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Bifurcational and dynamical analysis of a continuous biofilm reactor
Journal of Biotechnology 135 (2008) 295–303
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Bifurcational and dynamical analysis of a continuous biofilm reactor M.E. Russo, P.L. Maffettone, A. Marzocchella ∗ , P. Salatino Dipartimento di Ingegneria Chimica, Universit` a 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 2008 Accepted 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 means of immobilized cells of Pseudomonas sp. OX1 whose metabolic activity was previously characterized (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 by Pseudomonas stutzeri OX1. Journal of Biotechnology 123, 464–477). The model embodies the key processes relevant to the reactor performance, with a particular 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 of the suspended biophase. The co-operative/competitive effects of free cells and immobilized biofilm result in rich bifurcational patterns of the steady state solutions of the governing equations, which have been investigated in the phase plane of the process parameters. Direct simulation under selected operating conditions confirms the importance of the dynamical equilibrium establishing between the immobilized and 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 stimulated by 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 microbial aggregates/biofilms as effective tools for achieving enhanced productivity in bioprocesses has also been demonstrated (e.g. in the food 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) much larger than those commonly harvested in liquid broths (Nicolella et al., 2000; Qureshi et al., 2005). The possibility of achieving large cell densities is an attractive feature which can be exploited in a number of applications of fermentation to improve process intensification. Currently, immobilization of cells and membrane reactor technology are the two common pathways to achieve high-density confined cell cultures in either discontinuous or flow reactors. Immobilization of cells by adhesion on natural, typically inexpensive, 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). 0168-1656/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jbiotec.2008.04.003
choice of granular supports, together with careful selection of the microbial strain, are the keys to the successful design of multiphase biofilm reactors. The establishment of solid-supported biomass loading in continuous biofilm reactors results from the competition between cells adhesion/growth on the granular carrier and detachment of biofilm fragments from the granules. These phenomena have been the subject of extensive investigation with the aim of identifying the controlling parameters (Gjaltema et al., 1997a,b; Horn et al., 2003). The build-up of the biofilm results from the simultaneous progress of the following processes: (i) cells adhesion on the carrier surface, an event that is strongly influenced by the nature of the interactions between 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 the growing biofilm. Biofilm detachment, in turn, is related to the following mechanisms: (i) shear-induced ‘erosion’, resulting in continuous loss of cells, either isolated or in small aggregates, from the carrier surface; (ii) removal of large patches of biofilm, a random process which is usually referred to as “sloughing”; (iii) “abrasion” of biofilm associated with particle-to-particle friction and/or collisions (Nicolella et al., 1997). The relevance of erosion and abrasion to biofilm detachment has been highlighted in previous studies on multiphase biofilm reactors (Gjaltema et al., 1997a,b). Moreover, biofilm development is critically influenced by dilution rate, which
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Nomenclature A d D = Q/V Da Dc G ki , km Ka Kc Kd Ki , Km Q ra rc rd s S Sin Sh t V Vs Vl x X y Y Z
dimensionless adhesion coefficient particle diameter (m) dilution rate (1/h) ¨ Damkohler number diffusivity of suspended cells (m2 /s) dimensionless biofilm detachment coefficient dimensionless growth kinetic parameters adhesion rate coefficient (s−1 , h−1 ) mass transfer rate coefficient (m/s) biofilm detachment rate coefficient (h−1 ) parameters of cells growth kinetics (g/L) liquid volumetric flow rate (L/h) suspended cells adhesion rate (kg/(m3 s), g/(L h)) suspended cells transfer rate (kg/(m3 s)) biofilm detachment rate (g/(L h)) dimensionless substrate concentration substrate concentration (g/L) substrate feeding concentration (g/L) Sherwood number time (h) reaction volume (L) solid phase volume (L) liquid phase volume (L) dimensionless suspended cells concentration suspended cells concentration (g/L, kg/m3 ) dimensionless biofilm concentration biofilm concentration (g/L) 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 value DM dry matter ss steady states
regulates the competition between immobilized and free cells for the common substrates. Only thin biofilms can be grown with hydraulic residence times longer than the time-scale of substrate consumption (Tang and Fan, 1987; Tijhuis et al., 1994), due to the continuous subtraction of the substrate by suspended cells. Conversely, operating conditions corresponding to wash out of the free cells enhance biofilm growth by reducing the chance that immobilized cells are exposed to substrate-starving conditions and by improving the competition between the rates of biofilm production and detachment. Modelling of biofilm reactors has been addressed in the recent literature. Kiranmai et al. (2005) and Manuel et al. (2007) followed different approaches to modelling biofilm growth in different types of multiphase reactors. A few studies have also addressed the bifurcational/dynamical patterns of chemostats (Pavlou, 1999; Pinheiro et al., 2004; Zhang and Henson, 2001), though the complex interaction between free and immobilized biophases in biofilm reactors, which is the main focus of the present study, has never been analysed.
A mathematical model of a multiphase continuous biofilm reactor is hereby presented, aimed at characterizing the basic bifurcational patterns and dynamical behaviour of the system. The proposed model embodies the key features of the phenomenology of the granular-supported biofilm: adhesion of cells onto the carrier surface, growth of attached cells, biofilm detachment. Adhesion and detachment rates are modelled as linear functions of the suspended cells and biofilm volumetric concentrations, respectively (Gjaltema et al., 1997a,b; Rijnaarts et al., 1995; Telgmann et al., 2004). The kinetic of Pseudomonas sp. OX1 cell growth is assumed to be substrate-inhibited (Viggiani et al., 2006).
