing the optimal strategy for the bioreactor,such a simple model is considered sufficient. The processvariables of interest are: (i) the concentration of the sub-strate present in the wastewater; and (ii) the immobilizedbiomass concentration. Here, only a single type of substrate,

S, and one biomass population, B, are considered. Exten-sions to a more complex model with several substrates and

different biomass populations (e.g., nitrifying and denitri-fying bacteria) can be envisaged to gain more biologicalinsight (Wanner and Gujer, 1986). With the assumption ofradial homogeneity and negligible axial dispersion, the re-actor model considers only axial convection. In addition,temperature effects are assumed to be negligible, becausepublished experimental data have indicated only a smallinfluence of practically occurring temperature variations(Rusten, 1984). A similar model has been verified experi-mentally by Samb et al. (1996a).

Model Equations

Based on material balances over infinitesimal volume ele-ments, a model expressed as a set of partial and ordinarydifferential equations has been developed:

S

t =

Q

Al

S

z

rB

YBl (1)B

t =rB

with the initial and boundary conditions:

Sz,t=0=0, Sz=0,t= SinBz,t=0= B0

(2)

where S is the substrate concentration in the bulk liquidusually expressed in units of COD (chemical oxygen de-mand),B is the fixed biomass concentration with respect toreactor volume,Sinis the inlet substrate concentration, Q isthe volumetric feed flow rate, and Ais the cross-sectionalarea of the reactor. l represents the bed void fraction oc-cupied by the liquid phase.rBrepresents the biomass growthrate, and YB is the biomass yield. Substrate is transportedwith the liquid flow along the column, and this convection

is modeled by the partial derivative term with respect to theaxial coordinate z.Biomass grows by the consummation of the substrate by

active microorganisms that grow in the biofilm. It is as-sumed that the microbial activity is significant only in thebiofilm, and thus any liquid-phase substrate removal is ne-glected. Biomass growth kinetics have been chosen in ageneral structure that is capable of representing the differentcases of substrate- and diffusion-limited kinetics:

rB= maxS

KS+ S

B

KB+ B (3)

wheremaxis the maximal growth rate, KSis the saturation

constant for substrate, and KBis the saturation constant ofactive biomass. The Monod kinetic term, S/KS+ S,repre-sents the substrate limitation of a single substrate, S.Masstransfer limitations inside the biofilm result in decreasingmicrobial activity with increasing biofilm depth (Grasmicket al., 1979). It has been reported that the active biofilmlength is on the order of 100 m (Harris and Hansford,1976; Skowlund and Kirmse, 1989). The last term, B/KB+B, represents these diffusional limitations on the biomassFigure 1. Schematic view of the fixed-bed bioreactor.

BENTHACK ET AL.: OPTIMAL OPERATING STRATEGY FOR FIXED-BED BIOREACTORS 35


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growth and was used instead of complex calculation of aneffectiveness factor. For Monod kinetics, similar expres-sions that approximate the numerical solutions have beensuggested (Kobayashi et al., 1976; Samb et al. 1996a; Voset al., 1990; Yamane, 1981; Yamane et al., 1981). The ap-proach presented here considers, in a very simple manner,the limited influence of biomass concentration on the bio-mass growth.

The volume fraction occupied by the biomass is deter-mined as:

b=B

(4)

where denotes the local density of the biofilm. The par-tition of free column volume between the gas and liquidphases and hence the liquid hold-up lcan be expressed byparameter:

l (1 )(0 b) (5)

where ois a material-dependent constant that denotes thebed void fraction before inoculation. depends on theReynolds numbers of the liquid and gas phase; that is, thehydrodynamic conditions determined by the gas and liquidvelocities (Samb, 1997, Samb et al., 1996b). The followingexpression for the partition coefficient is used in this study(Achwal and Stepanek 1976):

=1

1+20.45QA0.13g0.563

(6)

where (Q/A) and vg(expressed in meters per hour) are theliquid and gas superficial velocities, respectively.

Simulation Study

The dynamic model given earlier is discretized in spaceusing finite differences and integrated in time using theAdamsBashforth algorithm (Hirsch, 1988). Tests on nu-merical stability have indicated that 41 spatial discretizationpoints are sufficient. The parameter values utilized for thesimulation are given in Table I. The wastewater feed com-

position, Sin is assumed to be constant in this simulationstudy.

