inﬂuence of microbial growth kinetics on steady state...

ARTICLE

Influence of Microbial Growth Kinetics onSteady State Multiplicity and Stability ofa Two-Step Nitrification (SHARON) Model

Eveline I.P. Volcke,1 Mihaela Sbarciog,2 Mia Loccufier,2 Peter A. Vanrolleghem,1,3

Erik J.L. Noldus2

1BIOMATH, Department of Applied Mathematics, Biometrics and Process Control,

Ghent University, Coupure links 653, B-9000 Gent, Belgium; telephone: þ32 9 264 59 37;

fax: þ32 9 264 62 20; e-mail: [email protected] of Electrical Energy, Systems and Automation, Ghent University,

Technologiepark 913, B-9052 Zwijnaarde, Belgium3modelEAU, Departement de genie civil, Pavillon Pouliot, Universite Laval,

G1K 7P4 Quebec, QC, Canada

Received 30 October 2006; revision received 27 February 2007; accepted 6 April 2007

Published online 26 April 2007 in Wiley InterScience (www.interscience.wiley.com). D
OI 10.1002/bit.21464
ABSTRACT: In this paper, the influence of microbial growthkinetics on the number and the stability of steady states for anitrogen removal process is addressed. A two-step nitrifica-tion model is studied, in which the maximum growth rate ofammonium oxidizers is larger than the one of nitrite oxidi-zers. This model describes the behavior of a SHARONreactor for the treatment of wastewater streams with highammonium concentrations. Steady states are identifiedthrough direct calculation using a canonical state spacemodel representation, for several types of microbial kinetics.The stability of the steady states is assessed and the corre-sponding phase portraits are analyzed. Practical operation ofa SHARON reactor aims at reaching ammonium conversionto nitrite while suppressing further conversion to nitrate.Regions in the input space are identified that result in thisdesired behavior, with only nitrite formation. It is demon-strated that not only the dilution rate plays a role, as iscommonly known, but also the influent ammonium con-centration. Besides, the type of microbial (inhibition)kinetics has a nonnegligible influence. While the resultsindicate that product inhibition does not affect the numberof steady states of a (bio)reactor model, it is shown thatsubstrate inhibition clearly yields additional steady states.Particular attention is devoted to the physical interpretationof these phenomena.

Biotechnol. Bioeng. 2007;98: 882–893.

� 2007 Wiley Periodicals, Inc.

KEYWORDS: modeling; nitrification; stability analysis;steady states; wastewater treatment

Correspondence to: E.I.P. Volcke

882 Biotechnology and Bioengineering, Vol. 98, No. 4, November 1, 2007

Introduction

In (bio)chemical reaction systems, steady state multiplicity,by which we mean that different initial reactor conditionsresult in different steady states reached for the same inputvariables, is often encountered. Experimental evidence ofsuch behavior has been reported by several authors, forexample, by Lei et al. (2003) during aerobic cultivation ofSaccharomyces cerevisiae. As good knowledge of processdynamics is essential for design and control purposes, it hasbeen attempted to rigorously formulate conditions underwhich steady state multiplicity occurs, for example, byAgrawal et al. (1982) for isothermal continuous stirredbioreactors. Whereas in a nonisothermal reacting system,the nonlinear dependence of the rate constant ontemperature may contribute to system multiplicity, inisothermal systems like biological wastewater treatmentsystems, this multiplicity may result from nonlinear kinetics(Tong and Fan, 1988). Besides, also the phenomenon ofmetabolic regulation, an omnipresent feature of biochemicalpathways that endows cells with the ability to make choicedecisions at metabolic crossroads, leads to nonlinearbehavior in bioreactors and is a source of steady statemultiplicity (Namjoshi and Ramkrishna, 2001).

This paper assesses steady state multiplicity and stabilityof a two-step nitrification model for ammonium removalfrom wastewater. The model can be used to describe the

� 2007 Wiley Periodicals, Inc.

Figure 1. Basic two-step nitrification reaction scheme. [Color figure can be

seen in the online version of this article, available at www.interscience.wiley.com.]

behavior of a SHARON (Single reactor High activityAmmonium Removal Over Nitrite) process (Hellinga et al.,1998), that is ideally suited to remove nitrogen fromwastewater streams with high ammonium concentration.The SHARON reactor is operated as a continuously stirredtank reactor (CSTR) without biomass retention. At theprevailing pH (about 7) and high temperature (30–408C),ammonium oxidizers grow faster than nitrite oxidizers. Forthis reason, it is possible to establish ammonium oxidationto nitrite only and prevent further oxidation of nitrite tonitrate by setting an appropriate dilution rate. In this way,substantial savings in aeration costs are realized, incomparison with oxidation of ammonium to nitrate.

In an attempt to identify regions in the input space forwhich a unique steady state exists, only the wash-out statehas been found—for very high values of the dilution rate(Volcke et al., 2006), which is clearly not an interestingoperating point. In this contribution, the steady states of atwo-step nitrification model are calculated directly in anumber of simplified cases. Note that here the term ‘‘steadystate’’ is used synonymous to ‘‘equilibrium’’ or ‘‘equilibriumstate,’’ as a solution of the steady state process model. Itshould be stressed that these steady states are not a prioristable. Particular attention is devoted to the effect ofbiological conversion kinetics on the number of steady statesand on their stability. Phase trajectories illustrate the processbehavior. It is studied which combinations of inputvariables, in particular the applied dilution rate for giveninfluent ammonium concentrations, result in ammoniumoxidation to nitrite only. The influence of changingparameter and input values, reflecting the uncertainty inthe description of biological systems as well in as influentcharacterization, is addressed briefly.

