bifurcacion 1

8/22/2019 bifurcacion 1

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Journal of Biotechnology 135 (2008) 295303

Contents lists available at ScienceDirect

Journal of Biotechnology

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j b i o t e c

Bifurcational and dynamical analysis of a continuous biofilm reactor

M.E. Russo, P.L. Maffettone, A. Marzocchella , P. Salatino

Dipartimento di Ingegneria Chimica, Universita degli Studi di Napoli Federico II, P.le V. Tecchio, 80 80125 Napoli, Italy

a r t i c l e i n f o

Article history:

Received 12 November 2007

Received in revised form 20 March 2008Accepted 8 April 2008

Keywords:

Bifurcation

Biofilm

Detachment rate

Dynamics

a b s t r a c t

A dynamical model of a continuous biofilm reactor is presented. The reactor consists of a three-phase

internal loop airlift operated continuously with respect to the liquid and gaseous phases, and batchwise

with respect to the immobilized cells. The model has been applied to the conversion of phenol by meansof immobilized cells ofPseudomonas sp. OX1 whose metabolic activity was previously characterized (Vig-giani, A., Olivieri,G., Siani, L., Di Donato, A., Marzocchella,A., Salatino,P., Barbieri,P., Galli, E., 2006. An airlift

biofilm reactor for the biodegradation of phenol by Pseudomonas stutzeri OX1. Journal of Biotechnology123, 464477). The model embodies the key processes relevant to the reactor performance, with a par-ticular emphasis on the role of biofilm detachment promoted by the fluidized state. Results indicate that

a finite loading of free cells establishes even under operating conditions that would promote wash out ofthe suspended biophase. The co-operative/competitive effects of free cellsand immobilized biofilm result

in rich bifurcational patterns of the steady state solutions of the governing equations, which have beeninvestigated in the phase plane of the process parameters. Direct simulation under selected operating

conditions confirms the importance of the dynamical equilibrium establishing between the immobilizedand the suspended biophase and highlights the effect of the initial value of the biofilm loading on the

dynamical pattern. 2008 Elsevier B.V. All rights reserved.

1. Introduction

The study of microbial cell aggregates has long been stimulatedby issues related to the formation of biofilm in medical devices,prevention and control of infection diseases and bio-fouling

in process plants. Conversely, the potential of using microbialaggregates/biofilms as effective tools for achieving enhanced pro-ductivity in bioprocesses has also been demonstrated (e.g. in thefood industry) and has stimulated additional research effort aimed

at expanding their use in bioreactor technology.Formation of aggregates makes it possible to obtain cultures

characterized by cell densities (biomass per unit volume) muchlarger than those commonly harvested in liquid broths (Nicolella

et al., 2000; Qureshi et al., 2005). The possibility of achieving largecell densities is an attractive feature which can be exploited in anumber of applications of fermentation to improve process inten-sification. Currently, immobilization of cells and membrane reactor

technology are the two common pathways to achieve high-densityconfined cell cultures in either discontinuous or flow reactors.Immobilization of cells by adhesion on natural, typically inexpen-sive, supports is the first step for the production of biofilm. Proper

Corresponding author. Tel.: +39 0817682541; fax: +39 0815936936.

E-mail address: [email protected](A. Marzocchella).

choice of granular supports, together with careful selection of the

microbial strain, arethe keys to the successful design of multiphasebiofilm reactors.

The establishment of solid-supported biomass loading in con-tinuous biofilmreactorsresults fromthe competitionbetween cells

adhesion/growth on the granular carrier and detachment of biofilmfragments from the granules. These phenomena have been thesubject of extensive investigation with the aim of identifying thecontrolling parameters (Gjaltema et al., 1997a,b; Horn et al., 2003).

The build-up of the biofilm results from the simultaneous progressof the following processes: (i) cells adhesion on the carrier surface,an event that is stronglyinfluencedby thenature of theinteractionsbetween the cell wall and the carrier surface; (ii) adhered biomass

growth and simultaneous production of extracellular matrix; (iii)detachment of biomass fragments and/or individual cells from thegrowing biofilm. Biofilm detachment, in turn, is related to thefollowingmechanisms:(i) shear-induced erosion, resulting in con-

tinuous loss of cells, either isolated or in small aggregates, from thecarrier surface; (ii) removal of large patches of biofilm, a randomprocess which is usually referred to as sloughing; (iii) abrasionof biofilm associated with particle-to-particle friction and/or colli-

sions (Nicolella et al., 1997). The relevance of erosion and abrasionto biofilm detachment has been highlighted in previous studies onmultiphase biofilm reactors (Gjaltema et al., 1997a,b). Moreover,biofilm development is critically influenced by dilution rate, which

0168-1656/$ see front matter 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.jbiotec.2008.04.003
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296 M.E. Russo et al. / Journal of Biotechnology 135 (2008) 295303