The simulation is performed for a complete operationcycle between two successive backwashings. A cycle isconsidered complete when:

maxzB(z,tf) = Bmax (7)

where tfis the final time at which the bioreactor has to be

backwashed, and Bmaxis the maximum allowable biomassconcentration throughout the reactor. In this work, Bmax 0.6 0, which means that 60% of the bed void fraction isoccupied by the biomass.

To illustrate conventional reactor operation, simulationsare performed with a constant value ofQ.Figures 2 and 3show typical concentration profiles along the column attime intervals of 5 h for Sand B, respectively. In this simu-lation study, the reactor operation cycle is 36.5 h.

The substrate concentration decreases with increasingaxial position in the column due to consumption by thebiomass. It can also be seen from Figure 2 that, at a givenaxial position,Sdecreases with time as the biomass presentin the reactor increases. In the beginning of the cycle, thebiomass is homogeneously distributed along the column.However, toward the end of the cycle, because more sub-strate is available at the bottom (entrance) than at the top ofthe reactor, the growth rate and the biomass concentrationare higher at the bottom.

The influence of the feed flow rate,Q,on the wastewatertreatment capability of the reactor is investigated in whatfollows. As a result of various simulation studies, the fol-lowing qualitative conclusions can be drawn: (i) a largerflow rate,Q,leads to a lower residence time, a concentrationprofile that is less steep along the reactor, and a higher

concentration of Sat the exit of the reactor, and (ii) theoperation time,tf, is almost insensitive to variations in feedflow rate. The time depends on the growth rate at the bottomof the reactor, which is almost independent ofQ.

Table I. Parameter values for the bioreactor model.

Symbol Value Units

A 1.0 m2

B0 0.67 kg/m3

Sin 0.4 kg (COD)/m3

max 0.18 kg/(m3h)

KS 0.16 kg (COD)/m3

KB 0.5 kg/m3

vg 40 m/hYB 0.4 kg/kg (COD)0 0.4

H 4.5 m 18.4 kg/m3 Figure 2. Substrate concentration profiles at time intervals of 5 h for

Q 3 m3/h, first profile at t 1 h.

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OPTIMAL OPERATION

In industrial (or municipal) practice, the biofilter is typically

operated with constant feed flow rates and an uncontrolledinlet concentration. However, this work investigates howbiofilter operation can be improved by dynamically adjust-ing the feed flow rate.

Problem Formulation

The performance criterion to be maximized is the treatmentefficiency defined as the amount of substrate removed perunit of cycle time. This can be expressed as the integral overtime of the difference in the amounts of substrate enteringand leaving the reactor divided by the cycle time. The op-timization problem can be formulated as follows:

maxQt,Sint

J=1

tf

0

tfQtSin Souttdt

s.t. S

t =F1

B

t =F2

maxz

Bz,tf= Bmax (8)

Soutt Slim

Qlb Qt Qub

where [Qlb, Qub] is the range of admissible values for thefeed flowrate, and Slim the maximum acceptable effluentconcentration. The limit, Slim, is prescribed by the legalrequirements for discharge of cleaned wastewater into theenvironment.F1, F2denote the right-hand sides of systemEq. (1).

This problem formulation reflects a possible economicobjective in that a time-efficient wastewater treatment,which guarantees a required effluent concentration quality,is sought. In other words, one seeks the type of operation

that achieves, with a given installation, the highest through-put. This performance criterion is independent of the spe-cific operational costs, because it was assumed that pump-ing the wastewater through the column does not vary sig-nificantly with the velocity as long as the flow rate is keptin realistic ranges. Other choices of optimization criteria arepossible and have been discussed elsewhere (Benthack etal., 1996).

Numerical Optimization

The optimal control problem [Eq. (8)] is first solved nu-merically using control vector parameterization (CVP)(Ray, 1989). The following lower and upper bounds arefixed for the numerical optimization: 1 m3/h Q 8 m3/h.The maximal outlet concentration is given by Slim 0.1 kg(COD)/m3. The input variable, Q(t), is parameterized usingpiecewise linear approximations on nelements. This resultsin a nonlinear programming problem in (n + 1) decisionvariables. At every iteration of the optimization algorithm,system Eq. (1) have to be integrated numerically. Sequential

quadratic programming (SQP) is used as the nonlinear pro-gramming method. It has been reported to be one of themost efficient algorithms for nonlinear optimization (Edgarand Himmelblau, 1988).

