Available online at www.sciencedirect.com

Bioresource Technology 99 (2008) 3665–3675

Anaerobic digestion model no. 1-based distributed parameter modelof an anaerobic reactor: I. Model development

S.J. Mu a, Y. Zeng a, P. Wu a, S.J. Lou b, B. Tartakovsky b,*

a Institute of High Performance Computing, 1 Science Park Road, #01-01, The Capricorn 117528, Singaporeb Biotechnology Research Institute, NRC, 6100 Royalmount Avenue, Montreal, Quebec, Canada H4P 2R2

Received 4 May 2006; received in revised form 16 July 2007; accepted 16 July 2007Available online 18 September 2007

Abstract

This work presents a distributed parameter model of the anaerobic digestion process. The model is based on the Anaerobic digestionmodel no. 1 (ADM1) and was developed to simulate anaerobic digestion process in high-rate reactors with significant axial dispersion,such as in upflow anaerobic sludge bed (UASB) reactors. The model, which was named ADM1d, combines ADM1’s kinetics of biomassgrowth and substrate transformation with axial dispersion material balances. ADM1d uses a hyperbolic tangent function to describe bio-mass distribution within a one compartment model. A comparison of this approach with a two-compartment, sludge bed – liquid abovethe bed, model showed similar simulation results while the one-compartment model had less equations. A comparison of orthogonal col-location and finite difference algorithms for numerical solution of ADM1d showed better stability of the finite difference algorithm.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Distributed parameter model; ADM1; UASB reactor; Anaerobic digestion

1. Introduction

Compared to its aerobic counterpart, anaerobic diges-tion of wastewater features advantages such as energyrecovery in the form of methane production, productionof low sludge volume, and a capacity for degradation ofhigh strength wastewater (Bernard et al., 2001). Whileproblems of instability are infrequent for anaerobic diges-tors treating primary sludge, high-rate upflow anaerobicsludge bed (UASB) reactors are highly sensitive to externaldisturbances such as hydraulic or organic shock loads,which lead to imbalances in the acidogenic and methano-genic populations. As a result, the anaerobic reactors treat-ing high-strength wastewaters are prone to instability andmay be subject to reactor failures. These problems havemotivated a number of studies on anaerobic digestionkinetics and the development of reliable reactor models

0960-8524/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.biortech.2007.07.060

* Corresponding author. Tel.: +1 514 496 2664; fax: +1 514 496 6265.E-mail address: Boris.Tartakovsky@cnrc-nrc.gc.ca (B. Tartakovsky).

to study ‘‘what-if’’ operational scenarios for improvingprocess stability and enhancing process control strategies.

A comprehensive structured model, known as the Anaer-obic digestion model no. 1 (ADM1), was proposed by theInternational Water Association task group on anaerobicdigestion (Batstone et al., 2000a,b, 2002). ADM1 takes intoaccount the processes of disintegration of complex solids,hydrolysis of particulate organic materials, substratedegradation, and biomass growth and decay. Since it wasfirst introduced in 2000, ADM1 has been widely used inmodeling of anaerobic processes (Batstone et al., 2005;Blumensaat and Keller, 2005; Jeong et al., 2005).

While the development of ADM1 is a significant steptowards comprehensive modeling of anaerobic digestion, ituses the assumption of ideal mixing. However, UASB reac-tors are often operated at high volumetric organic loads andlow recirculation-to-influent ratios. These operating condi-tions are suitable for the formation of biofilm granules,but they also lead to the existence of significant substrateand biomass gradients in the sludge bed, which have aprofound influence on long-term reactor performance and

Nomenclature

a, b regression coefficients (Eqs. (12) and (13))A, B orthogonal collocation method matricesAr reactor cross section area, m2

Bo Bodenstein numberc liquid phase component concentration, kg

COD m�3

D dispersion coefficient, m2 day�1

H total reactor height, mHs sludge bed height, mN number of internal collocation pointsOLR organic loading rate, kg COD m�3

R day�1

pgas;H2O partial pressure of water vapor, barPatm total gas pressure, barqgas biogas flow rate, m3 day�1

qin influent flow rate, m3 day�1

qrec external recirculation flow rate, m3 day�1

qs bypass flow rate, m3 day�1

Q total input flow, m3 day�1

r biotransformation rate, kg COD m�3 day�1

rT gas component transfer rate, kmol m�3 d�1

R gas law constant, bar m3 kmol�1 K�1

Rw sludge washout factors soluble matter concentration, kg COD m�3

sgas gas concentration, kg COD m�3

sion ion concentration, kg mole m�3

t time, dayT operating temperature, K

u linear upflow velocity, m day�1

v stoichiometric parameterV, Vliq reactor and liquid volumes, m3

Vgas reactor gas volume, m3

x suspended matter concentration, kg COD m�3

xat maximal attainable biomass concentration,kg COD m�3

xmax attainable concentration of microorganisms inthe sludge bed, kg COD m�3

xmin attainable concentration of microorganisms inliquid above sludge bed, kg COD m�3

xtotal total biomass concentration, kg COD m�3

z reactor axial position, mb positive constant in Eq. (11)s hydraulic retention time, dayg dimensionless reactor axial position

Subscriptsac acetateat attainablec compositesin influent flowmax maximalmin minimalout effluent flowr recirculation flows sludge compartment

sludge zone

liquid zone

gas phase

effluent

Off-gas

influent

reci

rcul

atio

n

bypa

ss

sludge

height

xmax xmin

Hs

H

Fig. 1. Schematics of an UASB reactor (left panel) and a qualitativeprofile of maximal attainable biomass concentration (xat) obtained using a

3666 S.J. Mu et al. / Bioresource Technology 99 (2008) 3665–3675

stability during shock loads (Sam-Soon et al., 1991;Kalyuzhnyi et al., 1996; Torkian et al., 2003).