Materials and Methods

Two-Step Nitrification Model Under Study

The simplified SHARON reactor model under studyconsiders two nitrification reactions: oxidation of ammo-nium to nitrite and subsequent oxidation of nitrite tonitrate, as schematically represented in Figure 1. Theconcentrations (xi(t)� 0) of the following four components,involved in the biological conversion reactions, areidentified as the system’s state variables: (total) ammonium(x1¼NHþ

4 þNH3), (total) nitrite (x2¼HNO2þNO�2 ),

ammonium oxidizers (x3), and nitrite oxidizers (x4). Theconcentration of nitrate, which is produced during nitriteoxidation, is not taken up as a state variable of the model, asit is directly related to the amount of nitrite oxidizers (x4)in the reactor. Although the two biological conversionreactions are essentially consecutive reactions (ammoniumis first oxidized to nitrite, which is subsequently oxidized tonitrate), ammonium is consumed in both reactions: duringammonium oxidation, it serves as both the electron donorand the nitrogen source for incorporation in biomass, while

a small amount of ammonium is also consumed duringnitrite oxidation, as the nitrogen source for incorporation inbiomass.

The reactor is modeled as a CSTR without biomassretention. The reactor liquid volume is assumed constant.The system’s state equations are given by the individual massbalances for ammonium, nitrite, ammonium oxidizers, andnitrite oxidizers (see Volcke, 2006, for details):

_x1 ¼ u0ðu1 � x1Þ � ar1ðxÞ � br2ðxÞ � f1ðxÞ_x2 ¼ �u0x2 þ cr1ðxÞ � dr2ðxÞ � f2ðxÞ

_x3 ¼ �u0x3 þ r1ðxÞ � f3ðxÞ_x4 ¼ �u0x4 þ r2ðxÞ � f4ðxÞ

(1)

in which x¼ [x1 x2 x3 x4]T, T indicating the transposed

vector. The model parameters a¼ 1/Y1, b, c¼ (1/Y1)� b,and d¼ 1/Y2 represent constant, strictly positive stoichio-metric coefficients, in which Y1 and Y2 denote the biomassyield coefficient on substrate during ammonium oxidationand nitrite oxidation, respectively, while b represents thenitrogen fraction in biomass. The input variables are thedilution rate (u0(t)� 0) and the influent concentration oftotal ammonium (u1(t)� 0). The influent is assumed tocontain neither nitrite nor biomass. Ammonium and nitriteoxidation rates are described by their respective reactionrates, r1 and r2, each proportional to the correspondingbiomass concentration:

r1 ¼ m1x3r2 ¼ m2x4

(2)

Many different expressions for the microbial growth ratesmi have been reported in literature, differing essentially inthe inhibition effects acting on the nitrifying biomass. Sinceearly reports, for example, by Anthonisen et al. (1976), it isgenerally accepted that ammonium (ammonia) and nitrite(nitrous acid) may inhibit both ammonium oxidation and

Volcke et al.: Influence of Microbial Growth Kinetics 883

Biotechnology and Bioengineering. DOI 10.1002/bit

nitrite oxidation. These inhibition effects become particu-larly important when treating wastewater streams with highammonium concentrations. However, many different valuesare found in literature with respect to the level of inhibition.Although some of the reported results even seem contra-dictory, they essentially reflect the strong capacity ofnitrifying organisms to adapt to different reactor andinfluent conditions. For instance, while Anthonisen et al.(1976) report on inhibition of ammonium oxidation byammonium rather than by nitrite, the model of Hellingaet al. (1999), which was based on experimental results,considers inhibition of ammonium oxidation by nitrite butnot by ammonium. The latter findings have recently beenexperimentally confirmed by Van Hulle et al. (2007).

Regarding the different inhibition effects on nitrificationreported by different literature sources, a more pragmaticapproach has been decided on in this contribution. Fourbasic types of microbial conversion kinetics (different mi)have been considered to evaluate their effect on the numberof steady states and their stability (Table I). This constitutesthe difference between the models evaluated in this study.

The first model assumes that no inhibition of any kindtakes place. Model II adds nitrite inhibition of ammoniumoxidation (with inhibition constant c1) to model I, whilemodel III considers ammonium inhibition of ammoniumoxidation (with inhibition constant d1). Model IV accountsfor nitrite inhibition of nitrite oxidation as the only form ofinhibition (with constant d2). The maximum growth rates ofammonium and nitrite oxidation (a1 and a2, respectively),the substrate affinity constants (b1 and b2) as well as theabove-mentioned inhibition constants are model para-meters, which have a constant value, at least for a SHARONreactor in which temperature and pH are controlled at afixed level, as is assumed here. Note that the terminology‘‘maximum growth rates’’ has been used to denote theparameters a1 and a2 throughout this manuscript, althoughtheir values do not strictly correspond to the maximumvalues of m1 in case of model III and m2 in case of model IV,respectively. It is further assumed that oxygen is alwayspresent in excess, or at least a constant oxygen level ismaintained (the influence of sub-optimal oxygen levels isthen comprised in the parameters a1 and a2), which is a

Table I. Microbial conversion kinetics for the models under study.

Ammonium oxidation

Model I m1 ¼ a1x1

b1 þ x1

Model II m1 ¼ a1x1

b1 þ x1

c1c1 þ x2

Model IIIm1 ¼ a1

x1b1 þ x1

d1d1 þ x1

; with b1 < d1

Model IV m1 ¼ a1x1

b1 þ x1


reasonable assumption for a SHARON reactor. Thenumerical values for the model parameters applied in thisstudy are summarized in Table II.

