where specific rates are modelled by Monod equations with double limitation of .... differential equations with double boundary condition (species in biofilm ...
MODELLING BIOFILMS WITH CELLULAR AUTOMATA
Cristian Picioreanu FINAL REPORT
for the period 20 January 1996 - 20 April 1996
Research funded by European Environmental Research Organisation
1.The continuous approach on biofilm modelling
2.The discrete approach on biofilm modelling
3.Experimental evaluation of the biofilm structures
4.Summary of activities achieved
1. The continuous approach on biofilm modelling
Modelling the effect of oxygen concentration on nitrite accumulation in a biofilm airlift suspension reactor
This paper will be presented at the “IAWQ Special Conference on Biofilm Systems”, 27-30 August 1996, Copenhagen and then will be submitted to Water Science and Technology
MODELLING THE EFFECT OF OXYGEN CONCENTRATION ON NITRITE ACCUMULATION IN A BIOFILM AIRLIFT SUSPENSION REACTOR C. Picioreanu*, M.C.M. van Loosdrecht and J.J. Heijnen Department of Biochemical Engineering, Delft University of Technology, Julianalaan 67, 2628 BC Delft, The Netherlands; *Department of Chemical Engineering, University Politehnica of Bucharest, Splaiul Independentei 313, 77206 Bucharest, Romania
ABSTRACT For an integrated nitrification-denitrification processes, nitrite formation in the aerobic stage leads to big savings. Recently experimental observations (Garrido et al., 1996) have shown that it is possible to obtain full ammonium conversion with approximately 50 % nitrate and 50 % nitrite in the effluent of a biofilm airlift suspension reactor. With oxygen concentrations between 1 and 2 mg/l a maximum nitrite accumulation of 50 % was reached. Here we give a simple diffusion-reaction model describing these results. All the kinetic and mass transfer parameters were taken from the literature, except the mass transfer coefficient around the biofilm surface, which was fitted. The proposed model describes very well the measured data, despite the assumptions made. Using this model the influence of operational parameters was evaluated in order to establish ways to affect the NO2- concentration. None of these (surface loading, kl, pH) had a significant effect on NO2- accumulation. Controlling the oxygen concentration seems to be the most practical method to obtain optimal nitrification to nitrite in BAS reactors since this can be done simply by varying the superficial gas velocity or by partial recirculation of the off-gas.
KEYWORDS Air-lift reactor; biofilm; nitrification; nitrite accumulation; mathematical modelling; orthogonal collocation.
INTRODUCTION Biofilm reactors are more efficient for nitrification than the classical activated sludge process since immobilisation of biomass enables to operate the reactor at a high sludge age and at high volumetric loading rate. However, in biofilters and in the fludized bed reactors clogging usually occurs. One way to overcome these problems is the use of a biofilm air-lift suspension reactor (Tijhuis et al., 1992). Full ammonia conversion to nitrate is possible for loading rates up to 5 kg N/m3 reactor/day. In an integral nitrogen removal process (nitrification + denitrification), intermediate nitrite formation is of interest because:
(1) 25 % lower oxygen consumption in the aerobic stage implies 60 % energy savings, (2) in the anoxic stage the electron donor requirement is lower, and (3) nitrite denitrification rates are 1.5 to 2 times higher than with nitrate. Three main strategies for obtaining nitrite as the main product were proposed. The use of pure Nitrosomonas cultures immobilised in gels was proposed by Kokufuta et al. (1988) and Santos et al. (1992), but it is expensive and not very practical. Suthersand and Ganczarczyk (1986) used an activated sludge system in which pH is raised to get Nitrobacter inhibition. The system is however difficult to control because adaptation of the bacteria to high free ammonia concentrations occurs. The third possibility would be to control the dissolved oxygen level in the reactor. At lower oxygen concentrations the nitrite oxidation rate decreased more than the ammonia oxidation rate (Tanaka and Dunn, 1981; Tanaka and Dunn, 1982), but without full ammonia conversion. Recently experimental observations (Garrido et al., 1996) have proved that it is possible to obtain full ammonium conversion with approximately 50 % nitrate and 50 % nitrite in the effluent. With oxygen concentrations between 1 and 2 mg/l a maximum nitrite accumulation was reached. Moreover, the nitrite formation was stable during several months, but also during a short term experiment. Therefore, it was suggested that the influence of dissolved oxygen on nitrite accumulation is an intrinsic characteristic of the biofilms in the BAS reactor. Here we try to give a simple model formulation describing these results. Using this model the influence of operational parameters was evaluated in order to establish ways to affect the NO2- concentration.
BASIS FOR MODEL • The BAS reactor is continuously operated at a constant flow rate and loading rate. After each switch in dissolved oxygen concentration a steady-state in the concentration of soluble compounds was achieved. • The average particle diameter and biofilm thickness are constant, because the time scale of the experiment with variable dissolved oxygen is much smaller than the characteristic time of growth (biofilm in “frozen state”). Moreover, the biomass concentration profile in the biofilm depth is also considered constant. • Retention time distribution experiments showed that the liquid phase of the biofilm airlift suspension reactor is completely mixed (Garrido et al. 1996). • The proportion between ammonia and nitrite oxidisers in the biofilm was assumed based on a review by Wiesmann (1994). • For the processes occurring inside the biofilm we used the classical model with diffusion and reaction in a catalyst bead. • External mass transfer resistance was taken into account for all components. • Decay and endogenous processes have been neglected because they have no large influence on ammonia conversion. All parameters of the model are in the Table 1.
Stoichiometry We can write the global stoechiometry for growth of ammonium oxidising bacteria as: 1 1 1 − ( NH 4+ ) − (O2 ) − ... + (C5 H 7 NO2 ) + ( NO2− ) + ... = 0 YXN1 YXO1 YXN 1 and for growth of nitrite oxidising bacteria as: 1 1 1 − ( NO2− ) − (O2 ) − ... + (C5 H 7 NO2 ) + ( NO3− ) + ... = 0 YXN 2 YXO 2 YXN 2 Kinetics
(1)
(2)
The substrate conversion kinetics can be expressed as: rS1 = − qS1 CX1 (3) rS 2 = − q S 2 C X2 (4) where specific rates are modelled by Monod equations with double limitation of donor species (ammonia S1 or nitrous acid S2) and acceptor (oxygen). If the nitrogen substrates are present in high concentrations, both ammonia and HNO2 could be inhibitors and this is represented by the following equations: C NH3 CO q S1 = qm, S1 ⋅ ⋅ (5) 2 C NH3 KO1 + CO K S1 + C NH3 + Ki1 CHNO2 CO q S 2 = q m, S 2 ⋅ ⋅ (6) 2 CHNO2 KO2 + CO K S 2 + CHNO2 + Ki 2 Concentrations of nitrogen neutral species can be related to the concentrations of ionic species with the equilibrium equations: CNH 4 10 pH ⎛ 6344 ⎞ CNH 3 = with K a = exp⎜ (7, 8) ⎟ Ka ⎝ 273 + t ⎠
CHNO2 =
CNO2 Kb 10
with
pH
⎛ − 2300 ⎞ K b = exp⎜ ⎟ ⎝ 273 + t ⎠
(9, 10)
Component balances • in bulk liquid - the mass balance over the liquid phase for soluble components: Ci, in − Ci, l VR (11) = Φ i av Qin with Φi the mass flux between liquid and biofilm. • in biofilm - for each relevant component i we can write a mass balance, taking into account diffusion processes in a spherical geometry (with an effective diffusion coefficient Di,eff) and an overall reaction rate in the catalyst particle (ri): ⎛ d 2 C 2 dCi ⎞ (12) Di, eff ⋅ ⎜ 2 i + ⎟ + ri = 0 r dr ⎠ ⎝ dr where the net reaction rates are: NH4 : (13) rNH 4 = − q S1 C X1 YXN1 Y q S1 C X1 − XN 2 q S 2 C X 2 YXO1 YXO2 = q S1 C X1 − q S 2 C X 2
rO2 = −
(14)
NO2 :
rNO2
(15)
NO3 :
rNO3 = qS 2 C X 2
O2
:
(16)
The system of second order differential equations needs two sets of boundary conditions, one at the inner limit of the biofilm (carrier-biofilm interface), and the biofilm surface (biofilm-diffusion film): dCi =0 (zero fluxes at the carrier surface) (17) dr r = rc and Di, eff
dCi dr
r = rc + δ
(
= kl Ci, l − Ci
MODEL SOLVING
r = rc + δ
)=Φ
i
(equality of fluxes at the biofilm outer surface)
(18)
Method The mathematical model of the biofilm air-lift suspension reactor consists of a system of coupled parabolic differential equations with double boundary condition (species in biofilm balances) and a system of algebraic equations (species in liquid balances). In order to solve the model we have applied an orthogonal collocation method, as described in Finlayson (1972). Firstly, the model was made dimensionless by a series of substitutions. Concentrations were related to a reference concentration, for instance concentration of ammonium in the inlet flow, CNH4,in, so that ci = Ci / CNH4,in. Dimensionless coordinate was X = (r - rc) / δ so that collocation points could have values between 0 and 1. Discretisation of the dimensionless model results in a non-linear system of N-2 equations for each component balance in the collocation points: N ⎛ N ⎞ 2 2 ⎜⎜ ∑ B j , k ci, k + A c j = 2 .. N-1, i = 1 .. 4 (19) ∑ j , k i, k ⎟ ⎟ + ϕ i Ri, j = 0 for / δ r + X ⎝ k =1 ⎠ c j k =1 and boundary conditions: N