Nomenclature

A dimensionless adhesion coefficientd particle diameter (m)

D = Q/V dilution rate (1/h)Da Damkohler numberDc diffusivity of suspended cells (m2/s)

G dimensionless biofilm detachment coefficientki, km dimensionless growth kinetic parametersKa adhesion rate coefficient (s1, h1)Kc mass transfer rate coefficient (m/s)Kd biofilm detachment rate coefficient (h

1)

Ki, Km parameters of cells growth kinetics (g/L)Q liquid volumetric flow rate (L/h)ra suspended cells adhesion rate (kg/(m3 s), g/(L h))rc suspended cells transfer rate (kg/(m3 s))

rd biofilm detachment rate (g/(L h))s dimensionless substrate concentrationS substrate concentration (g/L)

Sin substrate feeding concentration (g/L)

Sh Sherwood number

t time (h)V reaction volume (L)

Vs solid phase volume (L)

Vl liquid phase volume (L)x dimensionless suspended cells concentrationX suspended cells concentration (g/L, kg/m3)

y dimensionless biofilm concentration

Y biofilm concentration (g/L)Z biomass yield (gDM/g)

Symbols

specific phenol consumption rate (g/(gDM h))

max specific phenol consumption kinetic parameter(g/(gDM h))

dimensionless time dimensionless specific growth rate

Subscripts

0 initial valueDM dry matterss steady states

regulates the competition between immobilized and free cells for

the common substrates. Only thin biofilms can be grown withhydraulic residence times longer than the time-scale of substrateconsumption (Tang and Fan, 1987; Tijhuis et al., 1994), due to the

continuous subtraction of the substrate by suspended cells. Con-versely, operating conditions corresponding to wash out of the freecells enhance biofilm growth by reducing the chance that immo-bilized cells are exposed to substrate-starving conditions and byimproving the competitionbetween the rates of biofilmproduction

and detachment.Modelling of biofilm reactors has been addressed in the recent

literature. Kiranmai et al. (2005) and Manuel et al. (2007) followeddifferent approaches to modelling biofilmgrowth in different types

of multiphase reactors.A few studies have also addressed the bifur-cational/dynamical patterns of chemostats (Pavlou, 1999; Pinheiroet al., 2004; Zhang andHenson, 2001), though the complex interac-tion between free and immobilized biophases in biofilm reactors,

which is the main focus of the present study, has never been anal-

ysed.

A mathematical model of a multiphase continuous biofilmreactor is herebypresented, aimed at characterizingthe basic bifur-cational patterns and dynamical behaviour of the system. The

proposed model embodies the key features of the phenomenologyof the granular-supported biofilm: adhesion of cells onto the car-rier surface, growth of attached cells,biofilm detachment.Adhesionand detachment rates are modelled as linear functions of the sus-

pended cells and biofilm volumetric concentrations, respectively(Gjaltema et al., 1997a,b; Rijnaarts et al., 1995; Telgmann et al.,2004). The kinetic ofPseudomonas sp. OX1 cell growth is assumedto be substrate-inhibited (Viggiani et al., 2006).

2. Theory

2.1. General description of the model

A mathematical model was developed to analyse the dynamicand bifurcational behaviour of a continuous multiphase biofilmreactor. The model was specifically applied to the growth of

Pseudomonas sp. OX1 on phenol. The growth process is substrate-inhibited. Immobilization of Pseudomonas sp. OX1 on granular

materials of different nature (silica sand, pumice stone) is effec-tive and typically yields a patchy biofilm of moderate thickness

(Viggiani et al., 2006; Alfieri, 2006).The reactor flow pattern assumed in the model was that of an

internal loop airlift with pneumatic mixing of both the liquid andthe solid phases (Chisti, 1989), the last consisting in biofilm sup-

ported by granular solids. The phenol-bearing liquid stream wasfed continuously to the reactor at pre-fixed volumetric flow rate(Q) and phenol concentration (Sin).

Biofilm detachment in a three-phase airlift reactor has been

characterized by Gjaltema et al. (1997b). These authors sampledfree cells from the liquid phase of a bench scale airlift reactor oper-ated under wash out conditions with Pseudomonas putida biofilm.Once plated, cell colonies displayed phenotypical features equal

to those of the immobilized cells. In particular, a much enhancedextra-cellular polymer production was recorded. Based on thesefindings, the analysis of the long-term behaviour of a biofilm reac-tor is performed by assuming that the metabolic pathways (growth

kinetics andyield coefficients) of both freelysuspendedand immo-bilized cells conform to that of the immobilized phenotype.