Canonical State Space Model Representation

By defining new state space variables:

y1y2y3y4

2664

3775 �

x1 þ ax3 þ bx4x2 � cx3 þ dx4

x3x4

2664

3775 (3)

the two-step nitrification model defined by Equation (1) istransformed into a canonical form (Bastin and Dochain,1990; Bastin and Van Impe, 1995), consisting of a linear partof dimension 2, dynamically coupled with a nonlinear partof dimension 2:

_y1 ¼ u0ðu1 � y1Þ � g1ðyÞ_y2 ¼ �u0y2 � g2ðyÞ

_y3 ¼ ð�u0 þ l1Þy3 � g3ðyÞ_y4 ¼ ð�u0 þ l2Þy4 � g4ðyÞ

(4)

with

liðyÞ ¼ miðxÞ; i ¼ 1; 2 (5)

in which

x1 ¼ y1 � ay3 � by4x2 ¼ y2 þ cy3 � dy4

x3 ¼ y3x4 ¼ y4

(6)

The state variables xi (i¼ 1,. . .4) of the two-stepnitrification model represented by Equation (1) cannotbecome negative. Call Sy the image of<þ;4 � fx 2 <4 : xi � 0; i ¼ 1; . . . 4g under the transfor-mation x 7! y. Sy is the state space of the system defined byEquation (4). Every trajectory that starts at t¼ 0 in a point y0of Sy, stays in Sy for t� 0. It subsequently converges (for

Nitrite oxidation

m2 ¼a2

x2b2 þ x2

for x1 > 0

0 for x1 ¼ 0

(

m2 ¼a2

x2b2 þ x2

for x1 > 0

0 for x1 ¼ 0

(

m2 ¼a2 �

x2b2 þ x2

for x1 > 0

0 for x1 ¼ 0

(

m2 ¼a2

x2b2 þ x2

d2d2 þ x2

for x1 > 0

0 for x1 ¼ 0

8<: ; with b2 < d2

DOI 10.1002/bit

Table II. Numerical values of the SHARONmodel parameters (at pH¼ 7

at T¼ 358C).

Symbol Value Unit Reference

a1 2.1 day�1 Hellinga et al. (1999)

b1 4.73 Van Hulle et al. (2007)

1.32* mole m�3 Carrera et al. (2004)

c1 837 mole m�3 Van Hulle et al. (2007)

a2 1.05 day�1 Hellinga et al. (1999)

b2 0.4 mole m�3 Wiesmann (1994)

d1 3,406 Wiesmann (1994)

36* mole m�3 Carrera et al. (2004)

d2 106 mole m�3 Wiesmann (1994)

a ¼ 1Y1

16 mole mole�1 Y1 from Hellinga et al. (1999)

b 0.2 mole mole�1 typical textbook value

c ¼ 1Y1� b 15.8 mole mole�1 determined by the values of a and b

d ¼ 1Y2

58.6 mole mole�1 Y2 from Hellinga et al. (1999)

*This value has only been applied to model III (in Figs. 5 and 6), forcomparison with the default values.

positive constant input values u0 and u1) to the cross-sectionD of Sy with the plane {y1¼ u1, y2¼ 0}, as can be seen fromEquation 4. D is a bounded region, defined by theinequalities

u1 � ay3 � by4 � 0cy3 � dy4 � 0

y3 � 0y4 � 0

(7)

The process converges to a second order behavior that isdetermined by the dynamics of y3 (¼x3) and y4 (¼x4). In thesequel, a mathematical steady state solution yss of Equation(4) will be called a physical steady state if it lies in Sy.

Stability Assessment Of Steady States

The linearization principle is used to investigate the localasymptotic stability of steady states. A steady state xss of thetwo-step nitrification model (Eq. 1) is locally asymptoticallystable if the eigenvalues of the Jacobian matrix:

@fðxÞ@x

��x ¼ xss

�

@f1@x1

��x ¼ xss

� � � @f1@x4

��x ¼ xss

..

. ...

@f4@x1

��x ¼ xss

� � � @f4@x4

��x ¼ xss

2666664

3777775 (8)

are all in the open left half plane, that is, have a strictlynegative real part.

The stability of a steady state can also be assessed using thecanonical state space model representation. A steady state yssof the system represented by Equation 4 is locallyasymptotically stable if all eigenvalues of the system’sJacobian matrix, evaluated for this steady state,

JðyssÞ ¼@gðyÞ@y

��y¼yss

, are in the open left half plane. It can

easily be seen that JðyssÞhas a double eigenvalue �u0 andthat its remaining two eigenvalues are the eigenvalues of the2-dimensional matrix

J34ðyssÞ �

@g3@y3

��y ¼ yss

@g3@y4

��y ¼ yss

@g4@y3

��y ¼ yss

@g4@y4

��y ¼ yss

26664

37775 (9)

The linearization principle provides a necessary andsufficient condition for local asymptotic stability for so-called hyperbolic steady states, of which the Jacobian matrixhas no eigenvalues on the imaginary axis.

In case the system possesses only one asymptotically stablesteady state yss (besides potential unstable steady states),which is lying on the boundary of the canonical state spaceSy, this steady state is also globally asymptotically stablewithin the physical boundaries of the system. This resultsfrom the fact that every trajectory inside a bounded region,for a second order system, should converge to either a steadystate, or a limit cycle, taking into account that a limit cyclecannot surround a region containing no steady states and nolimit cycle can occur around a steady state that is lying on theboundary of Sy, since this would imply that a trajectory isleaving the system’s state space. In case the system possessesonly one asymptotically stable steady state yss, not lying onthe boundary of the canonical state space, the absence oflimit cycles should be verified through simulation.