X=0
∑ A1, k ci, k = 0
for
i = 1 .. 4
(20)
− Da1 = 0
for
i = 1 (ammonium)
(21)
− Dai = 0
for
i = 2 .. 4
(22)
k =1
X=1
1 − c1, l
N
∑ AN , k c1, k
k =1
− ci, l
N
∑ AN , k ci, k
k =1
where external diffusion resistance leads to : 1 ci, l = ci, N + Bii
N
∑ AN , k ci, k
k =1
(23)
The system is now described by a set of dimensionless numbers: Thiele modulus ϕ, Damkohler number Da, Biot number Bi, dimensionless affinity and inhibition constants, and a geometric parameter which is the rc/δ ratio. For O2 dissolved in the liquid phase we have a constant (and measured) value, C2,l = Cdissolved oxygen. The calculation of the steady-state profiles of dissolved components consists finally in the solution of a nonlinear system of coupled algebraic equations. The system was solved by a classical Newton-Raphson iterative procedure. The main difficulty was to choose an adequate set of start values, because the procedure diverges if the starting profiles are too far from the solution profiles. In order to avoid this problem, an analytic continuation method was chosen (Reichert et al., 1989). For high diffusion coefficients and low retention times the problem is solved easily, since the concentrations in the biofilm are constant and have the same values with the concentrations in liquid. Furthermore, these are equal with the inlet concentrations since the low retention times maintain constant values in the bulk liquid. This simplified problem is solved firstly and then hypotheses are gradually relaxed: the effective diffusion coefficient is decreased and the residence time is increased till they reach their correct values. Solved profiles for the first dissolved oxygen concentration are used then as initial approximation for another level of dissolved oxygen, till the entire domain is scanned. Our program runs showed that a grid of 8 to 10 collocation points is sufficient for an accurate solution of this problem. Calculations were performed with a source written in BPascal 7.0 on an IBM 486 computer. The computing time necessary for determination of bulk concentrations in the range of 0 to 4 mg dissolved O2 /L was approx. 60 seconds with 9 collocation points in the biofilm and 250 steady-states function of dissolved oxygen. Input parameters
All parameters used are summarised in table 1. Table 1. Parameter Kinetics Maximal activity of ammonia-oxidisers Maximal activity of nitrite-oxidisers Monod saturation constant of NH3 for ammoniaoxidisers Monod saturation constant of HNO2 for nitriteoxidisers Inhibition constant of NH3 for ammonia-oxidisers Inhibition constant of HNO2 for nitrite-oxidisers Monod saturation constant of oxygen for ammoniaoxidisers Monod saturation constant of oxygen for nitriteoxidisers Growth yield of ammonia-oxidisers on ammonia Growth yield of nitrite-oxidisers on HNO2 Growth yield of ammonia-oxidisers on oxygen Growth yield of nitrite-oxidisers on oxygen Mass transfer Diffusion coefficient of NH4+ in water Diffusion coefficient of O2 in water Diffusion coefficient of NO2- in water Diffusion coefficient of NO3- in water Liquid-solid mass transfer coefficient Porosity of biofilm Geometry and operation Total reactor volume Liquid input flowrate Ammonium concentration in input wastewater Specific area Carrier diameter Biofilm thickness pH Temperature Ammonia-oxidisers concentration in biofilm Nitrite-oxidisers concentration in biofilm Total density of biomass in biofilm
Model parameters Symbol
Value
Units
Reference
qm1 qm2
13.4 44.74
(a) (a)
KS1
2.8⋅10-2
mg N-NH3 / mg odm /day mg N-HNO2 / mg odm /day mg N-NH3 / L
KS2
3.2⋅10-5
mg N-HNO2 / L
(a)
Ki1 Ki2 KO1
540 0.26 0.3
mg N-NH3 / L mg N-HNO2 / L mg O2 / L
(a) (a) (a)
KO2
1.1
mg O2 / L
(a)
YXN1 YXN2 YXO1 YXO2
0.147 0.042 0.046 0.039
mg odm / mg N-NH3 mg odm / mg N-HNO2 mg odm / mg O2 mg odm / mg O2
(a) (a) (a) (a)
D1 D2 D3 D4 kl ε
1.86⋅10-9 2⋅10-9 1.7⋅10-9 1.7⋅10-9 3.5 .. 7.0 0.7
m2/s m2/s m2/s m2/s m/d m3 water / m3 biofilm
(b) (b) (b) (b) (b), (d), (e) assumed
VR Qin
3 72 200 3000 0.3 0.2 7 30 54 16 70
L L/d mg N-NH4 / L m2 interface / m3 reactor mm mm
measured, (c) measured, (c) measured, (c) calculated, (c) measured, (c) measured, (c) measured, (c) measured, (c) (a) (a) measured
CNH3,in av dc δ t CX1 CX2 CX
°C g odm / L g odm / L g odm / L
(a)
(a) Wiesmann (1994), (b) Tijhuis (1994), (c) Garrido et al. (1996), (d) Hunik et al. (1994), (e) Wijffels et al. (1991)
Kinetic coefficients. There are several reports which describe the kinetic behaviour of nitrifying bacteria. Wiesmann (1994) presented some averaged coefficients of widely scattered data collected from literature, together with own recent measurements. In this paper we have used their kinetic model and parameters, with the sole assumption that the decay and endogenic processes are not significant for the overall conversion of N-compounds. Maximum specific growth rates were corrected for 30 ºC. Because differentiation of the two nitrifying species is difficult we have assumed a constant ratio between ammonia and nitrite oxidisers, calculated supposing maximum conversion rates of substrates and complete conversion of nitrite intermediate: (Wiesmann, 1994) C X 2 qm1 = = 0.29 (24) C X1 q m 2 Since the measured density of biomass after 120 days was 70 g/l biofilm, 54 g/l biofilm NH4-oxydizers and 16 g/l biofilm NO2-oxydizers were assumed as densities in the biofilm during the experiment. Mass transfer coefficients.
Data for diffusion coefficients in gel beads and biofilms (Di,eff) are also
scattered (Hunik, 1994). We assume the gel porosity of biofilm equal to 0.7 (very probable according with our light microscopic photographs and the high biomass density in the biofilm), diffusion coefficients of O2 and ionic species in water should be halved. The mass transfer coefficient between liquid phase and solid biocatalyst in a BAS reactor was calculated as described by Tijhuis (1994). Assuming the liquid velocity around the particle estimated as the settling velocity of a single particle in a stagnant liquid, we can obtain a rough approximation of oxygen mass transfer coefficient kl = 8.7⋅10-5 m/s = 7.5 m/d. This value was chosen as starting point for the data fitting procedure. Another value, calculated as described by Wijffels et al. (1991) for an air-lift reactor was 3.5 m/d. Due to the close diffusion coefficients of nitrogen species and oxygen, kl was set equal for all components and fitted at the value 4.4 m/d. Geometric and operation parameters. All these parameters were determined experimentally in our laboratory. Ammonium loading rate was 5 kg N-NH4/m3d. The hydraulic retention time, determined experimentally with a pulse signal of Blue Dextran (not diffusable in biofilm), was 0.5 h. The volume fraction of solids was 0.41 and gas fraction about 0.07 (Garrido et al., 1996), the overall retention time is approx. 1 h. The temperature was 30 °C and the acidity was kept around pH 7.
RESULTS Model evaluation
Ammonium concentration, mmol/l
Oxygen concentration, mg/l
Figures 1a-d show oxygen, ammonium, nitrite and nitrate profiles calculated along the dimensionless radius of the biofilm. An arbitrary length of stagnant film (0.2 units) was chosen only for representation of external profiles on these graphs. It can be seen that oxygen is indeed the rate limiting substrate (fig. 1a). The penetration depth of O2 in the biofilm rises with an increasing oxygen concentration in the bulk volume, allowing the respiration of a greater amount of biomass and consequently a faster ammonia conversion. For CO2 = 3 mg/l in the bulk liquid, the penetration depth δO2 > 100 μm is sufficient so that all nitrite produced by ammonia-oxidisers can be converted by the slower metabolism of nitrite-oxidisers. As figures 1b-d show, neither internal nor external diffusional resistance affects significantly the penetration of ammonia in the pellet or the release of products in the bulk liquid. These results revealed that mass transfer affects mainly the oxygen conversion which is dissolved at low level in the reactor. 3 2.5 2 1.5 1 0.5 0 0
0.2 0.4 0.6 0.8 1 1.2 Dimensionless biofilm length, X
10 8 6 4 2 0 0
0.2 0.4 0.6 0.8 1 1.2 Dimensionless biofilm length, X
Figure 1a. Calculated profiles of oxygen in biofilm Figure 1b. Calculated profiles of ammonium in biofilm
Nitrate concentration, mmol/l
Nitrite concentration, mmol/l
8 6 4 2 0
14 12 10 8 6 4 2 0
0
0.2 0.4 0.6 0.8 1 Dimensionless biofilm length, X
1.2
0
0.2 0.4 0.6 0.8 1 1.2 Dimensionless biofilm length, X
Figure 1c. Calculated profiles of nitrite in biofilm Figure 1d. Calculated profiles of nitrate in biofilm Symbols: dissolved oxygen concentration 0.5, 1.0, 1.5, 2.0, z 2.5, { 3.0 mg/l Experimental and calculated steady-states are shown in figure 2. It appears that the proposed model fits very well the measured data, despite the assumptions made. A better validation of the model would be possible if the spatial distribution of two microorganisms were determined with available techniques (Hunik et al., 1993).