Gjaltema et al. (1997a) reported that the main contributionto biofilm detachment in a three phase airlift reactor was due to

particle-to-particle collisions and that the detachment rate can bemodelled as first-order with respect to particle concentration. Theseverity of collisions depends on the surface properties of theparti-cles involved (bare particles or bio-coated particles). In the present

study, the solid phase is characterizedby formation of a thin patchybiofilm (thickness much smaller than the bare particle diameter)

because this is the morphology observed for Pseudomonas sp. OX1adhering on granular solids (Viggiani et al., 2006). For this reason

the detachment can be related to collisions between particles hav-ing statistically similar surface characteristics (roughness, biofilmcoverage and mechanical strength). Accordingly the linear expres-sion proposed by Gjaltema et al. (1997a) for the detachment rate

(rd) can be safely assumed.

2.1.1. Model assumptions

The model relies on the following assumptions:

(1) the reactor is well-mixed with respect to liquid, gas and solidphases as a consequence of the typical flow pattern established

in airlift reactors (Chisti, 1989);


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M.E. Russo et al. / Journal of Biotechnology 135 (2008) 295303 297

(2) phenol is the rate-limiting substrate for both free and immobi-lized cells. Aeration of the medium in the airlift reactor is largeenough to make oxygen not a limiting substrate. Metabolism of

nitrogen is not considered in the model;(3) both free and immobilized cells are characterized by substrate-

inhibited growth kinetics. Parameters of the Andrews modelandthe biomass yield coefficienthavebeen taken from Viggiani

et al. (2006). Two different choices were made as regards theselection of the relevant growth yield/kinetic parameters forthe freely suspended phase:(a) the growth yield/kinetics of the freely suspended cells was

taken equal to that of the immobilized biophase in com-putations directed to assess the long-term performance ofthe biofilm reactor as well as in the bifurcational analysis.This choice was justified by the previously reported find-

ings due to Gjaltema et al. (1997b) as regards the prevailingphenotypical nature of freely suspended cells;

(b) the growth yield/kinetics of the freely suspended cells wastaken equal to that of planktonic cells in the analysis for the

purpose of computations aimed at simulating the short-term dynamics of the reactor start-up (with an inoculumof suspended cells and sterile carrier). This choice was jus-tified by the prevailingly unmodified phenotypical nature

of freely suspended cells over short-term operation of thebiofilm reactor.

(4) adhesion of suspended cells onto the carrier surface wasmodelled as a first-order kinetics with respect to the cells con-

centration in theliquid phase.The adhesion coefficient (Ka)wasestimated from studies available in the literature (Rijnaarts etal., 1993, 1995) as described in the next section.

(5) the biofilm detachment rate (rd) was modelled as a first-order

kinetics with respect to the biofilm volumetric concentration(Y).

(6) the biofilm detached fragments were assumed to be composedof free cells, in agreement with assumption 3);

(7) resistance to mass transfer in the boundary layer around sin-

glecells and biocoated particles was neglected when comparedwith intrinsic reaction kinetics;

(8) the solid carrier was impervious and, accordingly, biofilm

attachment could only take place at the external surface of

the carrier particles. The diffusional resistance to substratemass transfer across the biofilm was neglected when comparedwith intrinsic reaction kinetics. The validity of this hypothesis

was checked by an order-of-magnitude assessment of sub-stratemass transfer rate acrossthe biofilm. Theassessment wasreferred to biofilm loadings of 10 g/L and smaller with a biofilmapparent density of=1000kg/m3. The solid carrier consists of

particles of about 650m size dispersed at a volumetric frac-tion of about 3% in the liquid phase. With these figures, thebiofilm thickness turns out to be of the order of 10 m, thinenough to ensure even distribution of the substrate across the

biofilm.

2.1.2. Model equations

Mass balance equations onthe substrate,the biofilmand thefreecells extended to the volume of the liquid phase (Vl) are reportedin Table 1 together with the relevant constitutive equations. Themeaning of the symbols is reported in the Nomenclature section.

Modelling of adhesion was based on studies due to Rijnaarts

et al. (1993, 1995) and Chen and Strevett (2001). Rijnaarts et al.(1995) indicated that, for different Pseudomonas strains interacting

with either hydrophobic or hydrophilic surfaces, the adhesion fluxcan be taken equal to the diffusional flux of cells (under stagnant

conditions) toward the carrier surface, neglecting activation energybarriers. Accordingly, the rate of cells adhesion (ra) can be assumedequal to the rate of cells transfer (rc):

ra = rc = Kc6

d

VsVl

X (1)

where Kc is the mass transfer coefficient between the bulk of the

liquid phase and the carrier particle, X is the volumetric concen-tration of suspended cells and d the particle diameter. Kc can becalculated as

Kc =ShDc

d (2)

where Sh is the Sherwood number (Knudsen et al., 1999) and Dcthe diffusivity of cells in theliquid phase. From Eq. (1), an adhesion

Table 1

Model equations: mass balance, boundary conditions and constitutive equations

Mass balance

SubstratedS

dt= D (Sin S)(S) X(S) Y (T.I.1)

Free cellsdX

dt= D X+(S) ZX+ rd ra (T.I.2)

Immobilized cells (biofilm)dY

dt= (S) Z Y rd + ra (T.I.3)

Initial conditions t= t0

X=X0 0S = S0 0Y= Y0 0

(T.I.4)

Constitutive equations

Specific phenol consumption ratea

(S) =max S

Km + S + S2 /Ki(T.I.5)

max (g/(gDM h)) 0.14b 0.96c

Km (g/L) 0.041b 0.31c

Ki (g/L) 0.1b 0.13c

Suspended cells adhesion rate ra(X) = Ka X (T.I.6)

Biofilm detachment rate rd(Y) = Kd Y (T.I.7)

a Viggiani et al. (2006).b Biofilm.

c Free cells.


