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Wal. Sci. Tech. Vol. 36. No.5. pp. 285-292. 1997.© 1997 IAWQ. Published by Elsevier Science Ltd

Printed in Great Britain.0273-1223/97 SI7·00 + 0·00

DYNAMICAL MODELLING OF ANANAEROBIC DIGESTION FLUIDIZEDBED REACTOR

Benjamin Bonnet*, Denis Dochain* andJean-Philippe Steyer**

* Cesame. Universite Catholique de Louvain. Biitiment Euler. avo G. Lema/tre 4-6.B-1348 Louvain-La·Neuve. Belgium** l.JJE-INRA. Avenue des Etangs. 11100 Narbonne. France

ABSTRACf

One of the main difficulties in modelling a Fluidized Bed Biofilm Reactor (FBBR) is to take into accounthydraulic phenomena (such as bed expansion) and its interactions with the biological variables. In this paper,we shall present a dynamical model of the process, analyse the stability of the hydrodynamics and illustrateits performances in simulation. A key feature of the model is that it combines mass balance of the processcomponents with momentum balance equations in order to emphasise the different hydrodynamics of theliquid phase and of the solid phase. and the interactions between both phases. The model derivation fmallyleads to a set of partial differential equations (POE). This model is intended to be used as a basis for thederivation of controllers and for dynamical simulation. © 1997 IAWQ. Published by Elsevier Science Ltd

KEYWORDS

Dynamical modelling; anaerobic digestion; fluidized bed reactor

INTRODUCTION

Over the past decades a large number of biotechnological waste and wastewater treatment processes havebeen developed with the objective to meet the requirements of the industry and municipalities. Theiroperation is most of the time continuous. The evolution of the scientific knowledge and technology hasresulted in the development of processes whose increasing efficiency can be connected to increasing micro•organism concentrations. This can be achieved by re-injecting cells after settling (sludge recycle) or fixingthem on a carrier (immobilised cell processes). In fixed bed reactors, cells are growing on a carrier within thereactor. The wastewater then flows through the reactor where the matter to be treated is captured anddegraded by the immobilised cells. These reactors are also sometimes referred as biological filters. Thefluidized bed reactors are based on a similar concept except that now the supports are not fixed anymore andare allowed to move inside the reactor and put in suspension by the fluid upflow.

Figure 1. Biological Fluidised Bed Reactor285

286 B. BONNET et 01.

The FBBR under study is shown in Figure 1. The recirculation flow circuit is used to guarantee a sufficientfluidization flow rate Qin for the bioparticles (inert supports + biofibn). The settler is used to avoid theintroduction of bioparticles into the recirculation. The dynamics are characterised by two types ofphenomena : hydrodynamics (related to the fluidisation mechanisms) and biological reactions (via thedegradation of the organic matter). Note that there are two distinct zones in the reactor: the lower part (z e[0. HJ) is the fluidised zone with the bioparticles. and the upper part (z e ]Hj • H]) do not contain any solidparticle. In the following we assume that the biological activity is concentrated in the lower reactor part.

One of the main difficulties in modelling a FBBR is to take into account hydraulic phenomena (such as bedexpansion) and its interactions with the biological variables. In this paper. we shall present a dynamicalmodel of the process. analyse the hydrodynamics stability and illustrate its performances in simulation. Themodel derivation leads to a set of partial differential equations (POE). Le. a distributed parameter system(DPS).

A key feature of the model is that it combines mass balance of the process components with momentumbalance equations in order to emphasise the different hydrodynamics of the liquid phase and of the solidphase. together with the interactions between both phases. Our experience suggests that on one hand. theseare likely to play an important role in the process dynamics and in the process control performance. On theother hand. we have decided to limit the complexity of the model as much as possible: indeed our objectiveis to obtain a qualitatively good dynamical model of the process for control design and for process dynamicalsimulation. The (quantitative) calibration of the model parameters is already an almost hopeless task withsimple dynamical models in anaerobic digestion due the lack of some measurements (typically. microbialpopulations' concentrations). the scarcity and uncertainty of others (e.g. volatile fatty acids. COO....) and theuncertainty related to the process dynamics (mainly the reaction kinetics); it will most probably become acompletely desperate task with overcomplexified models. and it will be almost impossible to detect withoutany ambiguity the source of important dynamical phenomena and their connection with the experimentalobservations. This explains why some simplifying assumptions have been considered : e.g. onlyhydrodynamic interactions between liquid and solid phases are considered; the biomass is considered as asimple catalyst in the total solid and liquid balance equations.

