transient analysis of a wastewater treatment biofilter

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Transient analysis of a wastewatertreatment biofilter distributedparameter modelling and stateestimationA. Vande Wouwer a , C. Renotte a , I. Queinnec b & Ph. Bogaerts ca Facultéé Polytechnique de Mons, Boulevard Dolez 31, 7000,Mons, Belgiumb LAAS-CNRS, Toulouse, Francec Universitéé Libre de Bruxelles, Belgium

Available online: 16 Feb 2007

To cite this article: A. Vande Wouwer, C. Renotte, I. Queinnec & Ph. Bogaerts (2006): Transientanalysis of a wastewater treatment biofilter distributed parameter modelling and state estimation,Mathematical and Computer Modelling of Dynamical Systems, 12:5, 423-440

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Transient analysis of a wastewater treatment

biofilter – distributed parameter modelling

and state estimation

A. VANDE WOUWER*{, C. RENOTTE{, I. QUEINNEC{,and PH. BOGAERTSx

{Faculte Polytechnique de Mons, Boulevard Dolez 31, 7000 Mons, Belgium{LAAS-CNRS, Toulouse, France

xUniversite Libre de Bruxelles, Belgium

This paper is concerned with a pilot-scale fixed-bed biofilter used for nitrogen removal frommunicipal wastewater. Process modelling yields a set of mass balance partial differentialequations describing the evolution of the component concentrations along the biofilter. Basedon sets of experimental data collected over several months, unknown model parameters areestimated by minimizing an output-error criterion. The resulting distributed parameter modeland a few pointwise measurements of nitrate, nitrite, and ethanol concentrations are then usedto design observers allowing the unmeasured biomass concentrations to be reconstructed on-line. First, it is demonstrated that asymptotic observers are not applicable to the given modelstructure. Then, a receding-horizon observer is designed and tested, showing a very satisfactoryperformance.

Keywords: Mathematical modelling; Identification; State estimation; Distributed parametersystems; Biotechnology

1. Introduction

Nitrogen removal is an important step in the treatment of municipal wastewater. In therecent past biofilter systems have received considerable attention; see, for instance, theconference proceedings and journals of the International Water Association (IWA) [1].The main advantages of these wastewater treatment systems are their ease of use,compactness, efficiency, and low energy consumption. New biofilters, including dual-column systems, have recently been proposed to achieve pre-treatment, nitrogen and/or phosphorus removal [2,3].

Based on experimental data collected from a pilot-scale fixed-bed biofilter, theobjective of this paper is to develop and validate a dynamic model allowing the

*Corresponding author. Email: [email protected]

Mathematical and Computer Modelling of Dynamical SystemsVol. 12, No. 5, October 2006, 423 – 440

Mathematical and Computer Modelling of Dynamical SystemsISSN 1387-3954 print/ISSN 1744-5051 online ª 2006 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/13873950600723335

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evolution of the several component concentration profiles to be reproduced. This modelcan be used for simulation purposes (e.g. for system analysis and design) or as a basis forthe development of a software sensor (to estimate unmeasured variables on-line).The modelling task involves the selection of an appropriate reaction scheme and

kinetics and the derivation of mass balance partial differential equations (PDEs). Theunknown model parameters are estimated by minimizing an output-error criterionmeasuring the deviation between the experimental signals and the model prediction.Particular attention is paid to the assumptions on the measurement errors, and thecorresponding formulation of an output-error criterion.This work builds upon a previous modelling study reported in Bourrel [4] and

Bourrel et al. [5], which was carried out using the same pilot plant. However, this studywas based on the assumption that steady-state operations were achieved after a fewhours, which, as we will see later, is not in agreement with experimental observations.As a matter of fact, the biofilter experiences very long transient phases due tovariations in the input flow rate and concentrations. Consequently, the parameterestimation problem was not properly formulated in Bourrel [4] and Bourrel et al. [5],leading to model discrepancies. Here, a different model structure is proposed, and acriterion taking the measurement errors into account is minimized in order to estimatethe unknown model parameters.Together with a few pointwise measurements of nitrate, nitrite and ethanol