2. Theory 2.1. General description of the model A mathematical model was developed to analyse the dynamic and bifurcational behaviour of a continuous multiphase biofilm reactor. The model was specifically applied to the growth of Pseudomonas sp. OX1 on phenol. The growth process is substrateinhibited. Immobilization of Pseudomonas sp. OX1 on granular materials of different nature (silica sand, pumice stone) is effective 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 and the solid phases (Chisti, 1989), the last consisting in biofilm supported by granular solids. The phenol-bearing liquid stream was fed 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 sampled free cells from the liquid phase of a bench scale airlift reactor operated 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 enhanced extra-cellular polymer production was recorded. Based on these findings, the analysis of the long-term behaviour of a biofilm reactor is performed by assuming that the metabolic pathways (growth kinetics and yield coefficients) of both freely suspended and immobilized cells conform to that of the immobilized phenotype. Gjaltema et al. (1997a) reported that the main contribution to biofilm detachment in a three phase airlift reactor was due to particle-to-particle collisions and that the detachment rate can be modelled as first-order with respect to particle concentration. The severity of collisions depends on the surface properties of the particles involved (bare particles or bio-coated particles). In the present study, the solid phase is characterized by formation of a thin patchy biofilm (thickness much smaller than the bare particle diameter) because this is the morphology observed for Pseudomonas sp. OX1 adhering on granular solids (Viggiani et al., 2006). For this reason the detachment can be related to collisions between particles having statistically similar surface characteristics (roughness, biofilm coverage and mechanical strength). Accordingly the linear expression 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 solid phases as a consequence of the typical flow pattern established in airlift reactors (Chisti, 1989);
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(2) phenol is the rate-limiting substrate for both free and immobilized cells. Aeration of the medium in the airlift reactor is large enough 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 substrateinhibited growth kinetics. Parameters of the Andrews model and the biomass yield coefficient have been taken from Viggiani et al. (2006). Two different choices were made as regards the selection of the relevant growth yield/kinetic parameters for the freely suspended phase: (a) the growth yield/kinetics of the freely suspended cells was taken equal to that of the immobilized biophase in computations directed to assess the long-term performance of the biofilm reactor as well as in the bifurcational analysis. This choice was justified by the previously reported findings due to Gjaltema et al. (1997b) as regards the prevailing phenotypical nature of freely suspended cells; (b) the growth yield/kinetics of the freely suspended cells was taken equal to that of planktonic cells in the analysis for the purpose of computations aimed at simulating the shortterm dynamics of the reactor start-up (with an inoculum of suspended cells and sterile carrier). This choice was justified by the prevailingly unmodified phenotypical nature of freely suspended cells over short-term operation of the biofilm reactor. (4) adhesion of suspended cells onto the carrier surface was modelled as a first-order kinetics with respect to the cells concentration in the liquid phase. The adhesion coefficient (Ka ) was estimated from studies available in the literature (Rijnaarts et al., 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 composed of free cells, in agreement with assumption 3); (7) resistance to mass transfer in the boundary layer around single cells and biocoated particles was neglected when compared with intrinsic reaction kinetics; (8) the solid carrier was impervious and, accordingly, biofilm attachment could only take place at the external surface of
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the carrier particles. The diffusional resistance to substrate mass transfer across the biofilm was neglected when compared with intrinsic reaction kinetics. The validity of this hypothesis was checked by an order-of-magnitude assessment of substrate mass transfer rate across the biofilm. The assessment was referred to biofilm loadings of 10 g/L and smaller with a biofilm apparent density of ∼ =1000 kg/m3 . The solid carrier consists of particles of about 650 m size dispersed at a volumetric fraction of about 3% in the liquid phase. With these figures, the biofilm thickness turns out to be of the order of 10 m, thin enough to ensure even distribution of the substrate across the biofilm. 2.1.2. Model equations Mass balance equations on the substrate, the biofilm and the free cells extended to the volume of the liquid phase (Vl ) are reported in Table 1 together with the relevant constitutive equations. The meaning 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 flux can be taken equal to the diffusional flux of cells (under stagnant conditions) toward the carrier surface, neglecting activation energy barriers. Accordingly, the rate of cells adhesion (ra ) can be assumed equal to the rate of cells transfer (rc ): ra = rc = Kc
6 Vs X d Vl
(1)
where Kc is the mass transfer coefficient between the bulk of the liquid phase and the carrier particle, X is the volumetric concentration of suspended cells and ‘d’ the particle diameter. Kc can be calculated as Kc =
ShDc d
(2)
where Sh is the Sherwood number (Knudsen et al., 1999) and Dc the diffusivity of cells in the liquid phase. From Eq. (1), an adhesion
Table 1 Model equations: mass balance, boundary conditions and constitutive equations Mass balance Substrate
dS = D · (Sin − S) − (S) · X − (S) · Y dt
(T.I.1)
Free cells
dX = −D · X + (S) · Z · X + rd − ra dt
(T.I.2)
Immobilized cells (biofilm)
dY = (S) · Z · Y − rd + ra dt
(T.I.3)
X =X
t = t0 →
Initial conditions
Constitutive equations (S) = a
Specific phenol consumption rate
0 ≥0 S = S0 ≥ 0 Y = Y0 ≥ 0
(T.I.4)
max · S Km + S + S 2 /Ki
max (g/(gDM h)) Km (g/L) Ki (g/L)
(T.I.5) 0.14b 0.041b 0.1b
0.96c 0.31c 0.13c
Suspended cells adhesion rate
ra (X) = Ka · X
(T.I.6)
Biofilm detachment rate
rd (Y ) = Kd · Y
(T.I.7)
a b c
Viggiani et al. (2006). Biofilm. Free cells.
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Table 2 Dimensionless model equations: mass balance, boundary conditions and constitutive equations
Substrate
ds 1 · (1 − s) − ω · x − ω · y = Da d
(T.II.1)
Free cells
1 dx ·x+ω·x+G·y−A·x =− Da d
(T.II.2)
Immobilized cells (biofilm)
dy =ω·y−G·y+A·x d
(T.II.3)
x=x
= 0 →
Constitutive equations Specific phenol consumption rate
ω=
0 ≥0 s = s0 ≥ 0 y = y0 ≥ 0
s km + s + s2 /ki
sss sss 1 · (1 − sss ) − · xss − · yss Da km + sss + sss 2 /ki km + sss + sss 2 /ki 1 sss 0=− · xss + G · yss − A · xss · xss + Da km + sss + sss 2 /ki sss 0= · yss − G · yss + A · xss km + sss + sss 2 /ki (5) 0=
Mass balance
Initial conditions
For steady state conditions the set of ordinary differential Eqs. (T.II.1)–(T.II.3) reduces to a set of nonlinear algebraic equations:
in the unknowns sss , yss and xss . (T.II.4)
2.2. Computational methods (T.II.5)
Suspended cells adhesion rate
A·x
(T.II.6)
Biofilm detachment rate
G·y
(T.II.7)
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. Computations were performed using a continuation algorithm provided by the toolbox MATCONT associated with Matlab® . Analysis of the stability of steady solutions was carried out by evaluating the eigenvalues of the Jacobian matrix associated with the linearized form of system in Eq. (5) (Hale and Koc¸ak, 1991).
rate constant Ka of a first-order kinetics can be defined as Ka = Kc
3. Results and discussion
6 Vs d Vl
(3) 3.1. Analysis of steady state solutions
Eqs. (T.I.1)–(T.I.3) (Table 1) are a set of ordinary differential equations whose initial conditions are reported in Table 1. Eq. (T.I.5) describes the substrate-inhibited model (S) for phenol specific consumption rate, yielding a maximum rate of about 0.06 g/gDM h at S = 0.06 g/L for biofilm cells and 0.23 g/gDM h at S = 0.2 g/L for the planktonic cells used in the primary inoculum. Equations and initial conditions can be re-written in dimensionless form by introducing the 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)
(4) G = Kd /(max · Z)
Table 2 reports the material balance equations and the initial conditions in terms of dimensionless variables. The range of each dimensionless variable/parameter is as follows: • 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 variable y is taken positive and unbounded. The latter assumption becomes unrealistic in the limiting case G → 0 (a case that is outside the scope of the present study) as it would imply unlimited biofilm growth. In reality, biofilm growth is a self-regulating process, even in the absence of particle collisions and/or shear 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 of Da corresponding to wash out of free cells (Da = max /((km ki )1/2 ) in a chemostat would be 2.3, while it would be 4.1 when referred to the growth kinetics of planktonic cells used in the primary inoculum of Pseudomonas sp. OX1.