Several somewhat simplified distributed parameter mod-els of the anaerobic digestion process have already beenproposed. In the studies by Kalyuzhnyi et al. (2006) andSchoefs et al. (2004), relatively simple reaction kineticswere used. Batstone et al. (2005) developed a distributedparameter model by combining the ADM1 kinetics withthe Takacs clarifier model (Takacs et al., 1991), whichapproximates a UASB reactor using several layers, i.e.reactor hydrodynamics was simplified. In contrast, thisstudy presents a comprehensive distributed parametermodel, which combines the biotransformation kinetics ofADM1 with the axial dispersion transport model. The val-idation of this model with experimental data obtained in alaboratory-scale UASB reactor is presented in a subse-quent paper (Tartakovsky et al., 2007).

hyperbolic tangent function in Eq. (11) (right panel).

2. Modeling framework

2.1. Material balances

For an axially dispersed UASB reactor (Fig. 1a) thematerial balance of each component ci in the liquid phasetakes the following form (Kalyuzhnyi et al., 2006):

oci

ot¼ o

ozDiðz; tÞ

ociðz; tÞoz

� �� o

ozðuiðz; tÞciðz; tÞÞ

þ riðz; tÞ þ rT;iðz; tÞ ð1Þ

where Di is the dispersion coefficient, ui is the upflow veloc-ity, z is the axial position, t is the time, ri is the net trans-

S.J. Mu et al. / Bioresource Technology 99 (2008) 3665–3675 3667

formation rate of the ith component calculated as a sum ofspecific kinetic rates multiplied by corresponding stoichi-ometric coefficients, and rT,i is the transfer rate of gas com-ponents. The liquid phase components included in themodel are soluble organic matters (si), suspended particu-late matters (xi), and ions (sion,i). The distributed parametermodel takes into account the same 33 liquid phase compo-nents as ADM1. A detailed description of liquid phasecomponents and transformation rates ri(z, t) can be foundelsewhere (Batstone et al., 2002).

The Danckwerts boundary conditions (Danckwerts,1953) for the above liquid component material balanceequations are given in the following two equations:

Diðz; tÞoci

oz¼ uiðz; tÞðciðz¼0Þ � ci;inÞ; z ¼ 0 ð2Þ

oci

oz¼ 0; z ¼ H ð3Þ

where ci,in is the influent concentration of the ith compo-nent and H is the reactor height.

Because of biogas bubbles generated during the biodeg-radation of organic compounds in the liquid phase, gastransfer occurs within the liquid phase while gas transferat the gas–liquid interface at the top of the liquid can beneglected. Material balances for the gases are thereforeposition-dependent. The dispersion and convective trans-port of bubbles in the liquid are considered negligible incomparison with the gas transfer rate. The following equa-tion describes the concentration changes of gas-phasecomponents:

osgas;jðz; tÞot

¼ rT;jðz; tÞ �1

Ar

o

ozðqgasðz; tÞsgas;jðz; tÞÞ ð4Þ

where j is the index of biogas components, which are H2,CH4 and CO2. sgas denotes the concentration of the jth bio-gas compound, Ar is the reactor cross section, and qgas isthe volumetric flow rate of biogas, which is defined as

qgasðz; tÞ ¼RT

P atm � pgas;H2O

Ar

�Z H

0

rT;H2ðz; tÞ

16þ rT;CH4

ðz; tÞ64

þ rT;CO2ðz; tÞ

� �dz

ð5Þ

where Patm and pgas;H2O denote the total gas pressure andthe partial pressure of water vapor, respectively; T is theoperating temperature; and R is the gas constant.

The boundary conditions of the gas components aresgas,j = 0 at z = 0 and dsgas,j/dt = 0 at z = H.

The overall mass balances of the reactor shown in Fig. 1are given below

qin þ qrec ¼ qs þ Q ð6Þqinci;in þ qrecci;r ¼ qsci þ Qci;0 ð7Þ

where qin, qrec, and qs are flow rates of the influent, recir-culation, and bypass streams, respectively, Q is the totalinput flow, ci,in, ci,r and ci,0 are the concentrations of theith component in the corresponding flow. The bypass flow(qs) in Eq. (7) is expected to be dependent on factors suchas linear upflow velocity and organic load (Singhal et al.,1998; Garuti et al., 2004). However, in the absence of astrict theory the model does not use any specificdependence.

Overall, the distributed parameter model consists of 36partial differential equations (material balance equationsfor 12 soluble matters, 12 particulate matters, 9 ions and3 gases), 72 ordinary differential equations (two boundaryconditions for 36 components) and 33 algebraic equations(overall mass balances for 33 liquid components), whichwill be solved by a numerical algorithm.

For a completely stirred tank reactor (CSTR), materialbalances of liquid phase components are reduced to the fol-lowing form (Batstone et al., 2002):

dV liqciðtÞdt

¼ qinci;inðtÞ � qoutciðtÞ þ V liq

X19

j¼1

vi;jrj ð8Þ

Also, the CSTR gas phase component balance is given by

V gas

dSgas;j

dt¼ �qgassgas;j þ rT;jV liq ð9Þ

where

qgas ¼RT

P atm � pgas;H2O

V liqrT;H2

16þ rT;CH4

64þ rT;CO2

� �ð10Þ

All notations of the CSTR model are the same as in Eqs.(1)–(7).

2.2. Biodegradation kinetics

Kinetic dependencies describing the biotransformationof organic matter and the growth of microorganisms wereadopted from ADM1. Kinetic parameters and modelinputs are those used in ADM1 benchmark simulations(Rosen and Jeppsson, 2002). ADM1 kinetics is brieflyreviewed below, while a detailed description can be foundelsewhere (Batstone et al., 2002).

It is assumed that organic particulate polymers disinte-grate into inert materials, carbohydrates, proteins and fats,which are then hydrolyzed to sugars, amino acids and longchain fatty acids (LCFA). Sugars and amino acids are fer-mented to generate propionate, butyrate, valerate, acetateand hydrogen. Propionate, butyrate and valerate are fur-ther degraded to acetate and hydrogen. LCFA can bedirectly degraded to acetate and hydrogen. Methane is pro-duced by both the degradation of acetate and the reductionof carbon dioxide by hydrogen.

