Substrate counterdiffusion and reaction in membrane-attached biofilms: mathematical analysis of rate limiting mechanisms

Chemical Engineering Science 55 (2000) 1385}1398

Substrate counterdi!usion and reaction in membrane-attached bio"lms: mathematical analysis of rate limiting mechanisms Cristiano Nicolella, Prasert Pavasant, Andrew G. Livingston* Department of Chemical Engineering, Imperial College of Science Technology and Medicine, Prince Consort Road, SW7 2BY London, UK Received 14 April 1998; received in revised form 27 May 1999; accepted 5 July 1999

Abstract A mechanistic model of organic substrate biodegradation in membrane-attached bio"lms growing in extractive membrane bioreactors is presented and analysed to establish the rate-limiting steps. The model accounts for counterdi!usion and reaction of oxygen and organic substrate within the bio"lm, and predicts detailed substrate concentration pro"les and the evolution over time of bio"lm thickness. Good agreement was found between model predictions and organic substrate #ux and bio"lm thickness measured experimentally in a lab-scale single-tube extractive membrane bioreactor. Analysis using this model showed that, due to oxygen di!usion limitations, the reaction zone within the bio"lm is located at the bio"lm/biomedium boundary and constitutes a small fraction of the entire bio"lm volume. This allows a considerable simpli"cation of bio"lm modelling. A simple di!usion model was formulated as an alternative to the more complex full di!usion}reaction model for the calculation of organic substrate #ux. This simple model is based on the insight that the organic compound #ux is limited primarily by the bio"lm di!usion resistance. The di!usion model was combined to a yield-based expression for bio"lm accumulation to give the evolution over time of bio"lm thickness. The simpli"ed model predicts, as accurately as the full mechanistic model, the bio"lm thickness and organic substrate #ux. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Countercurrent di!usion; Bio"lm; Model; Mass transfer; Membrane bioreactor

1. Introduction Membrane bioreactors are applied for a variety of purposes, including waste gas and waste water management, as recently reviewed by Brindle and Stephenson (1996). In these systems, dense-phase membranes can be incorporated into bioreactors in applications where there are signi"cant advantages to be gained by keeping two distinct phases separated. The two most common applications are extractive membrane bioreactors (EMBs; Livingston, 1993a,b; Brookes & Livingston, 1994) and bubbleless aerated bioreactors for treatment of both gas streams (Reij, de Gooijer, de Bont & Hartmans, 1995) and water streams (Debus & Wanner, 1992; Rothemund, Camper & Wilderer, 1994). During operation of these dense-phase membrane systems, bio"lms may develop at the membrane/biomedium interface and participate in the separation/degradation

* Corresponding author. Tel: 44-171-58945-582; fax: 44-171-594-5604. E-mail address: [email protected] (A. G. Livingston)

process (Freitas dos Santos and Livingston, 1994; Reij et al., 1995; Rothemund et al., 1994). Bio"lms have hydraulic resistance and may lead to intolerable loss of performance (Mc Donogh, Schaule & Flemming, 1994). Moreover they reduce pollutant #ux across the membrane (Freitas dos Santos & Livingston, 1995). On the other hand, membrane-attached bio"lms (MABs) were shown to be e!ective in reducing air stripping of pollutants from the biomedium (Freitas dos Santos & Livingston, 1995). By considering positive and negative e!ects, an optimal thickness can be de"ned for membrane-attached bio"lms (Pavasant, Freitas dos Santos, Pistikopoulos & Livingston, 1996) and consequently control of membrane-attached bio"lm thickness plays a key role in the engineering design and operation of dense phase membrane bioreactors. In membrane-attached bio"lm systems, the membrane creates a structure which causes di!erent substrates, i.e. oxygen and organic pollutants, to di!use into the bio"lm from opposite directions (countercurrent di!usion of substrates). This is in contrast to conventional bio"lm systems where the substrates di!use through the bio"lm

0009-2509/00/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 4 1 7 - 0

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Nomenclature A C CD D K k ¸ N Q r R S t u < x >

surface area, m2 concentration, g m~3 drag coe$cient di!usivity, m2 s~1 partition coe$cient mass transfer coe$cient, m s~1 length, m #ux, g m~2 s~1 #ow rate, m3 s~1 radial coordinate, m radius, m dimensionless concentration time, s velocity, m s~1 volume, m3 dimensionless radial coordinate yield coe$cient

Greek letters d C

dimensionless thickness detachment or attachment rate, g m~2 s~1

in the same direction, usually inward at bio"lm/biomedium interface. There are several mechanistic models that describe conversion of soluble substrate by bio"lms and population dynamics, using a continuum approach and observing conservation principles (Wanner & Gujer, 1986; Characklis & Marshall, 1989; Skowlund, 1990; Arvin & Harraemoes, 1990; Wanner & Reichert, 1996; Wood & Whitaker, 1998). These concepts have been applied to the modelling of membrane-attached bio"lms in order to better understand the behaviour of extractive membrane bioreactors, and to be able to predict the e!ects of operating variables/parameters on system performance (Pavasant et al., 1996, Pavasant, Pistikopoulos & Livingston, 1997). Similar studies have also been performed for membrane-attached bio"lms growing in bubbleless membrane bioreactors (Wanner, Debus & Reichert, 1994; Debus, 1995). With rare exceptions, all mathematical models of bio"lms treat the bio"lm as a one-dimensional structure, whose porosity and shape have to be speci"ed as model input parameters. Only recently, Picioreanu, van Loosdrecht and Heijnen (1998a,b), by combining a discrete representation of the solid phase with classical continuous methods for soluble compounds, were able to predict bio"lm structure together with an accurate time evolution of concentrations, #uxes and conversion rates. However, even the classical one-dimensional approach is sometimes rather complicated for practical design pur-

c k h o q

shear stress, N m~2 kinetic rate, s~1 half rate constant, g m~3 density, g m~3 dimensionless time