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Table 2

Dimensionless model equations: mass balance, boundary conditions and constitu-tive equations

Mass balance

Substrateds

d=

1

Da (1 s) x y (T.II.1)

Free cellsdx

d=

1

Dax + x + G y A x (T.II.2)

Immobilized cells (biofilm) dyd

= y G y+A x (T.II.3)

Initial conditions = 0

x =x0 0

s = s0 0

y =y0 0

(T.II.4)

Constitutive equations

Specific phenol consumption rate =s

km + s+ s2/ki(T.II.5)

Suspended cells adhesion rate A x (T.II.6)

Biofilm detachment rate G y (T.II.7)

rate constant Ka of a first-order kinetics can be defined as

Ka = Kc6

d

VsVl

(3)

Eqs. (T.I.1)(T.I.3) (Table 1) are a setof ordinary differentialequa-

tions whose initial conditions are reported in Table 1. Eq. (T.I.5)describes the substrate-inhibited model (S) for phenol specificconsumption rate, yielding a maximum rate of about 0.06 g/gDM hat S= 0.06 g/L for biofilm cells and 0.23 g/gDM h at S= 0.2g/L for the

planktonic cellsused in the primary inoculum. Equationsand initialconditions can be re-written in dimensionless form by introducingthe following variables:

s = S/Sin x =X/(Sin Z) y = Y/(Sin Z)= t (max Z)

ki = Ki/Sin km = Km/Sin

Da = max Z/D A = Ka/(max Z) G = Kd/(max Z)

(4)

Table 2 reports the material balance equations and the initialconditions in terms of dimensionless variables.

The range of each dimensionless variable/parameter is as fol-

lows:

The dimensionless variables s and x vary in the interval [0,1]because both are normalized against the maximum value

accepted. In the framework of the present model the dimensionless vari-

able y is taken positive and unbounded. The latter assumptionbecomes unrealistic in the limiting case G0 (a case that is out-

side the scope of the present study) as it would imply unlimitedbiofilm growth. In reality, biofilm growth is a self-regulating pro-cess, even in the absence of particle collisions and/orshear stress,due to the onset of severe diffusional limitations to mass transfer

of substrates across the biofilm itself (see also Section 3.1). The Damkohler number is positive and unbounded. For the

growth kinetics of biofilm the value ofDa corresponding to washout of free cells (Da = max/((kmki)

1/2) in a chemostat would be

2.3, while it would be 4.1 when referred to the growth kineticsof planktonic cells used in the primary inoculum ofPseudomonas

sp. OX1.

For steady state conditions the set of ordinary differential Eqs.(T.II.1)(T.II.3) reduces to a set of nonlinear algebraic equations:

0 =1

Da (1 sss)

sss

km + sss + sss2/kixss

sss

km + sss + sss2/kiyss

0 = 1

Daxss +

sss

km + sss + sss2/kixss + G yss A xss

0 =sss

km + sss + sss2

/ki

yss G yss +A xss

(5)

in the unknowns sss, yss and xss.

2.2. Computational methods

Model computations were directed to the following goals: (i)identification of steady state solutions and analysis of bifurcations;

(ii) characterization of the dynamic behaviour of the system. Com-putations wereperformed usinga continuation algorithm providedby the toolbox MATCONT associated with Matlab. Analysis of thestabilityof steadysolutions was carried outby evaluating theeigen-

values of the Jacobian matrix associated with the linearized formof system in Eq. (5) (Hale and Kocak, 1991).

3. Results and discussion

3.1. Analysis of steady state solutions

Theadhesionrateof suspendedcellsontothe carrier surface wascalculated from Eqs. (1)(3) assuming d =650m and Vs/Vl =0.03.

Typical values of cells diffusivity (Dc) ofPseudomonas strains wasreported by Rijnaarts et al. (1993) to be about 41013 m2/s. TheSherwood number (Sh) was taken equal to2 (Knudsen et al., 1999).With these values:

Kad = 0.0012 0.0003h1 (6)

yielding a dimensionless adhesion rateA = 0.0110.003. The bifur-

cational patterns turned out to be negligibly affected by changes inthe adhesion coefficient within its confidence interval. This resultis in agreement with the findings reported by Tijhuis et al. (1994).