Results and Discussion

This contribution aims at identifying the number of steadystates, corresponding with constant input values u0 (thedilution rate) and u1 (the influent ammonium concentra-tion), as well as assessing the stability of these steady statesfor the simplified reactor models under study. It is howeverworthwile to mention that some general conclusions can bedrawn for extreme values of the dilution rate u0 and/or theinfluent ammonium concentration u1, without explicitlyknowing the expression of the microbial kinetics.

Steady State Behavior In Case u0U 0 Or u0UR11

In case u0¼ 0, that is, the reactor is operated as a batchreactor, Equation (1) yields an infinite number of steadystate solutions xss for the conversion kinetics of Equation (2)and Table I:

� x
ss1¼ 0; xss2, xss3, xss4 arbitrary: there is no ammonium(left) in the reactor;
� x
ss2¼ xss3¼ 0; xss1, xss4 arbitrary: neither nitrite norammonium oxidizers are present in the reactor;
� x
ss3¼ xss4¼ 0; xss1, xss2 arbitrary: no biomass is present inthe reactor.


The initial reactor conditions determine which steadystate will be reached.

On the other hand, for an infinitely large dilutionrate (u0!þ1), Equation (1) give rise to a uniquesteady state, corresponding with the influent conditions:xss ¼ ½ u1 0 0 0 �T, that is, the wash-out point where thereactor fails. Using the linearization principle, it followsstraightforwardly that this wash-out point for u0!þ1 isalways locally asymptotically stable (Volcke, 2006). Forrealistic values of u0 in between these limits, no generalconclusions can be drawn regarding the number of steadystates without explicitly defining the structure of themicrobial kinetics.

Steady State Behavior In Case u1U 0

In case the reactor influent does not contain ammonium(u1¼ 0), it follows from Equations (6) and (8) that

_y ¼ 0 , yss1 ¼ yss2 ¼ yss3 ¼ yss4 ¼ 0 ,xss1 ¼ xss2 ¼ xss3 ¼ xss4 ¼ 0

indicating a unique (trivial) steady state corresponding with

biomass washout.

Steady State Behavior in Case u0 > 0 and u1 > 0

As no general conclusions can be drawn on the number ofsteady states for dilution rates 0< u0<þ1, their values willbe calculated directly in four cases of simplified microbialkinetics (Table I). This calculation is substantially simplifiedusing the canonical state space model representation (Eq. 4).

In case u0> 0 and u1> 0, no steady state exists for whichthe ammonium concentration is zero (xss1¼ 0). Indeed,xss1¼ 0 implies that l1(yss)¼ l2(yss)¼ 0 (Table I and Eq. 6),so yss3¼ yss4¼ 0 (Eq. 4 for u0 6¼ 0) and consequently yss1¼ 0(Eq. 3), which requires u0u1¼ 0 (Eq. 4). As a result, inequilibrium the nontrivial expressions for m2 (m2 6¼ 0) inTable I hold, so the steady states fulfill the following set ofequations (see Eq. 4):

yss1 ¼ u1yss2 ¼ 0ð�u0 þ l1ðyssÞÞyss3 ¼ 0ð�u0 þ l2ðyssÞÞyss4 ¼ 0

(10)

From the definition of x2 (Eq. 6) where yss2¼ 0 and thephysical boundary conditions x2� 0 and y4� 0, it resultsthat yss3¼ 0 (no ammonium oxidizers) also implies yss4¼ 0(no nitrite oxidizers). This seems logical regarding the factthat, in a SHARON reactor, ammonium oxidizers growfaster than nitrite oxidizers. Consequently, three differenttypes of steady states are obtained:

1. y

88

ss3¼ yss4¼ 0 This steady state corresponds with com-plete biomass wash-out.


2. y
ss4¼ 0 while yss3 6¼ 0 In this case, no nitrite oxidizers arepresent and consequently no nitrate is formed, as istypically aimed for in a SHARON reactor. The dilutionrate equals the growth rate of ammonium oxidizers,u0¼l1, and one or more steady states with only nitriteformation result from this expression, depending on themicrobial kinetics (see further).
3. y
ss3 6¼ 0 and yss4 6¼ 0 In this case, both ammoniumoxidizers and nitrite oxidizers are present, so at least partof the formed nitrite is further oxidized to nitrate. Nowthe growth rates of ammonium oxidizers and nitriteoxidizers are equal and also equal the dilution rate:u0¼ l1¼ l2. Again, the number of corresponding steadystates depends on the microbial kinetics (see further).
In what follows, the number of steady states, theconditions under which they occur (i.e., are physical steadystates in the sense that they fulfill Eq. 7) as well as theirstability is assessed for the different models under study. Allsteady state values are expressed in the x-space, calculatedstraightforwardly from the obtained values in the y-spacethrough Equation (3). Detailed calculations can be found inVolcke (2006).

Model I: No Inhibition

In this case, up to three steady states occur, although in allcases only one steady state is (quasi) globally asymptoticallystable (Table III). While the wash-out state xA;Iss alwaysoccurs, the remaining two steady states only representphysical states under certain conditions. For a SHARONreactor, in which the maximum growth rate of ammoniumoxidizers is larger than the maximum growth rate of nitriteoxidizers (a1> a2), condition 12 for the occurrence of asteady state in which nitrate is formed (xC;Iss ) is clearly morestringent than condition 11 for the occurrence of a steadystate with only nitrite formation (xB;Iss ). Note that x

B;Iss1 ¼ x

C;Iss1,

which means that the ammonium concentration is the samefor both steady states.