Subscripts 0 a b bm d e in m o O ot s S t w X

initial attachment bio"lm biomedium detachment endogeneous decay inner membrane membrane overall oxygen outer membrane shell organic substrate tube wastewater biomass

poses, due to the high number of variables and parameters that it involves. Attempts have been made to simplify bio"lm models by using analytical solutions for steady state (Kim & Suidan, 1989) or approximate algebraic solutions (Golla & Overcamp, 1990) to derive substrate concentration pro"les and bio"lm thickness. However, these simpli"ed approaches have been exclusively applied to modelling conventional bio"lm systems featuring co-di!usion of substrates. No studies have so far focused on determining either the rate-limiting mechanisms or simpli"ed models for membrane-attached bio"lms where counter-di!usion of substrates is the norm. This work presents mathematical modelling of substrate counter-di!usion and biomass accumulation in membrane-attached bio"lms. The results of the simulations are compared to experimental data obtained in an extractive membrane bioreactor, to verify the accuracy of the model. Model equations are then analysed to identify the rate-limiting mechanisms in EMB processes. On the basis of this analysis, a simpli"ed di!usion model is proposed to predict the organic substrate concentration pro"le within the bio"lm, organic substrate #ux across the membrane, and bio"lm growth rate.

2. Materials and methods This study describes the modelling of physical and chemical processes involved in the biodegradation of

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Fig. 1. Schematic representation of a single-tube extractive membrane bioreactor. WT: wastewater tank; PP: peristaltic pumps; RT wastewater recycle tank; GP: gear pumps; MM: single tube membrane module; BR: perfectly mixed bioreactor; NT: nutrient tank; ET e%uent tank.

a volatile organic compound in a membrane-attached bio"lm growing in a single tube extractive membrane bioreactor (STEMB). The system considered, schematically represented in Fig. 1, incorporates a silicone rubber membrane tube "tted coaxially in a glass shell. Wastewater and biomedium streams #ow cocurrently in the membrane tube and in the shell, respectively, and are recirculated at high #ow rates. The biomedium is recirculated from the shell to a well-mixed bioreactor where temperature, pH and dissolved oxygen are controlled. A lab-scale single tube extractive membrane bioreactor was used to verify the proposed mechanistic model of membrane-attached bio"lm. A synthetic wastewater containing monochloro-benzene (MCB) was used throughout the experiments. A mixed microbial culture was used to degrade MCB. The composition of the nutrient media was: KH PO 0.4 g~-, K HPO 0.3 g~-, (NH ) SO 2 4 2 4 42 4 1 g~-, Hutner's salts 20 ml~- (Jorge and Livingston, 1999). Two experimental runs were performed at di!erent biomedium recycle #ow rates (run 1: Q "0.033] s 10~3 m3 s~1; run 2: Q "0.167]10~3 m3 s~1 corres sponding to Reynolds numbers in the shell side of 1750 and 8750, respectively). The operating conditions and the characteristics of the experimental system are shown in Table 1. During the experiments the bio"lm thickness was continuously monitored and recorded by the computer controlled video imaging technique (Zhang, Splendiani, Freitas dos Santos & Livingston, 1998). Enlarged images of the bio"lm attached to the silicone rubber membrane were captured by a video camera, displayed on a monitor, and recorded on tape by a video cassette recorder. Vertical (up and down) movements of the camera

Table 1 Experimental system characteristics and operating conditions Characteristics and conditions Membrane tube length, ¸ m Membrane tube inner radius, R */ Membrane tube outer radius, R ot Shell column radius, R s Waste feeding #owrate, Q w Biomedium recirculation #ow rate (run1), Q36/1 s Biomedium recirculation #ow rate (run2), Q36/2 s Waste recirculation #ow rate, Q t Waste feeding concentration, C Sw Shell dissolved oxygen concentration, C Os Bioreactor suspended biomass concentration, C Xs

Value

Unit

0.3 1.5]10~3 1.8]10~3 0.08 2.5 ) 10~8 3.3 ) 10~5

m m m m m3 s~1 m3 s~1

1.7 ) 10~4

m3 s~1

2.9 ) 10~5 100 7.5 200

m3 s~1 g m~3 g m~3 g m~3

were controlled by a microcomputer. Bio"lm thickness was measured at "ve vertical positions along the membrane tube by analysing the images online or after experiments. The organic substrate concentration in the inlet and outlet wastewater streams and in the biomedium were periodically measured by gas chromatography according to the procedure described by Zhang et al. (1998). The #ux of organic substrate across the membrane was then calculated as Q (C !C ) St , N " w Sw S nR ¸ */ m

