Physical stability and biological activity of biofilms under turbulent flow and low substrate concentration

Physical stability and biological activity of biofilms under turbulent flow and low substrate concentration L.F. Melo, M.J. Vieira Abstract The paper focuses on bio®lms subject to turbulent ¯ow and high liquid velocity (of the order of 1ms )1 ) which can be found in heat exchangers, water distribution systems and in some wastewater reactors. An overall model describing bio®lm development is presented, which includes the effects of biomass detachment due to the hydrodynamic forces. A methodology for estimating substrate consumption from data obtained through continuous monitoring of bio®lm growth is presented. Results show that the physical stability of the bio®lm increases with the liquid velocity, while the rate of substrate consumption decreases. List of symbols 1/b(s) mechanical strength of the bio®lm J p (kg bio®lm /m 2 s) bio®lm production ¯ux J r (kg bio®lm /m 2 s) bio®lm detachment or removal ¯ux K p (W m )1 K )1 ) thermal conductivity of the perspex wall K f (W m )1 K )1 ) thermal conductivity of the bio®lm m f (kg bio®lm /m 2bio®lm ) mass of bio®lm per unit surface area m f* (kg bio®lm /m 2bio®lm ) maximum mass of bio®lm per unit surface area m fa (kg active layer /m 2bio®lm ) mass of active layer per unit surface area R f (m 2 KW )1 ) thermal resistance of the bio®lm t(s) time T 1 (K) temperature of thermocouple 1 T 2 (K) temperature of thermocouple 2 T 3 (K) temperature of thermocouple 3 U(W m )2 K )1 ) overall heat transfer coef- ®cient Y X/S (kg biomass /kg substrate ) biomass yield Greek symbols (l p (kg bio®lm produced / bio®lm speci®c production kg bio®lm on the surface s) rate (l p ) a (kg bio®lm produced / bio®lm speci®c production kg bio®lm on the surface s) rate of the active layer q f (dry kg bio®lm /m 3bio®lm ) dry density of the bio®lm (dry mass per unit volume) q f (wet kg bio®lm /m 3bio®lm ) wet density of the bio®lm 1 Introduction A bio®lm is a highly hydrated biological structure attached to a support, composed of micro-organisms, extra cellular polymeric substances produced by them, as well as abiotic particles captured from the liquid medium and incorporated in the ®lm. The characteristics of such a structure depend not only on the microbial species involved and on the chemical composition of the ¯uid that is in contact with the bio®lm, but also on the hydrodynamic conditions under which the biological layer is formed. Much of the literature published so far on bio®lm properties, activity and modeling is focused on microbial ®lms formed under the operating conditions prevailing in most waste water treatment bioreactors: low or very low liquid velocities (of the order of 0.1 to 50 m h )1 ) and medium to high substrate concentrations, particularly in heterotrophic systems [10, 4, 8]. However, unwanted bio®lms formed in contact with water circulating at much higher velocities (of the order of 0.1 to 3 m s )1 ) and turbulent ¯ow, usually containing low organic substrate concentrations, are also quite important in many practical situations, such as in heat exchangers [3] and in drinking water distribution systems [18]. This so called ``biofouling'' phenomenon affects ¯uid ¯ow and heat transfer ef®ciency, as well as public health and equipment maintenance. Moreover, it indirectly causes environmental problems due to the addition of biocides to the water in order to prevent or reduce biofouling in industrial equipment. The effects of high liquid velocities on bio®lm systems have been reported in the last 20 years [1, 6, 20, 15]. Some authors advocate the use of high liquid velocities in bioBioprocess Engineering 20 (1999) 363 ± 368 ÓSpringer-Verlag 1999 363 Received: 5 May 1998 L.F. Melo, M.J. Vieira University of Minho, Centro de Engenharia Biologica Institute for Biotechnology and Fine Chemistry, 4700 Braga, Portugal Correspondence to: L.F. Melo The ®nancial support of Programme PRAXIS XXI, through Contract no. 2.1/BIO/37/94 and of Programme INTERREG, through Contract no. 01/REG II/6/96 is gratefully acknowledged. ®lm reactors, since such bio®lms are usually thinner, favouring the complete penetration of the substrate and reducing biomass wash-out [19, 5]. Moreover, the current trend for water re-utilisation will undoubtedly emphasize the need for treating liquid streams