Computational Fluid Dynamics Model of BioCASTMultienvironment Air-Lift BioreactorRyan S. D. Calder1; Laleh Yerushalmi2; and S. Samuel Li3

Abstract: A computational model was developed to study the hydrodynamic characteristics of a new multienvironment air-lift bioreactor.This model study considerably expands on the laboratory experiments by exploring the hydrodynamic characteristics of multiple combi-nations of geometries and operating conditions and by providing a visual illustration of the liquid-phase flow patterns. The model was firsttested against preliminary laboratory results to ensure its validity. This included comparing two simplified geometries for the three-discprototype air sparger assembly to determine which led to results closer to laboratory measurements. A torus geometry was found to betterrepresent the prototype than a single disc. The model was modified to evaluate the hydrodynamic characteristics of alternative operatingconditions and physical geometries beyond what would be possible in the laboratory. The flow pattern in the outer clarifier zone was shown tobe very sensitive to the geometry of the bioreactor wall separating the clarifier from the inner microaerophilic zone. Establishing a smooth,upward flow pattern in the clarifier was shown to be possible only when the clarifier was sufficiently shielded from the circulation inthe anoxic cone below. Further research is needed to quantify the effect of hydrodynamic characteristics on contaminant removal efficiency.DOI: 10.1061/(ASCE)EE.1943-7870.0000678. © 2013 American Society of Civil Engineers.

CE Database subject headings: Wastewater management; Water treatment; Computational fluid dynamics technique; Two phase flow;Simulation models; Reactors.

Author keywords: Wastewater management; Water treatment; Computational fluid dynamics; Two phase flow; Simulation model.

Introduction

A multienvironment wastewater treatment technology has recentlyemerged as a technology for the high-rate removal of carbonaceouscontaminants together with nitrogenous and phosphorous nutrients.The technology is known as BioCAST [Y. Yerushalmi and M. J.Ogilvie, U.S. Patent No. 7,785,479 (2010)] and consists of twointerlinked bioreactors (Fig. 1). The first bioreactor, explored in thisresearch, integrates aerobic, microaerophilic, and anoxic conditionsand is based on conventional air-lift designs. This system offershigh removal efficiencies of carbon and nitrogen under simpleoperation procedures while producing far less sludge than tradi-tional approaches that rely on unit processes in series (Yerushalmiet al. 2011). The liquid circulation within and between the environ-ments is driven by the inflow of air from a sparger at the base ofthe aerobic zone, located in the center of the first bioreactor.The microbiological activity in bioreactors is dependent on aerationlevels, mixing, and shear rates (Ma et al. 2003). In turn, these

hydrodynamic characteristics are dependent on bioreactor geom-etry and operating conditions, as investigated by authors who havedeveloped computational or laboratory-scale models (Camarasaet al. 2001).

The present research uses computational fluid dynamics (CFD)methods to model the principal hydrodynamic characteristics of thefirst bioreactor of the BioCAST system on a laboratory scale andexplores the effects of variable operating conditions and geometryon the liquid-phase flow patterns. This is the first study to use CFDto examine the hydrodynamic characteristics of this bioreactor.CFD allows for greater insight into the particularities of the flowpatterns in the bioreactor and allows for the study of many combi-nations of operating conditions and geometries. The time and costrequired to conduct a similar study on the physical prototype wouldhave been prohibitively high. This research first reproducesthe geometry and conditions studied in laboratory trials, both tovalidate model outputs against experimental observations, and toquantify flow patterns, pressure and gas holdup contours, and otherhydrodynamic information that could not be evaluated over theentire domain in the laboratory. Bioreactor and air sparger geom-etry, wastewater loading, and gas inflow rates are varied to assessthe effects of these parameters on the hydrodynamic characteristicsthat control bioreactor performance. One of the authors of thispaper supervised the laboratory experiments that generated the datathis study uses for purposes of comparison (Behzadian 2010).

The multienvironment bioreactor studied here has an outeranaerobic clarifier that is isolated from the air-driven flow circulat-ing between the interior aerobic zone (AEZ) and the microaero-philic zone (MZ), as shown in Fig. 1. The air is introduced viathe sparger at the bottom of the AEZ. As buoyancy brings theair bubbles to the surface, the bubbles transfer momentum to thesurrounding liquid phase, causing it to flow up in the AEZ and intothe MZ. By continuity, liquid enters the AEZ from the anoxic zonebelow. The displacement of water by air in the AEZ also creates a

1Doctoral Student, Dept. of Environmental Health, Harvard School ofPublic Health, 401 Park Dr. West, PO Box 15667, Boston, MA 02215.E-mail: ry.calder@mail.harvard.edu

2Adjunct Associate Professor, Dept. of Building, Civil and Environ-mental Engineering, Concordia Univ., 1455 De Maisonneuve Blvd. West,Montreal, Quebec H3G 1M8, Canada. E-mail: laleh@encs.concordia.ca

3Associate Professor, Dept. of Building, Civil and EnvironmentalEngineering, Concordia Univ., 1455 De Maisonneuve Blvd. West,Montreal, Quebec H3G 1M8, Canada (corresponding author). E-mail:sam.li@concordia.ca

Note. This manuscript was submitted on March 23, 2012; approved onNovember 7, 2012; published online on November 9, 2012. Discussionperiod open until November 1, 2013; separate discussions must be sub-mitted for individual papers. This paper is part of the Journal ofEnvironmental Engineering, Vol. 139, No. 6, June 1, 2013. © ASCE,ISSN 0733-9372/2013/6-849-863/$25.00.

