A three-dimensional numerical study on the correlation of...

*Corresponding author. Tel.: #031-15-278-1618; fax: #031-15-278-2355.

E-mail addresses: [email protected],[email protected] (M. C. M. van Loosdrecht).

Chemical Engineering Science 55 (2000) 6209}6222

A three-dimensional numerical study on the correlation of spatialstructure, hydrodynamic conditions, and mass transfer and conversion

in bio"lms

H. J. Eberl!,", C. Picioreanu", J. J. Heijnen", M. C. M. van Loosdrecht",*!GSF-National Research Center For Environment and Health, Institute for Biomathematics and Biometry, Pf 1129, 85758 Neuherberg, Germany

"Department of Biochemical Engineering, Kluyver Institute for Biotechnology, Delft University of Technology, Julianalaan 67,2628BC Delft, The Netherlands

Received 19 January 2000; received in revised form 10 May 2000; accepted 5 June 2000

Abstract

A three-dimensional model for convection, di!usion, and reaction in a porous, heterogeneous system has been implemented. It isused to analyse the in#uence of hydrodynamics and structural heterogeneities on mass transfer and conversion processes of solutes inbio"lm systems. The mathematical model comprises the full incompressible Navier}Stokes equations and mass transfer withnonlinear reactions in the bio"lm. It is found that increased bio"lm surface roughness means decreased mass conversion in the solidbio"lm. Secondly, a correlation between bulk #ow Reynolds number and the Sherwood number for mass transfer across the irregularliquid/solid interface is formulated. In a further study the contribution of convective transport to overall mass transfer from bulkliquid into the bio"lm was analysed. The main result was that the experimental observation of high convective #ux of solutes inbio"lm channels not necessarily is coupled with an equally high net convective contribution to mass transfer from bulk liquid intobio"lm. ( 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Bio"lm; Mass transfer; Hydrodynamics; Irregular surface; Mathematical modelling; Microstructure

1. Introduction

Bio"lms grow on surfaces in aquatic systems wherevernutrients are available to feed bacteria. They form almosteverywhere in our environment and on technical andtechnological systems. Biofouling, biocorrosion, and bac-terial infections are harmful impacts of bio"lms on theirhost system. On the other hand, engineers learned to usebio"lms in biotechnological applications like wastewatertreatment. Over the past few years, awareness stronglyincreased that bio"lms grow rather in highly irregularand heterogeneous spatial structures than in #at layers(e.g. Gjaltema, Arts, van Loosdrecht, Kuenen & Heijnen,1994, and the review paper of Costerton, Lewandowski,Caldwell, Korber & Lappin-Scott, 1995). They containvoids and channels and allow #ow through the bio"lm

system in addition to bulk liquid #ow over the structure.This also has been re#ected in mathematical modelling ofbio"lm processes and some papers have been publishedrecently taking multidimensional bio"lm descriptionsinto account (Wimpenny & Colasanti, 1997; Picioreanu,van Loosdrecht & Heijnen, 1998a, 1999; Hermanowicz,1999; Noguera, Pizarro, Stahl & Rittmann, 1999; Rit-tmann, Pettis, Reeves & Stahl, 1999). Most of thesepapers have been motivated from describing bio"lmgrowth mechanisms and many of them do not includehydrodynamics and often perform only inaccurate calcu-lation of mass transfer (see Morgenroth, Eberl & vanLoosdrecht, 2000, for a lineup). However, according toa hypothesis on bio"lm development presented in vanLoosdrecht, Picioreanu and Heijnen (1997), these factorsare of major importance for the formation of bio"lmstructure. Therefore, we will address two questions in thispaper:

A How do bio"lm structure and environmental condi-tions, like hydrodynamics and nutrient availability,a!ect mass transfer and conversion processes?

0009-2509/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 1 6 9 - X

Fig. 1. System description. Computational domain X is divided intotwo parts: liquid-phase X

1(consisting of bulk liquid and bio"lm chan-

nels) and solid-phase X2

(containing all the biomass). They are separ-ated by interface C.

B Does convective transport of nutrients in bio"lmchannels necessarily increase mass transfer from bulkliquid into the bio"lm?

Question A has been extensively investigated byPicioreanu, van Loosdrecht and Heijnen (2000) recently.Their study was based on two-dimensional modelling ofbio"lms with irregular surface shape. The authorspointed out that care must be taken while interpretingthe results: obstacles which might lead to backwatereddies in two-dimensional simulation can be bypassed inthree-dimensional simulation. Channel-and-cluster bi-o"lms cannot be simulated in two dimensions at all.Pores in two-dimensional bio"lm geometries have muchless wall friction than pores in three-dimensional geomet-ries and, thus, allow higher #ow rates. The present papertries to "ll this gap by comparing the results obtained inthe two-dimensional study with those from three-dimen-sional modelling. A comparative discussion is included.

From experimental work it was concluded thata quantitative numerical study of question B requiresthree-dimensional analysis (De Beer, Stoodley & Lewan-dowski, 1996). Rittmann et al. (1999) found "rst resultsfor an isolated colony with a strongly simpli"ed physicalmodel. Most of the restrictions of that study are given upin the present work: (a) genuinely three-dimensional clus-ter-and-channel structures are considered instead ofa single axisymmetric cylindrical colony, and (b) hydro-dynamics is taken into account explicitly instead ofa priori assumptions of boundary layer thickness. Thisallows a quanti"cation of the convective share to theoverall mass transfer process.