The model considers seven microbial trophic groups.Kinetic equations take into account the processes of

3668 S.J. Mu et al. / Bioresource Technology 99 (2008) 3665–3675

microbial growth and biomass decay. Biomass growthrates are proportional to degradation rates of organicmatter and are described by Monod-like dependencies.Also, the kinetic equations take into account the inhibitiveeffects of pH, hydrogen, ammonium, and LCFA.

2.3. Sludge bed modelling

Biomass granulation is a distinct feature of UASB reac-tors. Anaerobic biofilm granules, which have diameters of0.5–4 mm, are composed of a consortium of anaerobicmicroorganisms as well as of inert organic materials. Thesettling of the granules is affected by both the upwardliquid flow and the apparent settling velocities (Kalyuzhnyiet al., 2006) and results in the formation of a sludge bed.The liquid above the bed only contains fine suspended bio-mass particles. In this study, the sludge distribution in thereactor is modeled by a hyperbolic tangent function, whichdefines a maximal attainable biomass concentration, xat(z),at each reactor position

xatðzÞ ¼ðxmax � xminÞ

2� ð1� tanhðbðz� H sÞÞÞ þ xmin

� �ð11Þ

where xmax and xmin are the total attainable concentrationsof microorganisms in the sludge bed and the liquid abovethe sludge bed, respectively; Hs is the sludge bed height,and b is a positive constant. The resulting sludge distribu-tion is qualitatively shown in Fig. 1b. Furthermore, sludgebed parameters (xmax, xmin and Hs) are dependent on theupflow liquid velocity (u) and the organic loading rate(OLR), i.e. xmax = f1(u,OLR), Hs = f2(u,OLR). In the ab-sence of detailed experimental data and an appropriate the-ory, the following empirical linear dependences areproposed:

xmax ¼ a0 þ a1uþ a2OLR ð12ÞH s ¼ b0 þ b1uþ b2OLR ð13Þ

where a0–a2 and b0–b2 are the regression parameters. It isworth noting that volatile suspended solid (VSS) concen-trations in the sludge bed often decrease with increasingupflow velocity and that the sludge bed expands, i.e. itcan be expected that a1 < 0 and b1 > 0. Because increasingthe OLR results in a higher biogas production, the influ-ence of OLR is expected to be similar to that of upflowvelocity. Eqs. (12) and (13) can only be used following aparameter estimation procedure and its’ applicability islimited to a narrow range of operating parameters. Fur-thermore, regression constants should be periodically re-estimated.

The retention of biomass granules in the sludge bed isdescribed by multiplying the transport term (u) in Eq. (1)by a washout factor (Rw). The washout factor also uses ahyperbolic tangent function

RwðzÞ ¼ ½tanhðxtotalðzÞ � xatðzÞÞ þ 1�=2 ð14Þwhere xtotalðzÞ ¼

PixiðzÞ is the total biomass concentration

at reactor position z, xat(z) is defined by Eq. (11), and xi(z)is the concentration of the ith microorganism(i = 5, . . . , 11). Eq. (14) implies that if the total biomassconcentration exceeds the maximal attainable biomass con-centration, then Rw = 1, and the transport term in Eq. (1) isthe same as for all other liquid phase components. Other-wise, Rw = 0 and the transport term in Eq. (1) will be equalto zero, i.e. the biomass is retained in the reactor. This ap-proach uses continuous functions, which are capable ofdescribing biomass accumulation within the sludge bed,biomass washout in the liquid zone, and the existence ofa layer of fine biomass particles above the bed. Biomassconcentration profiles obtained using Eqs. (11)–(14) werein good agreement with the sludge profiles measured inUASB reactors (Kalyuzhnyi et al., 1996; Torkian et al.,2003).

By incorporating Eq. (14) into Eq. (1) the mass balanceof the microorganisms can be rewritten as follows:

oxi

ot¼ o

ozDx;iðz; tÞ

oxiðz; tÞoz

� �� o

ozðuxðz; tÞxiðz; tÞÞ þ rx;iðz; tÞ

ð15Þwhere ux(z, t) = ui(z, t)Rw(z) is the upflow biomass velocity.

The transport term of the above equation is extended asfollows:

d

dzðuxðz; tÞxiðz; tÞÞ ¼ uiðz; tÞxiðz; tÞ

dRwðzÞdz

þ RwðzÞxiðz; tÞduiðz; tÞ

dz

þ uiðz; tÞRwðzÞdxiðz; tÞ

dzð16Þ

This approach is essentially similar to other methods, suchas biomass distribution modeling using a fraction coeffi-cient a, which is defined as the ratio of bacteria in the liquidto that of the entire reactor (Bernard et al., 2001). The acan also be defined as the fraction of biomass transferredby the liquid flow (Vavilin et al., 2003).

2.4. Numerical methods

For the sake of comparison, the distributed parametermodel described above was solved by two different numer-ical methods, namely the orthogonal collocation method(OC) (Finlayson, 1980) and the finite-difference method(FD) (Rice and Do, 1995).

2.4.1. Orthogonal collocation method

For the OC method, the material balance (Eq. (1)), wasrewritten as follows:

dci;j

dt¼ 1

si

1

Boi

XNþ2

k¼1

Bj;kci;k �dDi

dg� ui

H 1

� �XNþ2

k¼1

Aj;kci;k þ f ðui;DRw;iÞ" #

þ ri;j þ rT;i;j

ð17Þ

Table 1Operating conditions in ADM1 benchmark simulations

Parameter Value Parameter Value

Vliq (m3) 1400 Ar, reactor area (m2) 93.33a

Vgas (m3) 100 T, operating temperature (K) 308.15qin (m3 d�1) 70 OLR (kg m�3

R d�1) 60Frecycle (m3 d�1) 280a Hs, sludge height (m) 10a

ba, dimensionless 0.25a

a Assumed for ADM1d simulations.