N-species concentrations, mmol/l
15
10
5
0 0
1 2 3 Dissolved oxygen concentration, mg/l
4
Figure 2. Experimental and calculated steady-states at different dissolved oxygen concentrations: { NH4+ , z NO2- , + NO3- . Evaluation of process conditions The influence of operational parameters was evaluated in order to establish procedures to maximise the NO2production. Firstly the influence of ammonia specific loading rate (ALR, kg N/m2biofilm d) was studied. The amount of
ammonia converted per biofilm area depends on the hydraulic retention time, specific area of biofilm, nitrogen concentration in the influent and the liquid hold-up and is defined as: C NH 3, in Qin (25) ALR = VR a v An increased ammonium loading rate has no effect on the maximum achievable NO2- accumulation. The only effect is that the maximum accumulation is reached at higher DO values (figure 3). This is in line with the expectation that oxygen diffusion becomes more limiting at higher ammonium loading rates. Changing the operating pH affects the ammonium oxidation rate since NH3 is the substrate. Diffusion itself is marginally affected. An increasing pH has indeed a positive effect on nitrite formation (figure 4). Finally the effect of mass transfer coefficients (reactor turbulence) is evaluated. In fact, a similar effect as for ammonium loading rate is observed. At high mass transfer rates a lower DO can be maintained for optimal NO2- accumulation. Overall it can be concluded that changing operational conditions will not directly result in a changed accumulation of nitrite. Only the DO at which this maximum is reached can be affected.
N-species concentration, mmol/l
14
NH4+
12
NO3NO2-
10 8 6 4 2 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
D isso lve d o x yg e n c o n c e n tra tio n , m g /l
Figure 3. Calculated concentrations of N-species for different ammonia surface loading rates (influent concentration is 14 mmol N-NH4+ / l):
0.8 10-3 __, 1.6 10-3 ---, 3.2 10-3 ... kg N/m2biofilm day
N-species concentration, mmol/l
14
NH4+
12
NO3-
10 8 6
NO2-
4 2 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
D isso lve d o x yg e n c o n c e n tra tio n , m g /l
Figure 4. Calculated concentrations of N-species for different medium pH: 6 ---, 7 __ , 8 ... pH
4
N-species concentration, mmol/l
14 12
NH4+
NO3NO2-
10 8 6 4 2 0 0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
D isso lve d o x yg e n c o n c e n tra tio n , m g /l
Figure 5. Calculated concentrations of N-species for different mass transfer coefficients: 2.5 ---, 5.0 __, 7.5 ...
m/day
CONCLUSIONS We could say that the proposed model describes the measured data properly, despite the assumptions made. All the kinetic and mass transfer parameters were taken from the literature, except the mass transfer coefficient around the biofilm surface which was fitted. Neither internal nor external diffusional resistance affects significantly the penetration of ammonia in the pellet or the release of products in the bulk liquid. These results revealed that mass transfer affects mainly the oxygen conversion which is dissolved at low level in the reactor. It was not needed to assume a distribution of nitrite oxidisers and nitrate oxidisers into the biofilm even if this could occur. Distribution of bacteria over biofilm depth depends on the concentration field of rate limiting substrate (oxygen in this case) and a better validation of the model would be possible if the spatial distribution of the two populations were determined with appropriate techniques. The influence of operational parameters was evaluated in order to establish ways to improve NO2production. The model was used to appreciate the effect of loading rate, mass transfer coefficient and pH on the nitrite concentration in the reactor. Influencing the oxygen concentration seems to be the most practical method to obtain partial nitrification to nitrite in BAS reactors since this can be done by varying the superficial gas velocity or by partial recirculation of the off-gas.
ACKNOWLEDGEMENTS We would like to thank Juan Garrido for the experimental determination of N-species concentrations in the BAS reactor at different dissolved O2 levels. We are also grateful to the European Environmental Research Organisation (EERO) for the short-term fellowship awarded, which enabled collaboration between laboratories located in different European countries.
REFERENCES Finlayson, B.A. (1972). The method of weighted residuals and variational principles, Academic Press, New York. Garrido, J.M., van Benthum, W.A.J., van Loosdrecht, M.C.M., Heijnen J.J. (1996). Influence of dissolved oxygen concentration on nitrite formation in a biofilm airlift suspension reactor (submitted). Hunik, J.H., van den Hoogen, M.P., de Boer, W., Smit, M., Tramper, J. (1993). Quantitative determination of the spatial distribution of Nitrosomonas europaea and Nitrobacter agilis cells immobilised in K- carrageenan gel beads by a specific fluorescent-antibody labelling technique, Appl. Environ. Microbiol., 9,1951-1954. Hunik, J.H. (1994). Engineering aspects of nitrification with immobilised cells, Ph.D.-dissertation, Wageningen, The Netherlands. Hunik, J.H., Bos, C.G., van den Hoogen, M.P., de Gooijer, C.D., Tramper, J. (1994). Coimmobilized Nitrosomonas europaea and Nitrobacter agilis cells: validation of a dynamic model for simultaneous substrate conversion and growth in Kcarrageenan gel beads. Biotechnol. Bioeng., 43, 1153-1163. Kokufuta, E., Shimohashi, M., Nakamura, I. (1988). Simultaneously occurring nitrification and denitrification under oxygen gradient by polyelectrolyte complex-coimmobilized Nitrosomonas europaea and Paracoccus denitrificans cells. Biotechnol.Bioeng., 31, 382-384. Reichert, P., Ruchti, J., Wanner, O. (1989). BIOSIM Interactive program for the simulation of the dynamics of mixed culture biofilm systems, Dübendorf, Switzerland. Santos, V.A., Tramper, J., Wijffels, R.H. (1992). Integrated nitrification-denitrification with immobilised microorganisms. In: L.F. Melo, T.R. Bott, M.Fletcher and B.Capdeville (ed.), Biofilms-Science and Technology, Kluyver Academic Publishers, Dordrecht, Boston and London. Suthersand, S., Ganczarczyk, J.J. (1986). Inhibition of nitrite oxidation during nitrification. Some observations. Water Poll.Res.Jour.Canada. , 21, 423-445. Tanaka, H., Dunn, I.J. (1981). Kinetics of nitrification using a fluidized sand bed reactor with attached growth. Biotechnol. Bioeng. , 23, 1683-1702. Tanaka, H., Dunn, I.J. (1982). Kinetics of biofilm nitrification. Biotechnol.Bioeng. , 24, 669-689. Tijhuis, L., van Loosdrecht, M.C.M., Heijnen, J.J. (1992). Nitrification with biofilms on small suspended particles in airlift reactors. Water Sci.Technol., 26, 2207-2211. Tijhuis, L. (1994). The biofilm airlift suspension reactor: biofilm formation, detachment and heterogeneity, p.123, Ph.D.dissertation, Delft, The Netherlands. Wiesmann, U. (1994). Biological nitrogen removal from wastewater. p. 113-154 In: Advances in Biochemical Engineering / Biotechnology, vol.51, A. Fiechter (ed.), Springer-Verlag, Berlin, Germany. Wijffels, R.H., Gooijer, C.D. de, Kortekaas, S., Tramper, J. (1991). Growth and substrate consumption of Nitrobacter agilis cells immobilised in carrageenan: part 2 model evaluation, Biotechnol. Bioeng., 38, 232-240.