The adhesion rate of suspended cells onto the carrier surface was calculated from Eqs. (1)–(3) assuming d = 650 m and Vs /Vl = 0.03. Typical values of cells diffusivity (Dc ) of Pseudomonas strains was reported by Rijnaarts et al. (1993) to be about 4 × 10−13 m2 /s. The Sherwood number (Sh) was taken equal to 2 (Knudsen et al., 1999). With these values: Kad = 0.0012 ± 0.0003 h−1
(6)
yielding a dimensionless adhesion rate A = 0.011 ± 0.003. The bifurcational patterns turned out to be negligibly affected by changes in the adhesion coefficient within its confidence interval. This result is in agreement with the findings reported by Tijhuis et al. (1994). The cell adhesion resulted to play a key role in the early stage of biofilm formation starting from sterile solid carriers. Fig. 1 reports results of base case computations. They refer to steady 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 conditions that promote wash out of the suspended cells (Tang and Fan, 1987; Tijhuis et al., 1994), base case computations corresponded to ¨ a value of the Damkohler number (Da = 1.5) at which the dilution rate exceeded the maximum specific rate of cell growth. The substrate concentration in the feeding, Sin , was fixed at 0.6 g/L. Fig. 1 shows that a multiplicity of steady states occurs, and the number of solutions is reported in the figure as roman numerals. A critical value for the parameter G limits the field of existence of non-trivial solutions, that is (sss < 1, xss > 0, yss > 0). For small values of G two solutions are physically consistent (region II): the triplet (sss = 1, xss = yss = 0) corresponding to the trivial case of no substrate consumption, and the solution corresponding to conversion of the substrate 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 characterized by substrate consumption. This behaviour is observed for values of G falling between the transcritical bifurcation and the upper limit (III). For even larger values of G 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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- Under the desirable conditions of large dilution rate (D > Dwash out ) and high substrate loading (Sin > (ks ki )1/2 ), the system approaches a regime characterized by stable biofilm concentration, provided that the detachment rate is smaller than a critical value. The steady biofilm concentration reflects the balanced effect of biofilm growth and detachment. The number and type of bifurcations occurring are primarily determined by the growth kinetic model. Once the functionality of growth kinetic is known bifurcational pattern assessment provides a tool for the selection of a proper value of the detachment coefficient. - The smaller the detachment rate, the larger the biofilm stable concentration, yss . However, a finite non-zero value of the detachment rate is always associated with operation of an airlift reactor. - Operational points falling in regions II or III of bifurcational maps correspond to different ways to approach the steady state. In region II, the establishment of stable biofilm can be achieved starting from a primary inoculum of planktonic cells. In region III, the establishment of stable biofilm requires a finite amount of biofilm as inoculum at the start-up of the reactor. This implies that for certain values of the detachment rates (corresponding to operation in region III) the start-up of the biofilm reactor cannot be simply achieved by exposure of sterile carriers to a primary inoculum of planktonic cells. Moreover, operating in this region implies that the steady regime would irreversibly be lost if, for some reason, biofilm concentration suddenly decreases below a threshold value. These feature, which have important implications on the stable and trouble-free operation of a biofilm reactor, will further be addressed in Section 3.3. It is remarkable that the stable high conversion solution shows a non-zero concentration of free cells (xss ), despite the base case investigated refers to a dilution rate larger than the maximum growth rate. This result reflects the fact that continuous renewal of 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 cells cannot be entirely ruled out by the choice of operating at large dilution rates. This feature affects the performance of the bioreactor along two paths. On one hand, the persistent establishment of a freely suspended biophase even under wash out conditions provides additional biological activity for the bioprocess to occur. On the other hand, the very same occurrence of a persistent free cell loading may hinder the build up of the immobilized biophase due to competition for the common substrate. It should be noted that the application of the continuation algorithm suggested that periodic regimes never establish within the investigated range of the parameters. 3.2. Bifurcational analysis
Fig. 1. Steady states as a function of the detachment coefficient G for Da = 1.5, Sin = 0.6 g/L, A = 0.011. Stable nodes and saddle-nodes are described by solid and dotdashed lines, respectively. Roman numerals indicate the number of steady solutions within the regions limited by dashed vertical lines.
concentrations and low substrate concentrations) are stable (stable nodes, solid lines) within their field of existence (II, III); (ii) low conversion solutions, appearing after transcritical bifurcation (III), are unstable (saddle-nodes, dot-dashed lines) and (iii) the trivial solution 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:
In this section the effect of Da and Sin on the bifurcational patterns is analysed. Fig. 2 reports the operating diagrams relative to the steady state values of the dimensionless variables s, x, and y as a function of G 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 correspond to dilution rates larger than the maximum value of the specific growth rate. Observing the diagram in the plane (yss , G), it appears that increasing Da up to 2.2 (i.e. decreasing the dilution rate until it approaches the wash out limit) the stable steady state shows a lower biofilm concentration. Furthermore, 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. 3. Steady states of biofilm concentration yss as a function of the detachment coefficient G for Da = 1.5, A = 0.011 for different values of Sin . Stable nodes and saddlenodes are described by solid and dot-dashed lines, respectively.
Fig. 2. Steady states as a function of the detachment coefficient G for Sin = 0.6 g/L, A = 0.011 for different values of Da. Stable nodes and saddle-nodes are described by solid and dot-dashed lines, respectively.