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bulk density gradient; the gas and liquid mixture in the AEZ is lessdense than the liquid in the other zones. This tends to create bulkmovement in the same direction, as caused by momentum ex-change; that is, water converges from the base of the MZ and flowsupwards in the AEZ and downwards in the MZ. The directionalityof bulk flow is represented in Fig. 1. Therefore, the substantiallyslower flow in the clarifier has the potential to be influenced by theeddies in the adjacent zones that exert shear forces on the bottom ofthe clarifier domain and by the wastewater throughput that, undersome operating conditions, may have a significant impact on theflow pattern in the clarifier. It is primarily these peculiarities ofthe outer clarifier that distinguish the model developed here frommodels of wastewater bioreactors developed over the past 10 to20 years. In contrast, these past studies have examined relativelysimple internal and external-loop geometries (Alhumaizi and Ajbar2006; Huang et al. 2009), typically with two-phase liquid/dispersedgas flow domains. The k − ε turbulence model has been usedalmost exclusively by CFD researchers, with a minority optingfor large eddy simulation (Tabib et al. 2008). Some studies expandthe flow domain to include a dispersed solid phase (Volker andHempel 2002; Heijnen et al. 1997). All of the work surveyedhas considered the flow to be driven entirely by air input.

The effectiveness of air-lift systems for wastewater treatmentderives from the use of rising gas, both for agitation and aeration.The system can be constructed to direct flow through variably aer-ated zones of different biological activity with greater turbulence,less shear stress, and lower energy costs than comparable impeller-driven technologies (Vial et al. 2002). The turbulent conditions thatallow these operational advantages come at the cost of modelingdifficulties. A general description of the challenges and previousresearch efforts on turbulence modeling is described by Chenand Jaw (1998). The present modeling research has specificallybeen motivated by the need to understand hydrodynamic character-istics of the treatment system under various combinations of geom-etries and operating conditions, and to enhance the hydrodynamiceffectiveness of the system; the subsequent findings will potentiallybe applied to its optimal design. In the following, details of themethods are given in the next section. The model results arepresented and discussed in the subsequent section. This is followedby conclusions in the final section.

Methods

Geometry and Mesh Generation

The bioreactor was created with ANSYS CFX software based on thedimensions of the laboratory-scale unit, as shown in Fig. 1. A 90°model with two planes of symmetry allowed for design, meshing,and computation to be conducted on one quarter of the bioreactorand extended to the full cylindrical shape. Therefore, the model isassumed to be symmetrical with respect to the two vertical faces ofthe 90° wedge. As described later and summarized in Table 1, thisgeometry was adjusted in some model runs to evaluate the resultingeffects on the flow pattern.

The horizontal spout between the AEZ and MZ was modeled asone 3-cm-radius, 3-cm-long extrusion breaching the wall (Fig. 1),centered at 45° from both planes of symmetry. The outlet from theclarifier is not modeled geometrically, but corresponds to the topsurface of the clarifier zone (CZ).

A small 5 cm wedge of radius was defined in the middle of thetop surface of the AEZ to allow for two boundary conditions: waterinlet through the central wedge and degassing around the wedge onthe remaining surface. The degassing condition is built into CFXand removes from the fluid domain all of the dispersed-phasebubbles that reach it. CFD models developed for bioreactors ofother geometries have tended to omit the hydrodynamic effectsof throughput (Volker and Hempel 2002), likely on account of theirmuch lesser influences on the flow pattern relative to the effect ofinflowing air. However, the inclusion of wastewater throughputallows the confirmation of the effects of varied wastewater flowon the hydrodynamic characteristics of the bioreactor, particularlyin the outermost CZ, which is isolated from the air-driven flow ofthe AEZ, MZ, and anoxic zone (ANZ). An unstructured grid withan element size of 9 mm was generated over the bioreactor domain.Including geometric protrusions for these inlet and outlet zonesfacilitated the definition of boundary conditions. Including themin the model domain appears to have very little effect on modeloutput (Ranade 2002).

The air sparger apparatus used in the laboratory prototypebioreactor was of complex geometry, approximating three ringscentered over a solid pump apparatus at the base of the AEZ.In the laboratory prototype, air was introduced through 63 circularholes of 1 mm diameter distributed over the surface of the bio-reactor. Because these inlet ports were smaller than the diameterof the bubbles in the fluid and because the two-phase dispersed

Fig. 1. Centerline cross section of the multienvironment bioreactor,showing various zones and their dimensions

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gas fluid has not been realized at the boundary of the air inlet ports,it was not realistic to model flow at the ports. Therefore, simplifiedsparger geometries were used: in the first eight runs (Runs 1–8,Table 1), the air sparger was modeled as a single 7.6-cm-radiuscylinder within the AEZ with its base at the bottom of the wallbetween the AEZ and MZ and extending upward for 6 cm. There-fore, the AEZ and MZ communicate through a 1.4-cm-thick circu-lar envelope around the edge of the sparger, down the base of thewall between the two zones. In the remaining 16 runs (Runs 9–25,Table 1), the sparger was modeled as a torus of 6.6 cm outer radiusand 2.4 cm inner radius (therefore, with a circular cross section of2 cm radius). In this case, the AEZ and MZ communicated throughthe center of the sparger, a circle of 4 cm diameter, and between theouter edge of the sparger and the base of the MZ, a ring of 9 cmouter radius and 7 cm inner radius. Six of the 16 runs with torussparger geometry (Table 1) corresponded to runs that had used acylindrical sparger and were conducted to test the sensitivity ofthe model on the different sparger assumptions. The conditionsused in all runs are specified in Table 1.