DYNAMICAL MODELLING OF THE FLUIDIZED BED REACTOR

In a fluidized bed reactor. the different phases (gas, liquid and/or solid) are capable of moving independentlywith their own hydrodynamics and more specifically with their own velocity..Here we consider two phases:one liquid phase with the substrates and intermediate products. and one solid phase with the biofibn (seeFigure 2). Although methane is obviously an important process component, w~ have not considered a third(gaseous) phase. basically in order to limit somewhat the model compleXity. Furth~rmore for similarsimplicity reasons. we neglect the dynamics of the settler. Finally we assume that there IS no dispersion inthe reactor.

The dynamical model of the fluidized bed reactor is base~ not onl~ on ~ss.balanc~s. for the processcomponents but also on the momentum equation in order to hnk the solid and liqUid velocloes. For the massbalances we have considered a two-step reaction scheme :

SI ~XI+S2+PI

S2 ~ X2+ PI + P2

(I)

(2)

where S to S2. X" X2. PI and P2 represent the organic matter. the volatile ~atty acids. the acidogenic bacteria.the methanogenic bacteria. the carbon dioxide and the methane. respecnvely. We have further assumed adeath/detachment for both biomass XI and X2. This leads to the following equations:

a(eSI) a~ = -d;:[S.eU.] - (l-e)k l J.11X I (3)

a(eS2) a~ = - az [S2eU.] + (I-e)(k21J..XI - k31J.2X2) (4)

a«(I-e)X.) aat =-d;:[X,(I-£)U.l + (I - e)(J.1IX I - K,UXI) (5)

a«(I-e)X~) aat ~ = -d;:[X2( l-e)U.l + (1 - e)(J.12X2- Kd2X2) (6)

Anaerobic digestion fluidized bed reactor

a(eP,) a---ar- = - az [P,eUll + (l-e)(k4J.1.,X, + k,J.l.2X2)-

a(eP2) ·a 1 aQ2---ar- =-a;[P2eUll +(l-e)~1l2X2 - 'Aai""

with the associated Danckwerts-type boundary conditions:

287

(8)

--

S1(O,t) =S!,in, S2(O.t) =S2,in. Xl (O,t) =0, X2(O,t) =0, PI(O.t) =PI,in. P2(O,t) = P2,in

In the above equations. S I. S2. PI. P2 are concentrations in the liquid phase. and Xl. X2, concentrations inthe solid phase. e is the void fraction (defined here as the ratio of the section of the liquid phase at theposition z, AI(Z). over the reactor section A); UI and Us are the superficial velocities of the liquid phase andof the solid phase. respectively; J.l.I and 112 are the specific growth rates of the acidogenesis and of themethanization. respectively; kit i = I to 6, are yield coefficients. I<dI and I<d2 are the death/detachmentcoefficient associated to Xl and X2 respectively, QI and Q2 are the gaseous outflow rates of C02 and Cf4,and SI,in. S2 in, PUn and P2.in are the influent flow rates (possibly partly via the recycle loop) of organicmatter, volatile fatty acids, CO2 and CH4. respectively. A quick look at the above equations shows that wehave 9 states (6 process components SI. 52. Xl. X2. PI and PI+ 3 hydraulic variables e. UI and Us> and only6 (partial) differential equations. Therefore we need more equations to be able to describe the processdynamics.

Figure 2. Motion of the liquid and solid phases

HYDRODYNAMICS

Let us consider that the fluidization is homogeneous (or particulate) (Batchelor, 1988) : this means that thevoid fraction e is constant in space in steady-state. and that variations of the fluid superficial velocity induceshomogeneous (without bubbles) variation of the bed height. The Richardson-Zaki (1954) equation thenapplies in steady state: en = Uo/UI where n. UI and Uo are the expansion coefficient. the particle terminalsettling velocity. and the (inlet) superficial fluid velocity, respectively. In transient conditions. the aboverelation is modified as follows (Poncelet et al, 1990) :

en=~ (10)UI

The model can then be completed by considering the liquid phase and the solid phase separately, and tocompute mass balance and momentum balance for each phase. This gives the following equations:

ae a(eU1)

dt=-aza(1 - e) a«1 - e)U.)at=- dZ

(11)