concentrations, the biofilter model can be used to design distributed parameterobservers of the unmeasured biomass concentration profiles. Asymptotic observers [6],which do not rely on the knowledge of the kinetic model and which have goodconvergence properties in the case of continuous systems, would a priori be a veryappealing solution. However, they appear unsuitable for the model structure, andattention is therefore focused on receding-horizon observers [7,8], which allow the stateestimation problem for nonlinear distributed parameter systems to be solved in anelegant way.This paper is organized as follows. In the next section, the experimental setup is

described. Section 3 deals with biofilter modelling, i.e. the derivation of a reactionscheme, reaction kinetics, and a system of mass balance PDEs. The unknown modelparameters are estimated from experimental data, and the model adequacy is checkedin direct and cross-validation (i.e. using independent data sets). The uncertainty in theparameter estimates is assessed by evaluating the Fisher Information Matrix (FIM). Insection 4, distributed parameter asymptotic observers and receding-horizon observersfor the unmeasured biomass concentration profiles are examined. Finally, section 5summarizes the main conclusions.

2. Process description

The pilot plant under consideration (figure 1) is a biofilter packed with lava rock(pouzzolane). The biofilter is fed with a mixture of raw municipal wastewater and aconcentrated nitrate solution.Several biological reactions take place inside the biofilter, e.g. removal of soluble

carbon and removal of nitrate and nitrite. The denitrification process consists of severalconsecutive reactions of oxydo-reduction and implies a transient accumulation ofnitrite in the biofilter

NO�3 !ð1Þ

NO�2 !ð2Þ

N2:

424 A. Vande Wouwer et al.

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Oxydo-reduction is achieved thanks to an organic carbon source (as donor ofelectrons). In this case, ethanol is used.

Eight sampling points are evenly distributed along the reactor axis, which allow theseveral component concentration profiles (ethanol, nitrate and nitrite) to be measured.The manipulated variables are the feed flow rate F(t) to the biofilter and the inletethanol concentration SC,in(t).

The experiments accomplished with the pilot plant aimed at sweeping the range ofoperating conditions (various C/N ratios and feed-flow rates) observed in a full-scalewastewater treatment plant located in Montargis, France. The experiments werecarried out at the Institut National des Sciences Appliquees de Toulouse (INSAT),France [9].

The biofilter start-up lasts about 3 weeks, which is the time necessary for thedevelopment of the colonizing biomass. After this period, an experimental protocol isfollowed in which the C/N ratio and F vary with time. Figure 2 shows the timeevolution of the nitrate and nitrite concentrations at the biofilter outlet, following threechanges in the feed flow rate F (the superficial velocity changes from 2 to 4 m/h, from 4to 6 m/h, and finally from 6 to 9 m/h). It is apparent that these changes in the feed flowrate are responsible for significant increases of the nitrite concentration at the biofilteroutlet (between two changes in the feed flow rate, outlet concentrations also vary to asmaller extend, in response to changes in the feed concentration, which are not detailedhere). We will attempt to explain this undesirable effect in the following section.

3. Model development

In this section, a system of mass-balance PDEs is derived, and the unknown modelparameters are estimated from experimental data.

Figure 1. Experimental setup.

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3.1 Reaction scheme

In contrast to the work of Bourrel [4] and Bourrel et al. [5], where a classicalreaction scheme based on the IAWQ model [10] is used, a macroscopic biolo-gical reaction scheme based on the concept of ‘pseudo-stoichiometry’ [6] is preferredhere:

n1carbonþn2a1

nitrate! biomassþ n2a2

nitrite ð1Þ

n3carbonþn4a2

nitrite! biomassþ n5nitrogen ð2Þ

active biomass! inactive biomass ð3Þ

where ni, i¼ 1, . . . , 5 are the pseudo-stoichiometric coefficients, and a1¼ 1.14, a2¼ 1.71are chemical oxygen demand (COD) conversion factors.Indeed, many biological reactions take place in the biofilter and the carbon source

could be used in other reaction pathways than the denitratation step (1) and thedenitritation step (2) (so that it is not possible to a priori relate n1 to n2 or n3 to n4through a balance on each elementary component). In addition, reaction (3)representing the biomass deactivation (biomass enclosed inside the lava rock and nolonger in contact with the bulk phase, biomass mortality, biomass washed out of thebiofilter) is introduced.Figure 3 shows simulation results (obtained with the model described in the next

sections), which illustrate the effect of a step change in the feed flow rate, resulting in astep change in the superficial velocity from 2 to 4 m/h. The increased feed flow movesthe nitrite concentration profile towards the reactor outlet, leading to larger outletnitrite concentrations as observed in figure 2. As the biomass has not completelydeveloped in the second half of the reactor yet, a relatively long time is required for theoutlet nitrite concentration to decrease (which corresponds to the time required forbiomass growth).