2. Numerical simulation of mass transfer and conversionin bio5lms

2.1. Mathematical model

The mathematical bio"lm system X consists of twocompartments, both treated as continua: the solid-phaseX

2containing all the biomass and the liquid-phase X

1,

separated by the interface C (see Fig. 1). According to thisde"nition, pores and channels in the bio"lm "lled withliquid are parts of X

1. In the sequel, the term bioxlm (or

bioxlm layer) refers to biomass clusters and liquid chan-nels, de"ned by a virtual plane parallel to the substratumat the maximum height of X

2(see Fig. 1). That is, bio"lm

channels and void are a part of the bio"lm layer. Theterm bulk liquid refers to the part of X

1above this plane.

Herewith, the last two terms are used in accordance withLewandowski, Stoodley and Altobelli (1995), and Stood-ley, Yang, Lappin-Scott and Lewandowski (1997). Theterm bioxlm system will refer to the whole system X.Bioxlm structure and bioxlm geometry refer to the actualshape of X

2.

In the liquid-phase X1, nutrients are transported due

to convection and di!usion. In the solid-phase X2, they

are transported by di!usion and consumed in biochemi-cal reactions. Inspecting characteristic time scales of bi-o"lm processes, Kissel, Carty and Street (1984) foundthat mass transfer and consumption in the bio"lm reachsteady state quickly compared to bio"lm growth. Fol-lowing this argument a steady-state analysis is con-sidered to be su$cient (Rittmann et al., 1999; Picioreanuet al., 2000). Thus, the convection}di!usion/di!usion}re-action model for nutrient concentration c reads

u ) +c"D1Dc in X

1, (1)

0"D2Dc!r(c) with r(c)"

k1c

k2#c

in X2. (2)

Across interface C, the nutrient concentration is con-tinuous and Fick's law must hold, i.e.

!D1

L c

Ln1

"!D2

L c

L n2

at C (3)

Here, n1@2

denote the normal vectors of C pointingfrom X

1@2into the outer direction.

D1

and D2

are the limiting nutrient di!usion coe$-cients in liquid- and solid-phase X

1@2, respectively. For

r(c) in Eq. (2), Monod kinetics with maximum reactionrate k

1and saturation constant k

2is assumed. The hy-

drodynamic #ow "eld u"(u1, u

2, u

3) entering Eq. (1) as

support of convective transport in the liquid-phase X1

isgiven by the incompressible Navier}Stokes equationsand the continuity equation for the conservation of liquidmass

(u )+ )u"!

1

o+P#vDu in X

1, (4)

+ ) u"0 in X1. (5)

As usual, constants o and l are #uid density and kin-ematic viscosity. The dependent variable P is the pressure"eld in X

1.

6210 H. J. Eberl et al. / Chemical Engineering Science 55 (2000) 6209}6222

Fig. 2. The seven-point stencil for the classical CDS method (left) andthe 19-point stencil of the HOC scheme (right).

2.2. Numerical methods

Many numerical algorithms have been proposed forincompressible steady-state Navier}Stokes equations (4)with (5) and mass transfer equations (1) and (2). Sincebio"lms grow in highly irregular morphologies, it is es-sential to use methods capable of dealing with virtuallyany geometrical structure. For simplicity, a regular gridis used and the interface C is discretized by mesh surfaces.This allows a very general description of computationalgeometries. Among "nite di!erence methods, schemeswith compact stencils are appropriate. They access onlydirect neighbours for the approximation of partial deriv-atives in one particular grid point. That is, to evaluate the"nite di!erences in a grid point of X

1only other grid

points in X1

or grid points on the interface C are used,but no grid points in X

2and vice versa.

For #ow "eld calculations the classical arti"cial com-pressibility method of Chorin (Chorin, 1967; Peyret& Taylor, 1990) has been applied using a staggered grid.This is su$cient to close the steady-state momentumequation (4) in a physical correct manner and circum-vents the well-known problems associated with the time-accurate pressure Poisson equation under Neumannboundary conditions (as mentioned in Hackbusch, 1986;Griebel, Dornseifer & Neunhoe!er, 1995).

The di!usion}reaction equation (2) is discretized in thesolid region X

2by the standard central diwerence scheme

(CDS) on the compact seven-point stencil. This method issecond-order accurate but it is known to become unsta-ble if applied to convection}di!usion equation (1). Stan-dard "nite di!erences for convection}di!usion problemson the compact seven-point stencil are of lower order andrequire a much "ner grid to achieve the same computa-tional accuracy. Thus, a high-order compact scheme(HOC) on a compact 19-point stencil (see Fig. 2) wasdeveloped for nutrient transport in U

1. It is fourth-order

accurate. The semilinear algebraic system resulting fromthe combined HOC/CDS discretization of Eqs. (1) and(2) in U

1and U

2is solved with an inexact Newton

method. A Jacobi preconditioned BiCGSTAB method isused to solve the linear subsystem in every Newtoniteration step. A detailed description of the computa-tional scheme is given in Eberl and van Loosdrecht(2000) (submitted).

Both algorithms, for #uid #ow and mass transfer, havebeen implemented in the data parallel high-performanceFortran (HPF) programming language for distributedmemory architectures and run on the CRAY T3E of theCentre for high-performance applied computing (HPaC)at Delft University of Technology.

3. System description

3.1. Physical system and boundary conditions

Due to the irregularity of bio"lm structures, thesmallest length scale of a heterogeneous bio"lm geometrycan be orders of magnitude smaller than the entire bi-o"lm system itself. The spatial resolution must be chosen"ne enough in the discretization of Eqs. (1) and (2). Thus,only small parts of an entire system can be taken intoconsideration for simulation. This means that assump-tions about the physical system outside of computationaldomain X must be made. These assumptions are for-mulated in terms of boundary conditions for di!erentialequations (1), (2), and (4) on the boundaries of X. In ourapplication, the computational domain X is chosen to berectangular. System size is ¸

1]¸

2]¸

3. The bio"lm is

attached to the bottom of X.Since for mass transfer processes only the concentra-

tion boundary layer is of importance, a fully developedhydrodynamic boundary layer is not necessary. Thethickness of the concentration boundary layer is muchsmaller than the one of the hydrodynamic boundarylayer in most bio"lm reactors. In an undisturbed #owregime (i.e., in a #at geometry) the #ow "eld will in goodapproximation develop a linear velocity pro"le close tothe substratum and the whole concentration boundarylayer can be contained in this linear region. It is assumedthat a rough bio"lm geometry induces a perturbation ofthe #ow "eld but this does not in#uence considerably theratio between the thickness of hydrodynamic and con-centration boundary layers. In Picioreanu et al. (2000)the reliability of this assumption, often referred to as theLeveque approximation, is discussed in more detail.