ble biofilm loading could be achieved by decreasing the dilution rate (while keeping D > Dwash out ). However, this strategy should be applied cautiously since operation with large detachment rate and small dilution rate is inherently more susceptible to perturbationinduced loss of stability (see map yss vs. G in Fig. 2). Fig. 3 reports operating diagrams yss vs. G for different substrate 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
3.1 remains unchanged (see Fig. 2). If Sin is set at the value 0.06 g/L only two solutions appear, whatever the detachment coefficient. The transcritical bifurcation occurs at the maximum G. Only two steady solutions are acceptable also for Sin < 0.06 g/L. This behaviour is related to the nonlinearity of the dependence of growth kinetics on S. It is recalled that 0.06 g/L is the substrate concentration corresponding to the maximum of (S) for biofilm cells. If the reactor is fed with a liquid stream containing Sin ≤ 0.06 g/L, because S < Sin , only one value of (S) would be acceptable and multiplicity of non-trivial solutions would be prevented. It is well known that continuous chemostats provide optimal solution for maximization of the productivity of substrate-inhibited cell growth processes (Villadsen, 1999), as large throughputs can be ensured also when inlet substrate concentration approaches the inhibition 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 that represents the upper limiting value for G above which only the trivial solution satisfies Eq. (5). The curves represent loci of transcritical bifurcations. Regions of the plane limited by these curves are characterized by different number of steady solutions (roman numerals). 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 condition of transcritical bifurcation is a function of Sin , showing a non-monotonic trend similar to that of (S). 3.3. Analysis of the reactor dynamics Model computations have been directed to the characterization of the transient response of the bioreactor to unsteady forcing. Fig. 5 reports results of transient simulations of the bioreactor in terms of the dimensionless variables s, x and y vs. dimensionless time . It must be noted that, consistently with the choice of unstructured microbial growth kinetics, the predicted transient effects are only those associated with the reactor dynamics and not on the dynamics of microbial metabolism in response to a change of the environment. A typical start-up operation of the biofilm reactor was first simulated. To this end, a likely actual start-up condition of an industrial biofilm reactor has been determined by a finite inoculum of previously developed granular biofilm (e.g. from laboratory or pilot scale-plant). The amount of the inoculum can be fixed, as function of detachment rate, on the basis of the solution diagrams 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 = 0.6 g/L, A = 0.011 and different 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 liquid stream has been hypothesized. So, values of Da and Sin have been 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. The set 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 the feeding stream, no inoculum of free cells and sufficient inoculum of attached biofilm. The simulation shows that a stable steady state is approached, consistently with results in Fig. 1. Increasing G up to 0.16, the slope of free cell concentration vs. time profile increases within a limited time interval with respect to the case of G = 0.07. This dynamic simulation provides a different way to show the colonisation of the liquid phase by the detached biofilm. A different start-up procedure has been simulated by assuming 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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lization of the operation of the reactor) would require taking into account the progress of phenotypical modifications of cells associated with immobilization and the corresponding change of growth yield/kinetic parameters. These features, which imply inclusion of a structured kinetic model of cell growth and of a population balance on a community of living cells, will be addressed in further developments of the present study. 4. Conclusions
Fig. 6. Transient behaviour of the reactor starting from free cells in liquid phase and sterile granular solids. Kinetic parameters adopted in dimensionless Eqs. 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 the bifurcational map. As already specified, the dynamic simulation has been performed in this case by assigning kinetic parameters for the growth of the suspended phase typical of planktonic cells from the primary inoculum (see Table 1). Results of the dynamic simulation are reported in Fig. 6. Notably, the validity of this simulation is limited to very early stage of biofilm build-up in a three-phase airlift reactor. Simulations extended over longer times (until full stabi-
A dynamical model has been set up to analyse the transient behaviour of a three-phase biofilm reactor belonging to the internal loop airlift typology. A specific feature of the model is represented by the close focus on biofilm detachment phenomena and their relevance to the establishment of a dynamic equilibrium between free and immobilized biophases. Substrate-inhibited consumption of phenol by Pseudomonas sp. OX1 has been considered. A linear dependence of biofilm detachment rate on biofilm loading was assumed to model the effect of abrasion due to particle collisions. Adhesion of suspended cells on the surface of the carrier has been modelled using a first-order kinetics with respect to free cells concentration, consistent with findings reported in the literature on adhesion of Pseudomonas cells. Bifurcational analysis of the steady state solutions indicates that no biofilm can be observed in the reactor at the steady regime and no substrate conversion can be obtained if the detachment coefficient exceeds a given threshold. Below this threshold multiple stable steady states can be established and the bifurcational patterns depend on values of the dilution rate and feed substrate concentration. The solution diagrams as a function of detachment coefficient suggest that, once the severity of biofilm abrasion is known, the larger the dilution rate and the substrate inlet concentration, 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 freely suspended biophase is always present as free cells are continuously renewed by detachment of immobilized biophase. On one hand, the persistence of a freely suspended biophase even under wash out conditions enhances the productivity of the bioreactor. On the other hand, this feature implies that competition of free and immobilized cells for the common substrate cannot be ruled out simply by operating at high dilution rate. This, in turn, might negatively affect the development of the stable and extensive biofilm loading that would be required for highly intensified operation of the bioreactor. These two features have to be carefully balanced by proper selection of operating conditions and, more specifically, by tuning the biofilm detachment rate. Dynamical simulations of the transient behaviour of the bioreactor confirm the basic features of the steady operation. The dynamical evolution of the freely suspended and of the immobilized biophases are linked to each other by the process of biofilm detachment. Results of simulations indicate that the dynamical patterns are dominated by the initial value of the biofilm loading. Growth of biofilm on sterile solids from an inoculum of suspended planktonic cells can only be achieved if the detachment rate is smaller than a given threshold. If this is not the case, a prescribed minimum amount of biofilm is needed as inoculum at the start-up of the reactor, a feature that makes addition of planktonic cells to the culture irrelevant. Acknowledgements The support to model computation of Mr. Domenico Saldalamacchia e Mr. Luca Bifulco is gratefully acknowledged. Financial
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support from both Centro Regionale di Competenza in Biotecnologie Industriali Bioteknet and MIUR Programmi di Ricerca Scientifica di Rilevante Interesse Nazionale is also acknowledged. References Alfieri, F., 2006. Bioconversione di composti aromatici mediante microrganismi liberi o immobilizzati [Ph.D.]. Universita` di Napoli ‘Federico II’, Napoli. Chen, G., Strevett, K.A., 2001. Impact of surface thermodynamic on bacterial transport. Environmental Microbiology 3 (4), 237–245. Chisti, M.Y., 1989. Airlift Bioreactors. Elsevier. Gjaltema, A., Vinke, J.L., van Loosdrecht, M.C.M., Heijnen, J.J., 1997a. Abrasion of suspended biofilm pellets in airlift reactors: importance of shape, and particle concentrations. Biotechnology and Bioengineering 53, 88–89. Gjaltema, A., van der Marel, N., van Loosdrecht, M.C.M., Heijnen, J.J., 1997b. Adhesion and biofilm development on suspended carriers in airlift reactors: hydrodinamic conditions versus surface characteristics. Biotechnology and Bioengineering 55 (6), 880–889. Hale, J.K., Koc¸ak, 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 and Bioengineering 81 (5), 608–617. Kiranmai, D., Jyothirmai, A., Murty, C.V.S., 2005. Determination of kinetic parameters in fixed-film bio-reactors: an inverse problem approach. Biochemical Engineering Journal 23, 73–83. Knudsen, J.G., Hottel, H.C., Sarofim, A.F., Wankat, P.C., Knaebel, K.S., 1999. Heat and mass transfer. In: Perry, R.H., Green, D.W., Maloney, J.O. (Eds.), Perry’s Chemical Enginners’ Handbook, seventh ed. McGraw Hill. Manuel, C.M., Nunes, O.C., Melo, L.F., 2007. Dynamics of drinking water biofilm in flow/non-flow conditions. Water Research (41), 551–562.