S.J. Mu et al. / Bioresource Technology 99 (2008) 3665–3675 3669

where si is the hydraulic retention time, s = uRw/H; Bo isthe Bodenstein number, Bo = D/uH; ci,j denotes the ithliquid component (i = 1, . . . , 33) at the jth collocation point(j = 1, . . . ,N), and N is the number of internal collocationpoints. A(N+2)·(N+2) and B(N+2)·(N+2) are the orthogonalcollocation matrices for first and second order derivates,respectively. The transport term f(ui,DRw,i) describedwashout of microbial species, hence for the microorgan-isms f ðui;DRw;iÞ ¼ uxi

dRw;i

dz and f(u,DRw,i) = 0 for all othercomponents.

The boundary conditions defined in Eqs. (2) and (3) arerewritten as follows:

1

Boi

XNþ2

k¼1

A1;kci;k þ ci;0 � ci;1 ¼ 0 ð18Þ

XNþ2

k¼1

ANþ2;kci;k ¼ 0 ð19Þ

The resulting set of differential-algebraic equations (DAE)consisted of 36 · N ordinary differential equations (ODEs)and 36 · 2 algebraic equations. The ODEs represented thematerials balances of 36 compounds at the N internal col-location points, and the algebraic equations described theboundary conditions. Since large differences in the magni-tudes of the bioreaction kinetic rates existed, the model sys-tem was highly stiff (Pind et al., 2002). Therefore, a stiffequation solver ‘ode23t’ of MATLAB? 7.01 (MathWorksInc., Natick, Massachusetts, USA) was used to solve Eqs.(17)–(19).

2.4.2. Finite difference method

The finite difference method, in contrast to the orthogo-nal collocation method, evaluates the derivative at a dis-crete point by using the information about discretevariables close to that point rather than from all discretevariables (Rice and Do, 1995). In this work, the centralfinite difference approximation with an error of order h2

was used for the internal finite difference points. The entryand exit boundary conditions were approximated by theforward and the backward approximations, respectively.

By specifying N internal finite difference points, i.e.N + 1 equally spaced intervals, this method resulted in36 · N ordinary differential equations (ODEs) and 36 · 2algebraic equations. Similarly to the collocation method,the model equations were solved using ‘ode23t’ equationsolver.

3. Results and discussion

3.1. Model parameters and initial values

This study used model parameters and operating condi-tions employed in ADM1 benchmark simulations (Rosenand Jeppsson, 2002) for comparison of the distributedparameter and ADM1 (CSTR) models. The values of stoi-chiometric, biochemical, and physiochemical parameters of

ADM1 used in the simulations are given in the Appendixand the operating conditions are given in Table 1. TheADM1d parameters used to calculate biomass profile inEqs. (11) and (14) were set at xmax = 15 kg COD m�3,xmin = 0.15 kg COD m�3, and b = 0.2. The values of xmax

and xmin were chosen to approximate sludge concentrationin benchmark simulations. For the sake of simplicity, azero bypass flow was assumed in ADM1d simulations.

In benchmark simulations composite materials com-prised 91% of total CODs, and acetate comprises 81% ofVFAs. Consequently, ADM1d and ADM1 were comparedby analyzing the concentrations of composites (xxc), ace-tate (sac), and methane. Meanwhile, there was no one pre-vailing microorganism in the consortium, hence the totalmicroorganism concentration was used.

At the startup of each simulation, steady-state outputsof the ADM1 benchmark simulations were used as initialvalues for ADM1. For ADM1d simulations the modelwas first integrated for a period of 200 days with anOLR of 3 kg COD m�3 d�1, which is the same as in bench-mark simulations, and the resulting steady state outputswere used as initial values for all subsequent simulations.In order to compare the performance of the two models,OLR was varied from 3 to 4 kg COD m�3 d�1. The ratioof each component concentration to the total influentCOD concentration remained equal to the original ratioin the benchmark simulation.

3.2. Comparison of numerical methods

The efficiencies of orthogonal collocation and finite dif-ference numerical methods for ADM1d solution were com-pared using the benchmark model parameters described inthe previous section and integrating the model for a periodof 20 days. Also, a volumetric (per reactor volume) OLR of3 kg COD m�3

R d�1 and a recirculation rate of 280 m3 d�1

were used. Model outputs obtained at the end of each inte-gration period were compared.

For the OC method, three runs were carried out by set-ting the number of internal points to 4, 8, and 16, respec-tively. Four runs were carried out using the finitedifference (FD) method with 10, 20, 30, and 50 internalpoints. The resulting spatial distributions of key state vari-ables at the end of the integration period are shown inFig. 2. The simulations were performed with MATLAB�

7.0.1 (Mathworks, USA) on a Pentium 4 PC with a

6.8

7

7.2

Position, mPosition, mpH

d

0

0.1

0.2

0.3

Ace

tate

, kg

CO

D m

-3

b

0

4

8

12

16

Com

posi

tes,

kg

CO

D m

-3 OC-04 OC-08

OC-16 FD-10

FD-20 FD-50

a

0

2

4

6

0 10 15

Bio

mas

s, k

g C

OD

m-3

c

5

0 10 155 0 10 155

0 10 155

Fig. 2. Spatial distribution of composite materials (a), acetate (b), total biomass (c), and pH (d) calculated using OC (solid gray line) and FD (solid blackline) algorithms. OC calculations were carried out with 4, 8, and 16 internal points. FD calculations were carried out with 10, 20, and 50 internal points.

3670 S.J. Mu et al. / Bioresource Technology 99 (2008) 3665–3675

2.8 GHz CPU. The required CPU times for the OC methodwere 27, 270, 367 and 1631 s for 4, 8, 12 and 16 internalpoints, respectively. The required CPU times for the FDalgorithm were 112, 19, 448, and 1596 s for 10, 20, 30,and 50 internal points, respectively.