2. The discrete approach on biofilm modelling
Cellular automata models for biofilm growth
This paper will be presented at the “Bioprocess Engineering Course”, 14-18 June 1996, Stockholm
CELLULAR AUTOMATA MODELS FOR BIOFILM GROWTH C. Picioreanu*, M.C.M. van Loosdrecht and J.J. Heijnen Department of Biochemical Engineering, Delft University of Technology, Julianalaan 67, 2628 BC Delft, The Netherlands; *Department of Chemical Engineering, University Politehnica of Bucharest, Splaiul Independentei 313, 77206 Bucharest, Romania
A. Introduction Cellular automata supply useful models for many investigations in natural science, they representing a natural way of studying the evolution of large physical systems. As Toffoli and Margolus, 1987 noted, a cellular automaton can be thought of as a stylised universe. Space is represented by a uniform grid, with each site or cell containing a few bits of data. Time advances in discrete steps, and at each step each cell computes its new state from that of its close neighbours. In this way, many simple components act together to produce complicated patterns. Neighbourhood structures considered for two-dimensional cellular automata are termed: (a) “five-neighbour square” for the rules which take into account only the nearest neighbours (von Neumann). (b) “nine-neighbour square” in which are considered both the nearest and next-nearest cells. (Moore).
a. five-neighbour square
b. nine-neighbour square
In the regular lattice of sites, each cell takes v possible values and it is updated in discrete time steps according to a rule φ that depends on the value of sites in some neighbourhood around it. The value ci,j of a site at position (i,j) in a square two dimensional cellular automaton evolves according to many possible rules (Packard and Wolfram, 1984): (a) the rule depends only on nearest neighbours:
[
ci(,tj+ 1 ) = φ ci(,tj) ,ci(,tj)+ 1 ,ci(+t )1, j ,ci(,tj)− 1 ,ci(−t 1) , j
]
(1)
(b) or often we consider the special class of totalistic rules, in which the value of a site depends only on the sum of the values in the neighbourhood:
[
ci(,tj+ 1 ) = φ ci(,tj) + ci(,tj)+ 1 + ci(+t 1) , j + ci(,tj)− 1 + ci(−t 1) , j
]
(2)
Despite the simplicity of their construction, cellular automata are found to be capable of very complicated behaviour. Direct mathematical analysis is sometimes of little utility in elucidating their properties, so that one must at first resort to empirical means (Packard and Wolfram, 1984). However, cellular automata were able to give alternative solutions for complicated systems of differential equations. Their advantage consist mainly on their simplicity, stability and ability to model complicated boundaries. In this respect were developed algorithms for problems encountered in a surprisingly large spectrum of natural sciences: hydrodynamic problems, plasma computations, magnetisation of solids, waves and optics, crystalline growth processes, reaction-diffusion systems, ecological communities, medicine, microbiology (for a comprehensive bibliography see Wolfram 1986, Doolen et al. 1990) Our interest is mainly focused on modelling the evolution of bacterial communities as are they in plate colonies, flocs, growth in gels or biofilms. Even if for bidimensional colonies and flocs are now many models proposed, for the development of biofilms grown on solid carriers are no valid models to date (according to our knowledge). We are trying to explain the problems associated with the spreading front of biofilm. Problems with complex boundaries (irregular geometry) cannot be solved with traditional continuous methods. Cellular automata are therefore valuable tools for investigating discrete systems.
B. Model construction I. The state system • The space for which the rules of the game were defined is a 2-D square lattice. The dimensions of the space are on x direction N points and on y direction M points. The maximum extension of the matrix is 1024 x 1024 points. The space can be also a rectangle (N ≠ M) with the maximum surface of 1048576 cells. • The boundary conditions are: a. At x=1 is a rigid wall (the carrier space), impenetrable both for microorganisms and for soluble reactants. b. The opposite x=N boundary is assumed an infinite reservoir (a linear perfect source), which replaces immediately the particles diffusing into the system. The concentration of substrate is all the time kept at a constant value cS0.
x 1
N
1 Infinite y
reservoir Periodic
Carrie M
Figure 1
The model space and boundary conditions
boundaries
c. In order to limit the effects caused by the finite size of the lattice, the system has periodic boundaries in the y direction. This means that the primary cell of NxM points is replicated periodically in the direction y to make an infinite space. A consequence of this construction is the behaviour experienced many times in the computer games: a particle leaving the system on a side, appears immediately in the same x position but on the opposite side. • The initial state of the system is a. At x=1 (attached on the carrier surface) are placed a number X0 < M of inoculum cells, randomly distributed between y=1 and y=M. The rest of matrix is empty, so the state assigned is cX=0. b. The substrate is initially uniform distributed in the whole space, a certain state cS0 being assigned in each cell. • At a certain moment each lattice-cell can be occupied by one and only one microorganism-cell. Contrary, a lattice-cell contains between 0 and cS0 units of substrate. In order to save memory in the computer, our model is an integer representation. II. The processes system To model the biofilm formation we included the following generic features: 1. Diffusion of nutrients and products 2. Microbial growth 3. Substrate consumption and product release 4. Attachment 5. Detachment 1. Diffusion of nutrients and products Substance diffusion between different cells in the matrix could be simulated in various ways. • The first problem is to choose from the NxM cells of the matrix the donor cell. This can be done in at least two ways: a. One is to pick randomly the source cell, and in this case the diffusion cycle will be complete when NxM diffusional events took place. A consequence of this mechanism is the fact that the same cell could be chosen many times as donor, while other cells were never suffered transport to the neighbours. b. The second possibility is to update in one diffusional step the whole matrix. Thus, every cell will be source once and only once time per complete update. This is in fact the characteristics of a real cellular automata. It is superior to the first way because the transfer events occur in the matrix in parallel rather than in a sequential scheme. • After one cell of the matrix was picked to dissipate substance to the neighbours (source cell or donor) there are many possibilities to chose the neighbour which will receive the amount of substance (destination cell or acceptor). But as a general rule, according to the second thermodynamic principle, mass will be transferred only to a cell which contain less substrate than the source (ck < ci). a. Choose the destination cell randomly from the neighbouring cells which have less substance (j cells). Of course this rule contains less physical meaning because the probability to distribute the substance should be dependent of the substrate content in the destination cell.
pD ,k
⎧1 0 ⎪ pD , =⎨ j ⎪0 , ⎩
for ck < ci ,
k = 1.. j
for ck ≥ ci ,
k = j + 1..4
(3)
b. Choose the destination cell as the neighbour containing the least amount of substance (Colosanti 1992). Even if this rule is more realistic than the first one, suggesting transport is the maximum gradient direction, it does not include the possibility of transfer to all cells containing less substrate.
p D ,k
0 ⎧ pD , ⎪⎪ = ⎨0 , ⎪ ⎪⎩
( )
for ck = min c j ,
k =1
for ck ≥ ci , c k < ci ,
k = j + 1..4 k = 2. . j
(4)
c. Substance may be transferred to the nearest neighbour site with a probability proportional with the difference in concentration between source and destination:
p D ,k
⎧ 0 ( ci − c k ) , ⎪ pD ⋅ j ⎪ =⎨ ∑ ( ci − c k ) ⎪ k =1 ⎪0 , ⎩
for c k < ci ,
k = 1.. j (5)
for ck ≥ ci ,
k = j + 1..4
In this mode all the neighbours which contain less substrate than the source can hope to receive it, but now the mechanism take in to account how big is the gradient in every direction (Barker and Grimson, 1993). This diffusion mechanism can be applied not only for the diffusion of substrates from the liquid zone inside the biofilm, but also for the diffusing products which are released in the medium. In the first case the matrix is completely filled with S0 units of substrate in each cell, while in the second case all matrix cells will contain P0 units of product (usually P0=0). 2. Microbial growth Division (microbial reproduction) is restricted by many factors: nutrient limitation, space limitation, ageing, toxic compounds, etc. In the present model we shall consider only the aspects generated by substrate depletion (important in the biofilm due to the diffusional limitation) and space limitation (occurring in a more or less compact matrix like the biofilm). The same two main problems as in the case of diffusion appear here too: which cell to choose to divide at a certain moment and where will it grow (the specific direction) ? • For the first question there are again many possibilities: completely randomness, selection by age, selection by substrate availability around the cell. a. Models which consider every cell having the same probability to divide: pG ,i =
1 0 ⋅ pG N⋅M
(6)
lead to a compact structure. The colony growing on a square lattice starts from one cell and each daughter occupies the nearest free square. A slightly irregular object with dented edges is formed (Eden, 1961). b. Because the probability of a growth event is obviously related to the substrate accessibility, the present model try to take this into account. Thus, one can take only the substrate existent in the every lattice cell and so, the probability to select the cell i will be: 0 pG ,i = pG ⋅
si Nc
(7)
∑ sk
k =1
where Nc is the number of cells living at a certain moment in the aggregate. We can also add to the substrate existent in the cell i every unit of substrate available in next neighbour cells, and the above probability becomes:
4
si + ∑ s j ,i pG ,i =
0 pG ⋅
j =1
Nc ⎛
⎞ ⎜ s + s ∑ ⎜ k ∑ j ,k ⎟⎟ ⎠ k = 1⎝ j =1 4
(8)
This last rule was used in the next simulations. However, this linear dependence of growth on the available substrate can be easily transformed in other usual functions: Monod-like, inhibited growth, logistic or even more structured models can be designed.