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Nicolella, C., Chiarle, S., Di Felice, R., Rovatti, M., 1997. Mechanisms of biofilm detachment in fluidized bed reactors. Water Science and Technology 36, 229–235. Nicolella, C., van Loosdrecht, M.C.M., Heijnen, J.J., 2000. Wastewater treatment with particulate biofilm reactors. Journal of Biotechnology 80, 1–33. Pavlou, S., 1999. Computing operating diagrams of bioreactors. Journal of Biotechnology 71, 7–16. Pinheiro, I.O., De Souza Jr., M.B., Lopes, C.E., 2004. The dynamic behaviour of aerated continuous flow stirred tank bioreactor. Mathematical and Computer Modelling 39, 541–566. Qureshi, N., Annous, B.A., Ezeji, T.C., Karcher, P., Maddox, I.S., 2005. Biofilm reactors for industrial bioconversion processes: employing potential of enhanced reaction rates. Microbial Cell Factory 4. (24). Rijnaarts, H.H.M., Norde, W., Bouwer, E.J., Lyklema, J., Zehnder, A.J.B., 1995. Reversibility and mechanism of bacterial adhesion. Colloids and Surfaces B 4, 5–22. Rijnaarts, H.H.M., Norde, W., Bouwer, E.J., Lyklema, J., Zehnder, A.J.B., 1993. Bacterial adhesion under static and dynamic conditions. Applied Environmental Microbiology 59, 3255–3265. Tang, W.T., Fan, L.S., 1987. Steady state phenol degradation in a draft-tube, gas–liquid–solid fluidized-bed bioreactor. AIChe Journal 33 (2), 239–249. Telgmann, U., Horn, H., Morgenroth, E., 2004. Influence of growth history on sloughing and erosion from biofilms. Water Research 38, 3671–3684. Tijhuis, L., van Loosdrecht, M.C.M., Heijnen, J.J., 1994. Formation and growth of heterotrophic aerobic biofilms on small suspended particles in airlift reactors. Biotechnology and Bioengineering 28, 595–608. 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 by Pseudomonas stutzeri OX1. Journal of Biotechnology 123, 464–477. Villadsen, J., 1999. On the optimal design and control of a biodegradation process with substrate inhibition kinetics. Industrial and Engineering Chemistry Research 38 (3), 660–666. Zhang, Y., Henson, M.A., 2001. Bifurcation analysis of continuous biochemical reactor models. Biotechnology Progress 17 (4), 647.
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Bifurcational and dynamical analysis of a continuous biofilm reactor M.E. Russo, P.L. Maffettone, A. Marzocchella ∗ , P. Salatino Dipartimento di Ingegneria Chimica, Universit` a 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 2008 Accepted 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 means of immobilized cells of Pseudomonas sp. OX1 whose metabolic activity was previously characterized (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 by Pseudomonas stutzeri OX1. Journal of Biotechnology 123, 464–477). The model embodies the key processes relevant to the reactor performance, with a particular 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 of the suspended biophase. The co-operative/competitive effects of free cells and immobilized biofilm result in rich bifurcational patterns of the steady state solutions of the governing equations, which have been investigated in the phase plane of the process parameters. Direct simulation under selected operating conditions confirms the importance of the dynamical equilibrium establishing between the immobilized and 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 stimulated by 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 microbial aggregates/biofilms as effective tools for achieving enhanced productivity in bioprocesses has also been demonstrated (e.g. in the food 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) much larger than those commonly harvested in liquid broths (Nicolella et al., 2000; Qureshi et al., 2005). The possibility of achieving large cell densities is an attractive feature which can be exploited in a number of applications of fermentation to improve process intensification. Currently, immobilization of cells and membrane reactor technology are the two common pathways to achieve high-density confined cell cultures in either discontinuous or flow reactors. Immobilization of cells by adhesion on natural, typically inexpensive, 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). 0168-1656/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jbiotec.2008.04.003
choice of granular supports, together with careful selection of the microbial strain, are the keys to the successful design of multiphase biofilm reactors. The establishment of solid-supported biomass loading in continuous biofilm reactors results from the competition between cells adhesion/growth on the granular carrier and detachment of biofilm fragments from the granules. These phenomena have been the subject of extensive investigation with the aim of identifying the controlling parameters (Gjaltema et al., 1997a,b; Horn et al., 2003). The build-up of the biofilm results from the simultaneous progress of the following processes: (i) cells adhesion on the carrier surface, an event that is strongly influenced by the nature of the interactions between 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 the growing biofilm. Biofilm detachment, in turn, is related to the following mechanisms: (i) shear-induced ‘erosion’, resulting in continuous loss of cells, either isolated or in small aggregates, from the carrier surface; (ii) removal of large patches of biofilm, a random process which is usually referred to as “sloughing”; (iii) “abrasion” of biofilm associated with particle-to-particle friction and/or collisions (Nicolella et al., 1997). The relevance of erosion and abrasion to biofilm detachment has been highlighted in previous studies on multiphase biofilm reactors (Gjaltema et al., 1997a,b). Moreover, biofilm development is critically influenced by dilution rate, which
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Nomenclature A d D = Q/V Da Dc G ki , km Ka Kc Kd Ki , Km Q ra rc rd s S Sin Sh t V Vs Vl x X y Y Z
dimensionless adhesion coefficient particle diameter (m) dilution rate (1/h) ¨ Damkohler number diffusivity of suspended cells (m2 /s) dimensionless biofilm detachment coefficient dimensionless growth kinetic parameters adhesion rate coefficient (s−1 , h−1 ) mass transfer rate coefficient (m/s) biofilm detachment rate coefficient (h−1 ) parameters of cells growth kinetics (g/L) liquid volumetric flow rate (L/h) suspended cells adhesion rate (kg/(m3 s), g/(L h)) suspended cells transfer rate (kg/(m3 s)) biofilm detachment rate (g/(L h)) dimensionless substrate concentration substrate concentration (g/L) substrate feeding concentration (g/L) Sherwood number time (h) reaction volume (L) solid phase volume (L) liquid phase volume (L) dimensionless suspended cells concentration suspended cells concentration (g/L, kg/m3 ) dimensionless biofilm concentration biofilm concentration (g/L) 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 value DM dry matter ss steady states
regulates the competition between immobilized and free cells for the common substrates. Only thin biofilms can be grown with hydraulic residence times longer than the time-scale of substrate consumption (Tang and Fan, 1987; Tijhuis et al., 1994), due to the continuous subtraction of the substrate by suspended cells. Conversely, operating conditions corresponding to wash out of the free cells enhance biofilm growth by reducing the chance that immobilized cells are exposed to substrate-starving conditions and by improving the competition between the rates of biofilm production and detachment. Modelling of biofilm reactors has been addressed in the recent literature. Kiranmai et al. (2005) and Manuel et al. (2007) followed different approaches to modelling biofilm growth in different types of multiphase reactors. A few studies have also addressed the bifurcational/dynamical patterns of chemostats (Pavlou, 1999; Pinheiro et al., 2004; Zhang and Henson, 2001), though the complex interaction between free and immobilized biophases in biofilm reactors, which is the main focus of the present study, has never been analysed.
