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Dynamic behaviour of a fixed-bed biofilm reactor: analysis of the role of the intraparticle convective flow under biofilm growth
ELSEVIER
Biochemical Engineering Journal
Biochemical Engineering Journal 2 ( 1998) l-9
Dynamic behaviour of a fixed-bed biofilm reactor: analysis of the role of the intraparticle convective flow under biofilm growth Anabela Leitao, Ali’rio Rodrigues * Department
of Chemical
Engineering,
Faculty
of Engineering,
Agostinho
Nero University,
P.O. Box I7.56, Luanda,
Angola
Received I8 November 1997; accepted 22 April 1998
Abstract The influence of the intraparticle convective flow on the dynamic response of a fixed-bedbiofilm reactoris evaluatedtheoreticallyfor a stepwise increasein substrate concentrationin thefeedline.Dimensionless transientmassbalanceequationsfor the substrate within the bulk liquid, biofilm andsupportphasesarederivedconsideringa substrateinhibitionkinetic model(Haldaneequation)andbiofilm growth. Governing equations are simultaneously solved for phenol biodegradation by finite element collocation methods to yield the bed response, in terms of dimensionless phenol concentration at outlet against a reduced time (breakthrough curves), for two different bulk flow rates. The results reported in this study, for different intraparticle velocities, show that the major beneficial effect of the intraparticle convective flow on the production of a good effluent quality is expressed by the occurrence of later breakthrough curves, due to the improvement of intraparticle mass transfer by convection, particularly when higher diffusional limitations are present inside the biofilm and support. Thus, bioreactors must be operated under such conditions that liquid flow occurs within biofilms. 0 1998ElsevierScienceS.A. All rightsreserved. Keywords:
Biolilm reactors; Biofilm
growth;
Intraparticle
convection
-
1. Introduction Applications of biological reactors,where porousparticles in fixed or fluidized bedsarecoatedwith a biofilm, arebecoming more numerous in water and wastewater treatment [21,18,11,5,13]. In these biofilm systems the overall rate of substrate removal may be limited by intraparticle masstransfer resistances. One of the ways of eliminating or reducing mass transfer resistancesinsideparticlesis by providing largepores for transport [ 171. Mass transport by convection inside ‘large-pore’ permeable materials, in addition to diffusive transport, is responsiblefor the improved performance of processesusing such materials. Experimental studiesconducted by De Beer et al. [3,4] on the masstransport in heterogeneousbiofilms, consistingof microbial cell clusters and interstitial voids, showedliquid movement in the voids dependenton the bulk liquid velocity. Consequently,in voids both diffusion andconvection may contribute to masstransfer and intrabiofilm convection is expected to be increasingly important asthe bulk velocity increases. * Corresponding author. Laboratory ing, School of Engineering, University 1369-703X/98/$ - see front matter 0 PIIS1369-703X(98)0001 l-4
of Separation and Reaction Engineerof Porte, 4099 Port0 Codex, Portugal. 1998 Elsevier
Several investigators (e.g., Refs. [ 19,15,18,21] ) dedicated their efforts to the dynamics of biofilm reactors with a growing biofilm, in the absenceof intraparticle convection. The growth of microorganismson activated carbon in an expanded-bedadsorption column was investigated by Tien and Wang [ 191. Their dynamic model described substrate uptake and growth by the microorganisms and predicted effluent breakthrough curves. Park et al. [ 151 analysed the dynamic behaviour of a continuous,fixed-film, fluidized-bed fermentor. The biofilm wasallowed to grow and its effect on reactor performancewasexamined.A rigorous mathematical model was developed in this study. Tang et al. [ 181 investigated experimentally the transient responseof a draft tube gas-liquid-solid fluidized bed biofilm reactor (DTFB) to a step increasein influent phenol concentration. A mathematical model, consideringthe external masstransfer resistance, the simultaneousdiffusion, adsorptionandreaction of phenol and oxygen inside the bioparticles, the dynamics of biofilm growth, was proposedto representthe experimental data. A dynamic mathematicalmodel, basedon a Monod kinetics for oxygen coupled with a Haldaneexpressionfor the inhibitory effect of phenol, was formulated by Worden and Donaldson [ 211 to analyse the dynamics of a biological fixed-film for phenol degradationin a fluidized-bed bioreactor.
Science S.A. All rights reserved.
2
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/ Biochemical
In the first part of our research [ 101, we developed a mathematical model for a fixed-bed biofilm reactor, packed with spherical particles, in order to investigate the roles of reaction, adsorption and mass transfer on substrate consumption. Mass transfer throughout the macroporous biofilm and support was described by simultaneous Fickian diffusion and convective how. A first order reaction was considered and biofilm growth was not included in the model. Numerical solutions were obtained for phenol biodegradation by implementing a double collocation on finite elements technique for the discretization of the particle radial and angular coordinates. In terms of substrate removal and production of a good effluent quality, the model results in the absence of a growing biofilm showed the beneficial role introduced by adsorption in a biodegradation process, as well as the beneficial effects of the convective flow through both the biofilm and support, particularly when a higher intraparticle convective velocity was assumed to occur through the biofilm. In the present study, the biofilm is allowed to grow and the role of the intraparticle convective flow is evaluated under such circumstances for a substrate inhibition kinetic model (Haldane equation). Some experimental results indicate that phenol, at high concentrations, has an inhibitory effect on microbial growth and that the Haldane equation represents adequately microbial growth [ 91. It is important to stress that this work is the first one which considers simultaneously a growing biofilm and the effects of intraparticle convection.
Engineering
2.1. Bioparticle
At time zero, the supportparticlesare surroundedby a very thin biofilm. As the biodegradationprogresses,the biofilm grows and its thicknessincreases.Mass transfer within both the macroporousbiofilm and the macroporous support is describedby simultaneousFickian diffusion and convective flow. The resistanceto substratemasstransfer between the bulk liquid and biofilm phasesis accounted for. From our previous work [ lo], assumingthat: ( 1) both the support particles and the bioparticles, i.e., support particles coated with biofilm, are sphericalin shape;(2) only one substrate is rate-limiting and is adsorbable; (3) simultaneousmass transfer and reaction occurs within the biofilm. The rate of substrateconsumption, - r,, follows an equation (Haldane equation) in the following form [ 91: -rs=
k llMXX”Cl k,+C,+C:/k,
1-9
(2)
(3) where (4)
(5) the transient dimensionlessmassbalancesfor the substrate within the biofilm and supportphasesare: (a) biojilm phase
ax,-N*, -+a2x, -ax,2 ae [ au*2 Li*+l/a +
model of a fixed-bed bioreactor is develthe role of the intraparticle convective growth. The model is based on unsteadywithin the bulk liquid, biofilm and support
dynamics
2 (1998)
where C, is the substrateconcentration in the biofilm pores, k,,, is the maximum specific reaction rate constant,X, is the concentrationof biomassin the biofilm or biofilm density, kp is the saturationconstantand kI is the inhibition constant.(4) Simultaneousmasstransfer and adsorptionoccurs within the support. The adsorption of substrateis instantaneousat the pore/wall interface and is representedby the Langmuir isotherm equation. (5) The biofilm is uniform in thickness, composition and reactivity; and using the fact that:
2. Model development A mathematical oped to investigate flow under biofilm state mass balances phases.
Journal
--a2x, aT2
-ax, a17 1-(2~-1)~ ax, -h, 2(0*+1/a) a7j ax,
1-(2~-1)~ 4(u*+l/a)’
-h,(29-1)s
-
au*
4:x,
277-i (/I*+l/a)2
(6)
U* aa 1
l+K,X,+K,K,X:
+TZiiG
(b) support phase
ax,-
ae -
N
E-K, l+tp (1+K,X,)2 + 1-(2~-1)~ 4ue2 -h2(24$-h2
a2x 2 ax 2+--L [ au*2 U* au*
a2x2 ---a7j2
27-i i.P2
ax, a7j
1-(2~-1)~
2u*
(7)
ax2 x
1
with the following initial conditions ( 8 = 0) : x,=0
vu*, oIu*11,
V7l. OI7j11, vz*, o
x,=0
vu*, o
and boundary conditions ( 8 > 0). ForVv,O
(f-3) (9)
A. Leikio,
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c*‘”
/ Biochemical
Engineering
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2 (1998)
flow dir&ion 1
(10)
I-9
rl=o I
4’1
I
,
LP=l N,(X-X,1,*=,)=3
(11)
X21uS=O=finite
(12)
x21rr*=l=xllu*w=o
(13)
ForVu*,O
X,=finite;
X,=finite
andVz*,OIz*Il: (14) Fig. I, Coordinates
77= 1: X, = finite; X, = finite
In the above equations, 8= t/rLO is a reduced time ( rLo is the initial bed space time), 4, * = (,&,X,/k,) (E, 8*/D,,) Kp = Co/k, and Kr= Co/k, are parameters char= K%,, acterizing the reaction inside the biofilm, where 6, and K will represent the Thiele modulus and the reaction rate constant for a first-order reaction, i.e., for very small liquid substrate concentrations [7], Kp is the dimensionless substrate saturation constant and K, is the dimensionless substrate inhibition constant. The equations were written in spherical coordinates, where u* and L’* defined by Eqs. (5) and (4) are the dimensionless radial coordinates in the support and biofilm and 17= (p’ + 1) /2,0 I 77< 1, is an angular coordinate increasing in the flow direction with p’ = costi’ and 0 is the angle between the z axis (in the flow direction) and the particle radial coordinate r. Fig. 1 shows a spherical particle with the coordinate system used. The dimensionless biofilm thickness, a( = SIR,), is in this work a function of time and bed axial coordinates, because the biofilm is growing; but, U* rangesfrom 0 to 1,regardlessof changesin the biofilm thickness.Thus, the moving spatial coordinate system was converted to a fixed coordinate system. A complete description of the bioparticle dynamics requires knowing the biofilm thicknessasa function of time for a given bed axial position. Following Tien [ 201, the dynamic change of biomassconstituting the biofilm on a single bioparticle (mr) can be describedby: dmfdt
-rgrowlh-rdecay
for the spherical
support coated with a biofilm.
(15)
(16)
Integration of the local specific growth rate, II, over the biofilm volume gives rise to the biomassgrowth rate as
Eq. (161, rdecayr is assumedto be linearly proportional to the total amount of cells presentin the biofilm. By applying the assumptionthat the net substrateuptake by the biofilm is consumedby the degradationreactiontaking place within the film [ 201, the following dimensionlessequation, expressing the variation of the dimensionlessbiofilm thickness with time, for a given bed axial position, can be derived:
aa -= do
y,cO x,
-(27)-1)&X2
) 1 dq
(18)
u* = 1
-- K, (l+U)3-l 3 (l+a>*
where, for simplification, Y, andXVwere taken to be constant in the derivation [ 201 and Kd is a dimensionlessdecay constant ( Kd = kdTLo). The initial condition is f3=0: a=ao &*
(19)
where a, is the initial dimensionlessbiofilm thickness. For a fixed-bed bioreactor, the presence of biological growth results in a decreaseof the bed porosity and so the pressure drop necessary to maintain a given flowrate increases.The changein bed porosity with biofilm thickness is expressedby
1 R,iS
(17)
rg’growth= ICI
0
R,
where the local specific growth rate is related with the local substrate consumption rate by )(L= ( -ra) Y,/X,, with Y, being the cell yield coefficient. The decay rate term in
(20) where E,,~is the initial bed porosity. The governing material balancesherein formulated for the bioparticle contain other parameterschanging with the biofilm thickness:
A. Leitcio, A. Rodrigues/Biochemical
4
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2 (1998)
1-9
3. Numerical technique
(23)
‘a0
where the subscript 0 the initial value for a given parameter. Eq. (23)) expressing the intraparticle Peclet number for the biofilm resulting from the convective flow in the biofilm, was derived assuming that the intraparticle convective velocity inside the biofilm macropores is constant, i.e., does not change with the biomass growth. The intraparticleconvective velocity inside the support macropores was also assumed to be constant. 2.2. Bioreactor
1 vz*, orz*<1
XI z*~o-=xlz*=o+-
g$
(24)
(251
p-o+
vo>o
ax =o t/o>0 az* z*= I
(26) (27)
where N,, number of masstransfer units for external film diffusion, expressedby
*
KfTLO &bO
and the solution X, in the kth subinterval of U* and nth subinterval of q was approximatedby: (32)
i=ILU=l
with the following initial and boundary conditions:
N,=3
(31)
dynamics
Assuming that: (1) the fixed-bed bioreactor is described by the axial dispersedplug flow model, with a constantvalue for the axial Peclet number; and (2) the superficial velocity is kept constantthrough the bed; the transientdimensionless massbalancefor the substratewithin the bulk liquid phaseis
x(e=o)=o
The bioreactor model was solved using the procedurepreviously reported by the authors [ IO]. (i) The support and biofilm massbalanceequations,with their associatedinitial and boundary conditions, were firstly discretized along the spatial coordinatesu*, v* and r], by implementing a double orthogonal collocation on finite elements technique with cubic Hermite polynomials asbasisfunctions [ 61. The intervals O
(28)
Rs(l+a)
varies with the biofilm thicknessaccording to (29)
J
where gj and g, are the collocation points within each subinterval of U* or u* and 7, respectively, with j= 1,2 and m = 1,2.The cubic Hermite polynomialsH,,( g,) andH,,( g,) with i= I,*,4 and C= 1,*,4 were defined over [vk*, v~+~*] or Ih*, Ukfl *I and 17711,qn+, I, respectively. (with k=l,*, NEB; n= l,*, NE) and Ci+2k-Z,Y+T!n-2’ Ci+2k-z.e+2n-z2 (with k= l,*, NES; n= l,*, NE) are the basisfunction coefficients to be determined.The problem of no specified boundary conditions at u* = 0, 77= 0 and 77= 1 was overcome by introducing additional equationsresulting from the satisfactionof the supportand biofilm equationsat theseboundary points. After the discretization process,we obtained a systemof (2NES + 2NEB -t 1) (2NE + 2) partial differential equationsin the basisfunction coefficients Cl’s and C2’sdependenton the bed axial coordinate, z*, andtime, 0. (ii) The systemof (2NES + 2NEB + 1) (2NE + 2) partial differential equations,resulting from the above discretization step, in conjunction with the massbalance equation in the bulk liquid phase,Bq. (24), and the biofilm growth equation, Eq. ( 18)) was further numerically integrated with the PDECOL package [ 121.
when the following correlation [ 81 2R,( l+a)K, D AW
4. Model results and discussion =4.58Sc”3Re1’3
(30)
is usedfor the external film masstransfer coefficient, I&. Eqs. (6)-(15), (18) and (19), (24)-(27) constitute the completedescription of the bioreactor model.
Model equations were solved for phenol biodegradation, using the following values for the kinetic parameters[ 2 1] : k max=1.6X lop4 g phenol/(g cells s), X,=0.012 g cells/ cm3,kp = 1.Omg/l, k, = 50 mg/l and Yg= 0.40 g cells/g phe-
A. L&Co,
A. Rodrigues
/Biochemical
nol. For these values of the kinetic parameters, the maximum rate of substrate consumption is achieved for a phenol concentration of 7.07 mg/l. Above this value, phenol has an inhibiting effect on microbial growth. Some other data used in the simulations were taken from our previous investigation [lo] and are summarized in Table 1. The values of the data d,, R, and U, listed in Table 1, were chosen in order to obtain a flat velocity profile for the bulk liquid and Re > 1. In this range of Reynolds numbers enhanced mass transport due to intraparticle convection occurs [ 11. Fan et al. [5] and Beyenal et al. [2] reported that the density of the biofilm, XV, is dependent upon the thickness of the biofilm and that the effective diffusivity through the biofilm, Deb, decreases with increasing biofilm density. However, for simplification of the mathematical model, X, and Deb were assumed to have constant values in our work. Monteiro [ 141 measured values of XV ranging from 0.090 to 0.0 12 g cells/cm3 as the biofilm thickness increased from 5 to 100 Frn. According to Eq. ( 18)) the rate of increase of the biofilm thickness is higher for lower values ofX,, leading us to choose a lower value for XV in order to investigate the effects of the intraparticle convective flow for higher values of the biofilm thickness. Model solutions, in response to a step input of a phenol concentration of 65 mg/l, are presented as plots of the dimensionless phenol concentration in the bulk fluid phase, X = Cl C,, at column outlet, against a reduced time, OS, (breakthrough curves), for two different bulk flow rates: 50 and 200 cm3/min. The reduced time, &, is obtained dividing the time, t, by a stoichiometric time, ts-r, defined in analogy to that for an adsorption process as
(33)
where
&2 (k-- I---EMI Et,0 (l+a,)36p is the bed adsorptive capacity parameter. 4.1. Lowerflow resistances)
rate (lower intraparticle
diffusional
The effect of a growing biofilm on substrate removal, in the absence of intraparticle convection, is shown in Fig. 2. Breakthrough curves are depicted for several values of the decay constant, Kdr and for an initial biofilm thickness, S,, of 5 km. For the lower values of the decay constant, the breakthrough curve initially increases with time until a maximum is reached, after which it decreases with time as the contri-
Engineering
Journal
2 (1998)
1-9
Table 1 Data used in the simulations Length of reactor, L Internal diameter of reactor, d, Support radius, R, Initial bed porosity, Q, Volumetric bulk flow rate, U Bed superficial velocity, U, Initial Reynolds number, Rq, Initial bed space time, 7Lu Biofilm porosity, E, Support porosity, E> Molecular diffusivity of phenol in pure water, D,, Effective diffusivity in biofilm, Deb Effective diffusivity in support, D,,
1.0
“0
S,-5pm PB - 100
147 cm 1.77 cm 0.0295 cm 0.4 50,200 cm3/min 0.34, 1.36 cm/s
X8 174, 43.5 s 0.7 0.7 8.47X 10-hcm2/s 3.5X IOmhcm2/s 3.5X lOmh cm’/s
constant biofilm tbiieas
1
Fig. 2. Effect of a growing intraparticle convection.