• Where to put the daughter cell is apparent an easy question, but only when there is at least one lattice space free around the cell. The classical model of Eden states that when a new cell happens to choose a site that is already occupied by another cell it dies (Eden, 1961, Markx and Davey, 1990). A second possibility is to proceed a random walk on neighbouring lattice sites. The walk could be accomplished if it reaches a free space in a maximum number of steps (called walk length). Longer walks increase the relative number of successful searches (Schindler and Rataj, 1992). Schindler and Rataj state that the result is the same as if the daughter cell would stick to the mother cell and neighbouring cells would be shifted to a free position. However, this is valid only when we model a pure colony. One can assume also as a necessary condition for growth in a lattice position, the existence of a certain neighbourhood density of population (having values between 0.2 and 0.8 in the model of Schindler and Rovensky, 1994). It is also possible to keep the cell alive for a certain number of iterations, in this time consuming substrate only for maintenance. So, first one has to determine the number and position of free neighbour places and then one can choose randomly the growth direction. In this case, the probability for a lattice cell to receive a daughter (a new cell) depends on the growth probabilities of the neighbours: 4
pG , k = ∑
j =1
1 pG , j fj
(9)
where fj is the number of free positions around the dividing cell j. This last possibility was accepted as a basis for our model. 3. Substrate consumption and product release In order to grow and divide, cells consume substrate. It is normal then to assume a certain conversion of substrate existent around the cell which will divide. a. When the cell chosen to grow has space around, it consumes a certain amount of substrate from the neighbour lattice cells. So, the probability of substrate consumption in a matrix cell is proportional with the probability of growth around it (j=1..4) plus growth exactly in that cell (j=0): 1 pS ,i = pG ,0 + f0
4
∑
j =1
1 pG , j fj
(10)
Corresponding to the related case of the growth probability we can also define the consumption of substrate only in the lattice-cell in which the microorganism is dividing:
pS ,i =
1 pG ,0 f0
(11)
For simplicity we have assumed that a dividing cell consumes the same amount of substrate from each of the five mentioned positions, corresponding to the yield of biomass on substrate (YS). b. But in the case when growth is not possible, all the four next positions being occupied, one can set up another rule so that the biomass consumes substrate for maintenance, proportional with the coefficient mS. In this case the probability for substrate concentration decrease is:
pSm,i = pG ,0 +
4
∑ pG , j
(12)
j =1
For product formation in a lattice-cell the rules are the same above, only YP will be added to the already existent amount of product. 4. Attachment of cells A model for biofilm formation must include also the attachment processes. For this purpose can be utilised one of the models already existent in the literature. Among them, the most famous is the diffusion-limited aggregation model (Witten and Sander, 1981). A random-walker is generated in the free space of the lattice and it joins the cluster if it orthogonally touches the biofilm.
Random walker
Biofilm
Generation point
Figure 2
The diffusion limited aggregation procedure
This technique was successfully applied for simulation of many natural processes, among them the deposits grown in a electrolytic cell (Voss, 1984) and crystallisation (Vicsek, 1984). Comprehensive reviews are available in many books (Kaye, 1989, Avnir, 1989). Only the cells situated in the front surface of the biofilm (defined later) have a non-zero probability to receive a random-walker. Different fractal dimensions, shapes of the aggregate and porosity can be obtained by varying the permitted motion of the walker, the size range and the number of individual wandering units and the probability of sticking to the biofilm.
⎛ sticking probability, density of walkers, ⎞ p A,i = p0A ⋅ f ⎜ ⎟ ⎝ outside exposure, probability cross of walker, ... ⎠
(13)
5. Detachment of biofilm cells The biofilm can lose pieces from it by many different mechanisms: erosion, abrasion, sloughing or grazing (Characklis and Marshall, 1990). In our opinion, erosion can be modelled the easiest. A flux of random-walkers is generated from a point of the square lattice. If a walker reaches the surface of the biofilm, the touched cell will be removed from the biofilm aggregate. In this mode, the long filaments which are growing faster on the biofilm surface will be eroded in time because they are more exposed to the flux of wandering particles. In the same time, the cells existent in the “bays” or “pits” are protected from the erosive action and can grow free, filling in the pores. a. Very important is the predominant direction of the walkers flux. We can imagine a frontal stream, generated from a random y point existent in the bulk liquid, on the interface with the infinite reservoir. Frontal flux
b. Also realistic is a parallel stream, generated from a random point situated on the periodic boundary, at a distance x > xmax, biofilm . If the stream has a single direction, the biofilm surface will be asymmetric eroded, becoming thicker in the opposite direction of the generation point. This fact can be avoided by choosing the starting point of walker at a random distance x and also random coordinate y. high probability
low
Parallel flux
probability
C. Characteristic variables There are two categories of measures employed in the characterisation of the simulated biofilm: superficial variables (which are related to the front properties) and volume variables (related to the inside structure of the biofilm). 1. Superficial variables We have to define first the biofilm front, called also in the literature “external perimeter”, “hull”, “shore” or “border”. According to Chopard and Droz, 1990 (who introduced the “diffusion front”) , the biofilm front is constructed as follows: 1. The first step is to determine the infinite cluster (“the ocean”) consists of those empty sites which are connected by the nearest neighbour to the infinite reservoir. 2. The complementary cluster is the biofilm (“the land”), defined as all the particles which are connected to the carrier by nearest neighbour particle-particle bonds. The biofilm includes also empty sites (pores, voids) which are finite empty clusters (“the lakes”). 3. Then, the biofilm front is the line of first neighbour particle-particle bonds in such way that each particle of the hull has at least one “liquid” (external) nearest or next-nearest neighbour (fig.3 a). There is also another possibility to define the front line: biofilm front particle is the particle which has as external neighbour only nearest points, situation drawn in fig. 3 b. biofilm front biofil
liquid
void
a.
b.
Figure 3 Construction of the biofilm front: a. nearest and next-nearest liquid boundary, b. only nearest liquid boundary permitted.
The following macroscopic quantities which characterise the biofilm can be introduced now (initially defined for diffusion front by Sapoval et al., 1985; Chopard and Droz, 1990): a. The average profile
The average concentration profile at distance x and time t is defined for our discrete system as: 1 M C( x , t ) = ⋅ ∑ c( x , y , t ) M y =1
(14)
This variable is used as well to average the concentration of dissolved substances (substrate, products) as to average biomass content of the biofilm at x. In this context, c(x,y,t) has values between 0 and c0 for substrate, while for biomass the possible states are 0 or 1. Analogous to the density of all points C(x,t), we can define the average density of biofilm front points, Cf(x,t): C f ( x ,t ) =
1 M ⋅ ∑ c X ( x , y ,t ) M y =1 f
(15)
where cXf is the state of lattice-cell: 0 for non-front point and 1 for a front point (so that 0 ≤ Cf ≤ 1). b. Front length The normalised front length, Lf , is the number of front points, given by: N
Lf (t ) =
N
∑ C f ( x ,t ) =
M
∑ ∑ c Xf ( x , y , t )
x =1 y =1
M
x =1
(16)
The connection of the front points is only by nearest neighbour bonds (the first option, fig. 3a). c. The average front position The average front position, xf, is the first moment of front particles distribution in relation with the origin: N
xf (t ) =
∑ x ⋅ C f ( x ,t )
x =1 N
(17)
∑ C f ( x ,t )
x =1
d. The average front width The average front width, σ2f , is defined as the second moment of the distribution, in rapport with the mean of distribution: N
σ2 f (t ) =
∑ ( x − x f ( t ))2 ⋅ C X f ( x , t )
x =1
N
(18)
∑ CX f ( x ,t )
x =1
but can also be defined the absolute mean front width σf(t): N
σ f (t ) =
∑ x − x f ( t ) ⋅ CX f ( x ,t )
x =1
N
∑ CX f ( x ,t )
x =1
(19)
e. Fractal dimension of the front The fractal dimension of biofilm front (or the “hull dimension”), Df, is defined in Chopard and Droz, 1990 as the exponent which satisfies the proportionality relation between the front length and the front width:
⎛ M D L f ∝ σ 2 ff ⋅ ⎜⎜ ⎝σ2 f
⎞ ⎟⎟ ⎠
(20)
If the plot of ln(Lf) versus ln(σ2f) satisfies a linear relation, the fractal dimension of biofilm front will be the (1 + slope of the plot).