(1)

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where Q is the feeding wastewater #ow rate, C is the w Sw organic substrate concentration in the feeding wastewater, C is the organic substrate concentration in the St membrane tube, R is the membrane tube inner diameter */ and ¸ is the membrane tube length. m 3. MAB mechanistic model = model validation A dynamic mathematical model of the system described above, including a full description of the #ow, mass transfer and reaction in the membrane tube, the membrane-attached bio"lm and the bioreactor, was developed by Pavasant et al. (1996,1997) and only the essential features are given below. The following mechanisms are considered (see Fig. 2): f di!usion of oxygen from the bio"lm/biomedium interface into the bio"lm; f di!usion of organic substrate from the membrane/bio"lm interface inward into the bio"lm; f convective mass transfer at the bio"lm/biomedium interface and at the wastewater/membrane interface; f reaction of oxygen and organic substrate within the bio"lm. 3.1. Assumptions f It is assumed that the EMB is operated at high wastewater and biomedium recirculation #ow rates (see Table 1), so that substrate concentration can be assumed uniform in the tube and in the shell, and equal to the concentration in the wastewater and biomedium vessel, respectively. f The rates of mass transfer at the biomedium/bio"lm and wastewater/membrane interfaces are proportional to the concentration di!erence between the bulk and the interface; the proportionality factors are mass transfer coe$cients.

f Di!usion across the membrane follows Fick's "rst law and organic substrate di!usion coe$cient is constant in the membrane phase. f The #ux of oxygen across the membrane from both wastewater and biomedium side is considered negligible. As shown by Fig. 1 and Table 1, in the experimental system used in this work, the wastewater is recirculated from the membrane module to a wellmixed vessel where a tiny amount of fresh feed is continuously added. Only a small amount of oxygen will therefore be present in this stream, and the oxygen #ux through the membrane from the wastewater side can therefore be considered negligible. Moreover, due to the low solubility of oxygen in water, when using air as oxygenating gas, the oxygen penetration in the bio"lm in the absence of reaction is limited to a layer of the order of 100 lm maximum (Bailey & Ollis, 1986), whose thickness is further reduced under reacting conditions. That means that the assumptions of negligible oxygen #ux through the membrane from the biomedium side is acceptable for thick bio"lms and only in the case of thin bio"lms (thickness of less than 100 lm) the assumption will result in a simpli"cation of the physical characteristics of the system considered. f The structure of a bio"lm plays a major role in the mass transfer and biodegradation processes that occur in the bio"lm. The heterogeneous structure of many bio"lms permits convective transport within voids and water channels permeating the bio"lm, whilst within the cell aggregates or clusters molecular di!usion is the predominant mode of mass transport (de Beer & Stoodley, 1995). However, a microcluster-type bio"lm presents formidable modelling complexities (Bishop, 1997). In this work, as in most bio"lm kinetic models (Characklis & Marshall, 1989; Rittmann & Manem, 1992; Wanner & Gujer, 1986), it is assumed that the bio"lm is continuous, and that the mass transfer through the mass boundary layer and within the bio"lm is di!usional and perpendicular to the bio"lm

Fig. 2. Concentration pro"les for oxygen and organic substrate in membrane attached bio"lms.

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surface. Mass transport of substrates inside the bio"lm is described using e!ective di!usion coe$cients, assumed to be a fraction (80%) of their values in water (Wij!els, de Goijer, Kortenkaas & Tramper, 1991; de Beer et al., 1997). The limits of this assumption will be of particular concern during the initial phase of bio"lm growth, when only a part of the membrane will be covered by patches of bio"lm, and heterogeneity will play a signi"cant role in the description of the system. In the case of thick bio"lms, the role of heterogeneity will be less important, and the thickness of the bio"lm will on average dampen the e!ects of local discontinuities and irregularities on the global behaviour of the system. f The bio"lm has uniform thickness along the length of the membrane and smooth surface. These assumptions were veri"ed by visual observations during the experiments described in the previous section.

where dimensionless time (q), radial coordinate (x), bio"lm thickness (d), organic substrate and oxygen concentration in the bio"lm (S and S ), are, respectively, Sb Ob de"ned by

3.2. Equations

3.3.1. Reaction kinetics Biological degradation of chlorinated organic compounds such as MCB is most often subjected to inhibition. In a recent paper, Jorge and Livingston (1999) showed that the growth of a microbial strain (JS150) capable of degrading MCB was strongly inhibited by high MCB concentrations. Their data were "tted by a piecewise function where the Monod model was applied until reaching the inhibitory concentration of 180 mg/l. Since the MCB concentrations considered in this work are lower than this values (see Table 1), it is assumed that substrate consumption for bacterial growth within the bio"lm can be described by Monod growth kinetics for two limiting substrates (oxygen and organic substrate):

These assumptions result in the following set of dimensionless equations: Organic substrate balance:

A

B

LS 1 1 L2S x dd LS k o R2 Sb " Sb # Sb # S b ot , # Lq D C d(xd#1) d dq Lx d2 Lx2 Sb St (2) Oxygen balance:

A

B

1 1 L2S D x dd LS D LS Ob " Ob # Ob Sb # Sb Lq D d(xd#1) D d dt Lx d2 Lx2 Ob Ob k o R2 # S b ot . (3) D C Ob Os Bioxlm accumulation:

P

1 R2 R dd ot " k (xd#1)d dx!C ot . X D o dq D (d#1) Sb 0 Sb b Boundary and initial conditions: k R 1 LS Sb "! Sm ot (1!S ), Sb D d Lx Sb x"0,∀qP LS Ob "0, Lx 1 LS k R Sb "! Ss ot (S !S ), Sb Ss d Lx D Sb x"1,∀qP 1 LS k R Ob "! Os ot (1!S ), Ob d Lx D Ob q"0,∀xPS "S "1, Sb Ob q"0Pd"d , 0

(4)

(5)

(6)

(7)

(8) (9) (10)

D r!R R !R C ot , d" b ot , S " Sb , q" Sb t, x" Sb R2 R !R R C ot b ot ot St C S " Ob . (11) Ob C Os In this work, the combination of di!usion}reaction model for substrate transfer and consumption within the bio"lm [Eqs. (2) and (3)], and moving boundary problem for bio"lm accumulation [Eq. (4)], together with inherent boundary and initial conditions [Eqs. (5)}(10)], is called &MAB mechanistic model'. 3.3. Model parameters

A

BA

B

C C Sb Ob k"k , (12) .!9 h #C h #C S Sb O Ob where l is the maximum growth rate, h and h are .!9 s O Monod's half-concentration constant for organic substrate and oxygen, respectively. Cell maintenance and death can be described using the concept of cell decay through an endogenous decay coef"cient, k , assumed to be constant: e k "k!k . (13) X e Speci"c reaction rates of organic substrate and oxygen are expressed as a function of k and their corresponding yield coe$cients (> and > ): XS XO k k "! , (14) S > XS k #> k , (15) k "! XOe e O > XO where > is the yield coe$cient for oxygen in the XOe endogeneous decay processes.

A

B

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3.3.2. Detachment and attachment kinetics The net removal rate of bio"lm is C"C !C . (16) d a The detachment and attachment rates, respectively, are expressed as (Rittmann, 1982) C [gm~2 s~1]"4.15]10~9o (R d)c0.58, (17) d b ot C [gm~2 s~1]"3.0]10~11C , (18) a Xs where o is the bio"lm dry density and C is the biomass b Xs concentration in the biomedium.The shear stress (c) exerted by the liquid on the bio"lm surface is calculated as c"1C o u2, (19) 2 D bm s where u is the liquid velocity in the shell, o is the s bm biomedium density and C is the drag coe$cient calD culated according to Perry and Green (1984). 3.3.3. Mass transfer coezcients The overall mass transfer #ux between the bulk liquid in the membrane tube and the membrane/bio"lm interface is described through a resistances-in series model: 1 1 R ln(R /R ) ot */ , " # */ (20) k k D K Sm St Sm Sm where k is the overall mass transfer coe$cient for the Sm #ux across the liquid "lm and the membrane, k is the St tube side "lm mass transfer coe$cient, D is the memSm brane di!usion coe$cient, K is the partition coe$cient Sm between the liquid and the membrane phase for the organic substrate, R is the inner radius of the mem*/ brane tube and R is the outer radius. ot The mass transfer coe$cients in the liquid "lms at the wastewater/membrane and biomedium/bio"lm interfaces are calculated according to the relationships proposed by Brookes and Livingston (1995). Under the

operating conditions of the experiments performed in this work (see Table 1), the Reynolds number was 12 000 in the membrane tube and varied from 1750 (run 1) to 8750 (run 2) in the shell. According to the de"nitions of #ow regimes in membrane tubes given by Gekas and Hallstrom (1987), the membrane tube is characterised by turbulent #ow, whilst the values of Reynolds in the shell are in the range of transition from laminar to turbulent #ow regime. The calculated tube-side liquid-"lm mass transfer coe$cient was 2.2]10~4 m s~1 for the organic substrate, whilst the calculated shell-side liquid "lm mass transfer coe$cients were 1.8]10~5 for the organic substrate and 2.5]10~5 for oxygen in run 1, and 7.8]10~5 for the organic substrate and 9.4]10~5 for oxygen in run 2. 3.3.4. Physical properties Physico-chemical parameters and kinetic rate constants used in the simulations are listed in Table 2. Unless otherwise speci"ed, the values of the operating conditions and system characteristics used in the simulations are those reported in Table 1. 3.4. Numerical solutions Due to bio"lm growth, bio"lm dynamic models are typically nonlinear moving boundary problems with mixed partial di!erential and integral equations. The solution was obtained by transforming the original moving boundary problem into a dimensionless "xed boundary problem [Eq. (11)] and by solving the resulting equations [Eqs. (2)}(20)] through a "nite representation of the derivatives using general PROcess Modelling System (gPROMS, Barton and Pantelides, 1994). 3.5. Experimental verixcation In order to verify the reliability of the mechanistic model assumptions, the results of the simulations were