with very low substrate concentrations which will probably result in the formation of thinner bio®lms in the reactors. However, the relationships between operating conditions, bio®lm thickness and substrate consumption are not yet quite clear: Peyton [14] found that the thickness of a Pseudomonas aeruginosa bio®lm in a rotating annular reactor increased with increasing glucose loading rates; nevertheless, the thickness was not signi®cantly affected by changes in the shear stress in the reactor. Tavares et al. [17] studied the effect of gas velocity on the growth of a heterotrophic bio®lm in an aerobic three-phase ¯uidized bed reactor: the authors concluded that higher gas velocities resulted in thinner bio®lms, although they did not seem to affect COD (Chemical Oxygen Demand) removal. The substrate consumption by bio®lms subject to low liquid velocities can be easily calculated from measured inlet and outlet substrate concentrations through axial mass balances to the reactor. However, in the case of heat exchangers and water transport systems, where turbulent ¯ow prevails, the on-line determination of the activity (and, consequently, the modeling) of bio®lms cannot rely upon the measurement of those substrate concentrations, since the residence time in the equipment is too short (a few seconds, in the case of heat exchangers) and/or the substrate concentration is too low. For that purpose, an overall mathematical model is presented below that takes into account the speci®c aspects of bio®lm formation in conditions similar to the ones just mentioned. The effect of the hydrodynamic forces on bio®lm detachment is much more intense in such cases than in laminar ¯ow situations and is included in the model. This simple model has to be applied to data describing the growth rate of bio®lms and, therefore, it demands the continuous monitoring of bio- ®lm accumulation from the beginning of the attachment process. The present paper shows how the measurement of bio®lm growth rate, together with the use of this overall model, can yield important information on the properties and behaviour of the bio®lm, namely on its biological activity and physical stability. 2 Overall model for biofilm development Let m f be the mass of attached bio®lm per unit surface area, at a given time t. The change in m f with time is the result of two contradictory phenomena: the production of biomass by the micro-organisms in the bio®lm and the removal of attached biomass (bio®lm detachment) caused by the hydrodynamic forces: dmf dtJpÿJr;1 J p ± ``bio®lm production ¯ux'' (increase in bio®lm mass per unit time and unit surface area, associated to the production of biomass ± cells plus extracellular polymers ± as the result of the microbial activity within the bio®lm), M L )2 t )1 . J r ± ``bio®lm detachment or removal ¯ux'' (decrease in bio®lm mass per unit time and per unit surface area, associated to the detachment of parts of the biological deposit caused by the ¯uid forces), ML )2 t )1 . The ``bio®lm detachment ¯ux'' is assumed to be proportional to the amount of biomass attached to the surface, since the probability of existing ``weak zones'' in a thick bio®lm is higher than in a thinner one. Therefore: Jrbm f;2 where bis proportional to the hydrodynamic forces acting upon the bio®lm surface and varies inversely with the cohesiveness of the bio®lm (i.e., 1/brepresents the ``mechanical strength'' or the ``resistance to detachment'' of the bio®lm). As regards the ``production ¯ux'' (J p ), the colonisation of the clean surface by bacteria coming from the ¯uid is an essential feature only in the ®rst hours of bio®lm formation. Experimental results have shown that the subsequent growth of the biolayer is mainly due to the activity of the micro-organisms located in the attached ®lm and not to the transport of new bacteria from the liquid to the bio®lm surface [2]. Due to this biological activity and to diffusional limitations, the substrate concentration may in some cases decrease down to zero within the bio®lm before reaching the surface of the support. Thus, modeling of the ``bio®lm production ¯ux'' (J p ) must take into account two different situations (named below as i and ii) during the build up of the bio®lm layer. A mono-species bio®lm will be considered here. i) Thick bio®lm, partially penetrated by the substrate In this case, there