The cell adhesion resulted to play a key role in the early stage ofbiofilm formation starting from sterile solid carriers.

Fig. 1 reports results of base case computations. They refer tosteady solutions for the dimensionless variables sss, xss and yss as

the value of the dimensionless detachment coefficient G is varied.Since biofilm growth is strongly favoured by operation under con-ditionsthat promote wash outof thesuspendedcells(Tangand Fan,1987; Tijhuis et al., 1994), base case computations corresponded to

a value of the Damkohler number (Da = 1.5) at which the dilutionrate exceeded the maximum specific rate of cell growth. The sub-strate concentration in the feeding, Sin, was fixed at 0.6g/L. Fig. 1

shows that a multiplicity of steady states occurs, and the num-ber of solutions is reported in the figure as roman numerals. Acritical value for the parameter G limits the field of existence ofnon-trivial solutions, that is (sss < 1, xss > 0, yss > 0). For small valuesofG two solutions are physically consistent (region II): the triplet

(sss = 1,xss =yss = 0) corresponding to the trivial case of no substrateconsumption, and the solution corresponding to conversion of thesubstrate coupled to suspended and attached biomass production(continuous line). Fig. 1 shows that as G increases, a transcritical

bifurcation marks the appearance of a third steady solution char-acterized by substrate consumption. This behaviour is observed forvalues of G falling between the transcritical bifurcation and theupper limit (III).For even larger values ofG only the trivial solution

is acceptable (I). The stability analysis of the steady states indicates

that: (i) high conversion solutions (branches with high biomass


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Fig. 1. Steady states as a function of the detachment coefficient G for Da = 1.5,

Sin =0.6g/L,A = 0.011. Stable nodesand saddle-nodes aredescribedby solidand dot-dashed lines, respectively. Roman numeralsindicate the number of steady solutions

within the regions limited by dashed vertical lines.

concentrations and low substrate concentrations) are stable (sta-

ble nodes, solid lines) within their field of existence (II, III); (ii) lowconversion solutions, appearing after transcritical bifurcation (III),are unstable (saddle-nodes, dot-dashed lines) and (iii) the trivialsolution is unstable for small values of G (II) and changes into a

stable solution beyond the transcritical bifurcation (III, I).

The reported findings can be interpreted as follows:

- Under thedesirable conditions of largedilution rate (D > Dwash out)and high substrate loading(Sin > (kski)

1/2), the systemapproachesa regime characterized by stable biofilm concentration, pro-

vided that the detachment rate is smaller than a critical value.The steady biofilm concentration reflects the balanced effect ofbiofilm growth and detachment. The number and type of bifur-cations occurring areprimarily determined by the growth kinetic

model. Once the functionality of growth kinetic is known bifur-cational pattern assessment provides a tool for the selection of aproper value of the detachment coefficient.

- The smaller the detachment rate, the larger the biofilm stable

concentration,yss. However, a finite non-zero valueof the detach-ment rate is always associated with operation of an airlift reactor.

- Operational points falling in regions II or III of bifurcational mapscorrespond to different ways to approach the steady state. In

region II, the establishment of stable biofilm can be achievedstarting from a primary inoculum of planktonic cells. In regionIII, the establishment of stable biofilm requires a finite amountof biofilm as inoculum at the start-up of the reactor. This implies

that for certain values of the detachment rates (corresponding tooperation in region III) the start-up of the biofilm reactor cannotbe simply achieved by exposure of sterile carriers to a primaryinoculum of planktonic cells. Moreover, operating in this region

implies that the steady regime would irreversibly be lost if, forsome reason, biofilm concentration suddenly decreases below athreshold value. These feature, which have important implica-tions on the stable and trouble-freeoperation of a biofilm reactor,

will further be addressed in Section 3.3.

It is remarkable that the stable high conversion solution showsa non-zero concentration of free cells (xss), despite the base case

investigated refers to a dilution rate larger than the maximumgrowth rate. This result reflects the fact that continuous renewalof free cells in the medium supplemented by biofilm detachment,either as isolated cells or as aggregates, effectively contrasts wash

out of suspended biomass (see Section 2.1). As a consequence,

the competition for substrate between immobilized and free cellscannot be entirely ruled out by the choice of operating at largedilution rates. This feature affects the performance of the bioreac-

tor along two paths. On one hand, the persistent establishment ofa freely suspended biophase even under wash out conditions pro-vides additional biological activity for the bioprocess to occur. Onthe other hand, the very same occurrence of a persistent free cell

loading may hinder the build up of the immobilized biophase dueto competition for the common substrate.

It should be noted that the application of the continuation algo-rithm suggested that periodic regimes never establish within the

investigated range of the parameters.

3.2. Bifurcational analysis

In this section the effect ofDa and Sin on the bifurcational pat-terns is analysed. Fig. 2 reports the operating diagrams relative tothe steady state values of the dimensionless variables s, x, and y asa function ofG taking Da as a parameter. Solid lines indicate stable

steady states (nodes) and dot-dashed lines unstable (saddle-type)steady states.