Fluid Domain

The bioreactor volume was modeled as a two-phase, isothermalmixture of air at 25°C dispersed in liquid water as bubbles witha uniform diameter. The CFD model developed here correspondsto the homogeneous bubbly flow regime described by Ishii andHibiki (2011). The prevailing flow regime in a real bubble columnvaries between that homogeneous condition and a heterogeneous

slug-and-churn turbulent flow (Taitel et al. 1980) as a functionof superficial gas velocity, gas rise velocity, and column diameter(Shah et al. 1982). However, the CFD model does support hetero-geneous flow conditions and in all cases will use bubbly flow as aninitial assumption. Therefore, the model inputs and outputs mustbe manually validated with respect to the conditions for bubblyflow to ensure that the homogeneous fluid domain model is anappropriate starting point for computing the hydrodynamic condi-tions of the bioreactor.

Let ug;s and ug;r denote the superficial gas velocity and gasrise velocity, respectively. Fair (1967) estimated the limits ofbubbly flow in terms of maximum ug;s as jug;sj < 0.05 ms−1.Levich (1962) gave the minimum and maximum ug;r as0.18 < jug;rj < 0.3 ms−1. For column diameters greater than10 cm, bubbly flow is confined to superficial gas velocities lowerthan 0.05 ms−1 and is transitional up to approximately 0.07 ms−1(Hills 1976).

The superficial and rising gas velocity fields predicted by thehydrodynamic model for different conditions must be comparedwith these guidelines to ensure the validity of the dispersed gasphase model, and hence, the model output in general. Experimentaltrials confirmed that bubbly flow exists at least within the range of10 to 70 Lmin−1 of gas input (Behzadian 2010).

The bubble diameter (db) was estimated using the correlationproposed by Akita and Yoshida (1974):

db ¼ 26Dc

�DcgρL

σ

�−0.5�gD3cjug;sj

v2LffiffiffiffiffiffiffiffigDc

p�−0.12

ð1Þ

Table 1. Geometric Characteristics and Operational Conditions of the Model Bioreactor

Run

Skirt

Coneheight(cm) Dma∶Da

Airflow rate(Lmin−1)

Wastewaterflow (L d−1)

Sparger

ExplanationLength(cm)

Angle(°) Shape

Height ofair inputa

(cm)

1 8.5 45 18 2 30 70 Cylinder 6 As-built lab scale at realistic wastewater flow rate.2 8.5 45 18 2 30 70 Cylinder 2 Run 2 tests the effect of lowering the air sparger

(as in the torus runs).3 8.5 45 18 2 10 70 Cylinder 6 Runs 3 to 5 test the effects of varying the air inflow

rate in the as-built dimensions with respect to Run 1.4 8.5 45 18 2 50 70 Cylinder 65 8.5 45 18 2 70 70 Cylinder 66 8.5 45 18 2 30 0 Cylinder 6 Runs 6 to 11 vary the wastewater throughput at a fixed

air inflow rate, from 0 to the rates used in the studyby Behzadian (2010).

7 8.5 45 18 2 30 1 Cylinder 68 8.5 45 18 2 30 720 Cylinder 69 8.5 45 18 2 10 70 Torus 2 Runs 9 to 12 are repeats of Runs 1 and 3 to 5, but with

a torus-shaped sparger. Run 2 demonstrates that the higherair sparger height of the cylindrical sparger does not producesignificant differences in the hydrodynamic characteristics.

10 8.5 45 18 2 30 70 Torus 211 8.5 45 18 2 50 70 Torus 212 8.5 45 18 2 70 70 Torus 213 8.5 45 18 2 30 0 Torus 2 Runs 13 to 18 vary wastewater throughput at a fixed air flow

rate to find the minimum throughput to eliminate circulationin the clarifier for the torus sparger configuration.

14 8.5 45 18 2 30 1 Torus 215 8.5 45 18 2 30 150 Torus 216 8.5 45 18 2 30 200 Torus 217 8.5 45 18 2 30 500 Torus 218 8.5 45 18 2 30 720 Torus 219 0 NA 18 2 30 70 Torus 2 Runs 19 to 23 vary the geometry of the skirt between

Zones 2 and 3 under constant conditions.20 15 45 18 2 30 70 Torus 221 4 90 18 2 30 70 Torus 222 12 0 18 2 30 70 Torus 223 8.5 0 18 2 30 70 Torus 224 8.5 45 30 2 30 70 Torus 2 Run 24 varies the height of the bottom cone.25 8.5 45 18 1.5 30 70 Torus 2 Run 25 varies the ratio of Dma=Da.

Note: Model is shown in Figs. 1 and 2. Characteristics and conditions include the length (L) and angle (α) of the skirt and the height (h) of the sparger’s airinput surface above the wall separating the AEZ and MZ. The cone has a height of 18 cm for all the runs, expect for Run 24, in which the height is 30 cm. Thediameter ratio between the MZ and AEZ is 2 for all runs, expect for Run 25 in which the ratio is 1.5. Dma indicates diameter of MZ; Da indicates diameter ofAEZ; NA indicates not applicable.aHeight of air input surface above wall separating aerobic and microaerophilic zones.

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whereDc = column diameter; g = gravity; ρ = density; subscripts Land g = liquid and gas phases, respectively; σ = interfacial tensionbetween liquid and gas phases (σ ¼ 0.072 Nm−1); ν = molecularkinematic viscosity.