Figure 2. Time evolution of nitrate and nitrite concentrations at the biofilter outlet.


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3.2 Reaction kinetics

The specific growth rates are assumed to have the following form

m1 ¼ m1;max

SNO3

SNO3þ KNO3

SC

SC þ KC1

1

1þ Xa=KXa1

ð4Þ

m2 ¼ m2;max

SNO2

SNO2þ KNO2

SC

SC þ KC2

1

1þ Xa=KXa2

ð5Þ

m3 ¼ m3;max ð6Þ

where the limiting substrates are the carbon source (SC), nitrate (SNO3) and nitrite

(SNO2).

The two growth-associated reactions (1) and (2) are auto-catalyzed by the biomass(without biomass these reactions would not take place). However, only a fraction ofthe biomass, denoted Xa, takes part in the nitrogen removal process and colonizes theactive sites of the porous bed (the rest dies or is removed from the porous bed andwashed out). This is represented by inhibition factors in the specific growth rates (4)and (5), which limit the growth rates at high biomass concentration levels. Thedeactivation process is assumed to have first-order kinetics (the simplest possible modelin the absence of detailed knowledge about this process).

In the experiments considered in this study, the carbon source is always in excess sothat its limiting effect cannot be quantified and equations (4) and (5) reduce to

m1 ¼ m1;max

SNO3

SNO3þ KNO3

1

1þ Xa=KXa1

ð7Þ

m2 ¼ m2;max

SNO2

SNO2þ KNO2

1

1þ Xa=KXa2

: ð8Þ

Figure 3. Evolution of the nitrite concentration profiles following a step change in the feed flow rate (stepchange in the superficial velocity from 2 to 4 m/h).


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3.3 Mass balances

Based on this reaction scheme and these kinetics, it is straightforward to derive thefollowing mass balance PDEs.

@SNO3

@t¼ �v @SNO3

@z� n02a1

m1Xa ð9Þ

@SNO2

@t¼ �v @SNO2

@zþ n02

a2m1 �

n04a2

m2

� �Xa ð10Þ

@SC

@t¼ �v @SC

@z� n01m1 � n03m2� �

Xa ð11Þ

@Xa

@t¼ ðm1 þ m2 � m3ÞXa ð12Þ

where plug-flow conditions are assumed,

v ¼ F

eA

is the fluid flow velocity (A is the cross-section area of the biofilter and e is the bedporosity), and

n0i ¼1� ee

ni:

These equations are supplemented by boundary conditions corresponding to the inletconcentrations.

PDEs (9) – (12) are solved numerically using a standard method of lines procedure,i.e. finite differences with about 30 nodes.

3.4 Parameter estimation

Pseudo-stoichiometry and kinetics involve 11 unknown model parameters (ni, i¼ 1, . . . ,4, mi,max, i¼ 1, . . . , 3, KNO3

, KNO2, KXa1

and KXa2), whose numerical values have to be

inferred from experimental data.Based on the assumption of constant (but unknown) relative errors on the

measurement data, the following output-error criterion is defined

J ¼X15k¼1

X3i¼1

X8m¼1½lnðyi;mesðzm; tkÞÞ � lnðyi;modðzm; tkÞÞ�2 ð13Þ

where

. yi,mes(zm,tk) and yi,mod(zm,tk) denote the concentration measurements ofcomponent i (i.e. SC, SNO3

and SNO2) at position zm and time tk, and the model

prediction, respectively;


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. constant relative errors on the measurements yi,mes(zm,tk) are equivalent toconstant absolute errors on the logarithms of the measurements ln(yi,mes(zm,tk));

. 15 sample times, at each of which three component concentrations (SC, SNO3and

SNO2) in 8 different spatial locations (see figure 1) are measured, representing a

total of 360 data points;. measurement data correspond to different feed conditions (time-varying feed-

flow rate and concentrations);. the component concentrations SNO3

, SNO2and SC are scaled by a factor 25, 5 and

100, respectively.