(12)

288 B. BONNET et 01.

a(p.eU,) d(PJeUt)en dz. + FGl - FI+ Fpl

a(ps(l-e)Uj) a(ps(l- e)U~)en = dZ + FGl + FI + Fp•

(3)

(4)

(9)

(17)

In the above equations, FOi, Fpi (i=l. s) and FI represent the contributions of the forces due to gravity,pressure and interaction between phases, respectively. PI and Ps are the density of the liquid phase and thesolid phase, respectively. Note also that strictly speaking the above mass balance equations assume that massexchange between the liquid phase and the solid phase (Le. the biomass mainly acts as a catalyst, or at leastthe mass absorbed by the biomass for its growth is exactly compensated at any time by the mass of detachedbiomass). In line with Foscolo and Gibilaro (1987), the expressions of the different forces are given by thefollowing relationships:

P.-P. deFGl = -ep18, Fos = -(l-e)ps8, Fp. = 3.2d(l-e)g'"""Ps""""di" FPl = -(epl + (l-e)p.)g (IS)

with d the solid particle diameter. Note that by elimination of oElen in equations (11)(12), we obtain :

iHeU.) 0(0 - e)U.)--+ =0dZ oz

Le. we can algebraically express UI as a function of the other two variables e and Us :

eUl + (l - e)U. = Uo (18)

Therefore only the mass balance and momentum POE's (e.g. in the solid phase) plus the above algebraicequation (18) are necessary to describe the dynamics of the three variables e, UI and Us. Finally the

following Oanckwerts type boundary conditions for e and Us are considered :1

(Uo(t} )ii

U.(O,t} =0, e(O,I) = u;-Finally note that the inlet superficial fluid velocity Ua is related to the inlet flow rate Qr. the recycle ratio rand the reactor cross section A as follows:

Qt<l +r) (20)Uo= A

State space formulation of the model

After some manipulations, the POE model of the process can be rewritten in the following state spaceformat:

(21)

(22)

(23)

(24)

(2S)

Anaerobic digestion flUidized bed reactor

aPl Uo- (I-e)u. aPl 1-£at =- £ az + e(~~lXl + kS~2X2)-

ap2 Uo- (I-e)U. ap2 1-£ 1 aQ2at £ dZ +e~2X2 - AE. dZ

with the boundary conditions (9)(19).

STABILITY ANALYSIS OF mE HYDRODYNAMICS

(27)

(28)

289

(35)

(32)

(36)

A quick look at the above equations shows that there are basically two blocks in cascade: fust, the twoequations of E and Us which concentrate the hydrodynamics and whose solutions do not require informationfrom the other six; secondly, the last six equations, which need information about E and Us to be solved.

Since one of motivation of this work is to design control schemes for running the process in stable conditionsand optimizing its operation, an important objective of this modelling exercise is to build a model which ispossibly capable of predicting when the bed may become unstable, Le. move from an homogeneous(particulate) bed to an heterogeneous (aggregate) bed. This implies to carry out a stability analysis of thehydrodynamics part of the model. In order to be as rigourous as possible. the analysis has to be performed ona version as close as possible to the original one. The analysis on the original nonlinear infmite dimensionalmodel (21)(22) did give workable results. Therefore we decided to analyze the linearized version of themodel around a steady state. There is indeed no multiple steady state for the hydrodynamics (21)(22), whichis then equal to :

1

(Uo)'R

U•.eq =0, Eeq = 14The linearized tangent model around this steady state is equal to :

OE' au~dt =(I - Eeq)az- (29)

au~ = 4.8g P.-PI [ __l_u~ _ E' + 2d ae' ] (30)dt Ps nUtE~l 3 az

with U: = Us - Us.eq and E' = E - Eeq. Here the theory of infinite-dimensional systems (as e.g. applied inDochain and Winkin, 1995) was not useful. We finally decided to use a tool (normal modes analysis) whichis largely considered in fluid dynamics and which can be briefly described as follows. Let us rust derive theequation (29) with respect to time and eliminate Us by using equation (30). This gives:

iE' 2 a 2E' aE' aE'- =a - - ~~ -1Yr--:- (31)at2 az2 at az

with :2 P.-PI 4.8 P.-PI 1 - Eeaa =3.2dg-- (l - Eeq), ~ =-Ug--Eeq•y= nUo--'-

P. n 0 Ps EeqLet us now consider a disturbance f(z,t) solution of (31) of the form :

[(z, t) =Aoeat + ik(z-vt) (33)

where AO, a, v and k are the initial amplitude, a damping coefficient, the velocity and the frequency of thedisturbance wave, respectively. The disturbance will be damped if a is negative (stability condition). By

introducing (33) in (31), we obtain: _a2(jk)2 + (a - ikv)2 + ~yik + ~(a - ikv) =0 By separation of the realpart and the imaginary part. we obtain two conditions for f to be solution of (31), i.e. :