In particular, a "xed velocity parallel to the bottom isprescribed at the top boundary, i.e. at x

3"¸

3. At the

out#ow boundary x1"¸

1, the velocity is forced to be

the same as at the in#ow boundary x1"0. This is

equivalent to a continuous repetition of the bio"lm ge-ometry in main #ow direction x

1. At the boundaries

parallel to main #ow direction (i.e. x2"0, x

2"¸

2),

a symmetric (mirrored) continuation of the bio"lm ge-ometry is assumed. As a consequence, there is no mo-mentum #ux across these boundaries. With theseassumptions, the computational domain for #uid #ow X

1is a small section of an in"nite bio"lm system witha continuously self-repeating pattern. At the interfaceC between liquid-phase X

1and solid-phase X

2a no-slip

H. J. Eberl et al. / Chemical Engineering Science 55 (2000) 6209}6222 6211

condition holds. The whole substratum is covered withbiomass. Therewith, no #ow conditions may be speci"edat the substratum.

For mass transfer, the repetition of the bio"lm ge-ometry is more di$cult. It is possible to apply the no-#uxmirror principle at x

2"0 and ¸

2; however, a cyclic

boundary condition at in#ow/out#ow boundaries is notfeasible. Nutrients are consumed in the bio"lm and theconcentration is expected to decrease in #ow directiondue to the reaction terms in the model. Therefore, weprescribe a constant nutrient concentration explicitly atx1"0. At the top boundary x

3"¸

3and at the down-

stream boundary x1"¸

1, no-#ux out#ow conditions

are speci"ed. The substratum (x3"0) is impermeable

and, hence, a no-#ux condition must hold there.With this system de"nition, the boundary conditions

for #ow "eld and nutrient concentration read in particu-lar

uDx1/0

"uDx1/L1

, (6a)

uDx3/L3

"(u501

,0,0)T, (6b)

uDC"0, (6c)

u2Dx2/0,L2

"0,Lu

1,3Lx

2Kx2/0,L2

"0, (6d)

cDX1WMx1/0N"c

0, (7a)

Lc

Ln K/XC(X1

WMx1/0N)

"0. (7b)

With this set of boundary conditions no a priori assump-tion on the thickness of the concentration boundary layeris made. The internal boundary condition connectingX

1and X

2across C is given by Eq. (3). Due to the

arti"cial compressibility approach for the Navier}Stokesequations and usage of a staggered grid, no boundaryconditions must be speci"ed for the pressure "eld Pin X

1.

3.2. Bioxlm structures in this study

Picioreanu et al. (2000) used randomly generated wavybio"lm structures in their extensive two-dimensionalstudy of mass transfer in bio"lms. In order to allowa quantitative comparison of results, three-dimensionalgeometries have been created with similar properties. Inparticular, it is important that the biomass volume is thesame in all cases. That is, the volume of the bio"lmgeometry related to its width ¸

3must equal the area

covered by biomass in the two-dimensional study, whichwas kept constant for all geometries. The number ofbio"lm peaks and a maximum bio"lm surface height(above the base"lm) have been speci"ed a priori for eachbio"lm geometry. The height of the structure (above thebase"lm) in the grid points has been determined usinga random number generator. The surface itself was cre-

ated by two-dimensional spline interpolation. Once thebio"lm surface is computed, the thickness ¸

B(and, hence,

the number of grid cell layers Nb

of the base"lm) of thehomogeneous base"lm layer can be determined in orderto obtain a bio"lm geometry with the a priori speci"edbiomass volume. The area enlargement factor Ae is usedas a measure for the bio"lm irregularity. It is de"ned inEq. (8a). The more irregular and geometrically hetero-geneous a structure is, the more Ae deviates from thevalue obtained for a #at bio"lm which has Ae"1. Asa general rule of thumb holds, the more peak pointsa bio"lm has, the more irregular its structure will be.Note that the area enlargement factor is not the onlycriterion for surface irregularity one can think of. SeePicioreanu, Van Loosdrecht and Heijnen (1998b) andPicioreanu et al. (2000) for a compilation of further pos-sible criteria and their interdependencies. Sample bio"lmgeometries are shown in Fig. 3. Thirteen structures of thiskind have been generated, including the trivial case,a completely #at bio"lm. The structures have beenchosen to cover the same range of Ae as in the two-dimensional case, i.e. 1)Ae)2.5. These geometries aredenoted by wb3d1}wb3d13. Table 1 summarises theproperties of the generated bio"lm geometries.