Figure 2 depicts the occurrence of steady states in terms ofthe dilution rate u0 and the influent ammonium concentra-tion u1. The corresponding phase trajectories (in D) aredepicted in Figure 3. For high dilution rates u0 or lowinfluent ammonium concentrations u1, that is, whenEquation (11) is not fulfilled (zone A), only the wash-outstate xA;Iss is a physical steady state and is globallyasymptotically stable within the physical boundaries. Formoderately high dilution rates u0 and somewhat highinfluent ammonium concentrations u1, that is, Equation(11) is fulfilled but not Equation 12 (zone AB), the systempossesses two physical steady states. The wash-out state xA;Iss

is now unstable. The steady state xB;Iss (only nitriteformation) is quasi globally asymptotically stable, in thesense that all trajectories within the physical boundaries ofthe system converge to this point, except the trivialtrajectory coinciding with xA;Iss (initially no biomass present

DOI 10.1002/bit

Table III. Values, occurrence and stability of steady states for a model without inhibition (model I).

Symbol Type Value Occurrence Stability

xA;Iss

Wash-out u1000

2664

3775

Always Zone A: g.a.s.

Zones AB and ABC: unstable

xB;IssOnly nitrite formation b1u0

a1 � u0

c

au1 �

b1u0a1 � u0

� �1

au1 �

b1u0a1 � u0

� �0

26666666664

37777777775

if u0 <a1u1

b1 þ u1

,if u1 >

b1u0a1 � u0

and u0 < a1

(11)

Zone AB: quasi g.a.s.

Zone ABC: unstable

xC;IssNitrate formation b1u0

a1 � u0b2u0

a2 � u0

dðu1 � xC;Iss1Þ þ bxC;Iss2

ad þ bc

cðu1 � xC;Iss1Þ � axC;Iss2

ad þ bc

26666666666664

37777777777775

if u0 <a2ðc=aÞðu1 � ððb1u0Þ=ða1 � u0ÞÞÞb2 þ c=aðu1 � ððb1u0Þ=ða1 � u0ÞÞÞ

,if u1 >

b1u0a1 � u0

þ a

c

b2u0a2 � u0

and u0 < a2

(12)

Zone ABC: quasi g.a.s.

in the reactor, which is clearly a purely theoretical case). Forsufficiently low dilution rates u0 and corresponding influentammonium concentrations u1, for which Equation 12 isfulfilled (zone ABC), three steady states occur. The wash-outstate xA;Iss is still unstable, as is now also the second steadystate, xB;Iss , corresponding with only nitrite formation: onlytrajectories on the x3-axis converge to xB;Iss , correspondingwith a reactor in which initially only ammonium oxidizersare present (the latter again being a purely theoreticalcondition). Under the same conditions, the third steadystate, xC;Iss , corresponding with nitrate formation, is quasi

Figure 2. Steady states for a model without inhibition (model I). [Color figure can

be seen in the online version of this article, available at www.interscience.wiley.com.]

globally asymptotically stable, in the sense that alltrajectories within the physical boundaries of the systemconverge to this point, except for the steady state xA;Iss and thetrajectories on the x3-axis.

Model II: Nitrite Inhibition of Ammonium Oxidation

Adding nitrite inhibition of ammonium oxidation to themodel does not affect the number of steady states, neithertheir stability, although the position of the steady statescorresponding with biomass growth changes, in the sensethat less ammonium is converted (see Table IV). Regardingthe steady state with only nitrite production, in caseammonium oxidation is inhibited by nitrite, the corre-sponding ammonium concentration will be higher(xB;IIss1 > x

B;Iss1) and the nitrite concentration will be lower

(xB;IIss2 < xB;Iss2) than in case no inhibition takes place. As for

the steady state with nitrate production, it is clear that lessammonium will be converted (xC;IIss1 > x

C;Iss1, taking into

account f(u0)> u0) but the same amount of nitrite will beformed (xC;IIss2 ¼ x

C;Iss2), so less nitrate will be produced

compared to the case without inhibition.It can also be noted that the ammonium concentration

corresponding with the steady state with nitrate formation islower than in case only nitrite is formed (xC;IIss1 < x

B;IIss1 ): if

some nitrite is further oxidized to nitrate, more ammoniumcan be converted in case of nitrite inhibition of ammoniumoxidation.

While nitrite inhibition of ammonium oxidation does notaffect the condition for the occurrence of a steady state withonly nitrite production (Eq. 11), the condition for the thirdsteady state to be a physical state becomes more stringent



Figure 3. Trajectory fields for a model without inhibition (model I). [Color figure can be seen in the online version of this article, available at www.interscience.wiley.com.]

(condition 13 compared to condition 12, as f(u0)> u0). As aresult, the region AB increases: only nitrite formation isobtained in a wider range, even for lower dilution rates andhigher influent ammonium concentrations than in case noinhibition takes place (although this effect is almost notvisible in Fig. 4 compared to Fig. 2). This effect increases forstronger inhibition values. In case there is no nitriteinhibition (c1¼þ1), conditions 12 and 13 are identical, soa continuous transient exists between model I and model II.

Model III: Ammonium Inhibition of Ammonium Oxidation

While product (nitrite) inhibition of ammonium oxidationleaves the number of steady states unchanged in comparisonwith a model without inhibition, substrate (ammonium)

able IV. Values, occurrence and stability of steady states for a model with nitrite inhibition of ammonium oxidation (model II).

ymbol Type Value Occurrence stability

A;IIss

Wash-out u1000

2664

3775


Zones AB and ABC: unstable

B;IIss

Only nitrite formation xB;IIss1c

aðu1 � xB;IIss1 Þ

1

aðu1 � xB;IIss1 Þ

0

2666664

3777775 (a)

if u0 <a1u1

b1 þ u1,

if u1 >b1u0

a1 � u0and u0 < a1

(11)


Zone ABC: unstable

C;IIss

Nitrate formation b1fðu0Þa1 � fðu0Þ

b2u0a2 � u0

d u1 � xC;IIss1

� �þ bxC;IIss2

ad þ bc

c u1 � xC;IIss1

� �� axC;IIss2

ad þ bc

266666666666664

377777777777775(b)

if u1 >b1fðu0Þ

a1 � fðu0Þþ a

c

b2u0a2 � u0

and u0 < u0 (c) (13)Zone ABC: quasi g.a.s.

n which xB;IIss1 is obtained from a1xB;IIss1

b1 þ xB;IIss1

c1

c1 þ ca ðu1 � xB;IIss1 Þ

¼ u0.