The output-error criterion (13) is minimized with respect to the unknown modelparameters using a Levenberg-Marquardt algorithm. Positivity constraints on theparameters are imposed through a logarithmic transformation.

In addition to the 11 structural model parameters, some of the feed conditions arealso not well known. This is particularly true for the inlet nitrate concentration, whichdepends on the wastewater composition (on the other hand, there is no nitrite at thebiofilter inlet and the inlet concentration of ethanol can be adjusted by the operator), aswell as for the fluid flow velocity, which depends on the biofilter clogging. As aconsequence, 30 feed conditions (corresponding to the 15 sample times) must beestimated together with the 11 model parameters.

In order to check that the numerical minimization procedure is not sensitive to localminima, a multistart strategy is used (i.e. minimization is repeated with different initialguesses). After each minimization run, the numerical values of the criterion and theparameters are examined and the model prediction is compared to the measured signals(direct validation illustrated in figures 6 to 8 of the next section). At an earlier stage inthe selection of a model structure and parametrization, it was also particularlyimportant to check the time evolution of the (non-measurable) biomass profiles forreproducibility (i.e. an over-parametrized model can lead to good prediction of themeasured signals for different values of the model parameters, corresponding todifferent evolutions of the biomass profiles). With the proposed model structure (1) to(12), all these tests are successful, which confirms the existence of a set of ‘optimal’parameter estimates corresponding to the data at hand. For illustration pur-poses, figure 4 shows the minimization process as a function of the number ofiterations and figure 5 illustrates the evolution of the biomass concentration profile; aspredicted by the identified model.

3.5 Model validation

To check the validity of the identified model, three tests are performed:

. Direct validation: The model prediction is compared to the measured signals usedin the criterion definition (13); see, for instance, figures 6 to 8, which show thespatial concentration profiles of nitrate, nitrite and carbon source, at a particularsample time. The model agreement is very satisfactory.

. Cross-validation: The model prediction is compared to independent data sets,which have not been used in the parameter estimation procedure yet; see, forinstance, figures 9 to 11 which show the spatial concentration profiles of nitrate,nitrite and carbon source, at a particular sample time. The model agreement,although not as good as in direct validation, is quite acceptable.


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. Error estimation: The covariance of any unbiased estimator satisfies the Cramer-Rao inequality

P � F �1ðy�Þ ð14Þ

where F is the Fisher Information Matrix (FIM), which can be approximated[11] as

ðN� nyÞJðyÞ

XNj¼1

@ lnðymodðjÞÞ@y

@ lnðymodðjÞÞ@yT

ð15Þ

where N is the total number of data points and ny the number of unknown

parameters y (here, y* denotes the ‘true’ values of the model parameters and yan estimate).

Figure 4. Evolution of the output-error criterion (13) as a function of the number of minimization steps.

Figure 5. Evolution of the biomass concentration profiles as predicted by the identified model (the arrowindicates increasing times).


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The first factor in (15) corresponds to the inverse of the variance of the measurementerrors. In our case, the constant relative error on themeasurements can be estimated as 12%.

An estimate of the FIM can also be obtained as a by-product of the minimizationprocedure. Indeed, the Levenberg-Marquardt algorithm computes an approximationof the Hessian, whose value can be used at the optimum. Table 1 lists the parametervalues and the associated 99% confidence intervals, which are satisfactory and confirmthe validity of the identified model.

In addition, the examination of the off-diagonal coefficients of the FIM gives someinsight into the correlation between the several model parameters. As a result, some

Figure 6. Spatial nitrate concentration profile (direct validation: solid line, model prediction; circles,measured values).

Figure 7. Spatial nitrite concentration profile (direct validation: solid line, model prediction; circles, measuredvalues).


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parameters, e.g. m1,max and KXa1, appear as being significantly correlated. It is not

possible, however, to eliminate some of these parameters from the process modelwithout loss of physically important terms. More informative experimental data wouldbe required to alleviate this problem. Further experimental studies were however notpossible within the framework of this study.