Although the bio"lm geometries created with this al-gorithm have irregular surfaces with valleys and peaks,they do not have pores and channels which are reportedin the experimental studies (e.g. Gjaltema et al. 1994, andthe review of Costerton et al., 1995). The `mushroomabio"lm concept is considered to be an excellent examplefor cluster-and-channel bio"lm structures (Rittmann etal., 1999). Taking this expression literally, a computa-tional bio"lm geometry with several mushroom-shapedbacterial colonies was created. These colonies are ar-ranged in a regular self-repeating manner. Its allows #owbetween clusters and, hence, convective transport of nu-trients there. The distance between the colonies is in thesame order of magnitude as mushroom heights and dia-meters, which are approximately 100}130 lm. The heightof the mushroom heads is approximately the same as theheight of the mushroom stem. The thickness of the ho-mogeneous base"lm layer is four grid cells. This structureis called cc3d1 in this study and shown in Fig. 4a.A cluster-and-channel bio"lm with a more irregularstructure was created using a cellular automaton bio"lmgrowth model for hydrostatic environments described inPicioreanu et al. (1998a). Fig. 4b shows this geometrycalled cc3d2. Table 2 summarises properties of com-putational cluster-and-channel geometries.

4. Results and discussion

4.1. Characteristic ratios

Physical con"gurations can be described by character-istic quantities of their #uxes, forces, and geometrical


Fig. 3. Examples for wavy bio"lm geometries: (a) wb3d9 and (b) wb3d3.

Table 1Thirteen wavy bio"lm geometries!

Geometry N1

N2

N3

Nb

Ae

wb3d1 256 32 64 17 2.32wb3d2 256 32 64 17 2.32wb3d3 256 32 64 17 1.87wb3d4 256 16 64 18 1.68wb3d5 256 32 64 19 1.47wb3d6 256 16 64 20 1.42wb3d7 256 32 64 19 1.40wb3d8 256 16 64 20 1.29wb3d9 256 32 64 19 1.29wb3d10 256 16 64 21 1.25wb3d11 256 16 64 21 1.17wb3d12 256 16 64 21 1.04wb3d13 256 4 64 22 1.00

!Niis the number of grid points in the x

i-direction, N

bthe thickness of

the base "lm layer in grid points. Area enlargement Ae indicates bio"lmirregularity. System length in #ow direction is ¸

1"1.6 mm.

Table 2Two channel-and-cluster bio"lm geometries!

Geometry N1

N2

N3

Nb

Nc

Ae

cc3d1 256 32 56 4 19 2.380cc3d2 256 32 56 2 12 2.284

!Niis the number of grid points in the x

i-direction, N

bthe thickness of

the base "lm layer in grid points. Ncis the maximum height of bacteria

clusters. Ae is area enlargement. System length in #ow direction is¸1"1.6 mm.

shapes (Becker, 1976). The processes under considerationcan be empirically formulated in terms of characteristicratios of these quantities. For the bio"lm system in ourstudy, characteristic ratios are area enlargement Ae,Sherwood number Sh, Reynolds number Re, and Schmidtnumber Sc (Picioreanu et al., 2000). The latter one de-scribes the ratio of kinematic viscosity and di!usivity inthe bulk. Both parameters are kept constant throughout

the study and, therefore, Sc and Re can be eliminatedusing Peclet number Pe instead. This is the ratio betweenconvective and di!usive mass #uxes.

In particular, the characteristic ratios are de"ned as

Ae":dC

¸1¸2

, (8a)

Sh"ks(¸

3!¸

B)

D1

, (8b)

Re"(¸

3!¸

B)u

501l

, (8c)

Sc"l

D1

, (8d)

Pe"Re Sc"(¸

3!¸

B)u

501D

1

. (8e)


Fig. 4. Cluster-and-channel bio"lm geometries: (a) the mushroom type geometry cc3d1 and (b) the irregular geometry cc3d2.

Table 3Model parameters; they are kept constant throughout numerical simu-lations

Parameter Notation Value Units

Bulk in#ow concentration c0

1]10~3 kg m~3

Reaction rate k1

2.272]10~4 kg m~3 s~1

Saturation constant k2

1]10~4 kg m~3

Di!usion coe$cient inliquid

D1

2.3]10~9 m2 s~1

Di!usion coe$cient inbio"lm

D2

2.3]10~9 m2 s~1

Liquid density o 1000 kg m~3

Kinematic viscosity l 1]10~6 m2 s~1

Ae refers to the geometrical structure of the system and isthe ratio between actual liquid}solid interface area andcovered substratum area. Note that Ae was also used toquantify the spatial heterogeneity of the bio"lm shape inTables 1 and 2. Re, Sc and Pe are #ux ratios and refer tothe environmental conditions, hydrodynamics and nutri-ent availability, in the liquid-phase. Only Sh connects theenvironment with the bio"lm. It quanti"es mass transferfrom liquid-phase X

1into the solid-phase X

2across their

interface C. The lumped mass transfer coe$cient ks

instandard de"nition (8b) is unknown, and available atmost for simple uniform situations like a #at bio"lm witha constant boundary layer thickness. This, however, dis-agrees with the spatial irregular bio"lm structures underconsideration and boundary condition (7). Taking thisinto account, a local Sherwood number is obtained afterelimination of k

s(Deen, 1998)

ShH"(¸

3!¸

B)

c"6-,

!c

Lc

Ln1

at C. (8f )

It is de"ned in all points at the interface C. In order toallow a quanti"cation of mass transfer in the entire sys-tem, ShH is lumped by averaging according to

Sh":ShHdC

:dC. (8g)

For convenience, the overbar will be suppressed in thesequel and the average Sherwood number will be de-noted by Sh as usual. In order to evaluate Eq. (8g), thelocally de"ned ShH must be determined. Therefore, a nu-merical solution of model equations (1)}(5) is required.Physical model parameters c

0, k

1, k

2, D

1, D

2, o and l for

Eqs. (1), (2) and (4) are kept constant throughout allnumerical experiments and are given in Table 3. TheSchmidt number Sc can be obtained herewith immediate-ly as Sc"434.78 according to de"nition (8d).