2 biofilm
3 @ST on substrate
4 removal
5
in the absence of
bution to substrate removal by biodegradation increases. Comparing with the case of constant biofilm thickness (bold line curve), we can see that, as the decay constant decreases, the steady state substrate conversion increases and a higher substrate removal is achieved. Fig. 3a and b illustrate the variation of the biofilm thickness with time for several values of the decay constant, at the inlet and outlet of the bioreactor, respectively. The lower phenol concentrations at outlet, due to consumption along the bed, cause closer dynamic responses in terms of biofilm thickness. Breakthrough curves predicted in the presence of intrapartitle convection, by assuming equal velocities through the biofilm and support (u,~ = u,~), are shown in Fig. 4a and b for u,,= 2.96X 10-j cm/s and I/‘,, = 1.18X lo-’ cm/s, respectively. For any value of the decay constant, Kd, the breakthrough occurs later in the presence of intraparticle convection, such as observed for constant biofilm thickness (bold line curves). As K,, decreases, the decrease in the amount of substrate removal is larger due to intraparticle convection. Fig. 5a and b illustrate the effects of the intraparticle convective flow for u,,=2.96X10e3 cm/s and u,,=l.l8x 1Oe2 cm/s, respectively, when a higher intraparticle velocity is assumed to occur through the biofilm (u,, > u,~). The case
A. L.eit&, A. Rodrigues
6
/Biochemical
Engineering
Joumal2
1.0
(1998)
6,=5jbm Pe- 100 K, - 1.2
l-9
*&,-cl Am-25
--
constant
If
0.8
b-0 A,,-0.42
biofilm
thkkness
0.6
20
.......""Kd
0" 3
,,,,,...'.' = o.oo3
____ --_----------&s.:$.:“: -.-__. -.-._.__ ,<
._.,..../.... .....,
Kd = 0.005 -.-.-.____
10
0.4
0.2
-~----.-______
0 I-
I
0
2
@ST
1.0 z’-
50
f
SO-5pm
-0
“.“I
1
2
3
4
5
thickness with time for several values of the inlet; (b) bioreactor outlet.
rate (higher intruparticle
;I,-0
a,,-0
I
1
2
3
4
5
@ST
of constant biofilm thickness (bold line curves) is also depicted in this figures for comparison. It is apparent the larger beneficial effect of the intraparticle convective flow on the production of a good effluent quality, expressed by the occurrence of more delayed breakthrough curves and the achievement of a higher substrate removal. From Figs. 4 and 5 we observe that a higher intrabiofilm velocity, due to a higher biofilm permeability, produces an increase in the breakthrough time, but this effect is of little significance, in this case on lower intraparticle diffusional resistances when u,, = L’,*. 4.2. Higherpow resistances)
4
g-100
“0
5
--
6,-5pm
@ST
Fig. 3. Variation of the biofilm decay constant. (a) Bioreactor
3 *ST
diffusional
The effects of the intraparticle convective flow with u,, = 1.18X 10m2 cm/s, for equal (u,, =o,~) and different (u,, > uo2) velocities through the biofilm and support, are shown in Fig. 6a and b, respectively. In the presence of higher intraparticle diffusional resistances, it is apparent the lower sensitivity of the breakthrough curves to biomass growth. In this case, it is also remarkable the beneficial effect of the intraparticle flow, in terms of the occurrence of later break-
Fig. 4. Effects of intraparticle convectlon (u,,, = u,~) on substrate removal in the presence of a growing biofilm and lower intraparticle diffusional resistances. (a) u,,, =2.96X IO-‘cm/s; (b) LI,,= 1.18X 10-*cm/s.
through curves and the achievement of a higher substrate removal, particularly when the biofilm convective velocity is higher than the support velocity, such as seen for the constant biofilm thickness (bold line curves). All the results herein presented for a growing biofilm were calculated by assuming that the intraparticle convective velocity does not change with the biomass growth. As suggested by Rodrigues et al. [ 161, the intraparticle velocity is estimated by the equality of the pressure drops across the particle and across the bed. In laminar flow and from Darcy’s law, the following relation between velocity inside pores and bed superficial velocity is derived:
where B, is the particle permeability and B, is the bed permeability given by Bb=
E3d; b 150( 1-&b)2
with dP being the bioparticle diameter. As the bioparticle diameter increases, as result of the biomass growth, the bed
A. L&t&,
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/ Biochemical
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1-9
”
- Pe.100
0.8 0.6
a’-
--&,-km
/
0
6,-Spm Pa = 100
*&,-I7
*,0-o i,,=1.69
I
0
““C
6,=51u” Pa- 100 K, = 1.2 100 K- 1.32s-’ era = 0.310 Kp-65 K, = 1.3 Ne - 0.25 Ndlo - 870 Nr, - 93.5
1
I
I
I
2
3
4
(a)
5
la-O kl=* -._._.&,-x3.3 l,o- 1.69 .-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-. constant biofilm thickness
Pe- 100 K, = 1.2 g-100 K- l.Wd (,,=0.310 Kp=6S K,- 1.3 Na - 0.25 Ndlo - 370 Nt, - 33.5 I
W
Fig. 5. Effects of intraparticle convection (u,, > c,*) on substrate removal in the presence of a growing biofilm and lower intraparticle diffusional resistances. (a) c,,, =2.96X 10m3 cm/s; (b) L,,, =1.18X IO-‘cm/s.
Fig. 6. Effects of intraparticle convection (u”, = 1.18 X 1O-2 cm/s) on substrate removal in the presence of a growing biofilm and higher intraparticle diffusional resistances. (a) u,,, = II,,; (b) I’~, > o,~.
porosity decreases,which produces a decreasein the bed permeability and an increase in the intraparticle velocity. Thus, h, and A2 should change with the biofilm thickness according to:
through curves for the lower valuesof Kd. However, the initial portion of the breakthrough curves is not affected by the variation in intraparticle velocity with biomassgrowth. The integration of the model equations,by the numerical technique previously described, is very difficult when the intraparticle velocity changeswith the biomassgrowth. A larger numberof collocation points is then required for accurate solutions, leading to a very hard computation program in termsof memory and time.
(37) B A2=AzoT
(38) b
where 5. Conclusions The breakthrough curves depicted in Fig. 7a and b for the lower and higher bulk flow rates usedin this work, respectively, were calculated by accounting for the variation in intraparticle velocity with biomass growth in our model. Comparing thesecurves with those in Figs. 4a and 6a, we note that the influence of the variable intraparticle velocity on the bed dynamic responsesis of little significance. Other results obtained with higher intraparticle velocities indicate that the variation in intraparticle velocity affects the break-
The influence of the intraparticle convective flow on the dynamic responseof a fixed-bed biofilm reactor to a step increasein influent phenol concentration was evaluated in this work. Model equationswere basedon a substrateinhibition kinetic model and biofilm growth. The simulation resultsreported in this work for two different bulk flow rates show that: (i) the major beneficial effect of the intraparticle convective flow is on the increaseof the breakthrough time, due to the improvement of intraparticle masstransfer by con-
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8
A. Rodrigues
/Biochemical
Engineering
c2 D AW 0.8
d
DC, D,, K
K, - 1.2 s,-100 K- 1.92s-’ q,, -0.310
0.6 K,n-65 i K,- 1.3
0
N&-l Ndr, = 3480 Nlo - 232.4
0.4
L NLil 1.0 Nd2 0.8 -
Nf
6,-5~m P9- 100
0.6 -
Pe 40
K, - 1.2 $,- 100 K- 1.92~~’
0.4
Oe2t
0
/
0
I Re
K,- 1.3 Na = 0.25 N,,,, - 870 Nb = 93.5
i
1
I
I
I
2
3
4
RS - rr
ON
5
SC
ST
Fig. 7. Effects of intraparticle convection on substrate removal in the presence of a growing biofilm: u,, = o02, variable velocity with biomass growth. (a) Lower intraparticle diffusional resistances; (b) higher intraparticle diffusional resistances.
t &T 43 u*
vection; (ii) a larger beneficial effect is achieved when the biofilm velocity is higher than the support velocity. From our investigation on the analysis of the effects of the intraparticle convection in biodegradation/adsorption combined processes, we finally conclude that intraparticle convection is an important design factor to be accounted for on the production of a good effluent quality, particularly when higher diffusional limitations are present inside the biofilm and support. Thus, bioreactors must be operated under such conditions that liquid movement occurs in the voids of the biofilm.