All Da values considered correspondto dilution rateslarger thanthe maximum value of the specific growth rate. Observing the dia-

gramin theplane(yss, G), it appears that increasing Da upto 2.2 (i.e.decreasing the dilution rate until it approaches the wash out limit)the stable steady state shows a lower biofilm concentration. Fur-thermore, an increase of the upper limiting value of G is observed

as Da is increased. From the operational standpoint this implies

that, for large values of the detachment rate, operation with a sta-


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Fig. 2. Steady states as a function of the detachment coefficient G for Sin =0.6g/L,

A = 0.011 for different values ofDa. Stable nodes and saddle-nodes are described by

solid and dot-dashed lines, respectively.

ble biofilm loading could be achieved by decreasing the dilutionrate (while keeping D > Dwash out). However, this strategy should beapplied cautiously since operation with large detachment rate and

small dilution rate is inherently more susceptible to perturbation-induced loss of stability (see map yss vs. G in Fig. 2).

Fig. 3 reports operating diagrams yss vs. G for different sub-strate inlet concentration Sin. Decreasing Sin from 0.5 to 0.06 g/L a

smaller steady biofilm concentration can be reached for any given

G. For 0.6 < Sin < 0.06 the bifurcational pattern described in Section

Fig. 3.Steady states of biofilm concentration

yss as a function of the detachmentcoefficient G for Da = 1.5,A = 0.011 fordifferent valuesofSin. Stablenodes andsaddle-

nodes are described by solid and dot-dashed lines, respectively.

3.1 remains unchanged (see Fig. 2). IfSin is set at the value 0.06 g/Lonly two solutions appear, whatever the detachment coefficient.The transcritical bifurcation occurs at the maximum G. Only two

steadysolutions areacceptablealso for Sin < 0.06g/L. This behaviouris related to the nonlinearity of the dependence of growth kinet-ics on S. It is recalled that 0.06 g/L is the substrate concentrationcorresponding to the maximum of(S) for biofilm cells. If thereac-

tor is fed with a liquid stream containing Sin 0.06 g/L, becauseS< Sin, only one value of(S) would be acceptable and multiplicityof non-trivial solutions would be prevented.

It is well known that continuous chemostats provide optimalsolution formaximization of the productivity of substrate-inhibitedcell growth processes (Villadsen, 1999), as large throughputs canbeensured also when inlet substrate concentration approaches theinhibition limit. This model suggests that the same conclusion can

be drawn for the attached cells.Fig. 4 reports the loci of bifurcations in the plane (Sin, G) for

Da =1.5 and Da = 2.2. Both diagrams show a horizontal line thatrepresents the upper limiting value for G above which only the

trivial solution satisfies Eq. (5). The curves represent loci of tran-scritical bifurcations. Regions of the plane limited by these curvesare characterized by different number of steady solutions (romannumerals). Increasing Da the upper limit of G increases and the

curve of transcritical bifurcation modifies accordingly. The effect

of substrate inhibition is reflected also by this diagram as a conse-


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Fig. 4. Bifurcation diagrams in the phase-plane (Sin, G). Roman numerals indicate

the number of steady solutions.

quence of the balance between adhesion/growth and detachment

of the biofilm. Therefore, the value of G which marks the con-dition of transcritical bifurcation is a function of Sin, showing anon-monotonic trend similar to that of(S).

3.3. Analysis of the reactor dynamics

Model computations have been directed to the characterizationof thetransient response of thebioreactor tounsteadyforcing. Fig.5

reports results of transient simulations of the bioreactor in termsof the dimensionless variables s, x and y vs. dimensionless time. It must be noted that, consistently with the choice of unstruc-tured microbial growth kinetics, the predicted transient effects areonly those associated with the reactor dynamics and not on the

dynamics of microbial metabolism in response to a change of theenvironment.

A typical start-up operation of the biofilm reactor was firstsimulated. To this end, a likely actual start-up condition of an

industrial biofilm reactor has been determined by a finite inocu-lum of previously developed granular biofilm (e.g. from laboratoryor pilot scale-plant). The amount of the inoculum can be fixed,as function of detachment rate, on the basis of the solution dia-

grams like those in Fig. 1. In order to keep the environment

favourable to biofilm growth, wash out of suspended cells (from

Fig. 5. Transient behaviour of the reactor (plots of Eq. (T.II) for Da = 1.5, Sin =0.6g/L,

A = 0.011 and dif ferent values of the detachment coefficient G. Dot-dashed lines

G = 0.07, solid lines G = 0.16. Initial conditions are s0 = 1, x0 = 0, y0 =0.5.

detached biofilm debris) is ensured by large dilution rate. For

the same reason, large substrate concentration in the inlet liq-uid stream has been hypothesized. So, values of Da and Sin havebeen fixed at, respectively, 1.5 and 0.6 g/L. Two values of G have

been considered for this computation, G = 0.07 and G = 0.16. Theset of initial conditions reproduces start-up of the reactor as s0 = 1,x0 = 0, y0 = 0.5, that is substrate concentration equal to that in thefeeding stream, no inoculum of free cells and sufficient inoculumof attached biofilm. The simulation shows that a stable steady

state is approached, consistently with results in Fig. 1. Increas-ing G up to 0.16, the slope of free cell concentration vs. timeprofile increases within a limited time interval with respect tothe case ofG = 0.07. This dynamic simulation provides a different

way to show the colonisation of the liquid phase by the detachedbiofilm.