Note that internal bio"lm processes are not taken intoaccount explicitly in Eq. (8a)}(8g) but only implicitly viamass transfer from bulk into "lm, and hence in Sh. Thus,the physical con"guration under consideration is the


liquid-phase X1

as the bio"lm environment. This is moti-vated by the hypothesis that bio"lm formation is stronglyin#uenced by external conditions like hydrodynamicsand nutrient availability (van Loosdrecht et al., 1997).A characteristic ratio describing internal bio"lm pro-cesses is Thiele number

¹h"¸&*-.S

k1

D2

.(8h)

It depends on model parameters reaction rate k1, di!u-

sion coe$cient D2

and again on a characteristic geomet-rical quantity ¸

&*-.of the bio"lm. Thus, it is a global

characteristic ratio not accessing the concentration"eld and, hence, simulation results are not required toevaluate (8h).

4.2. Evaluation of simulation results

Numerical simulation of #uid #ow in the de"ned spa-tially heterogeneous structures shows that two distinctregions are obtained. While in the bulk liquid the shapeof the #ow pro"le is not heavily in#uenced by irregularbio"lm structures, it is highly deformed in the bio"lmchannels and in a small layer between bio"lm and bulkliquid. With increasing bulk #ow velocity, the velocity inthe bio"lm channels also increases. The velocity gradi-ents in the bio"lm voids are directly related to the bulkliquid velocity. These qualitative observations are in ac-cordance with the experimental results reported byLewandowski et al. (1995).

The nutrient concentration c decreases in #ow direc-tion and from bulk liquid into bio"lm due to nutrientconsumption in the solid compartment X

2. The lowest

concentration values are obtained at the substratum inthe downstream part of the system, the highest ones atthe in#ow boundary due to the condition (7a). At higher#ow rates, the concentration boundary layer penetratesdeeper into the voids of the bio"lm and its thicknessbecomes smaller. This was to be expected and is qualitat-ively in accordance with the measurements reported byZhang (1994). Due to the concentration boundary condi-tion (7a) the boundary layer has to develop along x

1. In

any case it must be thicker at x1"¸

1than it is at

x1"0. At higher #ow rates, this e!ect is observed only in

a small entrance region and a fully developed boundarylayer is obtained after short distance. For low #ow rates,the nutrient concentration at the substratum is lowerthan at higher velocities, where higher nutrient loads areentering the system. The more the #ow velocity u

501va-

ries (and hence Re and Pe), the bigger are the observeddi!erences.

Fig. 5 illustrates simulation results for bio"lm ge-ometry wb3d3 in two di!erent #ow regimes, namelyu501

"0.016 m s~1 (i.e. Re"4.1, Pe"1913) and

u501

"0.1295 m s~1 (i.e. Re"32.6, Pe"14174). Thesevelocities are "xed approximately 200 lm above thehighest bio"lm peak. If we consider a linear approxima-tion of the Poisseuille #ow "eld at the wall of a pipe withdiameter 1 cm, then the speci"ed bulk #ow velocities onthe top of the computational system correspond to max-imum #ow velocities in the pipe of 0.13 and 1.1 m s~1,respectively.

4.3. Question A: How do bioxlm structure and environ-mental conditions awect mass transfer processes?

The set G"MAe,Pe,ShN is chosen to be a generatorfor con"gurational analysis. Obviously, Ae and Pe areindependent in our study. Specifying bulk velocity u

501,

we can achieve any Reynolds (and hence Peclet) numberand neither is the choice of di!usion coe$cient D

1re-

stricted by the bio"lm geometry. On the other hand, thebio"lm geometries have been created without being in-#uenced by u

501. In the sequel we will try to "nd out if and

how the mass transfer process in the system is correlatedto geometrical structure and environmental conditions.In terms of con"gurational analysis this question reads:Is Sh independent of Ae and Pe, and thus, is the setG independent? Two numerical experiments are carriedout for this purpose: one of the two independent charac-teristic ratios (Ae or Pe) will be kept constant as a "xedparameter, and the other will be varied.

4.3.1. Correlation between Sh and Ae for xxedhydrodynamics

In the "rst computer experiment, the independentcharacteristic ratio Ae is varied while Pe (resp. Re) is keptconstant. It is carried out with the bio"lm geometrieswb3d1}wb3d13. Flow conditions were speci"ed asu501

"0.016 m s~1 (i.e. Re"4.1, Pe"1913) andu501

"0.1295 m s~1 (i.e. Re"32.6, Pe"14174).Our simulations con"rm the result Picioreanu et al.

(2000) found in a two-dimensional study: Mass transfer inthe system decreases with increased bio"lm roughness.Qualitatively similar "ndings for liquid}solid mass trans-fer in bio"lm airlift suspension reactors have been re-ported by Nicolella, van Loosdrecht and Heijnen(1998). In their laboratory experiments, the masstransfer coe$cient for rather rough bio"lm-coated par-ticles was found to be smaller than the values for spheri-cal particles.

Since bio"lm geometries and model parameters havebeen chosen in order to allow direct quantitative com-parison with the two-dimensional simulations ofPicioreanu et al. (2000), those can be considered specialcases of the more general three-dimensional case.Fig. 6 shows that indeed the two-dimensional calcu-lations match the three-dimensional results with verygood accordance in both cases, at low and high bulk #owvelocities.


Fig. 5. Nutrient concentration c for bio"lm geometry wb3d3 with (a) Re"4.1 and (b) Re"32.6. c decreases with grey shade, bulk concentration isblack. Shown are x

2}x

3-slices along the x

1-axis (grid units). The interface C is blank.

Fig. 6. Sherwood number Sh in dependence of area enlargement Ae for low (left) and high (right) velocity. Bullets correspond to three-dimensionalsimulations, empty circles to two-dimensional results of Picioreanu et al. (2000).