is an ``active layer'' located in the outer part of the bio®lm, and a ``non-active'' layer that occupies the inner part of the bio®lm, close to the support. The latter contains microbial species with residual activity as regards the main substrate, plus polymeric substances and, possibly, other microbial species that do not use that substrate. If the bio®lm is partially penetrated, then the active layer will have a constant thickness equal to the maximum depth of substrate penetration. On the contrary, the thickness of the inner layer can increase with time due, for instance, to the production of polymeric material by the micro-organisms in the active layer, which will result in an overall increase of the total amount of attached biomass. Let l p be the ``bio®lm speci®c production rate'', that is, the mass of bio®lm produced by the active layer per unit time and per unit mass of total bio®lm. Thus, at a given time t: lplpa mfa mf ;3 where (l p ) a is the (constant) biomass speci®c production rate of the active zone, i.e., the mass of bio®lm produced per unit time and per unit mass of active layer, and (m f ) a is the mass of active layer per unit surface area (constant with time). 364 Bioprocess Engineering 20 (1999) Therefore : Jplpmflpamfaconstant:4 ii) Thin bio®lm, completely penetrated by the substrate In this case, the bio®lm is biologically active (as regards the main substrate) throughout its entire depth, i.e. : lplpa:5 The mass of this active layer increases with time as the bio®lm builds up, until its thickness reaches the maximum penetration depth. From this point on, case i) applies. It should be stressed, however, that the number of microorganisms in the bio®lm does not increase proportionally to the bio®lm mass, because the result of their activity is not only the production of new microbial cells but also of extra cellular substances (biopolymers). The latter, although not biologically active, can be a major fraction of the bio®lm mass. As a consequence, the rate of biomass produced per unit mass of bio®lm will decrease with time, meaning that the speci®c activity (i.e., per unit mass) of the active layer will get lower as its mass builds up. Therefore, in a completely penetrated bio®lm, it does not seem unreasonable to assume that l p is inversely proportional to the mass of active layer at each instant of time: lplpa1 mfa :6 Since all the bio®lm is active (m f m fa ), the following equation may be applied to case ii): Jplpamfaconstant:7 In both cases, i) and ii), J p is constant. Replacing J r in Eq. (1) by Eq. (2): dmf dtJpÿbm f;8 which, upon integration, results in the ®nal equation of the overall model: mfm f1ÿexpÿbt:9 where m f(J p /b) is the maximum mass of bio®lm, at steady state. Graphically, Eq. (9) represents a curve that tends to an asymptotic value of m f for tin®nite. J p can be modeled in more detail by taking into account the rates of mass transfer and biological reaction within the bio®lm, according to well known concepts of heterogeneous catalysis. This will be the subject of another paper which is focused on the study of concentration pro®les and bio®lm ef®ciency in terms of substrate consumption. In the present work, the overall model, Eq. (9), will be used to interpret the data related to the physical stability and the biological activity of bio®lms formed in a lab-scale heat exchanger under different operating conditions. 3 Materials and methods The build up of the bio®lm was monitored by measuring its heat transfer resistance, which is a non-invasive technique that can be applied during continuous operation. The micro-organism used as a bio®lm producer was Pseudomonas ¯uorescens, a gram-negative bacteria which commonly appears in natural waters. The cells were grown aerobically in a continuous culture at 27 °C and pH 7, using glucose as the limiting substrate. The growth medium was composed of glucose (0.5%), peptone (0.25%) and yeast extract (0.125%) in distilled water, sterilised at 120 °C. The microbial suspension was diluted in pre-®ltered tap water in a mixing tank (volume 12 l), at a dilution rate of 1.2 h )1 , which was high enough to prevent planktonic growth. The ®nal liquid (here called the ``test ¯uid''), containing around 6.10 7 cells ml )1 and 20 mg l )1 of glucose was pumped through the test sections where the bio®lms were formed under different ¯uid