Overall, the FD algorithm had better stability and itrequired somewhat shorter CPU time. Fig. 2 shows thatfor all four state variables the profiles obtained using theFD algorithm converged when the number of finite pointsreached 20. Increasing N from 20 to 30 and then to 50 didnot show significant differences between the profiles. Over-all, N = 20 can be considered as an acceptable compromisebetween numerical accuracy and computational time.

For the OC method, obvious differences between N = 4and N = 8 were observed for all state variables except xxc

(Fig. 2), indicating that N = 4 is insufficient. Althoughthe profiles obtained at N = 8 almost converged to thoseobtained by the FD method, unstable results were obtainedfor N = 16 (Fig. 2b–d).

The numerical instability of the orthogonal collocationmethod with a large number of collocation points can beexplained by the stiffness and nonlinearity of the distrib-uted parameter model. Rice and Do (1995) pointed outthat though the orthogonal collocation method was supe-rior to the finite difference method in some problems, thelatter algorithm was easily applicable to many problemswith complex geometries. In the collocation method, thecollocation points were non-uniformly distributed. Conse-quently, OC was less efficient in calculating fast changesin ion and gas balances and in modeling biomass distribu-tion, which exhibits a sharp concentration drop at theinterface of the sludge bed and liquid zones. Overall, the

finite difference method outperformed the orthogonal col-location method in solving the distributed parameter modelof the UASB reactor. In the subsequent simulations, a 20-point FD algorithm, which offered satisfactory numericalaccuracy within an acceptable computational time wasused.

3.3. Comparison of the one- and two-compartment models

An alternative to modeling sludge distribution using ahyperbolic tangent function (Eq. (14)) within a single com-partment model was to use a two-compartment model,which distinguishes between the sludge bed and the liquidzones of the reactor. To compare the two approaches, atwo compartment model was developed in which the sludgebed and liquid zones were described by distributed param-eter and ideal mixing submodels, respectively. For the sakeof simplicity, the sludge bed compartment was stilldescribed by Eqs. (1)–(7) and (11)–(16), but the value ofxmin was increased to 3 kg COD m�3 as compared to0.15 kg COD m�3 for a single compartment model. TheCSTR material balances of the bulk liquid above the sludgecompartment were those of ADM1 (Eqs. (8)–(10)).

Both single and two-compartment models were solvedusing the FD algorithm with N = 20. A period of 100 daysof reactor operation was simulated. In the simulations, theOLR was maintained at 3 kg COD m�3 d�1 for the first 20days, then the OLR was changed to 4 kg COD m�3 d�1

and maintained at this level to the end of the integrationperiod. Fig. 3a–c shows the resulting spatial distributionof composite materials, acetate, and pH at 20 d. It can beseen that the resulting profiles are similar. As expected,

7

7.1

7.2

Time, day

pH

f

0

5

10

15

0 5 10 15

a

0

0.1

0.2

0.3

0 5 10 15

b

7

7.1

7.2

7.3

0 5 10 15Position, m

pH

c

0

2

4

6

0 25 50 75 100

0 25 50 75 100

0 25 50 75 100

Com

posi

tes,

kg

CO

D m

-3

d

0

0.1

0.2

0.3

0.4

Ace

tate

, kg

CO

D m

-3

Com

posi

tes,

kg

CO

D m

-3A

ceta

te, k

g C

OD

m-3

e

Fig. 3. Comparison of spatial (a–c) and temporal (d–f) profiles of key state variables calculated using one-compartment (solid black line) and two-compartment (solid gray line) distributed parameter models. Spatial profiles correspond to t = 20 day. Temporal profiles show effluent concentration ofcomposite materials (d), acetate (e), and pH in effluent (f).

S.J. Mu et al. / Bioresource Technology 99 (2008) 3665–3675 3671

the outputs of the two-compartment model exhibited dis-continuity at the interface of the sludge bed and the over-lying liquid, while the single compartment modelprovided continuous outputs. A comparison of time pro-files of effluent concentrations also showed qualitativelysimilar results (Fig. 3d–f), although the two-compartmentmodel predicted lower concentrations of composite materi-als and acetate (Fig. 3d and e). Overall, owing to a smallernumber of model equations and continuous concentrationprofiles, the single compartment model was a better choicefor reactor modeling.

3.4. ADM1 versus ADM1d

An important feature of ADM1d was its ability to sim-ulate spatial distribution of substrates and microorganismsin the reactor, as shown in Fig. 2 where all state variablesexhibited distinct gradients. Also, the model was capableof at least qualitative modeling of the sludge bed, as shownin Fig. 2c. A more detailed analysis of spatial profiles ofstate variables in Fig. 2 showed that the concentration ofcomposite materials monotonously decreased with increas-ing reactor height (Fig. 2a), while there was a peak of ace-tate at around 2 m from the reactor inlet (Fig. 2b).Notably, the acetate peak coincided with a drop in pH

(Fig. 2d). These profiles were consistent with the existenceof acidogenic and methanogenic zones in an UASB reac-tor. Because of limited mixing and a relatively high resi-dence time, the disintegration and hydrolysis ofparticulate substrates as well as the production of degrada-tion intermediates, such as volatile fatty acids, mainly pro-ceeded in the bottom part of the sludge bed. Theproduction of volatile fatty acids was followed by the meth-anization process, which was more intensive in the upperpart of the sludge bed. The existence of an acidogenic zonein an UASB reactor has been observed experimentally(Sam-Soon et al., 1991). Notably, the stratification of met-abolic activities decreased with increasing mixing, in partic-ular due to increasing external recirculation.