“01
U”2
v*
X X” Xl
x2
6. Nomenclature rs z
a c co c,
Ratio of biofilm thickness to support radius ( = 6/ R,), dimensionless Substrate concentration in the bulk fluid phase, kg/m3 Substrate concentration in the feed, kg/m3 Substrate concentration inside biofilm pores, kg/m3
Z*
Journal
2 (1998)
1-9
Substrate concentration inside support pores, kg/m3 Substrate molecular diffusivity in water, m*/s Effective diffusivity of substrate in the biofilm, m’/s Effective diffusivity of substrate in the support, m2/s Reaction rate constant for very small substrate concentrations, s-’ Decay constant, dimensionless External film mass transfer coefficient, m/s Substrate inhibition constant, dimensionless Langmuir adsorption equilibrium constant, m’/kg Substrate saturation constant, dimensionless Parameter in the Langmuir isotherm equation ( = K,C,), dimensionless Bed length, m Number of mass transfer units for biofilm pore diffusion ( = TV/ TV,>, dimensionless Number of mass transfer units for support pore diffusion ( = TV/ TV*), dimensionless Number of mass transfer units for external film diffusion, dimensionless Peclet number based on bed length, dimensionless Solid phase concentration in equilibrium with the liquid phase concentration CO, kg/m3 Radial coordinate in the particle, m Reynolds number based on bed superficial velocity, dimensionless Spherical support radius, m Volumetric rate of substrate consumption within biofilm, kg/m3 s Schmidt number, dimensionless Time, s Stoichiometric time, s Bed superficial velocity, m/s Dimensionless radial coordinate in the support (=rfR,) Intraparticle convective velocity inside the biofilm macropores, m/s Intraparticle convective velocity inside the support macropores, m/s Dimensionless radial coordinate in the biofilm (=(r/R,-1)/a) Dimensionless substrate concentration in the bulk fluid phase ( = C/C,) Concentration of cells in biofilm, kg/m3 Dimensionless substrate concentration inside biofilm pores ( = C,IC,) Dimensionless substrate concentration inside support pores ( = C,/C,) Cell yield coefficient, kg cells/kg phenol Axial coordinate in the bed, m Reduced axial coordinate in the bed ( = z/L), dimensionless
Greek letters P
Ratio of effective diffusivities dimensionless
( = Des/Deb),
A. kikio,
A. Rodrigues
/ Biochemical
Biofilm thickness, m Fixed-bed porosity, dimensionless Internal porosity of the biofilm, dimensionless Internal porosity of the support, dimensionless Angular particle coordinate, dimensionless Thiele modulus for very small substrate concentrations, dimensionless Intraparticle Peclet number for the biofilm ( = au,, /Deb), dimensionless Intraparticle Peclet number for the support ( = Rsvo2/Des), dimensionless Time constant for biofilm pore diffusion ( = E, S2/D,,), s Time constant for support pore diffusion ( = ~zR,2/De,) > s Initial bed space time ( = .q,&/u,), s Reduced time ( = tl T& , dimensionless Reduced time ( = t/tsT), dimensionless Bed adsorptive capacity parameter, dimensionless Support adsorptive capacity parameter ( = ( 1 - c2) / .c2qo/CO), dimensionless
Engineering
Journal
2 (1998)
1-9
9
Liquid flow and mass [41 D. De Beer, P. Stoodley, Z. Lewandowski, transport in heterogeneous biofilms, Water Res. 30 (1996) 27612165. K. Wisecarver, B. Zehner, Diffusion of [51 L. Fan, R. Leyva-Ramos, phenol through a biofilm grown on activated carbon particles in a draft-tube three-phase fluidized-bed bioreactor, Biotechnol. Bioeng. 35 (1990) 279-286. [61 B. Finlayson, Nonlinear Analysis in Chemical Engineering, McGrawHill, 1980. 171 G. Froment, K. Bischoff, Chemical Reactor Analysis and Design, 2nd edn., Wiley, New York, 1990. [81 A. Karabelas, T. Wegner, T. Hanratty, Use of asymptotic relations to correlate mass transfer data in packed beds, Chem. Eng. Sci. 26 (1971) 1581-1589. [91 A. Lallai, G. Mura, Kinetics of growth for mixed cultures of microorganisms growing on phenol, Chem. Eng. J. 41 ( 1989) B55-B60. 1101 A. Leitao, A. Rodrigues, Modeling of biodegradation/adsorption combined processes in fixed-bed biofilm reactors: effects of the intraparticle convective flow, Chem. Eng. Sci. 5 1 (20) (1996) 45954604. [Ill A. Livingstone, H. Chase, Modeling phenol degradation in a fluidizedbed bioreactor, AIChE J. 35 (12) (1989) 198&1992. 1121 N.K. Madsen, R.F. Sincovec, PDECOL: general collocation software for partial differential equations, ACM Trans. Math. Software 5 (1979)
326.
R. Mendez, J. Lema, Biofilm reactors technology in wastewater treatment. In: L.F. Melo et al. (Eds.), Biofilms-Science and Technology. Kluwer Academic Publishers, 1992, pp. 409-419. A. Monteiro. Biological Degradation of Phenol with Bacteria Immo[I41 bilized in Large-Pore Support, PhD Thesis, University of Porte, 1997. 1151 Y. Park, M. Davis, D. Wallis, Analysis of acontinuous, aerobic, fixedfilm bioreactor: II. Dynamic behaviour, Biotechnol. Bioeng. 26 Cl31
Acknowledgements This work was partially developed at the School of Engineering of Port0 during a sabbatical leave of A. Leitao. The support from the Angolan Petroleum Ministry is gratefully acknowledged.
(1984)468-476.
1161 A. [I71
References
Rodrigues, B. Ahn, A. Zoulalian, Intraparticle-forced convection effect in catalyst diffusivity measurements and reactor design, AIChE J. 28 ( 1982) 541-546. A. Rodrigues, Z. Lu, J. Loureiro, Residence time distribution of inert and linearly adsorbed species in a fixed bed containing large-pore supports: applications in separation engineering, Chem. Eng. Sci. 46 (1991)
[ 1 ] L. Backer, G. Baron, Residence time distribution in a packed bed bioreactor containing porous glass particles: influence of the presence of immobilized cells, J. Chem. Technol. Biotechnol. 59 (1994) 297302. [2] H. Beyenal, S. Seker, A. Tanyolac, B. Salih, Diffusion coefficients of phenol and oxygen in a biofilm of Pseudomonasputida, AIChE J. 43 ( 1997) 243-250. [3] D. De Beer, P. Stoodley, Z. Lewandowski, Liquid flow in heterogeneous biofilms. Biotechnol. Bioeng. 44 ( 1994) 636641.
Ll81
2765-2773.
W. Tang, K. Wisecarver, L. Fan, Dynamics of a draft tube gas-liquidsolid fluidized bed bioreactor for phenol degradation, Chem. Eng. Sci. 42(9)(1987)2123-2134. [ 191 C. Tien, S. Wang, Dynamics of adsorption columns with bacterial growth outside adsorbents, Can. J. Chem. Eng. 60 ( 1982) 363-376. 1994. [201 C. Tien, Adsorption Calculations and Modeling, Butterworth, [211 R. Worden, T. Donaldson, Dynamics of a biological fixed film for phenol degradation in a fluidized-bed bioreactor, Biotechnol. Bioeng. 30(1987)398-412.