4. Biomass reconstruction

As has become apparent from the modelling study, the long transient phases observed inreal-life operations can be explained by biomass growth and deactivation processes, i.e.

Figure 8. Spatial ethanol concentration profile (direct validation: solid line, model prediction; circles,measured values).

Figure 9. Spatial nitrate concentration profile (cross validation: solid line, model prediction; circles, measuredvalues).


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the global rate at which biomass develops determines the overall dynamics of thebiofilter. The uniformity of the biomass distribution inside the porous bed is also aprimary determinant of the biofilter performance faced with large variations of the feedconditions. For process monitoring, it would therefore be interesting to visualize thebiomass concentration profiles on-line. However, these profiles are difficult to measurein practice, and it is required to resort to state estimation techniques. With regard tothe modelling uncertainties, particularly of the reaction kinetics, it is appealing to designan asymptotic observer [6]. We will see, however, that asymptotic observers are

Figure 10. Spatial nitrite concentration profile (cross validation: solid line, model prediction; circles,measured values).

Figure 11. Spatial ethanol concentration profile (cross validation: solid line, model prediction; circles,measured values).


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unsuitable for the model structure considered in this study. It will therefore be necessaryto select another state estimation technique, e.g. receding-horizon observers [7,8], whichprovide a particularly elegant solution for nonlinear distributed parameter systems.

4.1 Asymptotic observers

The mass balance PDEs (9) to (12) can be reformulated in a more compact form asfollows.

@n

@t¼ �v @n

@zþ Ku ð16Þ

or

@

@t

nfns

� �¼ �v � @nf@z

0

� �þ Kf

Ks

� �� u ð17Þ

where v is the velocity vector, K is the pseudo-stoichiometry matrix, u¼ lXa is thereaction rate vector, and the state n¼ [SNO3

SNO2SC Xa]

T is decomposed into thecomponents in solution in the fluid phase nf¼ [SNO3

SNO2SC]

T, which can bemeasured on-line in a few locations along the reactor axis, and the componentanchored on the solid phase ns¼ [Xa], which is not measured.

Following [6,12], the procedure to develop an asymptotic observer is to partition n

into two subvectors na and nb, n¼ [na nb]T, such that the corresponding partition of the

pseudo-stoichiometry matrix K¼ [Ka Kb]T with

K ¼

�n2=a1 0 0n2=a2 �n4=a2 0�n1 �n3 01 1 �1

2664

3775 ð18Þ

is of full row rank. Here, rank (K)¼M¼ 3 (M is the number of independent reactionsin the reaction scheme), so that rank (Ka) should be equal to 3.

Table 1. Estimated parameters and errors.

Parameters Value Error

n0

1 50.95 + 0.96n0

2 13.9 + 0.13n0

3 7.23 + 0.10n0

4 5.03 + 0.07m1,max 0.179 + 0.007m2,max 0.277 + 0.002m3,max 0.0000316 + 0.0000030KNO3

6.25 + 0.23KNO2

0.135 + 0.003KXa1

47.5 + 2.0KXa2

22.6 + 0.4


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This condition excludes the commonly used partition into measured and nonmeasured components, i.e. na¼ nf and nb¼ ns, as rank (Ka)¼ 2 only. Hence, thefollowing partition is selected

na ¼xafxas

� �¼

SNO3

SNO2

Xa

24

35 and nb ¼ ½Sc� ð19Þ

which leads to a full row rank sub-matrix Ka.It is then possible to define a new state vector z by

z ¼ A0na þ nb ¼ A0f A0s

� nafnas

� �þ nb ð20Þ

where the matrix A0 is the unique solution of

A0Ka þ Kb ¼ 0: ð21Þ

The evolution of z is given by

@z

@t¼ A0f �

@naf@tþ A0s �

@nas@tþ @nb@t

: ð22Þ

Substituting the time derivatives by their expressions (17), the equations of theasymptotic observer, from which the reaction kinetics are eliminated, are obtained:

@z

@t¼ �v � A0f �

@naf@zþ @nb@z

� �ð23Þ

nas ¼ A�10s � z� A0f � naf � nb

� �: ð24Þ

For this observer to be completely defined, the inverse A�10s (which, in the particular

case under consideration, is a scalar) is required. This information can be obtained bysolving equation (21), which gives A0s¼ 0!