~ a2+ ~aa=-(y-v) k2=_

2v ' v2 _a2

The linearized tangent model of the hydrodynamics will then be stable if a > y, Le. if Uo is such that:1 1

(I -E.q)2 [ P.-PI]2nUo < 3.2dg-Ecq P.

290

Model reduction

B. BONNET et al.

When a product give~ off.in the gaseous phase and has a low solubility, its dynamical mass balance equationc~ ~e .reduced by usmg smgular perturbation arguments (e.g. Bastin and Dochain (1990). Dochain (1994» :this IS mdeed the case for Cf4 (P2) and C02 (PI). Let us look at the reduction procedure for P2. let us firstrewrite P2 = 7t Psat with Psal the saturation concentration of P2. Then equation (28) becomes:

P dlt Uo-O-e)Us dlt l-e 1 el02,alai" = - e P'aI elz + -e-~Jl2X2 - AE. elz (37)

When Psat is very low. the preceding quation may be rewritten with Psat --+ O. Le.:elQ2dZ =~O - e)Jl2X2 (38)

Le. after integration with respect to z :

Q2 = Ak6foHO- e)JlzX2'!z (39)

The same argument can be applied to equation (27). This gives:

QI = AfO-e)(k4J.i.IXI + kSJl2X2)dz (39)

These equations have been considered in the simulation software of the FBBR dynamics.

Connection with fixed bed reactor models

In practice the dynamics of fluidized bed reactors are often considered to be close to those of fixed bedreactors, at least when the bed remains rather still. It is therefore important to check whether our dynamicalFBBR model can be reduced to that of a fixed bed bioreactor when the solid phase velocity Us is low. Indeedthis is what happens by assuming that Us « UI : in particular, the convection term in the mass balances of5 1,52 reduces to U<ye el5i/Clz (i=I.2) and the convection term of Xl. X2 disappears (Le. Xl. X2 do not move).

SIMULATION RESULTS

Kinetics models

For the simulation. we have considered the following models for the specific growth rates:51 52

JlI = JlI,max Tl" _ X + S ,Jl2 = Jl2,max S2 !K'~I I I Kc2X2+S2+ 2 I

with JlI.max and Jl2.max the maximum specific growth rates of JlI and Jl2, Kcl and Kc2 Contois constants andKI an inhibition coefficient The choice of the above expressions is basically arbitrary (It is one of the 60 or70 (heuristic) kinetic models available in the literature. e.g. Bastin and Dochain (1990)!). The choice is atleast motivated by the following points: the Contois type of saturation is introduced to allow to have spatialgradient along the reactor at least in steady state (kinetic models that only depend on the limiting substratewill result in flat spatial profile); and the introduction of the inhibition term in Jl2 is in line with theexperimental evidence that volatile fatty acids may have an inhibitory effect at high concentrations (andresult in unstable dynamical process behaviour).

Choice of the model parameter values

The relations given by Gidaspow (1994) and Richardson and Zaki (1954) have been considered to determinethe terminal particle velocity Ul and of the expansion coefficient n The different data useful for thecomputation of the other parameters of the hydraulic part of the model (densities. diameter. viscosity,...) havebeen provided by the Laboratoire de Biotechnologie de L'Environnement (LBE, Narbonne). The values ofthe parameters and initial variables of the biological part of the model has also been defined in coordinationwith the LBE.