σf
Lf
xf xf,max M⋅Cf(x)
x Figure 4
Variables used to characterise the biofilm surface
2. Volume related variables At this moment, it is interesting to calculate the pore fraction of the simulated structure. It can be defined as the volume of finite empty clusters included in the “solid” structure created by microorganism growth, related to the total volume of the biofilm:
N⋅M −
ε =
N
∑ C" land" ( x , t ) −
X =1
N
N
∑ C" ocean" ( x , t )
X =1
N ⋅ M − ∑ C" ocean" ( x , t ) x =1
N
=
∑ C" voids" ( x , t )
x =1 N
N
x =1
x =1
∑ C" voids" ( x , t ) + ∑ C" land" ( x , t )
(21)
D. Tests and simulations 1. Validation of diffusion mechanism The diffusion algorithm have been tested in the absence of reaction, attachment and detachment mechanisms. The aim is to compare the concentration profiles given by simulation with the analytical solutions of macroscopic balance equation: D
∂ 2 C( x , t ) ∂C( x , t ) = ∂t ∂x 2
(22)
Given the boundary conditions: x = 0,
C( 0 , t ) = c S 0
x=N,
C( N , t ) = 0
t = 0,
C( x ,0 ) = 0 for 1 < x < N
(23 a,b)
and initial condition: (24 a,b)
C( 0 ,0 ) = cS 0 for x = 0 the well-known analytical solution of eq.(19) is: ⎡N − x 2 C( x , t ) = c S 0 ⋅ ⎢ − π ⎢⎣ N
∞
⎛1
x
D t ⎞⎤
∑ ⎜⎝ n sin( nπ N ) ⋅ exp( − n 2π 2 N 2 ⎟⎠ ⎥⎥ ⎦
n =1
(25)
a. The first test considered the equilibrium diffusion between two parallel walls : a perfect line source at x=0 and a perfect sink at x=N (which will remove all arriving particles). As the macroscopic model predicted, for t → ∞ the concentration profile is a straight line (fig. 5):
⎡N − x⎤ C( x , t ) = cS 0 ⋅ ⎢ ⎣ N ⎥⎦
(26)
b. The second test compares theoretical and simulated concentration profiles obtained in unsteady-state, with the source at x = 0 only and a semiinfinite space (the boundary x = N is open): ⎡ 2 C( x , t ) = c S 0 ⋅ ⎢1 − π ⎢⎣
∞
⎛1
x
Dt ⎞⎤
∑ ⎜⎝ n sin( nπ N ) ⋅ exp( − n 2π 2 N 2 ⎟⎠ ⎥⎥ = ⎦
n =1
⎡ ⎢ 2 = cS 0 ⋅ ⎢1 − π ⎢ ⎢ ⎣
x 2 Dt
∫ 0
e−u
2
⎤ ⎥ du⎥ ⎥ ⎥ ⎦
(27)
C(x)
8
8
6
6
4
4
t >
2
2
0
0 0
20
40
60
80
100
x
Figure 5 Concentration profile of the substrate at two time moments and in steadystate (the straight line). A perfect source is assumed at x=0 and the ideal sink at x=100. Results obtained for different time moments are presented in fig. 6. The unknown diffusion coefficient D was substituted using the identity (Chopard and Droz, 1990): 2 Dt =
4
π
x
(28)
with the average substrate position x defined by eq.(15). 8
8
6
6
C(x)
Dt=25
4
4
Dt=200 Dt=400
2
2
0
0 0
20
40
60
80
100
x
Figure 6 Simulated (symbols) and analytical (lines) solutions for diffusion with a perfect source at x = 0 and open-end at x=100, for three D⋅t values.
C(x)
8
8
6
6
4
4
t >
2
2
0
0 0
10
20
30
40
50
x
Figure 7 Simulated concentration profiles in a biofilm with unsteady-state reaction and diffusion (+ and
points) and the steady state reached when detachment processes balance growth ( thick line ).
The conclusion arising from our simulations is that the proposed model for diffusion can represent well the expected behaviour of the system. c. In addition, a third test was performed to check if the algorithm is able to predict the steady - state in a diffusion - reaction system. For this purpose, the algorithm started in a 50x50 lattice, without detachment and with consumption of one unit of substrate both for growing and non-growing cells. Because the probability for a cell to be chosen to consume substrate is proportional with the sum of substrate units around it, the reaction could be considered of first order. After 1250 cells were formed (1/2 of the total matrix size), the detachment procedure begins with a probability equal to the dividing probability. In this way the biofilm becomes smooth and because the number of dividing cells is balanced with detached cells an approx. constant thickness is maintained. Using a ratio of 3:1 between diffusional update of matrix and growth events, the steady-state presented in fig. 7 was reached. At least qualitatively the curve shows the linear substrate decrease in the external film (only diffusion, 0 < x < 50) and the concave function for substrate consumption inside the biofilm (25 < x < 50). b. Biofilm structure b1. Superficial structure In order to characterise numerically the biofilm surface, a number of measures were introduced above. The following variables were changed :
• yields for substrate consumption in growth and maintenance processes • the ratio between characteristic times for reaction and diffusion • substrate concentration s0 • initial loading of the carrier with cells • We can observe that increasing the specific consumption of substrate for growth and maintenance one obtains higher hull dimensions (Df) in almost all cases, meaning that larger tips are formed in this case. This is explained by a faster consumption of substrate inside the biofilm, which leads to a lower growth probability for those cells. Only outside cells, exposed to sufficient substrate, can grow.
• When substrate concentration is enough both for inside and outside cells, the ratio between growth and diffusion is not influencing very much the surface structure. The same comment is valid also for cases with low specific substrate consumption. In this cases Df values around 1.1 and below are obtained (no. 1-15 in the table). A greater importance of diffusional uptake of substrate is observed comparing simulations 16-20. In the cases 16-18 growth is limited by diffusion resulting larger tips (Df = 1.15-1.17) than for cases 19-20 (with Df = 1-1.1).
Lf
10
Lf
10
1
1 1
10
100
1000
1
10
100
σ
σ
a.
b.
1000
Figure 7 Double logarithmic graphs of Lf as a function of σ2f : a. inoculum filling completely the area b. inoculum 0.5 of the total area.
s0 = 3 units, consumption for growth 2 units, maintenance 1 unit Gro : Dif =
10:1 , • 1:1, + 1:10
Lf
10
Lf
10
1
1 1
10
100
1000
σ
a.
1
10
100 σ
b.
Figure 8 Double logarithmic graphs of Lf as a function of σ2f : a. inoculum filling completely the area b. inoculum 0.5 of the total area.
s0 = 20 units, consumption for growth 2 units, maintenance 1 unit Gro : Dif =
5:1 , • 1:1, + 1:5
1000
• Df are sensible higher when s0 is lower, but this fact is important only for fast-growing cells. When the diffusional uptake of substrate is not limiting (cases 5,10,15 and 20), the hull dimension is always very close to 1, meaning smooth surface. • Of great importance seems to be the initial inoculum present on the carrier surface. A different series of simulations were performed considering only half of the surface initially occupied by cells. In almost all cases much higher deformations of the surface were observed than when the carrier is completely filled with cells. The filaments formed are bigger and porous structures appear. In each case, the fractal dimension Df is one minus the slope of the line between the points of plot ln(Lf) ln(σ2f). b2. Volume structure Analysis of volume structure consist in calculation of the void fraction of biofilm. The conclusion of our simulations is that porous structures appear only for biofilms grown from poor inoculum distribution on the surface.
0.15
0.15
0.1
0.1
ε
0.2
ε
0.2
0.05
0.05
0
0 0
500
1000 N cells
1500
2000
0
500
a. Figure 9
1000 N cells
1500
2000
b.
The fraction of voids as function of number of cells in the biofilm for :
a. s0 = 3 units b. s0 = 20 units, consumption for growth 2 units, for maintenance 1 unit Ratios between growth and diffusion probabilities:
• 10:1, 5:1, + 1:1, { 1:5,
1:10 In this case the void fraction depends on the substrate accessibility to the cells. We are expecting then that the growth : diffusion ratio is an important parameter in void formation. When consumption of substrate by growth and maintenance is not balanced by diffusion, many voids appear in the biofilm volume(fig. 9 a,b, plots and z). By contrary, when much substrate is transported inside the biofilm, the pores are filled rapidly with new cells so that the void fraction remains low (fig. 9 a,b, plots { and
). As we expected, when conversion is the determinant mechanism the cells situated inside the biofilm suffer due to substrate depletion. This exhaustion of substrate is of course faster when less units of substrate are added than when is very much available substrate (compare figures 9 for and z). What can be also seen is fig. 9 b is the general decreasing of void fraction with the number of cells, to an equilibrium value, due to the gradual penetration of substrate into the biofilm. For biofilms grown at a low substrate concentration, the biofilm-solution interface is more unstable, large open crevices being closed at nonregularly time intervals, this resulting also in an instability of void fraction (fig. 8).
The instability of the biofilm front can visualised in fig. 10, where just a few new cells in the system modify the porosity from 5 % to 12 %.
a. Figure 10
b.
c.
Three stages is the biofilm developing (s0 = 3 units, cX0 = 0.5⋅M cells, Growth:Diffusion = 10:1):
a. N cells = 877, Lf = 3.6, σ2f = 22, ε = 0.039 b. N cells = 1000, Lf = 4.4, σ2f = 32, ε = 0.048 c. N cells = 1053, Lf = 3.3, σ2f = 31, ε = 0.126
b3. Influence of detachment processes As we have seen above, the biofilm surface is in a dynamic equilibrium only if growth is balanced with detachment. We can see this from the following three simulations. The first considers 20 % excess growth related to detachment events. The biofilm thickness will increase continuously in time as we can seen from fig. 11. In the same time, the front presents a relatively smooth surface.
Figure 11 Evolution of the biofilm when growth dominates detachment (Gro:Dif:Det =5:1:4, s0 = 20, cX0 = 60)
A second simulation begins without detachment. After a certain time the detachment starts with a defavourable (4:5) ratio relative to growth. Just a few time is necessary to remove completely the biofilm from the carrier surface (fig. 12).
Figure 12 Evolution of the biofilm when after 2000 iterations detachment dominates the growth (Gro:Dif:Det = 4:1:5, s0 = 3, cX0 = 20).
The last two simulations were performed to see how the detachment influences the internal structure of the biofilm. When after 2500 iterations the detachment process became active, only the superficial pores were filled because of the low penetration depth of substrate (fig.13). A different situation appears when the detachment starts only after 1000 iteration, when the biofilm is thinner (fig. 14). The surface becomes rapidly smooth, while the voids formed in free growth are now completely filled with cells.