According to Eq. (1), db changes as a function of ug;s which isvariable with respect to space. Eq. (1) reveals that db is insensitiveto changes in ug;s. Furthermore, Van Baten et al. (2003) report thatvelocity is practically independent of db in the range of 3 to 8 mm.Therefore, it is acceptable to use one db over the entire modeldomain for a given gas inflow rate (Qa). The effects of bubblecoagulation and breakup were explored, but not incorporated inthe model. For the range of bubble sizes used in the model andfurther discussed later (6.2 to 7.2 mm), the rate of coalescenceis approximately equal to the rate of breakup. For the range ofsuperficial gas velocities produced by the air inflows used in themodel runs and further discussed later (0.8 to 5.3 cm=s), breakupand coalescence are expected to change the mean bubble diameterup the height of the reactor by between 0 and 1 mm, depending onthe gas flow rate (with higher flow rates; hence. bubble diametershave a lower difference in bubble size between the bottom and topof the column) (Blanch and Prince 1990). Considering that thebubble diameters used, db, are average values over the length ofthe column, the range of bubble diameters in the column willbe [db − 1=2, db þ 1=2] for the smallest flow rates (and hence,bubbles) and [db − 0=2, db þ 0=2] for the highest flow rates(and hence, largest bubbles). Therefore, the effect of coalescenceand breakup does not cause the bulk of bubble diameters to varyto the point of affecting velocity and momentum exchange; there-fore, it does not have an appreciable effect on the hydrodynamics ofthe model.

For a given Qa, db used over the flow domain was determinedby substituting a single point estimate for the order of the super-ficial gas velocity into Eq. (1). This estimated superficial gasvelocity was found by dividing Qa by the area of the risersection, producing values that agree well with experimental data(Behzadian 2010). Table 2 provides the bubble diameters usedunder different air inflow rates.

Effects of buoyancy between the gas and liquid phases were in-cluded, assuming the gravity equal to 9.81 ms−2 in the negative zdirection and a reference density of 997 kgm−3. A referencepressure of zero was used as the pressure datum, so all pressureswere absolute.

Hydrodynamics Model Equations

The motion of fluids is governed by the equations of conserva-tion of mass and momentum. If the instantaneous variables inthe equations are resolved into mean values and fluctuations,and if the Reynolds average is taken, the resultant equationswill be

∂ρ∂t þ

∂ðρujÞ∂xj ¼ 0; j ¼ 1; 2; 3 ð2Þ

ρ

�∂ui∂t þ uj

∂ui∂xj

�¼ − ∂p

∂xi þ∂∂xj

�ν∂ui∂xj

�− ρτ ij

þ Fi � ðFMEÞi; i ¼ 1; 2; 3 ð3Þwhere t = time; uj = jth component of the Reynolds-averagedvelocity vector u ¼< u1; u1; u3 >; xj = coordinates; p =Reynolds-averaged pressure; τ ij = specific Reynolds shear stress;Fi = sum of body forces other than momentum exchange in thexi-direction (ANSYS CFX Solver); ðFMEÞi = the ith componentof the sum of momentum exchange body forces. These equationsseparately describe the motions of liquid and gas phases. They arerelated through the FME term, which is equal and opposite betweenliquid and gas. In Eq. (3), τ ij is related to the bulk flow velocitythrough the concept of eddy viscosity using the well-knownBoussinesq approximation:

τ ij ¼ μt

�∂ui∂xj þ

∂uj∂xi −

2

3

∂uk∂xk δij

�− 2

3ρkδij

where δij ¼ 1 if i ¼ j; k = turbulent kinetic energy per unit mass.The standard k − ε model was used as turbulence closure for

the liquid phase. The k − ε model accounts for history effects ofturbulence by describing μt;L in terms of specific turbulent kineticenergy (k) and turbulence dissipation rate (ε). The equations fork and ε are

∂ðρkÞ∂t þ ∂ðρkujÞ

∂xj ¼ ∂∂xj

��μL þ μt;L

σk

� ∂k∂xj

�þ Pk − ρε ð4Þ

∂ðρεÞ∂t þ ∂ðρεujÞ

∂xj ¼ ∂∂xj

��μL þ μt;L

σε

� ∂ε∂xj

�þ εkðCε1Pk − Cε2ρεÞ

ð5Þ

μt;L ¼ Cμρk2=ε ð6Þwhere μL = molecular viscosity of the liquid phase; Cμ (0.09),σk (1.0), σε (1.3), Cε1 (1.44), and Cε2 (1.92) are five closurecoefficients; Pk = production of turbulence modeled on the basisof the Boussinesq approximation. Because the liquid phase isconsidered to be isothermal and incompressible, with a densitygradient of zero everywhere, there is no buoyancy and no buoyancyturbulence. Through its influence on the velocity field of theliquid phase, the gas phase also contributes to the production ofturbulence.

Model Equations for Bubble Motion

Whereas Eqs. (4)–(6) are solved numerically to approximate thestress tensor for the liquid phase, a simpler model was used forthe dispersed bubbles. A dispersed-phase model built into ANSYSCFXwas applied to the gas phase, which defines gas eddy viscosity(μt;g) in terms of liquid eddy viscosity as μt;g ¼ μt;L=σPr. Thisis related to the gas phase Reynolds stress tensor through the

Table 2. Estimates of Bubble Motion Parameters for Given Wastewater Flow Rates (Q)

Q (Lmin−1) ug;s (m s−1) db (mm) ε1g ε2g ug;r (m s−1) ug;r max (m s−1) τ (s) TD (s) Stokes number CTD

10 0.008 6.2 0.021 0.024 0.36 3.87 0.0025 0.0431 0.0583 8.8730 0.023 6.8 0.058 0.059 0.40 3.88 0.0030 0.0431 0.0696 7.3650 0.038 7.0 0.089 0.091 0.43 3.88 0.0033 0.0430 0.0756 6.7470 0.053 7.2 0.117 0.115 0.46 3.88 0.0034 0.0430 0.0799 6.35

Note: Estimates include the time scales for particle relaxation (τ ) and turbulent dissipation (TD). The superscripts 1 and 2 refer to calculated and experimentalvalues, respectively.