The asymptotic observer is therefore not applicable to the model structure (17) sincethe partition of the state vector leads either to a sub-matrix Ka which is not full rowrank or to a null-matrix A0s. The only way around would be to simplify the model andto abandon the equation describing the biomass growth and deactivation processes (aswas the case in the work of Bourrel [4] and Bourrel et al. [5]), which we know is notsatisfactory.

4.2 Receding-horizon observers

As the concentration measurements are rare and corrupted by noise, the concept of afull-horizon observer [8], which uses all the measurement information available up tothe current time, is extended to the distributed parameter model of the biofilter.

The prediction step (between samples tk5 t5 tkþ 1) corresponds to the solution ofthe model PDEs (16)

@n

@t¼ �v @x

@zþ KuðxÞ 0 � t < tkþ1 ð25Þ


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subject to initial conditions

nð0Þ ¼ n0=k ð26Þ

and boundary conditions corresponding to the inlet concentrations.The correction step (at sampling times) corresponds to the following optimization

problem:

n0=k ¼ Argminn0

Jkðn0Þ ð27Þ

with

Jkðx0Þ ¼1

2

Xkj¼1ðymesðtjÞ � ymodðtjÞÞ

TQðtjÞ�1ðymesðtjÞ � ymodðtjÞÞ ð28Þ

where ymes represents the vector of measurements (nitrate, nitrite and ethanolconcentrations in eight locations along the biofilter axis), ymod the correspondingmodel prediction, and Q the covariance matrix of the measurement errors.

In order to reduce the dimensionality of the optimization problem, equations (27)and (28), the vector of initial conditions n0=kis expressed as a set of exponential profiles,i.e.

fiðzÞ ¼ ai � exp ð�bi � zÞ ð29Þ

(i¼SNO3, SNO2

, SC, Xa) which leads to the on-line determination of eight parameters.The observer is first tested in simulation. The biofilter model is used to generate

simulation data, which are then corrupted by noise.Figures 12 to 14 compare the temporal evolution of the ‘real’ concentrations (i.e. the

concentrations generated by the simulation model) and estimated concentrations (i.e.

Figure 12. Temporal evolution of the real (dots) and estimated (solid lines) nitrate concentrations in the eightmeasurement locations, and concentration measurements (circles) together with their 99% confidenceintervals.


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the concentrations reconstructed by the receding-horizon observer) in the eightmeasurement locations distributed along the biofilter axis, as well as the measuredconcentrations (i.e. the ‘real’ concentrations corrupted by noise) together with their99% confidence intervals. Figure 15 compares the biomass estimates with their realvalues (which are not measured), whereas figure 16 illustrates the time evolution of thecorresponding spatial profiles.

Figure 13. Temporal evolution of the real (dots) and estimated (solid lines) nitrite concentrations in the eightmeasurement locations, and concentration measurements (circles) together with their 99% confidenceintervals.

Figure 14. Temporal evolution of the real (dots) and estimated (solid lines) ethanol concentrations in the eightmeasurement locations, and concentration measurements (circles) together with their 99% confidenceintervals.


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As it is apparent from figures 12 to 16, the performance of the full-horizon observeris very satisfactory. Experimental application confirms this observation, as depicted infigures 17 to 18, which show the initial nitrite concentration profiles (initial measuredprofile and initial exponential guess) and the same profiles after 404 hours. Theconvergence of the observer is satisfactory, despite the modelling errors and themeasurement noise. The observer performance cannot, however, be fully tested asbiomass measurements are not available.

Figure 15. Temporal evolution of the biomass concentration estimates (solid lines) and of the real, non-measured, concentrations (dots).

Figure 16. Temporal evolution of the biomass spatial profiles (solid lines) and of the real, non-measured,concentrations (dots).


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5. Conclusions

In this paper, a distributed parameter model of a fixed-bed biofilter used for nitrogenremoval in municipal wastewater treatment is derived. The unknown model parametersare estimated from experimental data collected over a period of several months. Directand cross-validation tests demonstrate the good model agreement, despite the relativelylarge errors corrupting some of the measurement samples. In addition, uncertaintiesassociated with the parameter estimates are assessed from an approximation of theFisher Information Matrix.