Fig. 7. Correlation between mass transfer and hydraulics for bio"lmgeometry (a) wb3d4, (b) cc3d1, (c) cc3d2. Circles represent detailedsimulation results, lines show the empirically found correlation (9). Sc is"xed at 434.78.

4.3.2. Correlation between Sh and Pe for a particularbioxlm geometry

In the second computer experiment, the bio"lm ge-ometry is kept constant and the #ow velocity is varied.Picioreanu et al. (2000) found in their extensive numericalexperiments on wavy bio"lms a good correlation be-tween the average Sherwood number Sh and Peclet num-ber Pe for "xed bio"lm geometries. It reads

Sh"aRe1@3Sc1@3"aPe1@3. (9)

Herein, a is a parameter that varies for di!erent bio"lmstructures. The experiment is repeated here with thethree-dimensional wavy bio"lm structure wb3d4 andboth cluster-and-channel bio"lms cc3d1 and cc3d2. Ineach case, correlation (9) could be con"rmed again alsofor three-dimensional simulations, as it is shown in Fig. 7.

Since mass conversion in the solid-phase X2

in#uencesmass transfer from the liquid into the bio"lm, the ques-tion arises if Eq. (9) only depends on bio"lm geometry oralso on internal bio"lm processes. Since model para-meters and bio"lm geometry are not changed in thesimulations, variations of Sh according to Eq. (9) are onlyinduced by changes in Re, and hence, hydrodynamics.The actual numerical value of a is constant for a particu-lar bio"lm geometry; however, it might depend on theThiele number ¹h as de"ned in Eq. (8h) which was keptconstant in our simulations, as well as on Ae. This can beseen from the de"nition (8f ) of the local Sherwood num-ber ShH. It heavily depends on the nutrient concentrationand its gradient, which are in#uenced by the reaction ratek1

and the di!usion coe$cient D2

in the solid phase.Indications for these in#uences already have been givenin Picioreanu et al. (2000).

Correlation (9) should not be extrapolated from therange of Re (and, hence, Pe) where it was found. ForRe"0 (i.e. hydrostatic conditions) it would induce thatno mass transfer takes place what obviously is wrong.Also, extrapolation to higher #ow velocities must beavoided.

4.3.3. Two or three-dimensional simulations?Obviously, the computational e!ort for three-dimen-

sional simulations is much higher than for two-dimen-sional simulations. For this reason, it is worthwhilediscussing the feasibility of two-dimensional studies.Comparison of simulation results obtained in this studywith those published in Picioreanu et al. (2000) showsthat system behaviour with respect to mass transfer isqualitatively and * where applicable * quantitativelythe same for two and three dimensional calculations.Since these results also seem to hold for cluster-and-channel bio"lm structures, which have no two-dimen-sional counterparts, we conclude that the mass transferrelations originally found and extensively discussed inPicioreanu et al. (2000) are generally valid in the Pe}Aerange under discussion.

In all simulations in this paper and in Picioreanu et al.(2000) bio"lm geometries have been de"ned a priori. Theresults found here do not say that a two-dimensionalapproach is also su$cient for realistic bio"lm growthmodelling or for other processes. In order to allow thisgeneralisation for bio"lm development it must be shownthat bio"lm irregularities generated in two and three-dimensional approaches are the same. This cannot beanswered now and will be discussed in a forthcomingstudy.


Fig. 8. Surface DucD"DudD in a small section of the cluster-and-channel bio"lm geometry cc3d1. Above the plotted surface convection is dominating,below di!usion. Four di!erent #ow cases are shown: (a) Re"32, (b) Re"0.32, (c) Re"0.16 and (d) Re"0.032.

4.3.4. How do bioxlm structure and environmental condi-tions awect mass transfer processes?

Computer experiments have shown that mass transferand conversion in bio"lms, expressed by the averageSherwood number Sh according to Eq. (8g), is stronglya!ected by surface roughness and hydrodynamic condi-tions. Sh decreases for coarse bio"lm structures under thesame hydrodynamic conditions. For an a priori speci"edbio"lm geometry, mass transfer in the system increaseswith the #ow velocity in the bulk liquid. These resultshold in the range of area enlargement Ae, Peclet numberPe, and bio"lm length scales covered in this study.

4.4. Question B: Does convective transport of nutrients inbioxlm channels necessarily increase mass transfer frombulk liquid into the bioxlm?

This question addresses the contribution of both trans-port mechanisms, convection and di!usion to masstransfer from the bulk liquid into the bio"lm. Both mech-anisms are described by their #uxes

uc"cu, ud"!Di+c, (10)

where Diis the di!usion coe$cient in the liquid (i"1) or

in the solid (i"2).With the development of a hydrodynamic #ow "eld

and the presence of nutrients in bio"lm voids, it is obvi-ous that convective #uxes contribute to mass transportinside bio"lm channels. Discussing simulation results, itwas found that with increased bulk #ow velocity theconcentration boundary layer becomes thinner and theconcentration in the bio"lm increases due to enhancedmass transfer from the liquid into the solid phase. More-over, increased bulk #ow velocity means also increasedlocal #ow velocity in the bio"lm voids. Therewith, it is tobe expected that the contribution of convection to masstransport in the bio"lm channels increases with the bulk#ow velocity and from a certain #ow velocity on it willeven dominate over di!usive transport. For a segment ofthe mushroom-type bio"lm structure cc3d1 this isshown in Fig. 8: This segment was selected in the down-stream part of the structure (between grid point 158 and194, i.e. 0.988 mm"x

*/)x

1)x

065"1.213 mm) in or-

der to exclude in#ow e!ects induced by boundary condi-tion (7a) and #ow frontally approaching the bio"lmstructure. For low bulk #ow velocities, mass transfer in


Fig. 9. De"nition of mass #ow rates into the control volume to beconsidered in the #ux analysis.

the bio"lm segment is di!usion dominated. With increas-ing Reynolds number, the convection-dominated zonepenetrates into the structure. Finally, for very fast bulk#ow, mass transfer in bio"lm voids is convection domin-ated and only the solid structure X

2and a very thin layer

around it where due to friction the local #ow velocity issmall remains di!usion dominated. This is in accordancewith the experimental "ndings reported by De Beer andSchramm (1999), who studied #ow through bio"lms withmicroelectrodes.