Because the problem is formulated as a free terminal timeproblem, the length of the simulation is not known a priori.This difficulty can be resolved as follows: (i) either thedynamic model is normalized with respect to time, and theterminal time becomes an additional decision variable; or(ii) the simulation terminal time is chosen as an upper boundon any possible terminal time. The latter approach has thedisadvantage that the problem can become numerically ill-conditioned because the last element might not contribute to

the objective function. Nevertheless, the second approachwas used here, and the problem of ill-conditioning was cir-cumvented by imposing that the last element be so long thatit will always start before the operation cycle is over.

The optimal input found by this method is shown in Fig-ure 4. The feed flow rate increases with time, which islogical as there is more biomass inside the reactor and hencethe treatment capacity increases. The substrate outlet con-centration is near its upper limit, Slim(Fig. 5).

Characterization

The numerical optimization results have been validated ana-lytically (Benthack, 1997) and can be summarized as fol-lows:

For the range of parameters used in this study, the optimal

solution is determined by the constraints: During the fill-

ing phase, the feed flow rate should be low enough such

that the first effluent to come out does not exceed the

concentration limit. Later, the feed flow rate is such that

Figure 3. Biomass concentration profiles forQ 3 m3/h, first profile att 1 h.



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the outlet substrate concentration is at its upper limit,

Slim.

The complete operation cycle can be divided in two opera-tional phases. These are: (i) the filling phase; and (ii) theoperation at Sout(t) Slim. During the filling phase, a sub-strate outlet concentration cannot be measured because noliquid has yet reached the reactor outlet. The feed flow rateis at a low value, so that the first liquid portion that leavesthe reactor will not exceed the upper limit, Slim.

In a typical dynamic optimization problem, the optimalinput is time-varying, which is also true in this example, dueto different biomass concentrations in the reactor. What is ofinterest is that the optimum is characterized by being on thepath constraint Sout Slim. This is the case despite varia-tions in substrate inlet concentration and biomass growthrate. Thus, the dynamic optimization problem can be trans-formed into a tracking problem, which can be implementedusing feedback. However, toward the end of the batch, itmay also be that the biomass concentration inside the reac-tor is so high that, even withQ(t) Qub, Sout(t)


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of the bioreactor. It is well known that PI controllers mighthave problems handling systems with significant delays(strom and Hagglund, 1995).

CONCLUSIONS

The dynamic model of an aerobic fixed-bed bioreactorbased on material balances for substrate and biomass wasdeveloped. Its main characteristics were analyzed throughsimulation. The dynamic optimization of this reactor wasaddressed with the objective of maximizing the wastewatertreatment efficiency of the reactor, which is equivalent tomaximizing the global substrate removal rate over an op-eration cycle. The feed flow rate was considered as themanipulated variable. The optimal control problem was

solved numerically using control vector parameterization. Itwas found that the optimal feed flow rate has to be adjustedsuch that the required effluent quality is met precisely.

A feedback implementation of the optimal trajectorieshas been proposed. After the filling period, the path con-straint was tracked by an optimizing feedback loop. It wasshown that PI control is a very efficient way of implement-ing the optimal biofilter operation. During the filling periodthe reactor is operated in open-loop manner (e.g. using theoff-line-calculated optimal profile). To render the operationless sensitive to model mismatch, it may be of interest toexplore robust optimization methods by explicitly account-ing for model uncertainties in the calculation of the optimalfeed profile during the filling period.

The aeration rate was considered sufficiently high to en-sure aerobic conditions throughout the reactor during thewhole operation cycle, because anaerobic conditions wouldsignificantly deteriorate the substrate removal capacity ofthe bioreactor. On the other hand, if the objective were tominimize the operational costs, it would be of interest toallow manipulation of the aeration rate, which represents themost cost-intensive part of the operation, in order to obtaina compromise between good performance and low operat-ing costs. The first steps toward the solution of this optimi-zation problem would be to include the material balance of

dissolved oxygen in the bioreactor model and to accountexplicitly for its influence on the substrate removal rate.

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Figure 7. Input profile determined by PI control.

Figure 8. Substrate outlet concentration profile obtained by PI control.



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