A mathematical model of a multiphase continuous biofilm reactor is hereby presented, aimed at characterizing the basic bifurcational patterns and dynamical behaviour of the system. The proposed model embodies the key features of the phenomenology of the granular-supported biofilm: adhesion of cells onto the carrier surface, growth of attached cells, biofilm detachment. Adhesion and detachment rates are modelled as linear functions of the suspended cells and biofilm volumetric concentrations, respectively (Gjaltema et al., 1997a,b; Rijnaarts et al., 1995; Telgmann et al., 2004). The kinetic of Pseudomonas sp. OX1 cell growth is assumed to be substrate-inhibited (Viggiani et al., 2006).
2. Theory 2.1. General description of the model A mathematical model was developed to analyse the dynamic and bifurcational behaviour of a continuous multiphase biofilm reactor. The model was specifically applied to the growth of Pseudomonas sp. OX1 on phenol. The growth process is substrateinhibited. Immobilization of Pseudomonas sp. OX1 on granular materials of different nature (silica sand, pumice stone) is effective 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 and the solid phases (Chisti, 1989), the last consisting in biofilm supported by granular solids. The phenol-bearing liquid stream was fed 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 sampled free cells from the liquid phase of a bench scale airlift reactor operated 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 enhanced extra-cellular polymer production was recorded. Based on these findings, the analysis of the long-term behaviour of a biofilm reactor is performed by assuming that the metabolic pathways (growth kinetics and yield coefficients) of both freely suspended and immobilized cells conform to that of the immobilized phenotype. Gjaltema et al. (1997a) reported that the main contribution to biofilm detachment in a three phase airlift reactor was due to particle-to-particle collisions and that the detachment rate can be modelled as first-order with respect to particle concentration. The severity of collisions depends on the surface properties of the particles involved (bare particles or bio-coated particles). In the present study, the solid phase is characterized by formation of a thin patchy biofilm (thickness much smaller than the bare particle diameter) because this is the morphology observed for Pseudomonas sp. OX1 adhering on granular solids (Viggiani et al., 2006). For this reason the detachment can be related to collisions between particles having statistically similar surface characteristics (roughness, biofilm coverage and mechanical strength). Accordingly the linear expression 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 solid phases as a consequence of the typical flow pattern established in airlift reactors (Chisti, 1989);
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(2) phenol is the rate-limiting substrate for both free and immobilized cells. Aeration of the medium in the airlift reactor is large enough 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 substrateinhibited growth kinetics. Parameters of the Andrews model and the biomass yield coefficient have been taken from Viggiani et al. (2006). Two different choices were made as regards the selection of the relevant growth yield/kinetic parameters for the freely suspended phase: (a) the growth yield/kinetics of the freely suspended cells was taken equal to that of the immobilized biophase in computations directed to assess the long-term performance of the biofilm reactor as well as in the bifurcational analysis. This choice was justified by the previously reported findings due to Gjaltema et al. (1997b) as regards the prevailing phenotypical nature of freely suspended cells; (b) the growth yield/kinetics of the freely suspended cells was taken equal to that of planktonic cells in the analysis for the purpose of computations aimed at simulating the shortterm dynamics of the reactor start-up (with an inoculum of suspended cells and sterile carrier). This choice was justified by the prevailingly unmodified phenotypical nature of freely suspended cells over short-term operation of the biofilm reactor. (4) adhesion of suspended cells onto the carrier surface was modelled as a first-order kinetics with respect to the cells concentration in the liquid phase. The adhesion coefficient (Ka ) was estimated from studies available in the literature (Rijnaarts et al., 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 composed of free cells, in agreement with assumption 3); (7) resistance to mass transfer in the boundary layer around single cells and biocoated particles was neglected when compared with intrinsic reaction kinetics; (8) the solid carrier was impervious and, accordingly, biofilm attachment could only take place at the external surface of
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the carrier particles. The diffusional resistance to substrate mass transfer across the biofilm was neglected when compared with intrinsic reaction kinetics. The validity of this hypothesis was checked by an order-of-magnitude assessment of substrate mass transfer rate across the biofilm. The assessment was referred to biofilm loadings of 10 g/L and smaller with a biofilm apparent density of ∼ =1000 kg/m3 . The solid carrier consists of particles of about 650 m size dispersed at a volumetric fraction of about 3% in the liquid phase. With these figures, the biofilm thickness turns out to be of the order of 10 m, thin enough to ensure even distribution of the substrate across the biofilm. 2.1.2. Model equations Mass balance equations on the substrate, the biofilm and the free cells extended to the volume of the liquid phase (Vl ) are reported in Table 1 together with the relevant constitutive equations. The meaning 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 flux can be taken equal to the diffusional flux of cells (under stagnant conditions) toward the carrier surface, neglecting activation energy barriers. Accordingly, the rate of cells adhesion (ra ) can be assumed equal to the rate of cells transfer (rc ): ra = rc = Kc
6 Vs X d Vl
(1)
where Kc is the mass transfer coefficient between the bulk of the liquid phase and the carrier particle, X is the volumetric concentration of suspended cells and ‘d’ the particle diameter. Kc can be calculated as Kc =
ShDc d
(2)
where Sh is the Sherwood number (Knudsen et al., 1999) and Dc the diffusivity of cells in the liquid phase. From Eq. (1), an adhesion
Table 1 Model equations: mass balance, boundary conditions and constitutive equations Mass balance Substrate
dS = D · (Sin − S) − (S) · X − (S) · Y dt
(T.I.1)
Free cells
dX = −D · X + (S) · Z · X + rd − ra dt
(T.I.2)
Immobilized cells (biofilm)
dY = (S) · Z · Y − rd + ra dt
(T.I.3)
X =X
t = t0 →
Initial conditions
Constitutive equations (S) = a
Specific phenol consumption rate
0 ≥0 S = S0 ≥ 0 Y = Y0 ≥ 0
(T.I.4)
max · S Km + S + S 2 /Ki
max (g/(gDM h)) Km (g/L) Ki (g/L)
(T.I.5) 0.14b 0.041b 0.1b
0.96c 0.31c 0.13c
Suspended cells adhesion rate
ra (X) = Ka · X
(T.I.6)
Biofilm detachment rate
rd (Y ) = Kd · Y
(T.I.7)
a b c
Viggiani et al. (2006). Biofilm. Free cells.