Table 2 Model parameters Parameter

Value

Unit

Source

D Sb D O D Sm K o b > XS > XO > XOe k .!9 k e h O h S

1.5]10~9 2.4]10~9 1.8]10~10 70 55 000 0.45 0.2 1.37 6.7]10~5 1]10~7 1]10~2 5]10~1

m2 s~1 m2 s~1 m2 s~1

Zhang et al. (1998) Perry and Green (1986) Brookes and Livingston (1995) Brookes and Livingston (1995) Freitas dos Santos and Livingston (1994) Jorge et al. (1998) Stoichiometry Pavasant et al. (1995) Jorge et al. (1998) Atkinson and Mavituna (1991) Freitas dos Santos and Livingston (1994) Jorge et al. (1998)

g m~3

s~1 g m~3 g m~3

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compared to experimental data obtained in the lab-scale single-tube extractive membrane bioreactor described above (see Fig. 1 and Table 1). Since, due to high volumetric biodegradation rate and low dilution rate in the bioreactor, the organic substrate concentration in the biomedium (C ) was found to be zero in all the experiSs ments, all calculations reported in the following were performed with C "0. Ss Fig. 3 shows the organic substrate #ux as a function of the bio"lm thickness as calculated through model equations and also measured during the experiments at Q "3.3]10~5m3 s~1 (run 1). Fairly good agreement S between the two set of data can be observed. It should be noticed that the organic substrate #ux across the membrane rapidly decreases with increasing bio"lm thickness and even for rather thin bio"lms (400 lm) mass transfer across the membrane is strongly reduced. Bio"lm thickness estimates used in Fig. 3 are the average of "ve measurements taken along the length of the membrane. Standard deviations of these measurements ranged from 3% for thin bio"lms (d(0.25 mm) to 11% for large bio"lms (d'0.75). A comparison between experimental and calculated bio"lm accumulation is made in Fig. 4, which shows the bio"lm thickness as a function of time for two values of biomedium recycle #ow rates as calculated by the mechanistic model and as measured in the lab scale STEMB. Also in this case, the model gives a good approximation of the experimental results.

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Fig. 3. Organic substrate #ux across the membrane as a function of bio"lm thickness. Symbols represent experimental data for run 1, the line corresponds to the values calculated using the mechanistic model.

4. MAB Mechanistic model = analysis of rate-limiting phenomena 4.1. Rate-limiting steps In membrane-attached bio"lms growing in extractive membrane reactors, the organic substrate di!uses into the bio"lm from the wastewater inside the membrane tube, while oxygen is delivered from the biomedium in the shell side (see Fig. 2). Reaction takes place in a region of the bio"lm where both substrates are present simultaneously. The potential rate limiting steps within the bio"lm are: (i) di!usional mass transfer of oxygen; (ii) di!usional mass transfer of organic substrate; (iii) reaction rate. The system is controlled by di!usion when there is not enough organic substrate or oxygen to reach the maximum utilisation rate (partial penetration of substrates). When full penetration of both substrates into the bio"lm occurs the system is limited by the rate of reaction. The rate-limiting step will depend on many factors such as the oxygen concentration and organic substrate concentrations at the bio"lm boundaries, the bio"lm thickness and the kinetic parameters. The in#uences of the oxygen concentration in the biomedium (C ) and Os organic substrate concentration in the wastewater (C ) St

Fig. 4. Comparison between the experimental data for bio"lm thickness (symbols) with the simulation result (lines) for two experimental runs.

are described by introducing the ratio between the two concentrations (C /C ). The analysis reported in the St Os following work is limited to evaluating the e!ects of the ratio C /C and of bio"lm thickness, since, for a given St Os system, kinetic parameters are "xed, whilst operational characteristics such as C or C might be varied and the St Os bio"lm thickness changes during normal operation of the system.

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Fig. 5. Concentration pro"les for oxygen and organic substrate in membrane-attached bio"lms as calculated by the mechanistic model for di!erent bio"lm thicknesses. (a) C /C "10; (b) C /C "1. st os st os

Fig. 5a and b shows the pro"le of oxygen and organic substrate concentration within the bio"lm as calculated using the model equations for di!erent bio"lm thicknesses, for oxygen concentration in the shell side of 7.5 g m~3 (corresponding to the saturation in water at 303C) and for organic substrate concentration in the tube side equal to 75 and 7.5 g m~3 (Figs. 5a and b, respectively) corresponding to C /C ratios of 10 and 1, respecSt Os tively. Reaction rate pro"les corresponding to the concentration pro"les of Fig. 5 are reported in Fig. 6. The model suggests that penetration of both oxygen and organic substrate strongly depends on the relative amount of oxygen with respect to organic substrate in the