velocities ranging between 0.34 m s )1 and 0.97 m s )1 (shear stresses between 3.4 N/m 2 and 9.7 N/m 2 , and Reynolds numbers between 4 200 and 12 000). Glucose concentration was determined colorimetrically by means of a Sigma Diagnostics Enzymatic Glucose Determinations Kit. Basically, the test sections were vertical heat exchangers at laboratorial scale, each one made of a 60 cm long semicircular duct (hydraulic diameter 1.08 cm) containing three measurement zones (A, B and C) along its axis (Fig. 1). The deposition surface was an aluminium plate, having one face in contact with the ``test ¯uid'' and the other one in contact with the perspex wall of a duct of rectangular cross section. Heat was provided to the ``test ¯uid'' by water circulating at 60 °C in this rectangular duct. A grease of high thermal conductivity was spread between the metal plate and the perspex wall in order to improve the thermal contact between them. In each one of the three zones A, B and C, two thermocouples were inserted in the perspex wall adjacent to the aluminium plate and a third one immersed in the ¯uid, as shown in Fig. 1. Therefore, the transversal heat transfer ¯ux at each point (A, B, C) could be calculated from the temperature data, at any instant of time during bio®lm formation. It can be easily shown that the overall heat transfer coef®cient (U) is given by: UKpT1ÿT2 ypT1ÿT3;10 where K p (0.19 W m )1 K )1 ) is the thermal conductivity of the perspex wall where thermocouples 1 and 2 are Fig. 1. Test section for continuous monitoring of bio®lm development 365 L.F. Melo, M.J. Vieira: Physical stability and biological activity of bio®lms located, and y p is the distance between these two thermocouples. The thermal resistance (R f ,m 2 KW )1 ) introduced by the bio®lm attached to the surface can be calculated from the values of the overall heat transfer coef®cient at time t0 (clean surface) and at any other time t>0. Changes in the convective heat transfer coef- ®cient of the water caused by the increase in the surface roughness during bio®lm development were taken into account in the calculation (see details in Vieira et al., [20]). Since the water content of bio®lms is often more than 90% (w/w), the values of some of their physical properties, such as the wet density (q0 f1000 kg m )3 ) and the thermal conductivity (K f 0.6 W m )1 K )1 ), can be considered similar to the ones of the water itself [7]. Therefore, the experimental results, here presented in mass units (m f , kg of wet bio®lm per m 2 of surface area), were obtained from the correspondent values in thermal resistance units, by using the following relationshiping: mfRfKfq0 f;11 where R f is the thermal resistance introduced by the bio- ®lm layer. 4 Results and discussion Bio®lm development is illustrated in Fig. 2 by three sets of data corresponding to the formation of microbial ®lms in contact with a ``test ¯uid'' ¯owing at different velocities, containing the same substrate concentration (20 ppm): higher velocities result in smaller amounts of attached biomass, which also means lower thicknesses (Vieira et al., 1993). The data represents the average values measured in zones A, B and C. The overall model given by Eq. (9) (full lines in Fig. 2) ®ts well to these data. The average thickness of the bio®lms formed in the tests varied between 1.290 mm (lower velocities) and 0.460 mm (higher velocities), at steady state. To test the physical stability of the biological deposits, a change of the ¯uid velocity was imposed on two steadystate bio®lms which had a different history of formation. The results summarized in Table 1 indicate that part of the mass of the bio®lm formed under lower hydrodynamic forces was washed out by the liquid when the velocity was increased to 1 m s )1 ; conversely, the biological deposit formed under a higher ¯uid velocity kept stable when the velocity was also increased to 1 m s )1 . The cohesiveness (and stability) of a deposit can be considered to be related to its dry density [22]. In the present study, the bio®lms subject to stronger hydrodynamic forces contained a higher amount of dry biomass per unit volume of wet bio®lm (Table 1). These results con®rm the direct relationship between liquid velocity, dry density and stability of the bio®lms. The ``dry density'' of the bio®lm (dry biomass per unit wet volume) has been reported