With fðu0Þ ¼ u0 þb2u

20

c1 a2 � u0ð Þ.

ith u0 from fðu0Þ ¼ a1; 0 < u0 < a2.

T

S

x

x

x

aI

b

cW


inhibition of the ammonium oxidation—combined withsubstrate limitation, resulting in Haldane kinetics—doesaffect the number of steady states. Up to five steady states areobtained: one wash-out point (xA;IIIss ), two steady statescorresponding with only nitrite formation (xB;IIIss and xC;IIIss )and two steady states corresponding with nitrate formation(xD;IIIss and xE;IIIss ) (see Table V).

Figure 5 depicts the occurrence of steady states in terms ofthe dilution rate u0 and the influent ammonium concentra-tion u1 for two different levels of ammonium inhibition. Thevalue of d1¼ 3,406 mole m�3 (Wiesmann, 1994) corre-sponds with weak ammonium inhibition. In this case,maximally three steady states simultaneously occur in therange of influent ammonium concentrations u1 thatcan reasonably be expected (typically up to 1.5 kgN m�3

DOI 10.1002/bit

Figure 4. Steady states for a model with nitrite inhibition of ammonium oxidation

(model II). [Color figure can be seen in the online version of this article, available at

www.interscience.wiley.com.]

� 100 mole m�3 for reject water, although other types ofwastewater streams such as thin manure fractions or landfillleachate may contain up to twice this value). In the regionu1¼ [0,100], the behavior is qualitatively the same as for themodels I (Fig. 2) and II (Fig. 4). However, if strongerammonium inhibition of ammonium oxidation wouldprevail, up to five steady states could occur at the same time.This has been depicted in Figure 5 for b1¼ 1.32 mole m�3;d1¼ 36 mole m�3, corresponding to the values reported byCarrera et al. (2004) for suspended cells, after conversion topH¼ 7 at T¼ 358C, as assumed in this study.

Figure 6 gives the phase trajectories for the distinguishedoperating regions (for b1¼ 1.32 mole m�3; d1¼ 36

Figure 5. Steady states for a model with ammonium inhibition of ammonium oxidation

inhibition (right, b1¼ 1.32 mole m�3; d1¼ 36 mole m�3). [Color figure can be seen in the o

mole m�3). In the operating zones A, AB, and ABD,the stability of the steady states is the same as for thecorresponding zones in the models I and II without nitriteinhibition—only one stable steady state occurs at a time(note however that zone ABD for model III correspondswith zone ABC for models I and II). In contrast, theoperating zones ABC, ABCD, and ABCDE, in which asecond steady state corresponding with only nitriteproduction (xC;IIIss ) occurs, possess two steady states thatare locally asymptotically stable: the wash-out state and anontrivial one. The stability boundary, that separates theirattraction regions, determines from which initial states theprocess will converge to which steady state. In the operatingzones ABC and ABCD, only trajectories parallel to the x4-axis and with x

C;IIIss3 as the x3-coordinate, converge to the

unstable state xC;IIIss (Volcke, 2006), meaning that this steadystate (with only nitrite formation) will only be reached if theinitial amount of ammonium oxidizers in the system isexactly the same as for xC;IIIss . These trajectories separate theregions of attraction of the two steady states that are bothlocally asymptotically stable: xA;IIIss (wash-out) and xB;IIIss

(only nitrite production) in zone ABC; xA;IIIss and xD;IIIss

(nitrate production) in zone ABCD. The remaining steadystate xB;IIIss in zone ABCD, also corresponding with onlynitrite formation, is unstable and will only be reached ifinitially only ammonium oxidizers are present in the system,in a sufficient amount: only trajectories on the x3-axis to theright of xC;IIIss converge to xB;IIIss . In case the system possessesfive steady states (operating zone ABCDE), only xA;IIIss

(wash-out) and xD;IIIss (nitrate formation) are locallyasymptotically stable. Their regions of attraction areseparated by the trajectories that converge to the unstablesteady state xE;IIIss (nitrate formation). The exact location ofthis stability boundary is not obvious as for zones ABC andABCD but should be estimated, for example, by a trajectory

(model III), for relative weak (left, b1¼ 4.73 mole m�3; d1¼ 3406 mole m�3) and strong

nline version of this article, available at www.interscience.wiley.com.]



Table V. Values, occurrence and stability of steady states for a model with ammonium inhibition of ammonium oxidation (model III).


xA;IIIss

Wash-out u1000

2664

3775


Zones AB and ABD:

unstable

Zones ABC, ABCD,

and ABCDE: l.a.s.

xB;IIIss and xC;IIIssOnly nitrite

formationxa;IIIss1

c

au1 � xa;IIIss1

� �1

au1 � xa;IIIss1

� �0

2666664

3777775(

d)

if u0 <a1d1ffiffiffiffiffi

b1p

þffiffiffiffiffid1

p� �2 (14)

and xa;IIIss1 < u1 (15)

xB;IIIss


Zone ABC: l.a.s.