Fig. 6. Reaction rate pro"les as calculated by the mechanistic model for di!erent bio"lm thicknesses: (a) C /C "10; (b) C /C "1. st os st os

bulk liquid phases (ratio C /C ), and on bio"lm thickSt Os ness. Organic substrate either fully or partially penetrates the bio"lm, whilst oxygen is always completely consumed in the bio"lm for all conditions simulated in Figs. 5a and b and 6a and b. Full oxygen penetration may be expected only for very low values of organic substrate concentration in the tube side or very high values of oxygen concentration in the shell side (C /C (1). St Os For high C /C thin bio"lms are completely penSt Os etrated by the organic substrate. The organic substrate which crosses the membrane can be completely degraded in the bio"lm only in the case of the thickest bio"lm in Fig. 5a. The oxygen penetration depth is shallow and reaction is either very slow (thin bio"lm) or con"ned to

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a thin layer at the bio"lm biomedium interface (Fig. 6a). For low values of the ratio C /C , oxygen penetrates St Os the bio"lm deeply (Fig. 5b). Organic substrate is completely degraded in the bio"lm, but the reaction rate is relatively low (see Fig. 6b) if compared to the values of Fig. 6a which correspond to higher C values. St It should be observed that, in most practical cases, the ratio C /C is much higher than 1, due to use of air as St Os the aerating gas and the necessity of maintaining a high driving force for the #ux of organic substrate across the membrane. The corresponding organic substrate concentration and reaction pro"les are those reported in Figs. 5a and 6a, respectively. Therefore, referring to these "gures, it can be concluded that for cases of practical interest reaction takes places in a thin layer at the bio"lm/biomedium interface. 4.2. Ewect of external mass transfer Apart from the mass transfer resistance in the bio"lm, external mass transfer might also limit the performance of membrane-attached bio"lms. Fig. 7 represents the dimensionless oxygen penetration depth (ratio between the oxygen penetration depth and the bio"lm thickness) as a function of bio"lm thickness for various liquid}solid mass transfer coe$cients at the bio"lm/biomedium interface. For thick bio"lms, the penetration depth is a signi"cant fraction of the bio"lm thickness only for very high mass transfer coe$cients. In other cases a limitation in the oxygen penetration is expected, due to both external and internal resistances. Fig. 8 shows a comparison between the actual #ux of organic substrate across the membrane covered by the bio"lm and the #ux across the clean membrane (without any bio"lm). In this "gure, the predicted ratio g , de"ned b as flux across the membrane covered by biofilm g " b flux across the clean membrane

Fig. 7. Calculated oxygen penetration depth as a function of bio"lm thickness for di!erent values of mass transfer coe$cient in the shell side.

(21)

is plotted as a function of the oxygen concentration in the shell side, for di!erent bio"lm thicknesses and constant organic substrate concentration in the biomedium (C "100 g m~3). For low C the #ux is almost unafSt Os fected by the oxygen concentration in the shell side, due to the limited penetration of oxygen within the bio"lm. When C increases, the #ux also increases to reach Os a maximum value which corresponds to the full oxygen penetration. After this point, the #ux does not change more with further increase of oxygen concentration in the shell side, due to organic substrate di!usion limitation. The value of maximum #ux depends on parameters such as the kinetic rate constant and the di!usion coe$cient of the organic substrate in the bio"lm, but it does not depend on the bio"lm thickness. It should be noted that in thick bio"lms the maximum #ux can be reached only

Fig. 8. Calculated organic substrate #ux across the membrane as a function of oxygen concentration in the shell side and bio"lm thickness. The shaded area represents unobtainable values of dissolved oxygen concentration, since the maximum solubility of oxygen in water at ambient temperature is 40 g m~3.

for oxygen concentrations much higher than the values usually obtainable in saturated water. For values of practical interest, the #ux is almost una!ected by the oxygen concentration in the bulk liquid. However, even in these cases the amount of organic substrate consumed in the bio"lm strongly increases with increasing oxygen concentration in the shell side, as shown by Fig. 9 which reports the fraction of organic substrate consumed in the

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assuming simple relationships between the bio"lm growth rate and the substrate #uxes. This approach to MAB modelling leads to a set of ordinary di!erential equations that can be easily solved to give the organic substrate concentration pro"les within the bio"lm, the organic substrate #ux across the membrane and the bio"lm, and the bio"lm accumulation over time. 5.1. Organic substrate yux

Fig. 9. Fraction of organic substrate consumed in the bio"lm for di!erent values of oxygen concentration in the shell side.

bio"lm, de"ned as organic substrate consumed in the biofilm g" r organic substrate crossing the membrane

The main assumption adopted in the MAB di!usion model is that the consumption term can be neglected in the mass balance equation for the organic substrate in the bio"lm. Further simpli"cation can be derived by considering that the characteristic times of substrate diffusion are several orders of magnitude smaller than those of microbial growth (Kissel, Mc Carty & Street, 1984). This implies that microbial growth may be modelled assuming steady-state substrate concentration pro"les (Wanner & Gujer, 1986). According to these assumptions the mass balance equation for the organic substrate in the bio-"lm is d dS r Sb "0 dr dr

(23)

(22)

as a function of bio"lm thickness for di!erent values of C and constant organic substrate concentration in the Os biomedium (C "100 g m~3). The organic substrate is St only partially consumed in the bio"lm for the values of parameters considered in Fig. 9. The exiting #ux reaches the biomedium where it is assumed to be consumed by the suspended biomass (C "0). Ss

with boundary conditions given by !D

dS Sb Sb dr dS Sb Sb dr

r/Rot

"k (1!S D ), Sm Sb r/Rot

(24)