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Boussinesq approximation, as described previously for the liquidphase. Eqs. (2) and (3) are computed numerically for the gas phase,as for the liquid phase. However, whereas the liquid eddy viscosityis calculated by numerical computation of Eqs. (4)–(6), the gaseddy viscosity is simply calculated by dividing the liquid eddyviscosity by the turbulent Prandtl number σPr. In all simulations,σPr was set equal to 1. Details of this calculation are providedin the Appendix.

The rise velocity of an air bubble, ur;g, within a continuousliquid phase is determined by its exchange of momentum withthe liquid phase. The governing equation for ur;g can be written as

mbdur;g

dt¼ FME ¼ FD þ FL þ FTD ð7Þ

where mb = mass of an individual bubble under analysis; FME =sum of momentum exchange forces. Eq. (7) is solved at each timestep for the term ur;g, which is applied to Eq. (3) for the gas phase.Additionally, the term FME is substituted directly into Eq. (3) forthe liquid and gas phases; as explained previously, it is the FMEterm that relates Eq. (3) for the two phases in the domain. Thesignificant components of FME include: (1) drag force, FD; (2) liftforce, FL; and (3) turbulent-dispersion-induced force, FTD. Theyare computed using empirical relationships described in thefollowing.

The drag force between a bubble and the continuous fluidsurrounding it is defined by

FD ¼ 0.5CDρLAður;g − uLÞ ð8Þ

where A = effective bubble diameter; CD = drag coefficient. Insteadof prescribing one value of CD for all grid points and time steps inthe simulation, the Grace model (Yeoh and Tu 2010) was applied.Thus, CD was calculated individually at every grid point and timestep from CD ¼ 4gdbð1 − ρg=ρLÞð3juT j2Þ−1, where uT is the termi-nal velocity. The magnitude of uT is calculated as juT j ¼μLðρLdbÞ−1½μ4

LgðρL − ρgÞðρ2Lσ3Þ−1�−0.149ðJ − 0.857Þ, with J ¼0.94H0.751 for 2 < H ≤ 59.3, and J ¼ 3.42H0.441 for H > 59.3;where H is a parameter defined as H ¼ ð4=3ÞgðρL − ρgÞd2bσ−1½μ4

LgðρL − ρgÞρ−2L σ−3�−0.149ðμL=μrefÞ−0.14. The reference viscosityμref is that of water at 25°C. Because the liquid phase is modeled aswater at 25°C, the quotient μL=μref equals 1.

To calculate the lift force, different models have been generatedby rotational movement of one fluid around the other; in this case,rotational movement of water around elements of air. The standardequation for lift is given by

FL ¼ ρLCLεgðug;r − uLÞ × ▿ × uL ð9Þ

where CL = lift coefficient. The model proposed by Legendre andMagnaudet (1998) for spherical bubbles in viscous linear shearflow was used to calculate CL at every time step. The authors pro-duce CL values for different strain rates in the range of 0.1 ≤ R ≤ 5and demonstrate independence of strain rate for higher R valuesin addition to a general flattening out of CL through R on the orderof 1,000. The empirical correlation between R and CL for the rangeof R > 5 is CL ¼ 0.5ð1þ 16R−1Þð1þ 29R−1Þ.

The turbulent-dispersion-induced force was calculated usingthe standard model of Lopez de Bertodano (1998), given by

FTD ¼ CTρLk▿ð1 − εgÞ ð10Þ

whereCT = turbulent dispersion coefficient [the same constant as inEq. (6)]; k = turbulent kinetic energy of the liquid phase for bothliquid and gas phase momentum exchange equations. Similar to theturbulent Prandtl number, CT depends on the particle relaxation

time of the dispersed gas phase, relative to the turbulent dissipationtime scale. Values for CT were obtained from the relationshipCT ¼ C1=4

μ TDτ−1ð1þ τ=TDÞ−1, where Cμ (0.09) is the sameconstant as in Eq. (6); some CT values are listed in Table 2.

Boundary Conditions

Imposed boundary conditions match the conditions of thelaboratory-scale bioreactor. Conditions at the air sparger (Fig. 1)were approximated by modeling uniform airflow through thesurface of the sparger at the base of the aerobic zone. Air inflowspeeds were calculated to correspond to bulk flows of 10, 30, 50,and 70 Lmin−1 across the disc area of 0.02 m2 in the case of thecylindrical sparger and 0.01 m2 in the case of the torus sparger(air inlet was defined as the surface of a 45° wedge cut into thetorus). The liquid phase inflow fraction was set to zero. This doesnot have the effect of setting the boundary condition to 0% water atthe sparger: The results were examined after the completion of eachsimulation and it was found that within one grid space (∼1 cm) ofthe sparger, the air fraction in the domain was close to the approxi-mate values listed in Table 1 and the experimental values reportedby Behzadian (2010).

The entire surface of the bioreactor, with the exception of asmall wedge used for wastewater inlet, was considered to be opento air at 101.325 kPa, allowing for exit of the gas phase, but notthe liquid phase.

A 5-cm-radius wedge in the middle of the bioreactor wasconsidered to be the liquid-phase inlet area. The inflow was variedbetween 0 and 720 L day−1 to study its effect on the hydrodynamiccharacteristics of the bioreactor.

At the top of the CZ, defined as the liquid outlet boundary,a pressure of 101.325 kPa and normal flow direction were speci-fied. A free-slip wall was considered for the liquid and gas phases,based on the coarseness of the mesh size relative to the dimension-less wall distance. Two planes of symmetry allowed a 90° wedge ofthe bioreactor to serve as a model for the entire cylindrical shape.