Zones ABD, ABCD,

and ABCDE: unstable

xC;IIIss

Zones ABC,

ABCD,and ABCDE:

unstable

xD;IIIss and xE;IIIss

Nitrate

formationxb;IIIss1b2u0

a2 � u0

d u1 � xb;IIIss1

� �þ bxb;IIIss2

ad þ bc

c u1 � xb;IIIss1

� �� axb;IIIss2

ad þ bc

26666666666664

37777777777775(e)

if u0 <a1d1ffiffiffiffi

b1p

þffiffiffiffid1

p� �2 (14)

and u0 < a2 (16)

and xb;IIIss1 < u1 �a

c

b2u0a2 � u0

(17)

xD;IIIss

Zone ABD: quasi g.a.s.

ABCD and ABCDE: l.a.s.

xE;IIIss

Zone ABCDE:

unstable

da denotes B or C, xa;IIIss1 is obtained from a1xa;IIIss1

b1 þ xa;IIIss1

d1

d1 þ xa;IIIss1

¼ u0, say xB;IIIss1 < xC;IIIss1 .

eb denotes D or E, xb;IIIss1 ¼ xa;IIIss1 , say xD;IIIss1 < xE;IIIss1 , so xD;III

ss1 ¼ xB;IIIss1 and xE;IIIss1 ¼ xC;IIIss1 .

reversing technique (Genesio et al., 1985; Loccufier andNoldus, 2000).

Comparing these results with the ones for model I and II,it can be concluded that the main impact of ammoniuminhibition of ammonium oxidation lies in the fact that, fornot too low influent ammonium concentrations, the wash-out steady state is asymptotically stable and will be reached ifthe initial amount of ammonium oxidizers is not sufficient,even for relatively low dilution rates (zones ABC, ABCD,and ABCDE). The region in the input space for which onlynitrite is formed, is smaller in case of ammonium inhibitionof ammonium oxidation than in case of no inhibition

Figure 6. Trajectory fields for a model with ammonium inhibition of ammonium oxida

online version of this article, available at www.interscience.wiley.com.]


(region AB, Fig. 2 vs. Fig. 5), due to a downward shift of theA/AB-boundary, and is gradually reduced as this inhibitionlevel increases, that is, as d1 decreases. This remains truewhen considering that only nitrite formation will also berealized in the input region ABC for model III starting fromappropriate initial reactor conditions, that is, provided thatsufficient ammonium oxidizers are present. The zone in theinput space for which nitrate is formed in case of noinhibition (model I, zone ABC, Fig. 2) more or lesscorresponds to the zone for which stable nitrate formation ispossible in case ammonium oxidation is inhibited byammonium (model III, zones ABD, ABCD, and ABCDE,

tion (model III, b1¼ 1.32 mole m�3; d1¼ 36 mole m�3). [Color figure can be seen in the

DOI 10.1002/bit

Table VI. Values, occurrence and stability of steady states for a model with nitrite inhibition of nitrite oxidation (model IV).


xA;IVss

Wash-out u1000

2664

3775


Zones AB, ABC,

and ABCD: unstable

xB;IVssOnly nitrite formation b1u0

a1 � u0

c

au1 �

b1u0a1 � u0

� �1

au1 �

b1u0a1 � u0

� �0

26666666664

37777777775

if u0 <a1u1

b1 þ u1,

if u1 >b1u0

a1 � u0and u0 < a1

(11)


Zone ABC: unstable

Zone ABCD: l.a.s.

xC;IVss and xD;IVss

Nitrate formation b1u0a1 � u0

xg;IVss2

d u1 � xg;IVss1

� �þ bxg;IVss2

adþbc

c u1 � xg;IVss1

� �� axg;IVss2

ad þ bc

2666666666664

3777777777775

if u0 <a1u1b1þu1

(11)

and if u0 <a2d2ffiffiffiffiffi

b2p

þffiffiffiffiffid2

p� �2 (18)

and if xg;IVss2 <c

aðu1 � xg;IVss1 Þ (19)

xC;IVss

Zone ABC: quasi g.a.s.

Zone ABCD: l.a.s.

xD;IVss

Zone ABCD:

unstable

fg denotes C or D, xg;IVss2 is obtained from a2xg;IVss2

b2 þ xg;IVss2

d2

d2 þ xg;IVss2

¼ u0, say xC;IVss2 < xD;IVss2 .

Fig. 5). However, in the latter case wash-out may also occurif insufficient ammonium oxidizers are initially present inthe reactor (for zones ABCD and ABCDE, Fig. 5).

Figure 7. Steady states for a model with nitrite inhibition of nitrite oxidation

(model IV). [Color figure can be seen in the online version of this article, available at


Model IV: Nitrite Inhibition of Nitrite Oxidation

In order to further test to which extent substrate inhibitioninfluences the number of steady states, a model withsubstrate inhibition of nitrite oxidation, that is, nitriteinhibition (model IV), has also been studied. It has beenfound that up to four physical steady states can occur (seeTable VI): the wash-out state (xA;IVss ), a steady statecorresponding with only nitrite formation and two steadystates in which nitrate is formed. The value of the steadystate corresponding with nitrite formation is exactly thesame as for model I, which does not consider inhibitioneffects. The steady states with nitrate formation correspondwith the same ammonium conversion as the one with onlynitrite formation for the same model, which is also the sameas for the model without inhibition (model I). It canhowever be shown that, in case of nitrite inhibition of nitriteoxidation (model IV), the steady state with nitrateformation corresponds with higher nitrite concentrations(so less nitrate formed) than in case of no inhibition (modelI): xg;IVss2 > x

C;Iss2 for both g¼C and D.