Biochemical Engineering Journal
Biochemical Engineering Journal 2 ( 1998) l-9
Dynamic behaviour of a fixed-bed biofilm reactor: analysis of the role of the intraparticle convective flow under biofilm growth Anabela Leitao, Ali’rio Rodrigues * Department
of Chemical
Engineering,
Faculty
of Engineering,
Agostinho
Nero University,
P.O. Box I7.56, Luanda,
Angola
Received I8 November 1997; accepted 22 April 1998
Abstract The influence of the intraparticle convective flow on the dynamic response of a fixed-bedbiofilm reactoris evaluatedtheoreticallyfor a stepwise increasein substrate concentrationin thefeedline.Dimensionless transientmassbalanceequationsfor the substrate within the bulk liquid, biofilm andsupportphasesarederivedconsideringa substrateinhibitionkinetic model(Haldaneequation)andbiofilm growth. Governing equations are simultaneously solved for phenol biodegradation by finite element collocation methods to yield the bed response, in terms of dimensionless phenol concentration at outlet against a reduced time (breakthrough curves), for two different bulk flow rates. The results reported in this study, for different intraparticle velocities, show that the major beneficial effect of the intraparticle convective flow on the production of a good effluent quality is expressed by the occurrence of later breakthrough curves, due to the improvement of intraparticle mass transfer by convection, particularly when higher diffusional limitations are present inside the biofilm and support. Thus, bioreactors must be operated under such conditions that liquid flow occurs within biofilms. 0 1998ElsevierScienceS.A. All rightsreserved. Keywords:
Biolilm reactors; Biofilm
growth;
Intraparticle
convection
-
1. Introduction Applications of biological reactors,where porousparticles in fixed or fluidized bedsarecoatedwith a biofilm, arebecoming more numerous in water and wastewater treatment [21,18,11,5,13]. In these biofilm systems the overall rate of substrate removal may be limited by intraparticle masstransfer resistances. One of the ways of eliminating or reducing mass transfer resistancesinsideparticlesis by providing largepores for transport [ 171. Mass transport by convection inside ‘large-pore’ permeable materials, in addition to diffusive transport, is responsiblefor the improved performance of processesusing such materials. Experimental studiesconducted by De Beer et al. [3,4] on the masstransport in heterogeneousbiofilms, consistingof microbial cell clusters and interstitial voids, showedliquid movement in the voids dependenton the bulk liquid velocity. Consequently,in voids both diffusion andconvection may contribute to masstransfer and intrabiofilm convection is expected to be increasingly important asthe bulk velocity increases. * Corresponding author. Laboratory ing, School of Engineering, University 1369-703X/98/$ - see front matter 0 PIIS1369-703X(98)0001 l-4
of Separation and Reaction Engineerof Porte, 4099 Port0 Codex, Portugal. 1998 Elsevier
Several investigators (e.g., Refs. [ 19,15,18,21] ) dedicated their efforts to the dynamics of biofilm reactors with a growing biofilm, in the absenceof intraparticle convection. The growth of microorganismson activated carbon in an expanded-bedadsorption column was investigated by Tien and Wang [ 191. Their dynamic model described substrate uptake and growth by the microorganisms and predicted effluent breakthrough curves. Park et al. [ 151 analysed the dynamic behaviour of a continuous,fixed-film, fluidized-bed fermentor. The biofilm wasallowed to grow and its effect on reactor performancewasexamined.A rigorous mathematical model was developed in this study. Tang et al. [ 181 investigated experimentally the transient responseof a draft tube gas-liquid-solid fluidized bed biofilm reactor (DTFB) to a step increasein influent phenol concentration. A mathematical model, consideringthe external masstransfer resistance, the simultaneousdiffusion, adsorptionandreaction of phenol and oxygen inside the bioparticles, the dynamics of biofilm growth, was proposedto representthe experimental data. A dynamic mathematicalmodel, basedon a Monod kinetics for oxygen coupled with a Haldaneexpressionfor the inhibitory effect of phenol, was formulated by Worden and Donaldson [ 211 to analyse the dynamics of a biological fixed-film for phenol degradationin a fluidized-bed bioreactor.
Science S.A. All rights reserved.
2
A. L.&io,
A. Rodrigues
/ Biochemical
In the first part of our research [ 101, we developed a mathematical model for a fixed-bed biofilm reactor, packed with spherical particles, in order to investigate the roles of reaction, adsorption and mass transfer on substrate consumption. Mass transfer throughout the macroporous biofilm and support was described by simultaneous Fickian diffusion and convective how. A first order reaction was considered and biofilm growth was not included in the model. Numerical solutions were obtained for phenol biodegradation by implementing a double collocation on finite elements technique for the discretization of the particle radial and angular coordinates. In terms of substrate removal and production of a good effluent quality, the model results in the absence of a growing biofilm showed the beneficial role introduced by adsorption in a biodegradation process, as well as the beneficial effects of the convective flow through both the biofilm and support, particularly when a higher intraparticle convective velocity was assumed to occur through the biofilm. In the present study, the biofilm is allowed to grow and the role of the intraparticle convective flow is evaluated under such circumstances for a substrate inhibition kinetic model (Haldane equation). Some experimental results indicate that phenol, at high concentrations, has an inhibitory effect on microbial growth and that the Haldane equation represents adequately microbial growth [ 91. It is important to stress that this work is the first one which considers simultaneously a growing biofilm and the effects of intraparticle convection.
Engineering
2.1. Bioparticle
At time zero, the supportparticlesare surroundedby a very thin biofilm. As the biodegradationprogresses,the biofilm grows and its thicknessincreases.Mass transfer within both the macroporousbiofilm and the macroporous support is describedby simultaneousFickian diffusion and convective flow. The resistanceto substratemasstransfer between the bulk liquid and biofilm phasesis accounted for. From our previous work [ lo], assumingthat: ( 1) both the support particles and the bioparticles, i.e., support particles coated with biofilm, are sphericalin shape;(2) only one substrate is rate-limiting and is adsorbable; (3) simultaneousmass transfer and reaction occurs within the biofilm. The rate of substrateconsumption, - r,, follows an equation (Haldane equation) in the following form [ 91: -rs=
k llMXX”Cl k,+C,+C:/k,
1-9
(2)
(3) where (4)
(5) the transient dimensionlessmassbalancesfor the substrate within the biofilm and supportphasesare: (a) biojilm phase
ax,-N*, -+a2x, -ax,2 ae [ au*2 Li*+l/a +
model of a fixed-bed bioreactor is develthe role of the intraparticle convective growth. The model is based on unsteadywithin the bulk liquid, biofilm and support
dynamics
2 (1998)
where C, is the substrateconcentration in the biofilm pores, k,,, is the maximum specific reaction rate constant,X, is the concentrationof biomassin the biofilm or biofilm density, kp is the saturationconstantand kI is the inhibition constant.(4) Simultaneousmasstransfer and adsorptionoccurs within the support. The adsorption of substrateis instantaneousat the pore/wall interface and is representedby the Langmuir isotherm equation. (5) The biofilm is uniform in thickness, composition and reactivity; and using the fact that:
2. Model development A mathematical oped to investigate flow under biofilm state mass balances phases.
Journal
--a2x, aT2
-ax, a17 1-(2~-1)~ ax, -h, 2(0*+1/a) a7j ax,
1-(2~-1)~ 4(u*+l/a)’
-h,(29-1)s
-
au*
4:x,
277-i (/I*+l/a)2
(6)
U* aa 1
l+K,X,+K,K,X:
+TZiiG
(b) support phase
ax,-
ae -
N
E-K, l+tp (1+K,X,)2 + 1-(2~-1)~ 4ue2 -h2(24$-h2
a2x 2 ax 2+--L [ au*2 U* au*
a2x2 ---a7j2
27-i i.P2
ax, a7j
1-(2~-1)~
2u*
(7)
ax2 x
1
with the following initial conditions ( 8 = 0) : x,=0
vu*, oIu*11,
V7l. OI7j11, vz*, o
x,=0
vu*, o
and boundary conditions ( 8 > 0). ForVv,O
(f-3) (9)
A. Leikio,
A. Rodrigues
c*‘”
/ Biochemical
Engineering
Journal
2 (1998)
flow dir&ion 1
(10)
I-9
rl=o I
4’1
I
,
LP=l N,(X-X,1,*=,)=3
(11)
X21uS=O=finite
(12)
x21rr*=l=xllu*w=o
(13)
ForVu*,O
X,=finite;
X,=finite
andVz*,OIz*Il: (14) Fig. I, Coordinates
77= 1: X, = finite; X, = finite
In the above equations, 8= t/rLO is a reduced time ( rLo is the initial bed space time), 4, * = (,&,X,/k,) (E, 8*/D,,) Kp = Co/k, and Kr= Co/k, are parameters char= K%,, acterizing the reaction inside the biofilm, where 6, and K will represent the Thiele modulus and the reaction rate constant for a first-order reaction, i.e., for very small liquid substrate concentrations [7], Kp is the dimensionless substrate saturation constant and K, is the dimensionless substrate inhibition constant. The equations were written in spherical coordinates, where u* and L’* defined by Eqs. (5) and (4) are the dimensionless radial coordinates in the support and biofilm and 17= (p’ + 1) /2,0 I 77< 1, is an angular coordinate increasing in the flow direction with p’ = costi’ and 0 is the angle between the z axis (in the flow direction) and the particle radial coordinate r. Fig. 1 shows a spherical particle with the coordinate system used. The dimensionless biofilm thickness, a( = SIR,), is in this work a function of time and bed axial coordinates, because the biofilm is growing; but, U* rangesfrom 0 to 1,regardlessof changesin the biofilm thickness.Thus, the moving spatial coordinate system was converted to a fixed coordinate system. A complete description of the bioparticle dynamics requires knowing the biofilm thicknessasa function of time for a given bed axial position. Following Tien [ 201, the dynamic change of biomassconstituting the biofilm on a single bioparticle (mr) can be describedby: dmfdt
-rgrowlh-rdecay
for the spherical
support coated with a biofilm.
(15)
(16)
Integration of the local specific growth rate, II, over the biofilm volume gives rise to the biomassgrowth rate as
Eq. (161, rdecayr is assumedto be linearly proportional to the total amount of cells presentin the biofilm. By applying the assumptionthat the net substrateuptake by the biofilm is consumedby the degradationreactiontaking place within the film [ 201, the following dimensionlessequation, expressing the variation of the dimensionlessbiofilm thickness with time, for a given bed axial position, can be derived:
aa -= do
y,cO x,
-(27)-1)&X2
) 1 dq
(18)
u* = 1
-- K, (l+U)3-l 3 (l+a>*
where, for simplification, Y, andXVwere taken to be constant in the derivation [ 201 and Kd is a dimensionlessdecay constant ( Kd = kdTLo). The initial condition is f3=0: a=ao &*
(19)
where a, is the initial dimensionlessbiofilm thickness. For a fixed-bed bioreactor, the presence of biological growth results in a decreaseof the bed porosity and so the pressure drop necessary to maintain a given flowrate increases.The changein bed porosity with biofilm thickness is expressedby
1 R,iS
(17)
rg’growth= ICI
0
R,
where the local specific growth rate is related with the local substrate consumption rate by )(L= ( -ra) Y,/X,, with Y, being the cell yield coefficient. The decay rate term in
(20) where E,,~is the initial bed porosity. The governing material balancesherein formulated for the bioparticle contain other parameterschanging with the biofilm thickness:
A. Leitcio, A. Rodrigues/Biochemical
4
Engineering
Journal
2 (1998)
1-9
3. Numerical technique
(23)
‘a0
where the subscript 0 the initial value for a given parameter. Eq. (23)) expressing the intraparticle Peclet number for the biofilm resulting from the convective flow in the biofilm, was derived assuming that the intraparticle convective velocity inside the biofilm macropores is constant, i.e., does not change with the biomass growth. The intraparticleconvective velocity inside the support macropores was also assumed to be constant. 2.2. Bioreactor
1 vz*, orz*<1
XI z*~o-=xlz*=o+-
g$
(24)
(251
p-o+
vo>o
ax =o t/o>0 az* z*= I
(26) (27)
where N,, number of masstransfer units for external film diffusion, expressedby
*
KfTLO &bO
and the solution X, in the kth subinterval of U* and nth subinterval of q was approximatedby: (32)
i=ILU=l
with the following initial and boundary conditions:
N,=3
(31)
dynamics
Assuming that: (1) the fixed-bed bioreactor is described by the axial dispersedplug flow model, with a constantvalue for the axial Peclet number; and (2) the superficial velocity is kept constantthrough the bed; the transientdimensionless massbalancefor the substratewithin the bulk liquid phaseis
x(e=o)=o
The bioreactor model was solved using the procedurepreviously reported by the authors [ IO]. (i) The support and biofilm massbalanceequations,with their associatedinitial and boundary conditions, were firstly discretized along the spatial coordinatesu*, v* and r], by implementing a double orthogonal collocation on finite elements technique with cubic Hermite polynomials asbasisfunctions [ 61. The intervals O
(28)
Rs(l+a)
varies with the biofilm thicknessaccording to (29)
J
where gj and g, are the collocation points within each subinterval of U* or u* and 7, respectively, with j= 1,2 and m = 1,2.The cubic Hermite polynomialsH,,( g,) andH,,( g,) with i= I,*,4 and C= 1,*,4 were defined over [vk*, v~+~*] or Ih*, Ukfl *I and 17711,qn+, I, respectively. (with k=l,*, NEB; n= l,*, NE) and Ci+2k-Z,Y+T!n-2’ Ci+2k-z.e+2n-z2 (with k= l,*, NES; n= l,*, NE) are the basisfunction coefficients to be determined.The problem of no specified boundary conditions at u* = 0, 77= 0 and 77= 1 was overcome by introducing additional equationsresulting from the satisfactionof the supportand biofilm equationsat theseboundary points. After the discretization process,we obtained a systemof (2NES + 2NEB -t 1) (2NE + 2) partial differential equationsin the basisfunction coefficients Cl’s and C2’sdependenton the bed axial coordinate, z*, andtime, 0. (ii) The systemof (2NES + 2NEB + 1) (2NE + 2) partial differential equations,resulting from the above discretization step, in conjunction with the massbalance equation in the bulk liquid phase,Bq. (24), and the biofilm growth equation, Eq. ( 18)) was further numerically integrated with the PDECOL package [ 121.
when the following correlation [ 81 2R,( l+a)K, D AW
4. Model results and discussion =4.58Sc”3Re1’3
(30)
is usedfor the external film masstransfer coefficient, I&. Eqs. (6)-(15), (18) and (19), (24)-(27) constitute the completedescription of the bioreactor model.
Model equations were solved for phenol biodegradation, using the following values for the kinetic parameters[ 2 1] : k max=1.6X lop4 g phenol/(g cells s), X,=0.012 g cells/ cm3,kp = 1.Omg/l, k, = 50 mg/l and Yg= 0.40 g cells/g phe-
A. L&Co,
A. Rodrigues
/Biochemical
nol. For these values of the kinetic parameters, the maximum rate of substrate consumption is achieved for a phenol concentration of 7.07 mg/l. Above this value, phenol has an inhibiting effect on microbial growth. Some other data used in the simulations were taken from our previous investigation [lo] and are summarized in Table 1. The values of the data d,, R, and U, listed in Table 1, were chosen in order to obtain a flat velocity profile for the bulk liquid and Re > 1. In this range of Reynolds numbers enhanced mass transport due to intraparticle convection occurs [ 11. Fan et al. [5] and Beyenal et al. [2] reported that the density of the biofilm, XV, is dependent upon the thickness of the biofilm and that the effective diffusivity through the biofilm, Deb, decreases with increasing biofilm density. However, for simplification of the mathematical model, X, and Deb were assumed to have constant values in our work. Monteiro [ 141 measured values of XV ranging from 0.090 to 0.0 12 g cells/cm3 as the biofilm thickness increased from 5 to 100 Frn. According to Eq. ( 18)) the rate of increase of the biofilm thickness is higher for lower values ofX,, leading us to choose a lower value for XV in order to investigate the effects of the intraparticle convective flow for higher values of the biofilm thickness. Model solutions, in response to a step input of a phenol concentration of 65 mg/l, are presented as plots of the dimensionless phenol concentration in the bulk fluid phase, X = Cl C,, at column outlet, against a reduced time, OS, (breakthrough curves), for two different bulk flow rates: 50 and 200 cm3/min. The reduced time, &, is obtained dividing the time, t, by a stoichiometric time, ts-r, defined in analogy to that for an adsorption process as
(33)
where
&2 (k-- I---EMI Et,0 (l+a,)36p is the bed adsorptive capacity parameter. 4.1. Lowerflow resistances)
rate (lower intraparticle
diffusional
The effect of a growing biofilm on substrate removal, in the absence of intraparticle convection, is shown in Fig. 2. Breakthrough curves are depicted for several values of the decay constant, Kdr and for an initial biofilm thickness, S,, of 5 km. For the lower values of the decay constant, the breakthrough curve initially increases with time until a maximum is reached, after which it decreases with time as the contri-
Engineering
Journal
2 (1998)
1-9
Table 1 Data used in the simulations Length of reactor, L Internal diameter of reactor, d, Support radius, R, Initial bed porosity, Q, Volumetric bulk flow rate, U Bed superficial velocity, U, Initial Reynolds number, Rq, Initial bed space time, 7Lu Biofilm porosity, E, Support porosity, E> Molecular diffusivity of phenol in pure water, D,, Effective diffusivity in biofilm, Deb Effective diffusivity in support, D,,
1.0
“0
S,-5pm PB - 100
147 cm 1.77 cm 0.0295 cm 0.4 50,200 cm3/min 0.34, 1.36 cm/s
X8 174, 43.5 s 0.7 0.7 8.47X 10-hcm2/s 3.5X IOmhcm2/s 3.5X lOmh cm’/s
constant biofilm tbiieas
1
Fig. 2. Effect of a growing intraparticle convection.