A different start-up procedure has been simulated by assum-ing that sterile granular solids are loaded into the reactor fed with

a culture of native (i.e. non-phenotypically modified) suspended

cells. In this case, according to the previous analysis, it is required


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Fig. 6. Transient behaviour of the reactor starting from free cells in liquid phase

andsterilegranular solids. Kinetic parametersadopted in dimensionlessEqs. T.II are

that of free cell reported in Table 1. Da = 2, Sin = 0.6 g/L, A = 0.002 and G = 0.1. Initial

conditions are s0 = 1, x0 =0.5, y0 = 0.

that G be small enough to make the system fall into region II of thebifurcational map. As already specified, the dynamic simulationhas

beenperformed in this case by assigning kinetic parameters forthegrowth of the suspended phase typical of planktonic cells from theprimary inoculum (see Table 1). Results of the dynamic simulationare reported in Fig. 6. Notably, the validity of this simulation is lim-

ited to very early stage of biofilm build-up in a three-phase airlift

reactor. Simulations extended over longer times (until full stabi-

lization of the operation of the reactor) would require taking intoaccount the progress of phenotypical modifications of cells associ-ated with immobilization and the corresponding change of growth

yield/kinetic parameters. These features, which imply inclusion ofa structured kinetic model of cell growth and of a population bal-ance on a community of living cells, will be addressed in furtherdevelopments of the present study.

4. Conclusions

A dynamical model has been set up to analyse the transientbehaviour of a three-phase biofilmreactor belonging to theinternalloop airlift typology. A specific feature of the model is represented

by the close focus on biofilm detachment phenomena and theirrelevance to the establishment of a dynamic equilibrium betweenfree and immobilized biophases. Substrate-inhibited consumptionof phenol by Pseudomonas sp. OX1 has been considered. A linear

dependence of biofilm detachment rate on biofilm loading wasassumed to model the effect of abrasion due to particle collisions.Adhesion of suspended cells on the surface of the carrier has beenmodelled using a first-order kinetics with respect to free cells con-

centration, consistent with findings reported in the literature onadhesion ofPseudomonas cells.

Bifurcational analysis of the steady state solutions indicates thatno biofilm can be observed in the reactor at the steady regime

and no substrate conversion can be obtained if the detachmentcoefficient exceeds a given threshold. Below this threshold mul-tiple stable steady states can be established and the bifurcationalpatterns depend on values of the dilution rate and feed substrate

concentration. The solution diagrams as a function of detachmentcoefficient suggest that, once the severity of biofilm abrasion isknown, the larger the dilution rate and the substrate inlet concen-tration, the more reliable the operation in terms of biofilm stability.

An important feature of the steady operation of the reactor is that,even when wash out is promoted by large volumetric feed rates, a

freelysuspended biophase is always present as free cells are contin-uously renewed by detachment of immobilized biophase. On one

hand, the persistence of a freely suspended biophase even underwash out conditions enhances the productivity of the bioreactor.On the other hand, thisfeature implies that competition of free andimmobilized cells for the common substrate cannot be ruled out

simply by operating at high dilution rate. This, in turn, might neg-atively affect the development of the stable and extensive biofilmloading that would be required for highly intensified operation ofthe bioreactor. These two features have to be carefully balanced by

proper selection of operating conditions and, more specifically, bytuning the biofilm detachment rate.

Dynamical simulations of the transient behaviour of the biore-actor confirm the basic features of the steady operation. The

dynamical evolution of the freely suspended and of the immobi-lized biophases are linked to each other by the process of biofilmdetachment.Results of simulationsindicate that the dynamicalpat-terns are dominated by the initial value of the biofilm loading.

Growth of biofilm on sterile solids from an inoculum of suspendedplanktonic cells can only be achieved if the detachment rate issmaller than a given threshold. If this is not the case, a prescribedminimum amount of biofilm is needed as inoculum at the start-up

of the reactor, a feature that makes addition of planktonic cells tothe culture irrelevant.

Acknowledgements

The support to model computation of Mr. Domenico Saldala-

macchia e Mr. Luca Bifulco is gratefully acknowledged. Financial


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support from both Centro Regionale di Competenza in Biotecnolo-gie IndustrialiBioteknetand MIUR Programmidi Ricerca Scientificadi Rilevante Interesse Nazionale is also acknowledged.