Figure 13 After 2500 iteration detachment balanced with growth generates a smooth surface but cannot lead to pores filling because the biofilm is already too thick.
Figure 14 If detachment begins only after 1000 iterations, the smoothnes of the surface is accompagned by a rigid internal structure of the biofilm (without voids).
E. General comments • The proposed diffusion mechanism is satisfactory. • We need the analytical link between diffusion and reaction. Starting from microscopic balance equation from each cell, a macroscopic balance equation should be deduced at the limit. • We can assign every lattice cell with different biomass states, not only 0 or 1. Thus, if the number of cells in a cluster (lattice cell) is higher than a maximal value, the cells can jump in a neighbour lattice position. This can also simulate growth of multispecies biofilm. • We have to check if the stationary profile found for the diffusion-reaction system is conform with the analytical solution of the traditional macroscopic equation, for a first order reaction. • A process of cell decay in time have to be introduced. The decay rate can be proportional with the age of cell or inverse proportional with the availability of substrate around the cell or with the exposing time to zero substrate concentration. • The proposed model can produce, at least qualitatively now, the same patterns experimentally observed in the biofilm particles grown in the airlift suspension reactor (see part 3 of the report). ACKNOWLEDGEMENTS We are very grateful to the European Environmental Research Organisation for funding this project
REFERENCES 1. Toffoli T., Margolus N., 1987, “Cellular Automata Machine: a new environment for modelling”, MIT Press 2. Colosanti R.L., 1992, “Cellular automata models of microbial colonies”, Binary 4, 191-193 3. Barker G.C., Grimson M.J., 1993, “A cellular automaton model of microbial growth”, Binary 5, 132-137. 4. Eden N., 1961, 4th Berkeley Symposium on Mathematics, Statistics and Probability, ed. F. Neyman, University of California Press. 5. Markx G., Davey C.L., 1990, “Applications of fractal geometry”, Binary 2, 169-175. 6. Schindler J., Rataj T., 1992, “Fractal geometry and growth models of a Bacillus colony”, Binary 4, 66-72. 7. Schindler J., Rovensky L., 1994, “A model of intrinsic growth of a Bacillus colony”, Binary 6, 105-108. 8. Witten T.A., Sander L.M. (1981), “Diffusion-limited aggregation, a kinetic critical phenomenon”, Phys. Rev. Lett. 47, 1400. 9. Kaye B.H. (1989), “A random walk through fractal dimensions”, Weinheim, VCH. 10. Avnir D. (1989), “The fractal approach to heterogeneous chemistry”, Chichester, John Wiley and Sons. 11. Chopard B., Droz M. (1990), “Cellular automata approach to diffusion problems” in Springer Proceedings in Physics, 46, 130-143. 12. Sapoval B., Rosso M., Gouyet J.-F. (1985) , J. Phys. (Paris) Lett. 46, L149. 13. Characklis W.G., Marshall K.C. (editors), 1990, “Biofilms”, John Wiley & Sons, New York. 14. Packard N.H., Wolfram S. ,1984, ”Two-dimensional cellular automata”, J.Stat.Physics 38 (5/6), 901. 15. Wofram S., 1986, “Theory and applications of cellular automata” 16. Doolen G.D. editor, 1990, “Lattice gas methods for partial differential equations”, Proceedings in the Santa Fe Institute Studies in the Sciences of Complexity.
3. Experimental evaluation of the biofilm structures Practical work done at Kluyver Laboratory for Biotechnology in February-March 1996.
EXPERIMENTAL EVALUATION OF BIOFILM STRUCTURES
INTRODUCTION When microorganisms come into contact with solid surfaces, biofilms often form. Biofilm formation is a natural immobilisation process, and can be exploited to human advantage in biotechnological processes, for example, in traditional vinegar production, microbial leaching, waste water treatment or animal tissue culture. Immobilisation enables reactor operation at high biomass concentrations, leading to high volumetric conversion capacities. Especially in waste water treatment, where relatively low substrate concentrations occur, use of biofilm processes can permit the use of smaller reactor volumes. Understanding the biofilm formation process is highly relevant for improving reactor design and performance. Biofilm developing is often studied in systems where relatively flat biofilms are formed, such as flow cells, rotating disks, tubular or annular reactors. In airlift reactors, however, spherical biofilms are formed. Moreover, these biofilms are not only subject to the turbulence and liquid shear that affect flat biofilms, but also to particle collisions. The formation process of natural, mixed population , spherical biofilms is currently under investigation. Carrier loading, substrate loading rate and air superficial velocity are important parameters that affect the biofilm formation and the structures obtained. The aim of this study is to find procedures to characterise the external structure of the biofilms, which is the first step in achieving a new non-conventional modelling of biofilms by cellular automata approach. This must be done in relation with the operational parameters above mentioned.
MICROORGANISMS, MEDIA AND REACTOR For this research, concentric-tube airlift reactors combined with a three-phase separator were used. The reactor had the total active volume of 2.6 l. The temperature was maintained at 30 °C by means of a thermostated water-jacket and the pH was controlled at 7 by the supply of 1N hydrochloric acid or 1N NaOH with an automated pH-control system. The superficial air velocity in the reactor was controlled by means of a mass flow controller system, and air was sparged in the reactor by means of a sintered glass stone, attached to a thin glass tube. The system configuration is presented in figure 1. Basalt was the carrier material, with a particle density of 3 kg/l and a mean diameter of 330 μm. The carrier concentration was 50 g/l at the beginning of the experiment and it was increased to 100 g/l after 4 days. Concentrated synthetic waste-water was diluted with tap water to the desired reactor loading rate. The concentrated medium contained: ethanol 40 mmol/l, NH4Cl 20 mmol/l, K2HPO4 2 mmol/l, MgSO4 ⋅7H2O 12 mmol/l and 1 ml/l of Vishniac trace element solution. The reactor was not inoculated with microorganisms. The tap water used to dilute the concentrated medium contained enough microorganisms to promote the attachment and growth on basalt. In this way, an ethanol loading rate of 5 kg COD/m3day and a dilution rate of 2 h-1 were established.
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1. Acid reservoir (HCl 1N) 2. Base reservoir (NaOH 1N) 3. pH controller 4. Base peristaltic pump 5. Acid peristaltic pump 6. pH electrode 7. Air mass-flow controller 8. Electromagnetic valve 9. Three phase separator 10. Reactor 11. Sludge settler 12. Effluent
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13. Surplus sludge 14. Temperature controller 15. Sampling port 16. Air sparger 17. Level controller 18. Electromagnetic valve 19. Tap water reservoir 20. Level probe 21. Concentrated medium reservoir 22. Medium peristaltic pump 23. Water peristaltic pump { air bubbles z basalt carrier and biofilms
Figure 1 The set-up of biofilm airlift suspension reactor and control devices
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RESULTS The reactor was started on 26 February 1996 in the above conditions. Every day, samples taken from the reactor with a syringe (5 mm entry diameter) were observed at an Image Analyser system. The general aspect of carrier introduced in the reactor is observed in figure 2. There are basalt particles (black and rough) and quartz particles (transparent and with a smoother surface). The small size fraction was washed out in 2-3 days, so that from an initial average size of 0.21 mm (measured as the average Feret diameter) a size of 0.33 mm was obtained after a few days (figure 3). After one week, microorganisms were present in the reactor only as observable white flocs. Probably, the carrier was already colonised with attached cells, but invisible at the 25x microscope magnification (fig. 3).
Basalt
Sand
Biomass flocs Figure 2 Basalt and sand particles at the beginning of the experiment (day 1)
Figure 3 Biomass flocs appeared in the seventh day
Biofilm apparition on the solid carrier was observed in the 15th day, but a large fraction of the basalt remained still bare (figure 4). One day later, almost all carrier was covered with a dense layer of biomass, the biofilm looking a strong and compact developed structure (figure 5).
Completely covered Free biomass Bare carrier Patchy biofilm Figure 4 Bare carrier, patchy colonies and completely formed biofilm (day 15)
Figure 5 A large fraction of the carrier was covered with biofilms (day 16)
The seventeenth day of the experiment marked the apparition of fluffy structures developing on the dense and smooth layer (figure 6). In figure 7 can be observed that many particles got outgrowing biofilms, while some of them are still round and relatively smooth. In the same time, due to the explosive growth of microorganisms the device situated in the top of the reactor could hardly separate the suspended biomass
from the biofilm-carrier aggregates. As a result, very much basalt was lost in the settler. The basalt wash-out leaded to a lower shear stress on the biofilms and also to a higher surface load with nutrients. Both factors acted in the same direction: less detachment of new formed cells and more growth, this meaning more irregular structures. As we could see, this vicious circle leaded to a fast lose of the separator capacity and an unstable reactor operation.
Dense layer
Outgrowing colonies Figure 6 Outgrowing structures on the dense inner layer (day 17, magnification 64x)
Figure 7 Fluffy biofilms developed due to the decreased shear (day 18)
Also development of filamentous microorganisms was noticed (figure 8 and 9). It is well-known that this type of organisms have a very bad effect on the settling capacity of the sludge (here biofilms). The large volume structures like that in fig. 9 provoke the sludge bulking in many waste water treatment plants. It is interesting to notice that in the same particle exist the dense layer, the porous layer, outgrowing colonies and also filamentous colonies !