2 biofilm
3 @ST on substrate
4 removal
5
in the absence of
bution to substrate removal by biodegradation increases. Comparing with the case of constant biofilm thickness (bold line curve), we can see that, as the decay constant decreases, the steady state substrate conversion increases and a higher substrate removal is achieved. Fig. 3a and b illustrate the variation of the biofilm thickness with time for several values of the decay constant, at the inlet and outlet of the bioreactor, respectively. The lower phenol concentrations at outlet, due to consumption along the bed, cause closer dynamic responses in terms of biofilm thickness. Breakthrough curves predicted in the presence of intrapartitle convection, by assuming equal velocities through the biofilm and support (u,~ = u,~), are shown in Fig. 4a and b for u,,= 2.96X 10-j cm/s and I/‘,, = 1.18X lo-’ cm/s, respectively. For any value of the decay constant, Kd, the breakthrough occurs later in the presence of intraparticle convection, such as observed for constant biofilm thickness (bold line curves). As K,, decreases, the decrease in the amount of substrate removal is larger due to intraparticle convection. Fig. 5a and b illustrate the effects of the intraparticle convective flow for u,,=2.96X10e3 cm/s and u,,=l.l8x 1Oe2 cm/s, respectively, when a higher intraparticle velocity is assumed to occur through the biofilm (u,, > u,~). The case
A. L.eit&, A. Rodrigues
6
/Biochemical
Engineering
Joumal2
1.0
(1998)
6,=5jbm Pe- 100 K, - 1.2
l-9
*&,-cl Am-25
--
constant
If
0.8
b-0 A,,-0.42
biofilm
thkkness
0.6
20
.......""Kd
0" 3
,,,,,...'.' = o.oo3
____ --_----------&s.:$.:“: -.-__. -.-._.__ ,<
._.,..../.... .....,
Kd = 0.005 -.-.-.____
10
0.4
0.2
-~----.-______
0 I-
I
0
2
@ST
1.0 z’-
50
f
SO-5pm
-0
“.“I
1
2
3
4
5
thickness with time for several values of the inlet; (b) bioreactor outlet.
rate (higher intruparticle
;I,-0
a,,-0
I
1
2
3
4
5
@ST
of constant biofilm thickness (bold line curves) is also depicted in this figures for comparison. It is apparent the larger beneficial effect of the intraparticle convective flow on the production of a good effluent quality, expressed by the occurrence of more delayed breakthrough curves and the achievement of a higher substrate removal. From Figs. 4 and 5 we observe that a higher intrabiofilm velocity, due to a higher biofilm permeability, produces an increase in the breakthrough time, but this effect is of little significance, in this case on lower intraparticle diffusional resistances when u,, = L’,*. 4.2. Higherpow resistances)
4
g-100
“0
5
--
6,-5pm
@ST
Fig. 3. Variation of the biofilm decay constant. (a) Bioreactor
3 *ST
diffusional
The effects of the intraparticle convective flow with u,, = 1.18X 10m2 cm/s, for equal (u,, =o,~) and different (u,, > uo2) velocities through the biofilm and support, are shown in Fig. 6a and b, respectively. In the presence of higher intraparticle diffusional resistances, it is apparent the lower sensitivity of the breakthrough curves to biomass growth. In this case, it is also remarkable the beneficial effect of the intraparticle flow, in terms of the occurrence of later break-
Fig. 4. Effects of intraparticle convectlon (u,,, = u,~) on substrate removal in the presence of a growing biofilm and lower intraparticle diffusional resistances. (a) u,,, =2.96X IO-‘cm/s; (b) LI,,= 1.18X 10-*cm/s.
through curves and the achievement of a higher substrate removal, particularly when the biofilm convective velocity is higher than the support velocity, such as seen for the constant biofilm thickness (bold line curves). All the results herein presented for a growing biofilm were calculated by assuming that the intraparticle convective velocity does not change with the biomass growth. As suggested by Rodrigues et al. [ 161, the intraparticle velocity is estimated by the equality of the pressure drops across the particle and across the bed. In laminar flow and from Darcy’s law, the following relation between velocity inside pores and bed superficial velocity is derived:
where B, is the particle permeability and B, is the bed permeability given by Bb=
E3d; b 150( 1-&b)2
with dP being the bioparticle diameter. As the bioparticle diameter increases, as result of the biomass growth, the bed
A. L&t&,
A. Rodrigues
/ Biochemical
Engineering
Journal
2 (1998)
1-9
”
- Pe.100
0.8 0.6
a’-
--&,-km
/
0
6,-Spm Pa = 100
*&,-I7
*,0-o i,,=1.69
I
0
““C
6,=51u” Pa- 100 K, = 1.2 100 K- 1.32s-’ era = 0.310 Kp-65 K, = 1.3 Ne - 0.25 Ndlo - 870 Nr, - 93.5
1
I
I
I
2
3
4
(a)
5
la-O kl=* -._._.&,-x3.3 l,o- 1.69 .-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-. constant biofilm thickness
Pe- 100 K, = 1.2 g-100 K- l.Wd (,,=0.310 Kp=6S K,- 1.3 Na - 0.25 Ndlo - 370 Nt, - 33.5 I
W
Fig. 5. Effects of intraparticle convection (u,, > c,*) on substrate removal in the presence of a growing biofilm and lower intraparticle diffusional resistances. (a) c,,, =2.96X 10m3 cm/s; (b) L,,, =1.18X IO-‘cm/s.
Fig. 6. Effects of intraparticle convection (u”, = 1.18 X 1O-2 cm/s) on substrate removal in the presence of a growing biofilm and higher intraparticle diffusional resistances. (a) u,,, = II,,; (b) I’~, > o,~.
porosity decreases,which produces a decreasein the bed permeability and an increase in the intraparticle velocity. Thus, h, and A2 should change with the biofilm thickness according to:
through curves for the lower valuesof Kd. However, the initial portion of the breakthrough curves is not affected by the variation in intraparticle velocity with biomassgrowth. The integration of the model equations,by the numerical technique previously described, is very difficult when the intraparticle velocity changeswith the biomassgrowth. A larger numberof collocation points is then required for accurate solutions, leading to a very hard computation program in termsof memory and time.
(37) B A2=AzoT
(38) b
where 5. Conclusions The breakthrough curves depicted in Fig. 7a and b for the lower and higher bulk flow rates usedin this work, respectively, were calculated by accounting for the variation in intraparticle velocity with biomass growth in our model. Comparing thesecurves with those in Figs. 4a and 6a, we note that the influence of the variable intraparticle velocity on the bed dynamic responsesis of little significance. Other results obtained with higher intraparticle velocities indicate that the variation in intraparticle velocity affects the break-
The influence of the intraparticle convective flow on the dynamic responseof a fixed-bed biofilm reactor to a step increasein influent phenol concentration was evaluated in this work. Model equationswere basedon a substrateinhibition kinetic model and biofilm growth. The simulation resultsreported in this work for two different bulk flow rates show that: (i) the major beneficial effect of the intraparticle convective flow is on the increaseof the breakthrough time, due to the improvement of intraparticle masstransfer by con-
A. L&b,
8
A. Rodrigues
/Biochemical
Engineering
c2 D AW 0.8
d
DC, D,, K
K, - 1.2 s,-100 K- 1.92s-’ q,, -0.310
0.6 K,n-65 i K,- 1.3
0
N&-l Ndr, = 3480 Nlo - 232.4
0.4
L NLil 1.0 Nd2 0.8 -
Nf
6,-5~m P9- 100
0.6 -
Pe 40
K, - 1.2 $,- 100 K- 1.92~~’
0.4
Oe2t
0
/
0
I Re
K,- 1.3 Na = 0.25 N,,,, - 870 Nb = 93.5
i
1
I
I
I
2
3
4
RS - rr
ON
5
SC
ST
Fig. 7. Effects of intraparticle convection on substrate removal in the presence of a growing biofilm: u,, = o02, variable velocity with biomass growth. (a) Lower intraparticle diffusional resistances; (b) higher intraparticle diffusional resistances.
t &T 43 u*
vection; (ii) a larger beneficial effect is achieved when the biofilm velocity is higher than the support velocity. From our investigation on the analysis of the effects of the intraparticle convection in biodegradation/adsorption combined processes, we finally conclude that intraparticle convection is an important design factor to be accounted for on the production of a good effluent quality, particularly when higher diffusional limitations are present inside the biofilm and support. Thus, bioreactors must be operated under such conditions that liquid movement occurs in the voids of the biofilm.