"k (S D b !S ). (25) Ss Sb r/R Ss r/Rb Eqs. (23)}(25) can be solved analytically for any bio"lm thickness to give the concentration and #ux pro"les within the bio"lm: !D

5. MAB Simpli5ed model The analysis performed in the previous section leads to the conclusion that, under conditions typical of normal operation of extractive membrane reactors (C /C ' St Os 10), the reaction zone within the bio"lm is located at the bio"lm/biomedium boundary and constitutes a small fraction of the entire bio"lm volume. This implies that most of the bio"lm is acting as a non reacting medium through which organic substrate di!uses, but does not react, due to a lack of oxygen. These results allow a considerable simpli"cation of bio"lm modelling since a simple di!usion model can be used to describe the organic substrate concentration pro"le within the bio"lm. In the simpli"ed MAB model proposed in this work, the organic substrate #ux across the membrane and the bio"lm is calculated through a single mass balance equation describing the di!usion of the organic substrate across the bio"lm, neglecting the reaction within the bio"lm. The bio"lm thickness is calculated by

K K

C

A B

D

R R R b # */ k (C !C ), (26) C "C # */ ln Sb Ss So St Ss D r R k Sb b Ss R N " */ k (C !C ), (27) Ss Sb r So St where k is the overall mass transfer coe$cient for the So organic substrate #ux between the two aqueous solutions in the membrane tube and in the shell, de"ned through a resistances-in-series approach as

C

A B

1 R R ot */ ln k " # So k R K D St */ Sm Sm R R R 1 ~1 */ ln b # */ # , (28) R K D R k ot Sb Sb b Ss where k is the tube-side "lm mass transfer coe$cient, St k is the shell-side "lm mass transfer coe$cient, D is Ss Sm the membrane di!usion coe$cient, D is the bio"lm Sb

A B

D

C. Nicolella et al. / Chemical Engineering Science 55 (2000) 1385}1398

di!usion coe$cient, K is the partition coe$cient beSm tween the liquid and the membrane phase for the organic substrate, K is the partition coe$cient between the Sb liquid and the bio"lm phase for the organic substrate (assumed equal to 1, according to Zhang et al. (1998)), R is the inner radius of the membrane tube, R is the */ ot outer radius and R is the radial coordinate of the biob "lm/biomedium interface. The accuracy and limits of applicability of the di!usion model are assessed by comparing the organic substrate concentration pro"le and the organic substrate #ux, as calculated through Eqs. (26) and (27), respectively, with the predictions of the more complex mechanistic model for the case of C "0, which, as previously discussed, Ss corresponds to the experimental results obtained in the EMB experiments of this work. Organic substrate concentration pro"les, corresponding to C "7.5 g m~3, C "100 g m~3 and k "2] Os St Ss 10~5m s~1, as calculated through the mechanistic model and di!usion model are plotted in Fig. 10 for di!erent values of bio"lm thickness. The simple di!usion model gives a good approximation to the results obtained with the more complete mechanistic model for thick bio"lms. The pro"les predicted by the mechanistic model are slightly steeper than those predicted by the di!usion model, due to the fact that the assumptions adopted for the di!usion model are not satis"ed in the proximity of the bio"lm/biomedium interface, where reaction takes place (see Fig. 6). The approximation made in calculating the concentration pro"le within the bio"lm with the di!usion model results in a small error in the prediction of the organic substrate #ux across the membrane as shown by Fig. 11, where the ratio g , de"ned as f

Fig. 10. Comparison between the organic substrate pro"le in the bio"lm as calculated by the mechanistic model (lines) and the simpli"ed (symbols) model for di!erent values of bio"lm thickness.

information on the kinetics of biochemical reactions within the bio"lm, can predict reasonably well the organic substrate #ux across the membrane This is a key parameter in the design and operation of membrane bioreactors. 5.2. Bioxlm accumulation The simple di!usion model [Eqs. (23)}(25)] gives a good approximation of the organic substrate #ux across the membrane and the bio"lm for any bio"lm

flux across the membrane as calculated by the diffusion model g " f flux across the membrane as calculated by the mechanistic model

is reported for the case of C "7.5 g m~3, C " Os St 100 g m~3 and k "1]10~5 m s~1. The #ux calculated ss by the di!usion model is less than 10% lower than the #ux calculated through the mechanistic model. The deviation is higher for thin bio"lms, and tends to decrease with increasing bio"lm thickness. This is due the fact that, according to the main hypothesis of the di!usion model, the reaction in the bio"lm is assumed to take place at the interface with the biomedium, whilst in thin bio"lms reaction can take place at a considerable depth within the bio"lm with a resulting enhancement of #ux with respect to the case where the reaction occurs at the bio"lm boundary. From the results of Fig. 11 it can be concluded that the simple di!usion model, which does not require any

1395

(29)

thickness, but it is not able to predict the evolution over time of the bio"lm thickness. A simpli"ed approach to describing bio"lm accumulation is proposed below, which can be coupled with the MAB di!usion model to provide a prediction of bio"lm accumulation and #ux. Biomass accumulation in the bio"lm is given by the di!erence between growth rate, and net detachment rate [Eq. (16)] and endogeneous decay. The growth rate can be limited by either the oxygen #ux or the organic substrate #ux. It is therefore calculated as the minimum of the growth rates corresponding to the complete conversion into biomass of (i) the organic substrate crossing the membrane/bio"lm interface, and (ii) the oxygen crossing the biomedium/bio"lm interface. Accordingly,

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bio"lm is calculated as

Fig. 11. The ratio between the organic substrate #ux across the membrane calculated by the full mechanistic model (lines) and the simpli"ed model (symbols) as a function of bio"lm thickness.