In order to compare the responses of ADM1 andADM1d to variations in organic loading rate (OLR) andthe influent-to-recirculation ratio, a simulation of UASBreactor operation at four different combinations of OLRsand external recirculation rates was carried out. In the sim-ulation, each set of operating conditions was maintainedfor a period of 100 days to obtain steady state outputs,i.e. a total simulation time of 400 days was used. At thestartup of simulation, steady state values of the benchmarkADM1 simulation and the ADM1d simulation shown inFig. 3 were used as initial values. The OLR was set at

3672 S.J. Mu et al. / Bioresource Technology 99 (2008) 3665–3675

3 kg COD m�3 d�1 for both models, which is the same asin the benchmark simulations. The recirculation rate wasset at 70 m3 d�1, which corresponded to an influent-to-recirculation ratio of 1 (phase #1, 0–100 days). Then therecirculation rate was increased to 280 m3 d�1, while theOLR was unchanged (phase #2, 100–200 days). In phase#3 (200–300 days), the recirculation rate remained thesame as in phase #2, but the OLR was increased to4 kg COD m�3 d�1. Finally, in phase #4 (300–400 days),the recirculation rate was increased from 280 m3 d�1 to560 m3 d�1 resulting in a recirculation to influent ratio of8, while the OLR remained at 4 kg COD m�3 d�1. Becauseof a high recirculation rate, this phase corresponded tonear ideal mixing and similar outputs of ADM1 andADM1d were anticipated.

A comparison of effluent concentrations predicted byADM1 and ADM1d is shown in Fig. 4. In phase #1 theupflow velocity was low. As the result, ADM1d outputs

0

2

4

6

8

0 200 400

Com

posi

tes,

kg

m-3 a

800

1000

1200

1400

0 100 200 300 400Time, day

CH

4 flow

, m3 d-1

c

Fig. 4. Dynamic simulations of an UASB reactor with external recirculation upanels show effluent concentration of composites (a) and acetate (b), as well a

3000

4000

5000

6000

7000

0 100 200 300 400Time, day

Tota

l bio

mas

s, k

g C

OD a

Fig. 5. Time profiles of total biomass amount computed by ADM1 (solid graymain trophic groups computed by ADM1d for t = 20 day (b).

for effluent composite materials and acetate were lowerthan those of ADM1 (Fig. 4a and b). Consequently,ADM1d predicted slightly higher effluent pH and methaneflow rate (Fig. 4c and d). When the recirculation rate wasincreased from 70 up to 280 m3 d�1 (phase #2), ADM1outputs did not change because the ideal mixing modelwas unable to reflect changes in hydraulic mixing. On theother hand, ADM1d outputs changed in response toincreasing recirculation rate. In particular, increased mix-ing resulted in higher effluent concentrations of compositematerials and acetate as shown in Fig. 4a and b. This trendwas in agreement with results obtained in an UASB reactor(Sam-Soon et al., 1991), an anaerobic filter (Yu et al., 2000)and an anaerobic packed-bed bioreactor (Mshandete et al.,2004).

The responses of both models to increased organic loadin phase #3 were similar. As expected, the effluent concen-tration of composite materials increased, while a decrease

6.9

7.1

7.3

0 100 200 300 400Time, day

pH

d

0

0.1

0.2

0.3

0 100 200 300 400

Ace

tate

, kg

CO

D m

-3 b

sing ADM1 (solid gray lines) and ADM1d (solid black line) models. Thes methane production (c) and effluent pH (d).

0

1

2

3

4

5

0 5 10 15Position, m

Trop

hic

grou

ps, k

g C

OD

m-3

acidogenssyntrophsmethanogens

b

line) and ADM1d (solid black line) models (a) and spatial distribution of

S.J. Mu et al. / Bioresource Technology 99 (2008) 3665–3675 3673

of pH was predicted (Fig. 4a and d). At the same time, theacetate concentration decreased after the transition period,which can be attributed to biomass accumulation in thereactor (Fig. 5a). An increase of the recirculation rate inphase #4 resulted in a near ideal mixing and causedADM1d outputs to almost overlap with those of ADM1,in particular for pH and methane outputs (Fig. 4c andd). As in phase #2, the recirculation rate increase resultedin higher predicted effluent concentrations of compositematerials and acetate for ADM1d, while ADM1 outputsremained the same.

The simulated concentrations of composites and acetatein the reactor effluent (Fig. 4a and b) of ADM1d were lessthan those predicted by ADM1. This can be explained bythe fact that in ADM1 biodegradation rates were the sameat all reactor positions, while in ADM1d the rates wereposition dependent. The Monod-like kinetic equations,which were used to describe biotransformations inADM1d, resulted in higher degradation rates predicted atthe bottom of the sludge bed, where substrate concentra-tions were the highest. Consequently, an overall volumetricrate of biodegradation was higher in ADM1d than inADM1 thus resulting in lower effluent concentrations ofcomposite materials and acetate.

ADM1d provided a qualitatively correct description ofthe external recirculation rate influence on total biomassconcentration in the reactor. Each increase in the recircula-tion rate was followed by a decrease in total biomass quan-tity, while the consequence of increasing organic load wasan accumulation of biomass (Fig. 5a). An examination ofthe distribution of microbial species in the reactor at day20 (phase 1) showed similar trends for all trophic groups.The ratio of fermentative microorganisms was slightlyhigher at the bottom of the sludge bed, while the densityof methanogenic microorganisms was predicted to increaseclose to the top of the sludge bed (Fig. 5b). These trends,however, were not well pronounced.

ADM1d simulations suggested that an increase of therate of external recirculation above a certain valuedecreased the removal efficiency. At the same time, thebiogas flow rate was predicted to increase slightly (resultsnot shown) with increasing external recirculation becauseof increased gas transfer due to higher liquid phase con-centration of carbon dioxide and methane. This trendagreed with the experimental results of Romli et al.(1994) in a two phase upflow anaerobic reactor and inan upflow packed-bed anaerobic reactor (Mshandeteet al., 2004). Romli et al. (1994) explained that theincrease in biogas production was due to increased strip-ping of dissolved CO2 from the liquid phase because ofimproved mixing. Overall, the ideal mixing (ADM1)model was unable to respond to changes in the type ofmixing such as changes of recirculation rate or upflowvelocity. In contrast, ADM1d was able to reflect the effectof both changes in OLR and recirculation rate and

showed axial concentration gradients. In addition,ADM1d simulations suggested that an optimum recircula-tion rate existed and this parameter can be used in processoptimization and control.