Figure 17. Initial measured profile (circles) and exponential initial condition (solid line) of nitriteconcentration.

Figure 18. Measured profile (circles) and estimate (solid line) of nitrite concentration after 404 hours.


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The main limitation of the proposed model lies in the lack of description of thelimiting effect of the carbon source on the two growth-associated reactions (1) and (2).However, this effect cannot be identified with the data at hand, where the carbonsource is always in excess.Besides the development and validation of a process model, the main contribution of

this modelling study is to show that the long transient phases observed in real-lifeoperations can be explained by biomass growth and deactivation processes, i.e. the globalrate at which biomass develops determines the overall dynamics of the biofilter. Theuniformity of the biomass distribution inside the porous bed is also a primary determinantof the biofilter performance in face of large variations of the feed conditions.As the biomass distribution cannot be measured in practice, it is appealing to design

a software sensor (or state observer) to reconstruct this information on-line. To thisend, two options are considered: (a) an asymptotic observer, and (b) an exponentialobserver.The asymptotic observer does not rely on the knowledge of the reaction kinetics,

which is a decisive advantage with regard to the model uncertainties. However, theasymptotic observer appears unsuitable for the considered model structure, and anexponential observer, e.g. an extended Kalman filter or an extended Luenbergerobserver, is the only feasible solution. In this latter class of observers, receding-horizon(or full-horizon, when measurements are rare and corrupted by noise) observersprovide a very simple, yet rigorous, solution to the nonlinear state estimation problemin distributed parameter systems with stochastic disturbances.

Acknowledgement

The authors are very grateful to the Laboratoire d’Ingenierie des Procedes de l’Envi-ronnement (LIPE) of INSA (Toulouse, France) for providing the experimental data.

References

[1] Oh, J., Yoon, S.M. and Park, J.M., 2001, Denitrification in submerged biofilters of concentrated-nitratewastewater. Water Science and Technology, 43, 217 – 223.

[2] Falkentoft, C.M., Harremoes, P., Moesbeck, H. and Wilderer, P., 2000, Combined denitrification andphosphorus removal in a biofilter. Water Science and Technology, 41, 493 – 501.

[3] Pak, D. and Chang, W., 2000, Simultaneous removal of nitrogen and phosphorus in a two-biofiltersystem. Water Science and Technology, 41, 101 – 106.

[4] Bourrel, S., 1996, Estimation et commande d’un procede a parametres repartis utilise pour le traitementbiologique de l’eau a potabiliser. PhD thesis, Universite Paul Sabatier, Toulouse, France.

[5] Bourrel, S., Dochain, D., Babary, J.P. and Queinnec, I., 2000, Modelling, identification and control of adenitrifying biofilter. Journal of Process Control, 10, 73 – 91.

[6] Bastin, G. and Dochain, D., 1990, On-line Estimation and Adaptive Control of Bioreactors (Amsterdam:Elsevier).

[7] Allgower, F., Badgwell, T.A., Qin, J.S., Rawlings, J.B. and Wright, S.J., 1999, Nonlinear predictivecontrol and moving horizon estimation –An introduction overview. In: P.M. Frank (Ed.), Advances inControl (Highlights of ECC’99), Springer-Verlag, pp. 391 – 449.

[8] Bogaerts, Ph. and Hanus, R., 2001, On-Line State Estimation of Bioprocesses with Full HorizonObservers. Mathematics and Computers in Simulation, 56, 425 – 441.

[9] Bascoul, M., 1995, Conduite Optimale d’un Biofiltre en Denitrification d’Eau a Potabiliser, Memoire deDEA, Institut National des Sciences Appliquees de Toulouse, France.

[10] Henze, M., Grady, L., Gujer, W., Marais, G.R. and Matsuo, T., 1987, Activated sludge model No. 1.Technical report, IAWPRC Science and Technology Reports No. 1, London.

[11] Walter E. and Pronzato L., Identification of Parametric Models from Experimental Data, Springer –Masson, 1997.

[12] Dochain, D. and Vanrolleghem, P.A., 2001, Dynamical Modelling and Estimation in WastewaterTreatment Processes (IWA Publishing).


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transient analysis of a wastewater treatment biofilter

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