Figure 7 shows the different operating regions for thegiven parameter set. The corresponding phase trajectoriesare displayed in Fig. 8. In case of low dilution rates andaccordingly high influent ammonium concentrations, up tofour steady states can occur at the same time (operating zoneABCD, Fig. 7), of which two are locally asymptotically stable

(Table VI): xB;IVss , corresponding with only nitrite formationand xC;IVss , in which nitrate is formed. It is interesting to notethe difference with model III, where in case of two locallyasymptotic steady states, the wash-out state is always one ofthem. In zone ABCD, the steady state xB;IVss is reached andthus nitrate formation is suppressed if the initial amount ofnitrite oxidizers (x4) in the system is sufficiently low. Thiscan be understood as a small amount of nitrite oxidizers willnot be able to convert the nitrite that is built up in the system



Figure 8. Trajectory fields for a model with nitrite inhibition of nitrite oxidation

(model IV). [Color figure can be seen in the online version of this article, available at


fast enough to prevent its own inhibition by nitrite, resultingin their wash-out from the reactor. In case of a high initialamount of nitrite oxidizers, the steady state xC;IVss withnitrate formation is reached. The regions of attraction of thestable steady states xB;IVss and xC;IVss are separated by thetrajectories which converge to the unstable steady statexD;IVss . Its exact location can only be estimated, for example,by a trajectory reversing technique, as applied by Sbarcioget al. (2006).

Overall, the operating region in which only nitrite isformed, as is the aim of a SHARON reactor, is larger in caseof nitrite inhibition of nitrite oxidizers (model IV) than incase of no inhibition (model I): the steady state with onlynitrite formation (xB;IVss ) is reached in operating zone AB aswell as in operating zone ABCD, provided the initial amountof nitrite oxidizers is sufficiently low.

Influence of Uncertainty in Parameter and Input Values

For biological systems, it is often difficult to accuratelydetermine the parameter values. Also, parameter values maychange in time, for example, because of biomass adaptation.Besides, the input values may be uncertain. The questionarises to which extent this uncertainty influences thepresented results.

From the analysis in the previous sections, it is clear thatthe model structure, in particular the type of inhibition thattakes place, affects the number of steady states. It shouldhowever be noted that the exact value of the correspondinginhibition constants will not qualitatively influence thisbehavior. Even between different model structures, smoothtransitions exist: the behavior of a reactor with relativelysmall biomass inhibition effects is comparable to the one of a


reactor without inhibition. As a result, it is sufficient toknow parameter values within a certain range of accuracy.

Regarding the uncertainty in input values, the influence ofchanging dilution rates u0 and influent ammoniumconcentrations u1 has been studied explicitly through thediagrams presented in previous sections (Figs. 2, 4, 5, and 7)and does not require further discussion here. On the otherhand, the influent biomass concentration has been tacitlyassumed to be zero in Equation (1), while in practice therewill always be some ammonium oxidizers and nitriteoxidizers (‘‘at least one’’) present in the influent, of whichthe concentrations constitute additional model inputvariables. Biomass presence in the influent will particularlyaffect the occurrence of steady states that are located on theboundary of the system’s state space. It has been shown formodel I and II (Volcke, 2006; Volcke et al., 2006) as well asfor model IV (Sbarciog et al., 2006) that, in case ammoniumoxidizers are present in the influent, the wash-out steadystate xAss is only a physical steady state in case it is the onlysteady state, that is, in zone A. If nitrite oxidizers are presentin the influent, the steady state xBss, corresponding with onlynitrite formation, is only a physical steady state for model Iand II in operating zone AB, that is, in case the steady statexCss, corresponding with nitrate formation, does not occur(Volcke, 2006), or in operating zone AB or ABCD for modelIV (Sbarciog et al., 2006). Nevertheless, these effects are notvery drastic, as the steady states involved are notasymptotically stable in the concerned input regions, whichmeans they are usually not reached anyway for thecorresponding combinations of influent ammonium con-centrations and dilution rates.

Conclusions

In this paper, the influence of microbial growth kinetics onsteady state multiplicity and stability of a nitrogen removalprocess has been addressed. The model under study is a two-step nitrification model, in which the maximum growth rateof ammonium oxidizers is larger than one of nitrite oxi-dizers, which describes the behavior of a SHARON reactor.The results agree with well-known operating rules: in orderto realize only nitrite formation, the applied dilution rateshould not be too high to prevent wash-out of ammoniumoxidizers, but not too low to keep out nitrite oxidizers. Inaddition, the obtained results clearly show that the giveninfluent ammonium concentration also plays a role.

Inhibition of ammonium oxidation by its product nitritedoes not affect the number of steady states of the reactormodel, neither their stability, although the position of thesteady states corresponding with biomass growth changes.In contrast to product inhibition, substrate inhibition(combined with substrate limitation, resulting in Haldanekinetics) clearly is a source of additional steady states:ammonium inhibition of ammonium oxidation results inthe occurrence of up to five steady states, while nitriteinhibition of nitrite oxidation gives rise to up to four steadystates.

DOI 10.1002/bit

The region in the input space for which only nitrite isformed, increases in case ammonium oxidation is inhibitedby nitrite; it becomes smaller in case this reaction is inhibitedby ammonium. Inhibition of nitrite oxidation by nitriteoverall results in a larger operating region leading to onlynitrite formation.

The results in this contribution clearly demonstrate howthe microbial (inhibition) kinetics affect the number and thestability of steady states of a bioreactor model. Thedescription of microbial kinetics with respect to inhibitionappears a crucial step in modeling biological processes as itmay lead to very different steady state behaviors of theresulting model. Nevertheless, smooth transitions occurbetween different model structures. In the same sense,slightly changing parameter values only will have smalleffects. If a priori knowledge on microbial kinetics isavailable—even within a certain range of uncertainty for theparameter values—this insight can be used to predict thesteady states that can be reached for given input values,dependent on the initial conditions. Vice versa, it also allowselucidating the type of microbial kinetics from experimentalobservations.

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