to vary along its depth, from lower values in the zone near the bio®lm/liquid interface to higher values near the support [7, 14], which suggests that the less dense outer layers will tend to be more easily removed by the hydrodynamic forces than the inner layers. Looking at the results presented in Table 1 in relation to the stability of the bio®lms, one can conclude that the outer loose layers occupy a higher fraction of the bio®lms when the latter are formed under lower ¯uid velocities. A simpler way of assessing the bio®lm ``resistance to detachment'' is to compare the values of 1/bobtained by ®tting the overall model, Eq. (9), to the bio®lm growth curves. In the present study, 1/bincreased linearly from 1.5 ´10 5 s, for a liquid velocity of 0.34 m s )1 ,to 2.7 ´10 5 s, for a liquid velocity of 0.72 m s )1 . Therefore, when the goal is to have a ®rmly attached biomass (such as in waste water bio®lm reactors), the formation of bio®lms under very low velocities may not be advisable, since it can lead to highly unstable bio®lms, with negative direct consequences on the reactor ef®- ciency, in terms of the rate of substrate consumption and of the microbial contamination of the outlet stream. However, although higher velocities will result in thinner and more stable bio®lms, there are still doubts on whether the amount of active cells in such bio®lms and their biological activity will be lower or higher than in thicker bio®lms. The following discussion may shed some light on this subject. By ®tting Eq. (9) to the data in Fig. 2, values of J p (bio®lm production ¯ux) can be obtained. The substrate consumption ¯ux can be calculated from J p by dividing it by the biomass yield (biomass produced per unit mass of substrate consumed). The latter was measured in suspended cell cultures and its value is 0.93 kg kg )1 [21]. Table 2 shows the substrate consumption ¯ux, which is a measure of the biological activity of the bio®lm, obtained in various tests. Fig. 2. Illustrative curves of bio®lm formation (substrate concentration = 20 mg/l) Table 1. Effect of liquid velocity on bio®lm density and physical stability Liquid velocity during bio®lm formation (m/s) Dry density of bio®lm (kg/m 3 ) Fraction of bio®lm removed when the liquid velocity was increased to 1 m/s 0.35 14 40% 0.62 28 0% 366 Bioprocess Engineering 20 (1999) These data indicate that the thicker bio®lms (those formed in contact with lower liquid velocities) are more active, which means that they contain a greater amount of active cells. This results from the fact that the total biomass of the thicker bio®lms is greater than the mass of the thinner ones. Therefore, in turbulent ¯ow, lower velocities favour the formation of thicker, less stable, but more active bio®lms. Another type of tests was performed to obtain information on the fraction of active biomass within the bio- ®lms of Pseudomonas ¯uorescens. First, the bio®lms were grown until they reached steady-state (maximum thickness); then, the substrate (glucose) was suppressed from the ¯owing liquid. This lead to the detachment of part of the bio®lm, due to lack of nutrients, as can be observed in the illustrative curves of Fig. 3. The fraction of biomass detached from the bio®lm after the supression of the substrate depended on the hydrodynamic conditions under which the bio®lm was formed, as shown in Table 3. The bio®lms subject to lower velocities appear to be less affected by the lack of substrate than those subject to higher velocities. Although this seems to be in contradiction with Table 1, the two sets of data are not comparable: Table 1 reports a simple physical phenomenon, while Table 3 refers to a metabolical one. In fact, since the substrate is able to penetrate more deeply in thinner bio- ®lms, a higher fraction of their mass will be more dependent on the availability of that substrate. This fraction is thus related to the bio®lm mass that detaches from the deposit after the substrate concentration in the liquid is reduced to (practically) zero. Therefore, it can be concluded from Tables 2 and 3 that lower velocities contribute to the formation of bio®lms where greater amounts of substrate are consumed by a smaller percentage of their mass (the upper layer). This means that the density of active cells in the upper zones of these bio®lms is probably higher than in thinner bio®lms of