Ps = 1200 kg m-3, PI = 1000 kg m-3, H = 3.5 m, DR = 25 cm. A = 490 cm2, Ul = 77 m h· l• n = 3.392kl=O.036, JlI.max=O.05 h- I.KcI=5.7 lO-s,I<dI=O.OI h-l. Jl2.max=O.OI416 h-I. Kc2=O.0047.I<d2=O.00416 h-l

k2=O.oJ88. k2=O.029. KI=5 gl-I, £e=O.55. Uo=IO.147 mh- I, Q =0.1 ~h·l. r=4, St.r=I5 gl-I}, Hi 2.5 m

Anaerobic digestion fluidized bed reactor 291

Discretizalion and choice of the time and space sampling intervaJs

The implementation of the simulation software of the dynamical model (21)(26) has been performed via theuse of backward fInite differences for approximating both time and space derivatives:

ax(z,t) x(z,t} - x(z,t-I) ax(z,t) x(z,t) - x(z-l,t)~= ~t ,~= ~

The choice of the sampling intervals 6t and t.z may be critical for having stable numerical simulation(independently of the intrinsic stability properties of the dynamical model (21)(26». Here again the mostcritical part of the model is the hydrodynamics. The choice of 6t and 6z is based on the linearized tangentmodel of the discretized version of the model (21)(22) :. ... - .

~k+1 = AK!;k + B Kllk

f= -£1-£eq ,A K= 0 0 ~t bl 0 0

0 -~t bl ~tbl

£K-€eq0 0

VsI-V•.eq0 0 I 0 0 ~tbl

-------V.K-Vs.eq -6t~ 0 0 1-6t b4 0 0

-~tb3 -~t~ 0 1-6t b4

0 0

0 0 -~t~ 0 0 I-~t b4where:

I-£eq _( 2d) 3.2dg 4.8geeq 4.88Eeq( 2d) - Ps-PIbl =-X;:-, ~=4.8g 1-'3&' ~=-xz-. b4 = nV • bh no 1-'3& ,g =g-P-a 0 s

1S

~10

Ul 5:;)

01

11

tIImax 0 0 zmax tIImax 0 0

3.5

40

~30 g5 20 :f

101

lItmax 0 0 Z1zmax 0.5max

Figure 3. Response of the hydraulic variables to a step of the recycle rate

292 B. BONNET er al.

In ?r~er to ha,,:e numeri~al stability for the above linear system, the eigenvalues have to remain within theunit crrcle. This results 10 the following criteria:

2d 2nUoL\z> T (=416 ~m), L\t < -_- (= 4.4 ms)4.8ge"l

In the following, L\t and L\z have been fixed to the following values : L\t= 1 rns, L\z = 5 cm.

Results

The following figures show two sets of simulated data. For space reasons. only partial graphical results areexhibited. Figure 3 show the hydraulic variables after a step of the recycle rate r (from 4 to 9). Figure 4 showthe biological variables after a step of the influent flow rate Qc (from 0.1 to 0.15 m3 h- I ).

'0

_ 00

3

Figure 4. Response of the biological variables to a step of the influent flow rate

ACKNOWLEDGEMENT

This paper presents research results of the Belgian Programme on Inter-University Poles of Attractioninitiated by the Belgian State, Prime Minister's office for Science, Technology and Culture. The scientificresponsibility rests with its authors.

REFERENCES

Bastin. G. and Dochain. D. (1990). On-line Estimation and Adaptive Control of Bioreactors. Elsevier,Amsterdam.

Batchelor, G.K. (1998). A new theory of the unstability of a unifonn fluidized bed. J. Fluid. Mech., 193:75•110.

Dochain. D. (1994). Contribution to the Analysis and Control of Distributed Parameter Systems withApplication to (Bio)chemical Processes and Robotics. These d'Aggregation de l'EnseignementSuperieur, VCl, Belgium.

Dochain. D. and Winkin, 1. (1995). Dynamical analysis of a class of distributed parameter fixed bedreactors. Proc. CDC'95, 3225--3230.

Foscolo, P.U. and Gibilaro. L.G. (1987). Fluid dynamic stability of f1uidised suspensions: the particle bedmodel. Chemical Engineering Science, 42(6): 1489-1500.

Gidaspow, D. (1994). Multiphase Flow and Fluidization. Continuum alld Kinetic Theory Descriptions.Academic Press, Boston.

Poncelet. D.• Naveau. H., Nyns, E.-J. and Dochain, D. (1990). Transient response of a solid-liquid modelbiological f1uidised bed to a step change in fluid superficial velocity. 1. Chern. Tech. Biotechnol.•48:439-452.

Richardson. J.F. and Zaki. W.N. (1954). Sedimentation and f1uidisation : part I. TrailS. Instil. Chem.Engrs., 32:35-53.