Is it now possible to conclude from the detection ofincreased convective transport in bio"lm channels, that itenhances mass transfer from the bulk liquid into thebio"lm? The segment shown in Fig. 8 is an example thatthis is not necessarily true. This will be shown by meansof #ux and mass #ow rates analysis in what follows.

Opposed to question A where the mass transfer acrossthe liquid/solid interface C was investigated, we are nowconcerned with the mass transfer from the bulk liquidinto the bio"lm. That is, we consider the mass transferacross the virtual plane parallel to the substratum andtangent to the highest biomass colony, i.e. at a height ofx3"¸

c. Together with x

*/and x

065this virtual plate

de"nes the control volume for the subsequent analysis, assketched in Fig. 9. The #uxes uc,d across the surfaces ofthe rectangular control volume de"ne mass #ow ratesU and the average #uxes J according to

Uc,d*/"P

Lc

0P

L2

0

uc,d1

(x*/

,x2, x

3) dx

2dx

3, Jc,d

*/"

Uc,d*/

¸c¸2

,

(11a)

Uc,d065

"PLc

0P

L2

0

uc,d1

(x065

,x2,x

3) dx

2dx

3, Jc,d

065"

Uc,d065

¸c¸2

,

(11b)

Uc,d065

"Px065

x*/P

L2

0

uc,d3

(x1, x

2,¸

c) dx

2dx

1,

Jc,d501

"

Uc,d501

(x065

!x.)¸

2

. (11c)

Note, that due to boundary conditions (6) and (7) mass#uxes uc,d across the three remaining surfaces of thecontrol volume, i.e. at x

2"0, x

2"¸

2, x

3"0, vanish.

Hence, also the corresponding mass #ow rates and aver-age #uxes equal zero there. Also note, that the sign of thevertical velocity component in x

3"¸

cmight change due

to #ow entering and leaving the control volume throughthis boundary. Hence also uc

3changes its sign. Therefore,

Eq. (11c) has positive and negative parts contributing toconvective net mass #ow rate and net #ux. Summing upthe convective and di!usive mass #ow rates we have

Uc,d"Uc,d*/!Uc,d

065!Uc,d

501(12a)

and herewith the total mass #ow rate into the controlvolume

U/%5"Uc#Ud. (12b)

4.4.1. Comparison of net convective and diwusive masstransfer from bulk liquid into bioxlm

The results of the #ux analysis are summarised inFig. 10. In our example, Uc

501(Ud

501holds for all bulk

#ow velocities, even if the mass transport in the bio"lmchannels is convection dominated (as seen in Fig. 8). Onlyfor the fastest bulk #ow velocities the net-convectivecontribution Uc

501is in the same order of magnitude as the

di!usive Ud501

. This is explained through comparingUc

501with its positive and negative parts. While Uc

501"

6.670]10~17 kg s~1, there is a convective transportfrom bulk liquid into the bio"lm channels of 8.4723]10~15 kg s~1 and a transport of 8.4056]10~15 kg s~1

back from the bio"lm channels into the bulk liquid. Fora lower Re the contribution of the net-convective #ux ismarginal but also here the in and out #uxes are consider-able (e.g at Re"0.32,Uc

501"2.87]10~19 kg s~1, while

its positive and negative parts are obtained as5.5275]10~17 and 5.4988]10~17 kg s~1, respectively).Thus, the convective transport of solutes into and out ofthe bio"lm through the top surface more or less balanceeach other. The net-convective transport is less than 1%of the convective transport towards the bio"lm.

These observations show that interpreting bio"lm pro-cesses from local measurements (e.g. with microelec-trodes) may be misleading. Generally, it is concluded thatconvection dominates the mass transport, as it is alsoshown in Fig. 8. For the overall bio"lm processes, themass transport from bulk liquid to bio"lm phase is how-ever equally important. It seems that to a large extent thiscan be considered a di!usion-dominated process. Ina study by Lewandowski et al. (1995), it was shown thatin real bio"lm pores the liquid velocity can be10}20 lm s~1. Similar velocities in bio"lm channels oc-cur in our study at Re"0.16 and 0.32. Although theoverall geometries are not identical, it shows that ouranalysis was performed in realistic conditions. All ouranalyses are based on the transport of solutes. For the


Fig. 10. Mass #ow rates into the bio"lm segment of Fig. 8. Shown are Uc501

and Uc501

from the bulk liquid into the bio"lm (empty symbols) and the totalconvective and di!usive mass #ow rates Uc and Ud (black symbols). All mass #ow rates are related to the net mass #ow rate U/%5 into the segment whichincreases with bulk #ow velocity.

transport of particles (with much lower di!usion coe$-cients) convective transport will play a more dominantrole.

The analysis was here performed based on mass #owrates. Since both average #uxes Jc,d

501are related to the

same entry surface, the results are the same if they areused instead of mass #ow rate.