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Table 2 Dimensionless model equations: mass balance, boundary conditions and constitutive equations
Substrate
ds 1 · (1 − s) − ω · x − ω · y = Da d
(T.II.1)
Free cells
1 dx ·x+ω·x+G·y−A·x =− Da d
(T.II.2)
Immobilized cells (biofilm)
dy =ω·y−G·y+A·x d
(T.II.3)
x=x
= 0 →
Constitutive equations Specific phenol consumption rate
ω=
0 ≥0 s = s0 ≥ 0 y = y0 ≥ 0
s km + s + s2 /ki
sss sss 1 · (1 − sss ) − · xss − · yss Da km + sss + sss 2 /ki km + sss + sss 2 /ki 1 sss 0=− · xss + G · yss − A · xss · xss + Da km + sss + sss 2 /ki sss 0= · yss − G · yss + A · xss km + sss + sss 2 /ki (5) 0=
Mass balance
Initial conditions
For steady state conditions the set of ordinary differential Eqs. (T.II.1)–(T.II.3) reduces to a set of nonlinear algebraic equations:
in the unknowns sss , yss and xss . (T.II.4)
2.2. Computational methods (T.II.5)
Suspended cells adhesion rate
A·x
(T.II.6)
Biofilm detachment rate
G·y
(T.II.7)
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. Computations were performed using a continuation algorithm provided by the toolbox MATCONT associated with Matlab® . Analysis of the stability of steady solutions was carried out by evaluating the eigenvalues of the Jacobian matrix associated with the linearized form of system in Eq. (5) (Hale and Koc¸ak, 1991).
rate constant Ka of a first-order kinetics can be defined as Ka = Kc
3. Results and discussion
6 Vs d Vl
(3) 3.1. Analysis of steady state solutions
Eqs. (T.I.1)–(T.I.3) (Table 1) are a set of ordinary differential equations whose initial conditions are reported in Table 1. Eq. (T.I.5) describes the substrate-inhibited model (S) for phenol specific consumption rate, yielding a maximum rate of about 0.06 g/gDM h at S = 0.06 g/L for biofilm cells and 0.23 g/gDM h at S = 0.2 g/L for the planktonic cells used in the primary inoculum. Equations and initial conditions can be re-written in dimensionless form by introducing the 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)
(4) G = Kd /(max · Z)
Table 2 reports the material balance equations and the initial conditions in terms of dimensionless variables. The range of each dimensionless variable/parameter is as follows: • 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 variable y is taken positive and unbounded. The latter assumption becomes unrealistic in the limiting case G → 0 (a case that is outside the scope of the present study) as it would imply unlimited biofilm growth. In reality, biofilm growth is a self-regulating process, even in the absence of particle collisions and/or shear 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 of Da corresponding to wash out of free cells (Da = max /((km ki )1/2 ) in a chemostat would be 2.3, while it would be 4.1 when referred to the growth kinetics of planktonic cells used in the primary inoculum of Pseudomonas sp. OX1.
The adhesion rate of suspended cells onto the carrier surface was calculated from Eqs. (1)–(3) assuming d = 650 m and Vs /Vl = 0.03. Typical values of cells diffusivity (Dc ) of Pseudomonas strains was reported by Rijnaarts et al. (1993) to be about 4 × 10−13 m2 /s. The Sherwood number (Sh) was taken equal to 2 (Knudsen et al., 1999). With these values: Kad = 0.0012 ± 0.0003 h−1
(6)
yielding a dimensionless adhesion rate A = 0.011 ± 0.003. The bifurcational patterns turned out to be negligibly affected by changes in the adhesion coefficient within its confidence interval. This result is in agreement with the findings reported by Tijhuis et al. (1994). The cell adhesion resulted to play a key role in the early stage of biofilm formation starting from sterile solid carriers. Fig. 1 reports results of base case computations. They refer to steady 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 conditions that promote wash out of the suspended cells (Tang and Fan, 1987; Tijhuis et al., 1994), base case computations corresponded to ¨ a value of the Damkohler number (Da = 1.5) at which the dilution rate exceeded the maximum specific rate of cell growth. The substrate concentration in the feeding, Sin , was fixed at 0.6 g/L. Fig. 1 shows that a multiplicity of steady states occurs, and the number of solutions is reported in the figure as roman numerals. A critical value for the parameter G limits the field of existence of non-trivial solutions, that is (sss < 1, xss > 0, yss > 0). For small values of G two solutions are physically consistent (region II): the triplet (sss = 1, xss = yss = 0) corresponding to the trivial case of no substrate consumption, and the solution corresponding to conversion of the substrate 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 characterized by substrate consumption. This behaviour is observed for values of G falling between the transcritical bifurcation and the upper limit (III). For even larger values of G 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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- Under the desirable conditions of large dilution rate (D > Dwash out ) and high substrate loading (Sin > (ks ki )1/2 ), the system approaches a regime characterized by stable biofilm concentration, provided that the detachment rate is smaller than a critical value. The steady biofilm concentration reflects the balanced effect of biofilm growth and detachment. The number and type of bifurcations occurring are primarily determined by the growth kinetic model. Once the functionality of growth kinetic is known bifurcational pattern assessment provides a tool for the selection of a proper value of the detachment coefficient. - The smaller the detachment rate, the larger the biofilm stable concentration, yss . However, a finite non-zero value of the detachment rate is always associated with operation of an airlift reactor. - Operational points falling in regions II or III of bifurcational maps correspond to different ways to approach the steady state. In region II, the establishment of stable biofilm can be achieved starting from a primary inoculum of planktonic cells. In region III, the establishment of stable biofilm requires a finite amount of biofilm as inoculum at the start-up of the reactor. This implies that for certain values of the detachment rates (corresponding to operation in region III) the start-up of the biofilm reactor cannot be simply achieved by exposure of sterile carriers to a primary inoculum of planktonic cells. Moreover, operating in this region implies that the steady regime would irreversibly be lost if, for some reason, biofilm concentration suddenly decreases below a threshold value. These feature, which have important implications on the stable and trouble-free operation of a biofilm reactor, will further be addressed in Section 3.3. It is remarkable that the stable high conversion solution shows a non-zero concentration of free cells (xss ), despite the base case investigated refers to a dilution rate larger than the maximum growth rate. This result reflects the fact that continuous renewal of 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 cells cannot be entirely ruled out by the choice of operating at large dilution rates. This feature affects the performance of the bioreactor along two paths. On one hand, the persistent establishment of a freely suspended biophase even under wash out conditions provides additional biological activity for the bioprocess to occur. On the other hand, the very same occurrence of a persistent free cell loading may hinder the build up of the immobilized biophase due to competition for the common substrate. It should be noted that the application of the continuation algorithm suggested that periodic regimes never establish within the investigated range of the parameters. 3.2. Bifurcational analysis
Fig. 1. Steady states as a function of the detachment coefficient G for Da = 1.5, Sin = 0.6 g/L, A = 0.011. Stable nodes and saddle-nodes are described by solid and dotdashed lines, respectively. Roman numerals indicate the number of steady solutions within the regions limited by dashed vertical lines.
concentrations and low substrate concentrations) are stable (stable nodes, solid lines) within their field of existence (II, III); (ii) low conversion solutions, appearing after transcritical bifurcation (III), are unstable (saddle-nodes, dot-dashed lines) and (iii) the trivial solution 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:
In this section the effect of Da and Sin on the bifurcational patterns is analysed. Fig. 2 reports the operating diagrams relative to the steady state values of the dimensionless variables s, x, and y as a function of G 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 correspond to dilution rates larger than the maximum value of the specific growth rate. Observing the diagram in the plane (yss , G), it appears that increasing Da up to 2.2 (i.e. decreasing the dilution rate until it approaches the wash out limit) the stable steady state shows a lower biofilm concentration. Furthermore, 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. 3. Steady states of biofilm concentration yss as a function of the detachment coefficient G for Da = 1.5, A = 0.011 for different values of Sin . Stable nodes and saddlenodes are described by solid and dot-dashed lines, respectively.