It is possible to set the values of k and ε manually at thewastewater inlet (Fig. 1), or to automatically scale the intensityof turbulence based on the ratio between the magnitudes of the fluc-tuating and average parts of the Reynolds averaged Navier-Stokesequations. This ratio was set to the default of 5%.

Fig. 2 presents top and section views of the reactor, showing theCFD mesh and boundary conditions.

Setup of Model Runs

In total, 25 model runs (Table 1) were conducted under varyingoperating conditions and geometries to study the resulting differ-ences in hydrodynamic characteristics of the bioreactor. The runswere roughly grouped to examine the effects of modifying the valueof a single input parameter while holding the others constant. Thisrelatively large number of runs was needed to explore the effect ofalternate geometries and operating conditions on the hydrodynamiccharacteristics of the reactor to an extent not feasible with a physi-cal prototype. Runs 1 to 8 modeled the air sparger as a solidcylinder and Runs 9 to 25 modeled the sparger as a torus. Runs1 and 3 to 5 (cylinder sparger) and 9 to 12 (torus sparger) variedthe air inflow rate while holding the wastewater flow rate (Q) con-stant. Runs 6 to 8 (cylinder sparger) and 13 to 18 (torus sparger)varied the wastewater flow rate while holding the air inflow rateconstant at 30 Lmin−1. Runs 13 to 18 further variedQ to determinewhether higher flows had a smoothing effect on the velocity field inthe clarifier. The geometry in Runs 21 and 22 is modified variouslyto partially shield the entrance to the clarifier from the shearing

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Fig. 2. A 90° wedge model domain for hydrodynamics simulations (the wedge model represents a quarter of the multienvironment bioreactor shownin Fig. 1)

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horizontal eddies in the anoxic zone. These were the only tworuns that were found to smooth the flow pattern in the clarifier.Fig. 3 illustrates the flow patterns predicted for Run 10 (without

adjustments to skirt) and Run 22 (with adjustment to skirt). Runs19 to 25 varied the geometry in other ways while holding both airand wastewater flow rates constant. Runs 8 (cylinder sparger) and

Fig. 3. Comparison of flow patterns between Runs 10 and Runs 22 around the base of the clarifier zone

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18 (torus sparger) used the geometry and operating conditionsfrom the bioreactor setup, including a wastewater flow rate of720 L day−1. The values of all parameters for each run are indicatedin Table 1.

Results and Discussion

The outputs of the runs (1 and 3 to 18) most closely approximatingthe experimental setup, as described in Table 1, were compared toexperimental results, which are compiled in Table 3. Liquid circu-lation time around the AEZ and MZ was calculated from bioreactoroutput data for these runs and compared with the experimentallycalculated circulation time. In the study by Behzadian (2010),the experimental circulation time was measured by the acid tracerresponse technique. A pH sensor was installed in the microaero-philic zone and a pulse of hydrochloric acid was injected at thetop of the AZ. The mean circulation time was measured as the aver-age length of time between pH minima. The experimental circula-tion time was also calculated by converting pH to hydrogen ionconcentration and assessing the mean residence time of each cycleby taking the first moment of the residence time distribution func-tion for each peak and then taking the average of the first moments.The values of circulation time calculated from the peak and firstmoment methods differed by less than 5%. The 95% confidence in-terval for the circulation time was estimated to be [26.1, 30.9] s inthe case of air inflow of 50 L=min and [34.5, 44.5] s in the case of40 L=min. The values reported by Behzadian (2010) for the cir-culation time are representative and the differences between themappear statistically significant. Therefore, it can be concluded thatthey are an adequate quantitative basis for comparison between thephysical model and the CFD model in this study, in which the cir-culation time was determined using particle tracking methods.

The experimental liquid circulation time at air inflow rate of30 Lmin−1 was 33 s, and the dependence on wastewater flow ratewas not studied in the laboratory (Behzadian 2010). A circulationtime of 90.2 s was calculated from the output of Run 8 (cylindersparger) and 48.9 s from Run 18 (torus sparger), both of whichmodeled wastewater flow rates of 720 L day−1. Run 6 (cylindersparger) and Run 13 (torus sparger) yielded circulation times of31.3 and 25.10 s, respectively. The effects on the circulation time

Table 3. Comparisons of Predicted Liquid Velocities and Circulation Time with Experimental and Theoretical Values

Run

Computational model Reference values

Mean z velocity (m=s) Circulation time (s) Mean z velocityin Zone 1 (m=s)

(Chisti et al. 1988)

Mean z velocityin Zone 1 (m=s)(Bello et al. 1984)

Circulation time:experimental (s)(Behzadian 2010)Zone 1 Zone 2 Zone 1 Zone 2 Total