the same species. However, since the thicker bio®lms are much less stable from a physical standpoint (Table 1), their outer layers are easily removed by the liquid ¯ow, which means that most of their active cells can be washed out from the system when detachment occurs. This has an important practical signi®cance in industrial systems, where unexpected detachment happens much more frequently than in carefully controlled laboratory experiments. Finally, Fig. 4 con®rms that the adhesion of new bacteria coming from the ¯uid is not relevant for the development and stability of the microbial ®lm, as assumed when deriving the overall model: removing the bacteria from the ¯owing ¯uid did not affect the amount of biomass attached to the surface, after the initial surface conditioning period. A ®nal comment regarding the physical structure of bio®lms. Recently, several authors [9, 15] have shown pictures obtained with confocal laser scanning microscopy techniques, where bio®lms seem to be composed of isolated clusters of cells and biopolymers within a network of liquid channels. This structure is highly heterogeneous and does not adjust to the concept of a more uniform matrix with superimposed active and inactive layers associated to one-dimensional growth and a ¯at bio®lm surface. It is still a matter of controversy whether all bio- ®lms, even those thinner and more compact layers formed under highly turbulent conditions, display such an open structure with so many channels, mainly in the case of ``older'' bio®lms, i.e., with several weeks, as it happens in Table 2. Substrate consumption by bio®lms formed under different liquid velocities Liquid velocity during bio®lm formation (m s )1 ) Substrate consumption ¯ux ´10 8 (kg substrate m )2 s )1 ) 0.28 9.8 0.35 8.2 0.46 6.7 0.62 5.5 Fig. 3. Effect of suppressing the substrate from the liquid stream Table 3. Fraction of bio®lm mass detached after supression of the substrate Liquid velocity during bio®lm formation m s )1 ) Fraction of bio®lm mass detached after substrate supression 0.34 21.4% 0.54 79.4% 0.72 90.7% Fig. 4. Effect of suppressing the bacteria from the liquid stream 367 L.F. Melo, M.J. Vieira: Physical stability and biological activity of bio®lms heat exchangers and wastewater bioreactors. It should be noted that each ``cluster'' can also have outer layers with active cells and inner inactive layers where the substrate does not penetrate. Anyhow, the macroscopic results presented in this work are not inconsistent with the traditional image of a layer structured ®lm, and they should be interpreted as average data for the whole bio®lm, regardless of local heterogeneities. Although they are not able to shed light upon the detailed structure of the bio®lm matrix, these average parameters and properties are certainly most useful for the purpose of de®ning the design and operating conditions of industrial equipment, be it heat exchangers or bioreactors. 5 Conclusions The experimental study of bio®lms formed under turbulent ¯ow conditions and low organic substrate concentration has a particular interest in the case of heat exchanger biofouling. In such a case, due to the very low hydraulic residence time in the equipment, the accumulation and activity of the bio®lm cannot be followed by mass balances based on the measurement of inlet and outlet substrate concentrations. Instead, direct monitoring techniques, such as the local measurement of the additional heat transfer resistance introduced by the attached biomass, must be used. Appropriate mathematical models for this case should include explicit terms describing the development of the bio®lm with time, as well as provisions for evaluating the effects of biomass detachment on bio®lm growth. A simple ``overall model'' was established and ®tted to the data obtained by continuously monitoring the development of the microbial ®lm from the initial stages. The model yields information about the stability of the bio®lm (values of 1/ b) and also about the so-called ``bio®lm production rate'' (J p , amount of biomass produced per unit surface area and per unit time) and its substrate consumption capacity. The physical and biological behaviour of bio®lms should be always interpreted taking into account their ``past history'', i.e., the conditions under which they were formed. In the present work, where turbulent ¯ow