“01
U”2
v*
X X” Xl
x2
6. Nomenclature rs z
a c co c,
Ratio of biofilm thickness to support radius ( = 6/ R,), dimensionless Substrate concentration in the bulk fluid phase, kg/m3 Substrate concentration in the feed, kg/m3 Substrate concentration inside biofilm pores, kg/m3
Z*
Journal
2 (1998)
1-9
Substrate concentration inside support pores, kg/m3 Substrate molecular diffusivity in water, m*/s Effective diffusivity of substrate in the biofilm, m’/s Effective diffusivity of substrate in the support, m2/s Reaction rate constant for very small substrate concentrations, s-’ Decay constant, dimensionless External film mass transfer coefficient, m/s Substrate inhibition constant, dimensionless Langmuir adsorption equilibrium constant, m’/kg Substrate saturation constant, dimensionless Parameter in the Langmuir isotherm equation ( = K,C,), dimensionless Bed length, m Number of mass transfer units for biofilm pore diffusion ( = TV/ TV,>, dimensionless Number of mass transfer units for support pore diffusion ( = TV/ TV*), dimensionless Number of mass transfer units for external film diffusion, dimensionless Peclet number based on bed length, dimensionless Solid phase concentration in equilibrium with the liquid phase concentration CO, kg/m3 Radial coordinate in the particle, m Reynolds number based on bed superficial velocity, dimensionless Spherical support radius, m Volumetric rate of substrate consumption within biofilm, kg/m3 s Schmidt number, dimensionless Time, s Stoichiometric time, s Bed superficial velocity, m/s Dimensionless radial coordinate in the support (=rfR,) Intraparticle convective velocity inside the biofilm macropores, m/s Intraparticle convective velocity inside the support macropores, m/s Dimensionless radial coordinate in the biofilm (=(r/R,-1)/a) Dimensionless substrate concentration in the bulk fluid phase ( = C/C,) Concentration of cells in biofilm, kg/m3 Dimensionless substrate concentration inside biofilm pores ( = C,IC,) Dimensionless substrate concentration inside support pores ( = C,/C,) Cell yield coefficient, kg cells/kg phenol Axial coordinate in the bed, m Reduced axial coordinate in the bed ( = z/L), dimensionless
Greek letters P
Ratio of effective diffusivities dimensionless
( = Des/Deb),
A. kikio,
A. Rodrigues
/ Biochemical
Biofilm thickness, m Fixed-bed porosity, dimensionless Internal porosity of the biofilm, dimensionless Internal porosity of the support, dimensionless Angular particle coordinate, dimensionless Thiele modulus for very small substrate concentrations, dimensionless Intraparticle Peclet number for the biofilm ( = au,, /Deb), dimensionless Intraparticle Peclet number for the support ( = Rsvo2/Des), dimensionless Time constant for biofilm pore diffusion ( = E, S2/D,,), s Time constant for support pore diffusion ( = ~zR,2/De,) > s Initial bed space time ( = .q,&/u,), s Reduced time ( = tl T& , dimensionless Reduced time ( = t/tsT), dimensionless Bed adsorptive capacity parameter, dimensionless Support adsorptive capacity parameter ( = ( 1 - c2) / .c2qo/CO), dimensionless
Engineering
Journal
2 (1998)
1-9
9
Liquid flow and mass [41 D. De Beer, P. Stoodley, Z. Lewandowski, transport in heterogeneous biofilms, Water Res. 30 (1996) 27612165. K. Wisecarver, B. Zehner, Diffusion of [51 L. Fan, R. Leyva-Ramos, phenol through a biofilm grown on activated carbon particles in a draft-tube three-phase fluidized-bed bioreactor, Biotechnol. Bioeng. 35 (1990) 279-286. [61 B. Finlayson, Nonlinear Analysis in Chemical Engineering, McGrawHill, 1980. 171 G. Froment, K. Bischoff, Chemical Reactor Analysis and Design, 2nd edn., Wiley, New York, 1990. [81 A. Karabelas, T. Wegner, T. Hanratty, Use of asymptotic relations to correlate mass transfer data in packed beds, Chem. Eng. Sci. 26 (1971) 1581-1589. [91 A. Lallai, G. Mura, Kinetics of growth for mixed cultures of microorganisms growing on phenol, Chem. Eng. J. 41 ( 1989) B55-B60. 1101 A. Leitao, A. Rodrigues, Modeling of biodegradation/adsorption combined processes in fixed-bed biofilm reactors: effects of the intraparticle convective flow, Chem. Eng. Sci. 5 1 (20) (1996) 45954604. [Ill A. Livingstone, H. Chase, Modeling phenol degradation in a fluidizedbed bioreactor, AIChE J. 35 (12) (1989) 198&1992. 1121 N.K. Madsen, R.F. Sincovec, PDECOL: general collocation software for partial differential equations, ACM Trans. Math. Software 5 (1979)
326.
R. Mendez, J. Lema, Biofilm reactors technology in wastewater treatment. In: L.F. Melo et al. (Eds.), Biofilms-Science and Technology. Kluwer Academic Publishers, 1992, pp. 409-419. A. Monteiro. Biological Degradation of Phenol with Bacteria Immo[I41 bilized in Large-Pore Support, PhD Thesis, University of Porte, 1997. 1151 Y. Park, M. Davis, D. Wallis, Analysis of acontinuous, aerobic, fixedfilm bioreactor: II. Dynamic behaviour, Biotechnol. Bioeng. 26 Cl31
Acknowledgements This work was partially developed at the School of Engineering of Port0 during a sabbatical leave of A. Leitao. The support from the Angolan Petroleum Ministry is gratefully acknowledged.
(1984)468-476.
1161 A. [I71
References
Rodrigues, B. Ahn, A. Zoulalian, Intraparticle-forced convection effect in catalyst diffusivity measurements and reactor design, AIChE J. 28 ( 1982) 541-546. A. Rodrigues, Z. Lu, J. Loureiro, Residence time distribution of inert and linearly adsorbed species in a fixed bed containing large-pore supports: applications in separation engineering, Chem. Eng. Sci. 46 (1991)
[ 1 ] L. Backer, G. Baron, Residence time distribution in a packed bed bioreactor containing porous glass particles: influence of the presence of immobilized cells, J. Chem. Technol. Biotechnol. 59 (1994) 297302. [2] H. Beyenal, S. Seker, A. Tanyolac, B. Salih, Diffusion coefficients of phenol and oxygen in a biofilm of Pseudomonasputida, AIChE J. 43 ( 1997) 243-250. [3] D. De Beer, P. Stoodley, Z. Lewandowski, Liquid flow in heterogeneous biofilms. Biotechnol. Bioeng. 44 ( 1994) 636641.
Ll81
2765-2773.
W. Tang, K. Wisecarver, L. Fan, Dynamics of a draft tube gas-liquidsolid fluidized bed bioreactor for phenol degradation, Chem. Eng. Sci. 42(9)(1987)2123-2134. [ 191 C. Tien, S. Wang, Dynamics of adsorption columns with bacterial growth outside adsorbents, Can. J. Chem. Eng. 60 ( 1982) 363-376. 1994. [201 C. Tien, Adsorption Calculations and Modeling, Butterworth, [211 R. Worden, T. Donaldson, Dynamics of a biological fixed film for phenol degradation in a fluidized-bed bioreactor, Biotechnol. Bioeng. 30(1987)398-412.