1 0.0852 −0.0175 10.73 52.39 63.12 0.15 0.50 332 NC NC NC NC NC 0.15 0.50 333 0.0501 −0.0108 18.24 84.62 102.86 0.10 0.35 674 0.1080 −0.0212 8.47 43.15 51.62 0.20 0.55 28.55 0.1275 −0.0237 7.17 38.59 45.76 0.25 0.60 266 0.1572 −0.0358 5.82 25.51 31.32 0.15 0.50 337 0.0857 −0.0177 10.67 51.79 62.45 0.15 0.50 338 0.0570 −0.0123 16.03 74.16 90.20 0.15 0.50 339 0.0567 −0.0164 16.13 55.77 71.90 0.10 0.35 6710 0.0826 −0.0229 11.07 39.86 50.94 0.15 0.50 3311 0.1068 −0.0284 8.56 32.18 40.74 0.20 0.55 28.512 0.1238 −0.0317 7.39 28.87 36.25 0.25 0.60 2613 0.1560 −0.0475 5.86 19.24 25.10 0.15 0.50 3314 0.0843 −0.0235 10.84 38.95 49.80 0.15 0.50 3315 0.1511 −0.0460 6.05 19.87 25.92 0.15 0.50 3316 0.0817 −0.0227 11.19 40.26 51.45 0.15 0.50 3317 0.0854 −0.0238 10.70 38.36 49.06 0.15 0.50 3318 0.0858 −0.0239 10.65 38.27 48.92 0.15 0.50 3319 0.0864 −0.0253 10.58 36.18 46.76 0.15 0.50 NA20 0.0829 −0.0236 11.03 38.68 49.71 0.15 0.50 NA21 NC NC NC NC 0.15 0.50 NA22 0.0820 −0.0232 11.16 39.48 50.64 0.15 0.50 NA23 0.1328 −0.0408 6.89 23.92 0.15 0.50 NA24 0.0807 −0.0235 11.34 38.89 50.22 0.15 0.50 NA25 0.0490 −0.0149 18.64 61.52 80.16 0.15 0.50 NA

Note: For some runs, results are not applicable (NA) or not calculated (NC).

Fig. 4. Correlation between wastewater inflow rate and z-direction ve-locity predicted in the aerobic zone for Runs 6 to 8 and Runs 13 to 18

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of varying the wastewater throughput rate were further studied byconducting Runs 6 and 7 for the cylinder sparger and Runs 13 to 17for the torus sparger. Although runs conducted with no wastewaterflow yield circulation times that agree better with experimentalresults, no correlation was observed between wastewater flow rateand circulation time.

Experimental results showed that liquid circulation time is aparameter that is very sensitive to geometry and air inflow rates.For example, reducing the diameter of the port between theAEZ and MZ by 50% leads to an increase in liquid circulation timeof between 174 and 205% for the range of air inflow rates studied.The variations in calculated circulation time among runs with avariable wastewater flow rate are all less than �50% at approxi-mately the mean values of 61.3 s (cylinder sparger) and 41.7 s(torus sparger). Because the variations in circulation time calculatedare smaller than the variations caused by minor changes inbioreactor geometry, and because they show no trend with respectto the wastewater flow rate, the variations are considered to be

artifacts of the computational model. Furthermore, the circulationtime calculated from the model compares very well with the exper-imental results, particularly where the sparger is modeled as a torus:for these runs, the circulation times measured by the CFD modeland in the laboratory vary by between 7 and 43% with an average of23%. For the runs in which the sparger is modeled as a cylinder, thecirculation times vary by between 5 and 93% with an average of52%, comparing CFD model output to laboratory measurements.All circulation time data are compiled in Table 3.

Further comparisons were made between the average verticalliquid velocities in the AEZ calculated from the model and theo-retical values suggested by Chisti et al. (1988) and Bello et al.(1984). The average velocities for Runs 6 and 13, which had nowastewater flow, air inflow of 30 Lmin−1, and used cylinderand torus sparger geometry, respectively, are both within 5% ofthe value of 0.15 m=s suggested by Chisti et al. (1988). The otherruns agree reasonably well in the same way, and all but threeshow average vertical velocities within 50% of the theoretical

Fig. 5. Distribution of the vertical (or z-direction) velocity with radial distance (r) from the centerline of the bioreactor (Fig. 1) at three selectedelevations in the bioreactor modeled in Run 10; the elevations are 0.12, 0.43 and 0.73 m above the sparger, respectively

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values suggested by Chisti et al. (1988) for the relevant air inflowrates. The theoretical velocities suggested by Bello et al. (1984)are roughly an order of magnitude higher than the model output.No correlation was observed between average vertical velocity inthe aerobic zone and prescribed wastewater inflow rate, and thedifferences between Runs 6 to 8 (cylinder sparger) and 13 to 18(torus sparger) are likely artifacts of the computational model. Thisis demonstrated in Fig. 4, which plots the relationship betweenwastewater inflow rate and the z-direction velocity in the aerobiczone calculated by the CFD model for these two series of runs.

Liquid circulation time and mean velocities are important forjudging the accuracy of the developed model and for evaluatingmicrobiological treatment and wastewater characteristic require-ments implied by other designs that the model can be adaptedto simulate. Similarly, hydrodynamic characteristics such as turbu-lence, circulation pattern, and flow direction have implications formodel performance and loading and treatment requirements underdifferent operating conditions. This study only qualitatively exam-ines these other hydrodynamic characteristics.

In all runs, upward flow is observed in the core of the AEZ withdownward flow near the wall. Even at maximum wastewater flowrates from the top of the aerobic zone, the flow direction below theoutlet to the MZ is not affected. Flow in the MZ is downward, withsome circulation at and above the height of the outlet from the AEZ.At the base of the MZ, the flow separates. Part of the flow fieldfrom the MZ is continuous with the flow field leading upwards intothe AEZ past the sparger and part disperses into large eddies in theconical anoxic zone. These eddies appear to have a large impact onthe flow pattern in the CZ.