Filaments Detaching fragment Carrier
Dense biofilm Porous biofilm Outgrowth Figure 8 Dark-field image of a biofilm particle with porous layer on a dense core (day 18, magnification 64x)
Figure 9 Phase-contrast image of a strongly heterogeneous biofilm particle
Life of biofilms is even more complex due to the grazing caused by protozoa. They attached on the biofilm surface and consumed bacteria. Sometimes this phenomenon can cause significant lose on biofilm thickness. Two species of protozoa are presented in images 10 and 11.
Figure 10 Protozoan attached by a long flagellum to the biofilm
Figure 11 Two protozoa grazing the biofilm surface
Growth of biofilm continued in the third week of the experiment as it can be seen in figures 12 and 13. Round and smooth structures together with filamentous ones were observed in the same sample. In the 23rd day many basalt particles were covered with “star-like” biofilms (fig. 13).
Figure 12 Biofilm particles with increased thickness (day 21)
Figure 13 “Star-like” biofilms (day 23, magnification 15x)
Figure 14 Bar histogram of average Feret diameters of complete particles (biofilm+carrier)
Figure 15 Bar histogram of average Feret diameter of basalt only (obtained by thresholding the 256 graylevel image) For the biofilms sampled in the 21st day, average Feret diameter was measured, both for basalt particles and for covered particles. Average sizes of 0.34 mm for “only basalt” and 0.57 mm for “basalt and biofilm” particles were obtained (in figures 14 and 15 are presented the results for 1600, respective 450 particles counted) Another characteristic phenomenon for aged biofilms is sloughing. This means the incidental loss of large patches of biofilm. It can be caused by liquid shear, but depletion of nutrients or dissolved oxygen at the biofilm base or a sudden increase of nutrient concentration in the bulk liquid are also indicated as causes. Sloughing starts from fissures or cracks formed in the biofilm (figure 16). It continues with partially detachment of a biofilm layer (figure 17). The final result is a particle covered with an incomplete biofilm, as it can be easily observed in figures 18 and 19. It can be also seen that the dense and strong initial layer was not removed, and new cells grew on the discovered surface (fig. 18).
Detaching layer (sloughing)
Figure 16 Fissure in the biofilm (day 23)
Figure 17 Detachment of a big biofilm fragment (day 26)
Thick old layer
New biomass The dense oldest layer Figure 18 , 19 Partially covered biofilm particles (sloughing occurred) (day 23) The experiment was stopped in the 26th day due to the impossibility to settle the biofilm particles in the top of the reactor. The explosive overgrowth of biofilms created fluffy and floc-like structures (figure 20). As in the activated-sludge processes, is very difficult to prevent wash-out of these particles. At the end of the experiment more than 50 % of the carrier was washed from the reactor.
Figure 20 Fluffy biofilms and a big amount of suspended biomass at the end of the experiment (day 26)
Figure 21 Fissures in the dense layer of biofilm were filled with new (less dense) biomass
CONCLUSIONS Natural biofilms are usually very complex and heterogeneous with respect to the types of microorganisms present, the physicochemical interactions occurring, and the internal and external structure of the biofilm. This work was conducted to study qualitative and quantitative biofilm heterogeneity in a biofilm airlift suspension reactor. Biofilm apparition started quickly after only one week, which is a characteristic time for heterotrophic populations of microorganisms. Biofilm accumulation is the net result of a number of physical, chemical and biological processes, each leading to either an increase or decrease of the amount of biomass accumulated on the carrier. Due to the large bare carrier concentration existent at the beginning of the experiment, biofilm abrasion was very important. The collisions with suspended particles leaded to a smooth and round surface and a high biomass density. During the second week of the experiment, more and more basalt particles were covered with biofilms this affecting the softness of collisions between suspended particles. Biomass growth became predominant to the detachment processes and the biofilms appeared more rough. Fluffy biofilms leads to a hindered settling and a difficult separation of attached biomass from the suspended biomass. Due to this fact more and more carrier was lost, and outgrowing structures developed on the biofilm surface. The reactor operation had to be stopped after one month due to the large suspended and attached biomass amount that clogged the downcomer and the separator of the reactor. A new series of experiments will be performed with a modified design of the three-phase separator. In the same time, methods to quantify the biofilm surface will be introduced. In this way, the experimental work will be used to validate the cellular automaton model.
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Summary of activities achieved
1. An article entitled “Modelling the effect of oxygen concentration on nitrite accumulation in the biofilm airlift reactor” was finished. It will be presented at the “IAWQ Special Conference on Biofilm Systems”, 27-30 August 1996, Copenhagen and then will be submitted to Water Science and Technology. The article is presented in the first part of this report. It represents one of the classical ways (the continuum approach) in biofilm modelling. 2. A second article “Cellular automata models for biofilm growth” was also finished. It will be presented as a poster at “Bioprocess Engineering Course”, 14-18 June, Stockholm and then submitted in a revised form to Biotechnology and Bioengineering. This work is the subject of the second part of the report. It represents the new approach proposed to model the heterogeneous structure of biofilms. 3. A short-term experiment (one month) was carried in the biofilm airlift suspension reactor to observe the structure of biofilms formed under feeding with ethanol containing waste-water. The system set-up was built in February, and the experiments were done in March. The biofilm shapes were observed at an image analyser. The third part of the report presents some aspects of this research. This experiments will furnish data for cellular automaton model validation and will be the link between the new models and the classical theory of biofilms. 4. In parallel with the experiments a new mathematical model of the biofilm growth and development was built up. It is based on a continuous representation of the dissolved compounds field (substrate, products) and a discrete representation of the particulate compounds (biomass, polymers, carrier). The partial differential equations of substrate balance are integrated in time and two-dimensional in space through the iterative method of implicit alternating directions (ADI). Depending on the resulted substrate field, the microbial cells develops and their spreading is modelled by a cellular automaton mechanism. The whole computer program is integrated on a Windows platform and this makes it very easy to use. Until now, the results are very promising and also the quantitative representation was successful. This work is not finished yet, and it was not presented in the present report. However it will be the subject of new publications. 5. In order to develop the complete approach of biofilm modelling through cellular automata methods, a Ph.D. grant was awarded. Cristian Picioreanu will study this problem at TU Delft from 15 June 1996 to 15 June 1999. In this respect, EERO opened the way for new research activities which will be systematically carried out in the area of discrete modelling of biofilm reactors used for water treatment.
When microorganisms come into contact with solid surfaces, biofilms often form. Biofilm formation is a natural immobilisation process, and can be exploited to human advantage in biotechnological processes, for example in waste water treatment. Immobilisation enables reactor operation at high biomass concentrations. Especially in waste water treatment, where relatively low substrate concentrations occur, biofilm processes permit the use of smaller reactor volumes. Understanding and modelling the biofilm formation process is then highly relevant for improving reactor design and performance. The classical method of modelling microbial systems is to derive a set of differential equations that describe the spatial distribution and development in time of particulate (microbial cells, extracellular polymers, organic and inorganic solids) and dissolved (nutrients, products) components in the biofilm. This method works well for homogeneous systems with a limited number of components, but most real microbial ecosystems are complex and heterogeneous. The description of this type of system mathematically requires use of partial differential calculus and solving the equations needs powerful numerical techniques. Cellular automata techniques offer an alternative method. They supply useful models for many investigations in natural science, they representing a natural way of studying the evolution of large physical systems. A cellular automaton can be thought of as a stylised universe. Space is represented by a uniform grid, with each site or cell containing a few bits of data. Time advances in discrete steps, and at each step each cell computes its new state from that of its close neighbours. In this way, many simple components act together to produce complicated patterns. Despite the simplicity of their construction, cellular automata are found to be capable of very complicated behaviour. However, cellular automata were able to give alternative solutions for complicated systems of differential equations. Their advantage consist mainly on their simplicity, stability and ability to model complicated boundaries. In this respect were developed algorithms for problems encountered in a surprisingly large spectrum of natural sciences: hydrodynamic problems, plasma computations, magnetisation of solids, waves and optics, crystalline growth processes, reaction-diffusion systems, ecological communities, medicine, microbiology. The biochemical engineering community hope to use the new method to gain new insight into the relation between microbial biofilms and their environment. Firstly, we are trying to explain the problems associated with the spreading front of biofilm. As numerous experiments proved, the external structure of biofilms can be very complex, varying from a nicely round shape to fluffy filaments. Problems with complex boundaries (irregular geometry) become computationally unmanageable for traditional continuous methods (which deal mainly with the sphere, the cylinder and the flat slab). Then we can change very easily the carrier shape. Irregular substratum surfaces were implemented without difficulty. In the same time, simulation with different carrier roughness confirmed the hypothesis that initial steps of biofilm formation can be governed by the balance between growth in cracks and fissures, and cells detachment due to the shear stress. And finally the internal structure of microbial aggregates can be modelled