N D b "k C , (31) Ob r/R Os Os where k is the mass transfer coe$cient for oxygen in the Os shell and C is the oxygen concentration in the shell side. Os In this work, the combination of di!usion model for the calculation of organic substrate #ux [Eqs. (27) and (28)] and the yield-based expression for the calculation of bio"lm accumulation [Eq. (30)] is called &MAB simpli"ed model'. The MAB simpli"ed model allows the bio"lm thickness to be calculated as a function of time. A comparison with the values predicted by the full mechanistic model is made in Fig. 12, where bio"lm thickness calculated through both models is plotted as a function of time for di!erent external mass transfer conditions (C " Os 7.5 g m~3 and C "100 g m~3). The agreement between St the two sets of simulations is rather good, con"rming that the assumptions of MAB simpli"ed model allow the problem to be solved with far fewer equations, variables and parameters than are required by the mechanistic model. A last comparison between the predictions of simpli"ed and mechanistic models is made in Fig. 13, which shows the organic substrate #ux across the membrane over time as calculated by both models for C " Os 7.5 g m~3, C "100 g m~3 and k "2]10~5m s~1. St Ss Also in this case a good agreement is obtained between the two sets of data. Since the di!usion model considers the sole rate-limiting mechanisms to be the organic substrate di!usion across the bio"lm, it can be concluded from the results of Fig. 13 that the insight behind the #ux decrease over time, observed experimentally and predicted theoretically, is the resistance to organic substrate transfer caused by the bio"lm. 5.3. Practical implications

Fig. 12. Comparison between the bio"lm thickness over time as calculated by the full mechanistic model (lines) and the simpli"ed model (symbols) for di!erent values of mass transfer coe$cient in the shell side.

the bio"lm accumulation is given by d< b "min[> (AN ) , > (AN ) b ] o XO Ob r/Rb XS Sb r/R b dt !CAD b !k o < . (30) r/R e b b The organic substrate #ux at the bio"lm/biomedium interface can be calculated through Eq. (27). The oxygen transferred from the bulk liquid in the shell side to the

In EMBs the membrane phase is used to separate organic substrates from wastewater into a controlled composition biomedium, where they are biodegraded. The organic substrate #ux across the membrane is therefore a key design parameter: a reduction in the #ux will lead to a larger membrane area being necessary to achieve a "xed degree of pollutant removal from a given wastewater #ow. The di!usion model provides an analytical expression for the organic substrate #ux across the membrane, which can be easily used for the design of EMB processes. The removal e$ciency in EMB processes is de"ned as organic substrate crossing the membrane g " e organic substrate entering the system NA m , " Q C w Sw

(32)

C. Nicolella et al. / Chemical Engineering Science 55 (2000) 1385}1398

1397

The e!ect of bio"lm formation on removal e$ciency is represented in Fig. 14, which reports the removal e$ciency, calculated through Eq. (33) for the design and operating parameters considered in Table 1 (run 2), as a function of bio"lm thickness. It can be observed in this "gure that the organic substrate removal e$ciency decreases with increasing bio"lm thickness. Therefore, the formation of bio"lm at the membrane/biomedium interface is, in the case considered here, detrimental for the process performance.

6. Conclusions

Fig. 13. Comparison between the organic substrate #ux over time as calculated by the full mechanistic model (line) and the simpli"ed model (symbol) for C "7.5 g m~3, C "100 g m~3 and k "2]10~5m s~1. Os St Ss

For most cases of practical interest in the operation of extractive membrane bioreactors, biochemical reaction within the bio"lm developing at the membrane surface is con"ned to a thin layer at the bio"lm/biomedium interface due to oxygen di!usion limitation. Membrane-attached bio"lms can therefore be modelled by a simpli"ed model which only allows for organic substrate di!usion and which is proved to predict correctly the organic substrate concentration pro"le within the bio"lm, the organic substrate #ux across the membrane and the evolution over time of the bio"lm thickness with a remarkable reduction of equations, variables and parameters needed to describe the system.

References

Fig. 14. E!ect of bio"lm thickness on organic substrate removal e$ciency as calculated through the MAB simpli"ed model [Eq. (33)].

g can be calculated using Eq. (27), provided that the #ow e pattern in the tube side of the EMB is given for the calculation of the organic substrate concentration (C ). St For the simple case of a well-mixed compartment, which applies to the experimental system used in this work, the following design equation is obtained: 1 g " , (33) e 1#(Q /k A ) w So m where the overall mass transfer coe$cient is given by Eq. (28).

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