4.4.2. Comparison of internal and external convective masstransfer into the bioxlm

After a comparison of convective and di!usive contri-bution to mass transfer from bulk liquid into the bio"lm,we compare the convective contribution from neighbour-ing bio"lm segments (i.e. Uc,d

*/!Uc,d

065) with the one from

the bulk liquid (i.e. Uc501

). It turns out that the increase ofconvective contribution to mass transfer with increasingbulk #ow velocity (see Fig. 10) is mainly due to horizontalmass transfer, i.e. from neighbouring bio"lm elements.This component increases with Re much faster than thevertical bulk-bio"lm transfer. In Fig. 10, this can beconcluded from the di!erence between Uc and Uc

501with

de"nition (12a).Since in our example (x

065!x

*/)'¸

c, the di!erence

between vertical and horizontal convective contributionappears even bigger if average #uxes Jc

*/,Jc

065and Jc

501are

used instead of mass #ow rates for quanti"cation.

4.4.3. Does convective transport of nutrients in bioxlmchannels necessarily increase mass transfer from bulkliquid into the bioxlm?

In Fig. 8, we have seen that with increased bulk #owvelocity mass transport in bio"lm channels becomesconvection dominated rather than di!usion dominated.The evaluation of simulation results revealed, this doesnot necessarily induce a strong net-convective mass

transfer of solutes from the bulk liquid into the bio"lmnor vice versa. Instead, we found for our example twoother major contributions to high convectice transport inbio"lm channels:

f Mass is convectively transported through the bio"lmchannels along the streamlines. Since the #ow "eldfollows the bio"lm geometry this means there is con-vective transport into the bio"lm but also convectivetransport out of the bio"lm in a similar order ofmagnitude. Hence, for a large part the observed con-vective #ux in bio"lm channels is a mere transporta-tion through the bio"lm.

f Horizontal internal convective transfer can be domi-nating over vertical external transfer if the bio"lmis frontally approached by #ow like the cluster-and-channel geometry under consideration here (seeFig. 4(a) or also the wavy structure in Fig. 3(b)). Inbio"lm structures with a protecting biomass barrier(like in Fig. 4(b) or the two-dimensional structures inPicioreanu et al. (2000)), this e!ect might be muchsmaller. The #ow velocities and, hence, the convectivetransport in bio"lm channels then are also lower.However, cavity #ow will develop and the afore men-tioned e!ect of high convective transport in and out ofthe bio"lm will still be observed.

5. Conclusions

Mass transfer and conversion properties of bio"lmshave been analysed using fully three-dimensional math-ematical simulations. The focus was on the in#uence ofhydrodynamic conditions and of surface irregularity.


In the "rst aspect studied, the major results found ina former extensive two-dimensional study (Picioreanuet al., 2000) have been con"rmed for the more generalthree-dimensional case, namely, the increase of mass con-version with increasing #ow rates in the bulk liquid andits decrease with increasing surface irregularity.

A second aspect studied was the contribution of con-vection induced by bulk #ow (i.e. the contribution ofhydrodynamics) to mass transfer into a cluster-and-chan-nel bio"lm. It was found that convection is in mediumand fast #ow systems the dominating mechanism formass transport in the channel, but it does not necessarilycontribute signi"cantly to mass transfer from the bulkliquid into the bio"lm. It rather appears to be a mecha-nism transporting mass through the bio"lm channels.Since irregular cluster-and-channel bio"lms are genu-inely three dimensional, these results could not have beenfound with a lower-dimensional approach.

Notation

Ae area enlargement, 1c nutrient concentration, kg m~3

c"6-,

nutrient concentration, kg m~3

D1,2

di!usion coe$cients for nutrienttransport in X

1and X

2, m2 s~1

Jc,d convective and di!usive average #uxesacross control, volume surfaces (at in-#ow, out#ow, top surfaces), kg m~2 s~1

k1

reaction rate, kg m~3 s~1

k2

half-saturation constant, kg m~3

¸1, ¸

2, ¸

3length of computational domain in thex1, x

2and x

3directions, m

¸B

thickness of homogeneous base"lmlayer, m

¸c

height of highest biomass colony, m¸&*-.

characteristic length scale of the bi-o"lm, m

n outward normal unit vector of X,n1,n

2outward normal unit vectors ofC pointing from X

1,2,

P pressure "eld, kg m~1 s~2

Pe Peclet number, 1r(c) nutrient consumption rate, kg m~3 s~1

Re Reynolds number, 1Sc Schmitt number, 1Sh, ShH Sherwood number, local Sherwood

number, 1u"(u

1, u

2, u

3)T liquid #ow velocity vector and its

components, m s~1

u501

velocity at the top of the computa-tional domain (x

3"¸

3), m s~1

x1,x

2, x

3coordinate directions, m

x*/

, x065

in#ow and out#ow boundaries of con-trol volume, m

Greek letters

a parameter factor in empirical rela-tions, 1

C interface between X1

and X2, dimen-

sionlessD"(L2/Lx2

1,

L2/Lx22,L2/Lx2

3) Laplace operator (linear di!usion op-

erator),+"(L/Lx

1,

L/Lx2,L/Lx

3) gradient operator,

LX boundary of X, dimensionlessl kinematic viscosity, m2 s~1

uc,d"(uc,d1

,uc,d2

, uc,d3

)convective and di!usive

#ux vectors with components,kg m~2 s~1

Uc,d total convective and di!usivemass #ow rates into, controlvolume surfaces, kg s~1

U/%5 net mass #ow rateinto control volume, kg s~1

Uc*/,065,501

,Ud

*/,065,501convective and di!usivemass #ow rates acrossin#ow, out#ow, top surfaces ofcontrol volume, kg s~1

X computational domain,dimensionless

X1

liquid region of X,"lled with liquid, dimensionless

X2

solid region of X,"lled with biomass, dimensionless

Acknowledgements

This work was supported by the European Commun-ity under contract ERBFMRX-CI 97-0114, TMR Net-work Biological Removal of Nitrogen, From Bioxlms toBioreactors. Simulations have been carried out on theCRAY T3E parallel computer of the Centre for HighPerformance applied Computing, Delft University of Tech-nology (HPaC).

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