Fig. 2. Steady states as a function of the detachment coefficient G for Sin = 0.6 g/L, A = 0.011 for different values of Da. Stable nodes and saddle-nodes are described by solid and dot-dashed lines, respectively.
ble biofilm loading could be achieved by decreasing the dilution rate (while keeping D > Dwash out ). However, this strategy should be applied cautiously since operation with large detachment rate and small dilution rate is inherently more susceptible to perturbationinduced loss of stability (see map yss vs. G in Fig. 2). Fig. 3 reports operating diagrams yss vs. G for different substrate 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
3.1 remains unchanged (see Fig. 2). If Sin is set at the value 0.06 g/L only two solutions appear, whatever the detachment coefficient. The transcritical bifurcation occurs at the maximum G. Only two steady solutions are acceptable also for Sin < 0.06 g/L. This behaviour is related to the nonlinearity of the dependence of growth kinetics on S. It is recalled that 0.06 g/L is the substrate concentration corresponding to the maximum of (S) for biofilm cells. If the reactor is fed with a liquid stream containing Sin ≤ 0.06 g/L, because S < Sin , only one value of (S) would be acceptable and multiplicity of non-trivial solutions would be prevented. It is well known that continuous chemostats provide optimal solution for maximization of the productivity of substrate-inhibited cell growth processes (Villadsen, 1999), as large throughputs can be ensured also when inlet substrate concentration approaches the inhibition 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 that represents the upper limiting value for G above which only the trivial solution satisfies Eq. (5). The curves represent loci of transcritical bifurcations. Regions of the plane limited by these curves are characterized by different number of steady solutions (roman numerals). 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 condition of transcritical bifurcation is a function of Sin , showing a non-monotonic trend similar to that of (S). 3.3. Analysis of the reactor dynamics Model computations have been directed to the characterization of the transient response of the bioreactor to unsteady forcing. Fig. 5 reports results of transient simulations of the bioreactor in terms of the dimensionless variables s, x and y vs. dimensionless time . It must be noted that, consistently with the choice of unstructured microbial growth kinetics, the predicted transient effects are only those associated with the reactor dynamics and not on the dynamics of microbial metabolism in response to a change of the environment. A typical start-up operation of the biofilm reactor was first simulated. To this end, a likely actual start-up condition of an industrial biofilm reactor has been determined by a finite inoculum of previously developed granular biofilm (e.g. from laboratory or pilot scale-plant). The amount of the inoculum can be fixed, as function of detachment rate, on the basis of the solution diagrams 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 = 0.6 g/L, A = 0.011 and different 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 liquid stream has been hypothesized. So, values of Da and Sin have been 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. The set 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 the feeding stream, no inoculum of free cells and sufficient inoculum of attached biofilm. The simulation shows that a stable steady state is approached, consistently with results in Fig. 1. Increasing G up to 0.16, the slope of free cell concentration vs. time profile increases within a limited time interval with respect to the case of G = 0.07. This dynamic simulation provides a different way to show the colonisation of the liquid phase by the detached biofilm. A different start-up procedure has been simulated by assuming 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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lization of the operation of the reactor) would require taking into account the progress of phenotypical modifications of cells associated with immobilization and the corresponding change of growth yield/kinetic parameters. These features, which imply inclusion of a structured kinetic model of cell growth and of a population balance on a community of living cells, will be addressed in further developments of the present study. 4. Conclusions
Fig. 6. Transient behaviour of the reactor starting from free cells in liquid phase and sterile granular solids. Kinetic parameters adopted in dimensionless Eqs. 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 the bifurcational map. As already specified, the dynamic simulation has been performed in this case by assigning kinetic parameters for the growth of the suspended phase typical of planktonic cells from the primary inoculum (see Table 1). Results of the dynamic simulation are reported in Fig. 6. Notably, the validity of this simulation is limited to very early stage of biofilm build-up in a three-phase airlift reactor. Simulations extended over longer times (until full stabi-
A dynamical model has been set up to analyse the transient behaviour of a three-phase biofilm reactor belonging to the internal loop airlift typology. A specific feature of the model is represented by the close focus on biofilm detachment phenomena and their relevance to the establishment of a dynamic equilibrium between free and immobilized biophases. Substrate-inhibited consumption of phenol by Pseudomonas sp. OX1 has been considered. A linear dependence of biofilm detachment rate on biofilm loading was assumed to model the effect of abrasion due to particle collisions. Adhesion of suspended cells on the surface of the carrier has been modelled using a first-order kinetics with respect to free cells concentration, consistent with findings reported in the literature on adhesion of Pseudomonas cells. Bifurcational analysis of the steady state solutions indicates that no biofilm can be observed in the reactor at the steady regime and no substrate conversion can be obtained if the detachment coefficient exceeds a given threshold. Below this threshold multiple stable steady states can be established and the bifurcational patterns depend on values of the dilution rate and feed substrate concentration. The solution diagrams as a function of detachment coefficient suggest that, once the severity of biofilm abrasion is known, the larger the dilution rate and the substrate inlet concentration, 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 freely suspended biophase is always present as free cells are continuously renewed by detachment of immobilized biophase. On one hand, the persistence of a freely suspended biophase even under wash out conditions enhances the productivity of the bioreactor. On the other hand, this feature implies that competition of free and immobilized cells for the common substrate cannot be ruled out simply by operating at high dilution rate. This, in turn, might negatively affect the development of the stable and extensive biofilm loading that would be required for highly intensified operation of the bioreactor. These two features have to be carefully balanced by proper selection of operating conditions and, more specifically, by tuning the biofilm detachment rate. Dynamical simulations of the transient behaviour of the bioreactor confirm the basic features of the steady operation. The dynamical evolution of the freely suspended and of the immobilized biophases are linked to each other by the process of biofilm detachment. Results of simulations indicate that the dynamical patterns are dominated by the initial value of the biofilm loading. Growth of biofilm on sterile solids from an inoculum of suspended planktonic cells can only be achieved if the detachment rate is smaller than a given threshold. If this is not the case, a prescribed minimum amount of biofilm is needed as inoculum at the start-up of the reactor, a feature that makes addition of planktonic cells to the culture irrelevant. Acknowledgements The support to model computation of Mr. Domenico Saldalamacchia e Mr. Luca Bifulco is gratefully acknowledged. Financial
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