and low substrate concentrations were used, higher ¯uid velocities resulted in thinner bio®lms which were more compact and more stable, that is, more resistant to detachment. However, their biological activity was lower than in the thicker bio®lms, because the latter have a greater amount of active biomass (concentrated in their upper layers). A balance must be sought between enhancing physical stability (higher velocities) and enhancing biological activity (lower velocities). For instance, in wastewater bioreactors, although thicker bio®lms may lead to a higher substrate consumption rate, the time-averaged reactor ef®ciency may be low because of frequent detachment and wash-out of the active biomass. In heat exchangers, although thinner bio®lms introduce a smaller resistance to heat transfer, they can be sometimes more dif®cult to remove from the surface during the cleaning periods since they are strongly attached and they can also offer a greater resistance to mass transport of biocides through the biological matrix. References 1. Bott, T.R.; Pinheiro, M.M.V.P.S.: Biological fouling-velocity and temperature effects, Canad. J. of Chem. Eng. 55 (1977) 473±474 2. Bott, T.R.; Miller, P.C.: Mechanisms of bio®lm formation on aluminium tubes, J. Chem. Techn. and Biotech., 33 B (1983) 177±184 3. Bott, T.R.: Fouling of Heat Exchangers. Chap. 12, Elsevier Science B.V. Amsterdam 1995 4. Bryers, J.D.; Characklis, W.G.: Bio®lms in water and wastewater treatment, In: Characklis, W.G., Marshall, K.C. (Eds.): Bio®lms pp 671±696. N.Y.: John Wiley and Sons 1990 5. Capdeville, B.; Nguyen, K.M.; Rols, J.L.: Bio®lm modelling: structural, reactional and diffusional aspects. In: Melo, L.F., Bott, T.R., Fletcher, M. and Capdeville, B. (Eds.) Bio®lmsScience and Technology pp 251±276 Dordrecht: Kluwer Academic Publishers, 1992 6. Characklis, W.G.: Bio®lm development: a process analysis, Biotech. Bioeng. 23 (1981) 1923±1960 7. Christensen, B.E.; Characklis, W.G.: Physical and chemical properties of bio®lms. In: Characklis, W.G., Marshall, K.C. (Eds.): Bio®lms pp 93±130. N.Y.: John Wiley and Sons 1990 8. Coelhoso, I.; Rodrigues A.: Modeling of bio®lm reactors with consecutive reactions, Bioprocess Engineering 12 (1995) 187± 192 9. de Beer, D.; Stoodley, P.; Lewandowski, Z.: Liquid ¯ow in heterogeneous bio®lms Biotech. Bioeng. 44 (1994) 636±641 10. Harremo Èes, P.: Bio®lm kinetics. In: Mitchel, R. (Ed.): Water Pollution Control Microbiology vol. 2 pp 82±109, N.Y.: John Wiley and Sons, N.Y., 1978 11. Kern, D.; Seaton R.: A theoretical analysis of thermal surface fouling, Brit. Chem. Eng. 4 (1959) 258±262 12. Lewandowski, Z.; Stoodley P.: Flow induced vibrations, drag force and pressure drop in conduits covered with bio®lms Wat. Sci Tech. 32 (1995) 11±18 13. Melo, L.F.; Pinheiro, J.D.: Particulate fouling: controlling processes and deposit structure, 8 th International Heat Tansfer Conference, paper HX-16, 2781±2786, S. Francisco, 1986 14. Peyton, B.M.: Effects of shear stress and substrate loading rate on Pseudomonas aeruginosa bio®lm thickness and density. Wat. Res., 30 (1996) 29±36 15. Stoodley, P.; de Beer, D.; Lewandowski, Z.: Liquid ¯ow in bio®lm systems. Appl. Environ. Microb. 60 (1994) 2711±2716 16. Taborek, J.; Aoki, T.; Ritter, R.B.; Palen, J.W.; Knudsen, J.G.: Predictive methods for fouling behaviour. Chem. Eng. Pogr. 68 (1972) 69±78 17. Tavares, C.R.G.; Santana, G.L.Jr.; Capdeville, B.: The effect of air super®cial velocity on bio®lm accumulation in a threephase ¯uidized-bed reactor. Wat. Res 29 (1995) 2293±2298 18. van der Kooij, D.; Vrouwenvelder, H.S.; Veenendaal, H.R.: Kinetic aspects of bio®lm formation on surfaces exposed to drinking water, Wat. Sci. Techn. 32 (1995) 61±65 19. van Loosdrecht, M.C.M.; Eikelbom, D.; Gjaltema, A.; Mulder, A.; Tijhuis, L.; Heijnen J.J.: Bio®lm structures, Wat. Sci. and Techn. 32 (1995) 35±44 20. Vieira, M.J.; Melo, L.F.; Pinheiro, M.M.: Bio®lm formation: hydrodynamic effects on internal diffusion and structure. Biofouling 7 (1993) 67±80 21. Vieira, M.J.: Study of the formation of biological ®lms by Pseudomonas ¯uorescens and of the effects associated to internal mass transfer and incorporation of kaolin particles, PhD thesis (in portuguese), Univ. Minho, 1995 22. Wanner, O.; Gujer, W.: A multispecies bio®lm model. Biotech. Bioeng 28 (1986) 314±328 368 Bioprocess Engineering 20 (1999)