4. Conclusions

High rate UASB-type reactors exhibit significant gradi-ents of composite materials, volatile fatty acids, pH andvolatile suspended solids over the height of the reactor,which can be only described by a distributed parametermodel. In this study, Anaerobic Digestion Model No. 1was used as a basis for the development of a comprehen-sive distributed parameter model, named ADM1d.ADM1d used a hyperbolic tangent function to describebiomass distribution within a one compartment model.A comparison of this approach with a two-compartmentmodel, which consisted of a sludge bed and a liquid abovethe bed compartments, showed similar simulation resultswhile the one-compartment model had less equations. Itshould be noted, that the hyperbolic tangent model ofthe sludge bed does not take into account physical pro-cesses of granule settling and washout, but provides anonlinear regression model of the experimentally mea-sured sludge distribution. This regression model shouldbe substituted with more complex models (e.g. Kalyuzhnyiet al., 2006) if some insight on granular sludge dynamics isdesired.

A comparison of orthogonal collocation and finite-dif-ference algorithms for numerical solution of the distributedparameter model showed that both algorithms can be used,however FD algorithm showed better stability. A compar-ison of ADM1 and ADM1d outputs showed that ADM1dwas better suited for modeling anaerobic reactors with lim-ited mixing and high organic load. ADM1d was able tosimulate the impacts of both organic loading and externalrecirculation on process performance while ADM1, whichuses an assumption of ideal mixing, only simulated theimpact of organic loading. The capacity of ADM1d forpredicting spatial distribution of state variables can be veryuseful in developing a systematic approach for the designof UASB reactors.

Acknowledgements

This work was funded by NRC-A*STAR CollaborativeResearch Program for BRI-NRC, Canada, and by Singa-pore-NRC Joint Research Programme (A*STAR Grant)for IHPC, Singapore. The authors are thankful to D. Bat-stone, U. Jeppsson, and C. Rosen for providing the Matlabcode of ADM1 and to M.F. Manuel and D. Lyew for sug-gestions on improving the manuscript. This is NRC paper#49082.

Parameter Value Unit

3674 S.J. Mu et al. / Bioresource Technology 99 (2008) 3665–3675

Appendix. ADM1 parameters

Parameter Value Unit

A. Stoichiometric parametersfsI,xc 0.1 – fpro,su 0.27 –fxI,xc 0.2 – fac,su 0.41 –fch,xc 0.2 – Nbac 0.08/14 kmol C (kg COD)�1

fpr,xc 0.2 – Cbu 0.025 kmol C (kg COD)�1

fli,xc 0.3 – Cpro 0.0268 kmol C (kg COD)�1

Nxc 0.0376/14 kmol N (kg COD)�1 Cac 0.0313 kmol C (kg COD)�1

NI 0.06/14 kmol N (kg COD)�1 Cbac 0.0313 kmol C (kg COD)�1

Naa 0.007 kmol N (kg COD)�1 Ysu 0.1 –Cxc 0.03 kmol C (kg COD)�1 fh2,aa 0.06 –CsI 0.03 kmol C (kg COD)�1 fva,aa 0.23 –Cch 0.0313 kmol C (kg COD)�1 fbu,aa 0.26 –Cpr 0.03 kmol C (kg COD)�1 fpro,aa 0.05 –Cli 0.022 kmol C (kg COD)�1 fac,aa 0.4 –CxI 0.03 kmol C (kg COD)�1 Cva 0.024 kmol C (kg COD)�1

Csu 0.0313 kmol C (kg COD)�1 Yaa 0.08 –Caa 0.03 kmol C (kg COD)�1 Yfa 0.06 –ffa,li 0.95 – Yc4 0.06 –Cfa 0.0217 kmol C (kg COD)�1 Ypro 0.04 –fh2,su 0.19 – Cch4 0.0156 kmol C (kg COD)�1

fbu,su 0.13 – Yac 0.05 –Yh2 0.06 –

B. Biochemical parameterskdis 0.5 d�1 KS,pro 0.1 kg COD m�3

khyd,ch 10 d�1 KI,h2,pro 3.5E�6 kg COD m�3

khyd,pr 10 d�1 km,ac 8 d�1

khyd,li 10 d�1 KS,ac 0.15 kg COD m�3

KS,IN 1E�4 M KI,nh3 0.0018 Ma

km,su 30 d�1 pHUL,ac 7KS,su 0.5 kg COD m�3 pHLL,ac 6pHUL,aa 5.5 km,h2 35 d�1

pHLL,aa 4 KS,h2 7E�6 kg COD m�3

km,aa 50 d�1 pHUL,h2 6KS,aa 0.3 kg COD m�3 pHLL,h2 5km,fa 6 d�1 kdec,Xsu 0.02 d�1

KS,fa 0.4 kg COD m�3 kdec,Xaa 0.02 d�1

KI,h2,fa 5E�6 kg COD m�3 kdec,Xfa 0.02 d�1

km,c4 20 d�1 kdec,Xc4 0.02 d�1

KS,c4 0.2 kg COD m�3 kdec,Xpro 0.02 d�1

KI,h2,c4 1E�5 kg COD m�3 kdec,Xac 0.02 d�1

km,pro 13 d�1

C. Physiochemical parametersR 0.08314 bar M�1 K�1 kA,Bpro 1E+8 M�1 d�1

Kw 2.08E�14 M kA,Bac 1E+8 M�1 d�1

Ka,va 1.38E�5 M kA,Bco2 1E+8 M�1 d�1

Ka,bu 1.50E�5 M kA,BIN 1E+8 M�1 d�1

Ka,pro 1.32E�5 M pgas,h2o 0.0557 barKa,ac 1.74E�5 M Patm 1.013 barKa,co2 4.94E�7 M kLa 200 M bar�1

Ka,IN 1.11E�9 M KH,co2 0.0271 M bar�1

kA,Bva 1E+8 M�1 d�1 KH,ch4 0.00116 M bar�1

kA,Bbu 1E+8 M�1 d�1 KH,h2 7.38 M bar�1

a M = kmol m�3.