References

Alfieri, F., 2006. Bioconversione di composti aromatici mediante microrganismiliberi o immobilizzati [Ph.D.]. Universita di Napoli Federico II, Napoli.

Chen, G., Strevett, K.A., 2001. Impact of surface thermodynamic on bacterial trans-port. Environmental Microbiology 3 (4), 237245.

Chisti, M.Y., 1989. Airlift Bioreactors. Elsevier.Gjaltema, A., Vinke, J.L., van Loosdrecht, M.C.M., Heijnen, J.J., 1997a. Abra-

sion of suspended biofilm pellets in airlift reactors: importance ofshape, and particle concentrations. Biotechnology and Bioengineering 53,8889.

Gjaltema, A., vander Marel, N.,van Loosdrecht, M.C.M.,Heijnen, J.J.,1997b.Adhesionandbiofilmdevelopment onsuspended carriersin airliftreactors:hydrodinamicconditions versus surface characteristics. Biotechnology and Bioengineering 55(6), 880889.

Hale, J.K., Kocak, H., 1991. Dynamics and Bifurcations. Springer Verlag .Horn, H., Reiff, H., Morgenroth, E., 2003. Simulation of growth and detachment in

biofilm systems under defined hydrodynamic conditions. Biotechnology andBioengineering 81 (5), 608617.

Kiranmai,D., Jyothirmai,A., Murty,C.V.S.,2005.Determinationof kinetic parametersin fixed-film bio-reactors: an inverse problem approach. Biochemical Engineer-ing Journal 23, 7383.

Knudsen, J.G., Hottel, H.C., Sarofim, A.F., Wankat, P.C., Knaebel, K.S., 1999. Heat andmass transfer. In: Perry, R.H., Green, D.W., Maloney, J.O. (Eds.), Perrys ChemicalEnginners Handbook, seventh ed. McGraw Hill.

Manuel, C.M., Nunes, O.C., Melo, L.F., 2007. Dynamics of drinking water biofilm inflow/non-flow conditions. Water Research (41), 551562.

Nicolella, C.,Chiarle, S.,Di Felice, R.,Rovatti,M., 1997. Mechanisms of biofilm detach-ment in fluidized bed reactors. Water Science and Technology 36, 229235.

Nicolella, C., vanLoosdrecht, M.C.M., Heijnen, J.J.,2000. Wastewater treatmentwithparticulate biofilm reactors. Journal of Biotechnology 80, 133.

Pavlou, S., 1999. Computing operating diagrams of bioreactors. Journal of Biotech-nology 71, 716.

Pinheiro, I.O., De Souza Jr., M.B., Lopes, C.E., 2004.The dynamic behaviour of aeratedcontinuous flowstirredtank bioreactor. Mathematicaland ComputerModelling39, 541566.

Qureshi, N., Annous, B.A., Ezeji, T.C., Karcher, P., Maddox, I.S., 2005. Biofilm reac-tors for industrial bioconversion processes: employing potential of enhancedreaction rates. Microbial Cell Factory 4. (24).

Rijnaarts, H.H.M.,Norde, W.,Bouwer,E.J., Lyklema,J., Zehnder, A.J.B., 1995. Reversibil-ity and mechanism of bacterial adhesion. Colloids and Surfaces B 4, 522.

Rijnaarts, H.H.M., Norde, W., Bouwer, E.J., Lyklema, J., Zehnder, A.J.B., 1993. Bac-terial adhesion under static and dynamic conditions. Applied EnvironmentalMicrobiology 59, 32553265.

Tang, W.T., Fan, L.S., 1987. Steady state phenol degradation in a draft-tube,gasliquidsolid fluidized-bed bioreactor. AIChe Journal 33 (2), 239249.

Telgmann, U.,Horn, H., Morgenroth, E., 2004.Influence of growthhistoryon slough-ing and erosion from biofilms. Water Research 38, 36713684.

Tijhuis, L., van Loosdrecht, M.C.M., Heijnen, J.J., 1994. Formation and growth ofheterotrophic aerobic biofilms on small suspended particles in airlift reactors.Biotechnology and Bioengineering 28, 595608.

Viggiani, A., Olivieri,G., Siani, L., Di Donato,A., Marzocchella,A., Salatino,P.,Barbieri,P., Galli, E., 2006. An airlift biofilm reactor for the biodegradation of phenol byPseudomonas stutzeri OX1. Journal of Biotechnology 123, 464477.

Villadsen, J., 1999. On the optimal design and control of a biodegradation pro-

cess with substrate inhibition kinetics. Industrial and Engineering ChemistryResearch 38 (3), 660666.Zhang,Y., Henson,M.A.,2001. Bifurcation analysis ofcontinuousbiochemicalreactor

models. Biotechnology Progress 17 (4), 647.

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