S.J. Mu et al. / Bioresource Technology 99 (2008) 3665–3675 3675

References

Batstone, D.J., Keller, J., Newell, R.B., Newland, M., 2000a. Modellinganaerobic degradation of complex wastewater. I: model development.Bioresour. Technol. 75, 67–74.

Batstone, D.J., Keller, J., Newell, R.B., Newland, M., 2000b. Modellinganaerobic degradation of complex wastewater. II: parameter estima-tion and validation using slaughterhouse effluent. Bioresour. Technol.75, 75–85.

Batstone, D.J., Keller, J., Angelidaki, I., Kalyuzhnyi, S.V., Pavlostathis,S.G., Rozzi, A., Sanders, W.T.M., Siegrist, H., Vavilin, V., 2002.Anaerobic Digestion Model No. 1 (ADM1). IWA Publishing, London,UK.

Batstone, D.J., Gernaey, K.V., Steyer, J.-P., Schmidt, J.E., 2005. Aparticle model for UASB reactors using the Takacs Clarifier Model.In: Proceedings of the First International Workshop on the IWAAnaerobic Digestion Model No. 1, Copenhagen, Denmark, pp. 129–136.

Bernard, O., Hadj-Sadok, Z., Dochain, D., Genovesi, A., Steyer, J.-P.,2001. Dynamical model development and parameter identification foran anaerobic wastewater treatment process. Biotechnol. Bioeng. 75,424–438.

Blumensaat, F., Keller, J., 2005. Modelling of two-stage anaerobicdigestion using the IWA anaerobic digestion model no. 1 (ADM1).Water Res. 39, 171–183.

Danckwerts, P.V., 1953. Continuous flow systems. Chem. Eng. Sci. 2, 1–13.

Finlayson, B.A., 1980. Nonlinear Analysis in Chemical Engineering.McGraw-Hill, New York.

Garuti, G., Leo, G., Pirozzim, F., 2004. Experiments and modeling onbiomass transport inside upflow sludge blanket reactors intermittentlyfed. Water S.A. 30, 97–106.

Jeong, H.-S., Suh, C.-W., Lim, J.-L., Shin, H.-S., 2005. Analysis andapplication of ADM1 for anaerobic methane production. BioprocessBiosyst. Eng. 27, 81–89.

Kalyuzhnyi, S.V., Sklyar, V.I., Davlyatshina, M.A., Parshina, S.N.,Simankova, M.V., Kostrikina, N.A., Nozhevnikova, A.N., 1996.Organic removal and microbiological features of UASB-reactor undervarious organic loading rates. Bioresour. Technol. 55, 47–54.

Kalyuzhnyi, S.V., Fedorovich, V.V., Lens, P., 2006. Dispersed plug flowmodel for upflow anaerobic sludge bed reactors with focus on granularsludge dynamics. J. Ind. Microbiol. Biotechnol. 22, 221–237.

Mshandete, A., Murto, M., Kivaisi, A.K., Rubindamayugi, M.S.T.,Mattiasson, B., 2004. Influence of recirculation flow rate on theperformance of anaerobic packed-bed bioreactors treating potato-waste leachate. Environ. Technol. 25, 929–936.

Pind, P.F., Angelidaki, I., Ahring, B.K., Stamatelatou, K., Lyberatos, G.,2002. Monitoring and control of anaerobic reactors. Adv. Biochem.Eng./Biotechnol. 82, 135–182.

Rice, R.G., Do, D., 1995. Applied Mathematics and Modeling forChemical Engineers. John Wiley & Sons.

Romli, M., Greenfield, P.F., Lee, P.L., 1994. Effect of recycle on a two-phase high-rate anaerobic wastewater treatment system. Water Res.28, 475–482.

Rosen, C., Jeppsson, U., 2002. Anaerobic COST benchmark modeldescription. Department of Industrial Electrical Engineering andAutomation, Lund University, Lund, Sweden.

Sam-Soon, P., Loewenthal, R.E., Wentzel, M.C., Moosbrugger, R.E.,1991. Effects of a recycle in upflow anaerobic sludge bed (UASB)systems. Water S.A. 17, 37–45.

Schoefs, O., Dochain, D., Fibrianto, H., Steyer, J.P., 2004. Modeling andidentification of a partial differential equation model for an anaerobicwastewater treatment process. In: Proceedings of the 10th WorldCongress on Anaerobic Digestion, Montreal, Canada, vol. 1, pp. 343–347.

Singhal, A., Gomes, J., Praveen, V.V., Ramachandran, K.B., 1998. Axialdispersion model for upflow anaerobic sludge blanket reactors.Biotechnol. Prog. 14, 645–648.

Takacs, I., Patry, G.G., Nolasco, D., 1991. A dynamic model of theclarification-thickening process. Water Res. 25, 1263–1271.

Tartakovsky, B., Mu, S.J., Zeng, Y., Lou, S.J., Guiot, S.R., Wu, P., 2007.Anaerobic digestion model no 1-based distributed parameter model ofan anaerobic reactor: II. model validation. Bioresour. Technol. 99,3676–3684.

Torkian, A., Eqbali, A., Hashemian, S.J., 2003. The effect of organicloading rate on the performance of UASB reactor treating slaughter-house effluent. Resour. Conserv. Rec. 40, 1–11.

Vavilin, V.V., Rytov, S.V., Lokshina, S.Y., Pavlostathis, S.G., Barlaz,M.A., 2003. Distributed model of solid waste anaerobic digestion:effects of leachate recirculation and pH adjustment. Biotechnol.Bioeng. 81, 66–73.

Yu, H., Wilson, F., Tay, J.H., 2000. Prediction of the effect ofrecirculation on the effluent quality of anaerobic filters by empiricalmodels. Water Environ. Res. 72, 217–224.