The mean vertical velocity in the CZ was calculated to be manyorders of magnitude less than those in the AEZ and MZ. Therefore,the flow is easily perturbed by upstream activity around the edgesof the anoxic cone. Flow in the anoxic zone just below the clarifieris roughly radial and results from the broader pattern of eddies inthat area. In Runs 1 to 20, 23, and 25, the 7-cm-wide clarifier isfully open to the perpendicular flow below. In these runs, the flowpattern in the clarifier demonstrates pronounced circulation. Thegeometry in Runs 21 and 22 is modified variously to partially

Fig. 6. Distribution of the vertical (or z-direction) velocity with radial distance (r) from the centerline of the bioreactor (Fig. 1) at three selectedelevations in the bioreactor modeled in Run 22; the elevations are the same as in Fig. 5

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shield the entrance to the clarifier from the shearing horizontal ed-dies in the anoxic zone. The resulting flow pattern is very steady,uniform upward flow (Fig. 3 shows a comparison of the flowpatterns between Runs 10 and 22). Ensuring this flow pattern inthe clarifier is important from a treatment perspective to allowfor easy settling of solids. Runs 23 to 25 also tested modified geom-etry, but were found to not achieve any improvement in the flowpattern in the outer CZ.

It is interesting to compare Runs 10 and 22 (Table 1), whichdiffer only with respect to the geometry of the skirts around thebase of the clarifier. Figs. 5–8 demonstrate the similarity of flowcharacteristics within the AEZ and MZ, whereas Fig. 9 demon-strates the significant differences in the clarifier. These are the dif-ferences that are represented visually in Fig. 3, described earlier.Figs. 5 and 6 plot z-direction velocities versus radial distance(r) at three heights: 0.12, 0.43, and 0.73 m above the sparger(Fig. 1). The vertical position of each point on the graph representsthe velocity in the z-direction in the reactor of that point. Therefore,the vertical spread of points at a given horizontal position on

the graphs represents the extent to which the z-direction velocityin the flow field changes at a given position on the r-axis (i.e., howvariable vertical velocity is with respect to angle theta around thecore of the reactor).

The velocity distributions in the AEZ and MZ are very similar.Figs. 7 and 8 are set up in the same way as Figs. 5 and 6, butthe velocity data correspond to normalized vectors in the x- andy-directions. The velocity magnitudes in Figs. 7 and 8 aretwo-dimensional projections of the three-dimensional velocityvectors onto the xy-plane. A value of zero indicates that the three-dimensional velocity vector points entirely in the z-direction(unless it, too, is zero). Because the principle of continuity dictatesthat the average direction of fluid motion is in the positive directionof the z-axis in the AEZ and in the negative direction of the z-axis inthe MZ, the magnitude (and range of magnitudes) of the xy-planeprojection of the velocity vector indicates that the circulation occursat the studied point. Again, substantial similarity is shown betweenRuns 10 and 22. Additionally, the greatest circulation in the AEZoccurs nearest to the sparger, and the greatest circulation in the MZ

Fig. 7. Distribution of the magnitude of the velocity vector projected onto the xy-plane, with radial distance (r) from the centerline of the bioreactor(Fig. 1) at three selected elevations in the bioreactor modeled in Run 10; the elevations are the same as in Figs. 5 and 6

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occurs nearest to the spout connecting the two zones. Circulation inthe MZ dissipates rapidly as fluid moves down. The spread ofvelocities observed at the wall between the MZ and the clarifieris likely to be an artifact of the computational model.

Finally, Figs. 9(a–c) are unit histograms (i.e., the area undereach histogram equals 1) of the magnitudes of the xy-plane projec-tions of the velocity vectors in the clarifier. If flow in the clarifierwere ideal; that is, if the velocity vectors only had z-directioncomponents, there would be no nonzero xy-plane projections.By extension, flows that demonstrate predominantly small xy-planeprojections of velocity vectors in the clarifier are smoother thanflows that demonstrate larger xy-plane projections. Figs. 9(a–c)compare the statistical distribution of the magnitudes of thexy-plane projection of the velocity vectors in the clarifier at differ-ent heights between Runs 10 and 22. In this case, the figures areseparated according to height above the sparger (0.12, 0.43, and0.73 m) with Runs 10 and 22 compared on each. The vertical axisrepresents the probability density of each bin of velocity ranges.

In reality, velocity vectors in the clarifier fluctuate in time andvary in space. These fluctuations and variations were not resolved

in the computational model because relatively large time step andgrid sizes were used to achieve high computational efficiency. Forthe modeling perspective, it is important to allow for the effects ofsubgrid motions on the average flow, not to resolve the fluctuationsand variations themselves. The computational model did take sucheffects into account; it was adequate to resolve the average flowpattern in the clarifier and to allow a comparison between Runs10 and 22.

It was observed that the average flow pattern conformed topredictions about the impact of changing geometry and wastewaterflow rate. To roughly represent the circulation observed in Run 10(without clarifier isolation) and Run 22 (with clarifier isolation), thedistribution of the xy-plane projection of the velocity vector wasplotted. Indeed, the bulk of the distribution is significantly lowerin Run 22 than in Run 10, particularly at a height of 0.43 m abovethe sparger (Fig. 7, z = 0.43 m). The reduction of circulation is lesssignificant at the bottom (Fig. 7, z = 0.73 m) and the top (Fig. 7, z =0.12 m) of the clarifier, although this might be an artifact of thecomputational model. Therefore, although Figs. 4–7 demonstratecomparability between Runs 10 and 22 in the MZ and AEZ, Fig. 9

Fig. 8. Distribution of the magnitude of the velocity vector projected onto the xy-plane, with radial distance (r) from the centerline of the bioreactor(Fig. 1) at three selected elevations in the bioreactor modeled in Run 22; the elevations are the same as in Figs. 5 and 6

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Fig. 9. Unit histograms of the magnitude of the velocity vector projections onto the xy plane in the clarifier for Runs 10 and 22 at three selectedelevations above the sparger (Fig. 1): (a) 0.12 m; (b) 0.43 m; (c) 0.73 m

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demonstrates significant differences in the clarifier. This is repre-sented visually in Fig. 3, as described previously.

A principal finding of this research was the pronounced effect ofbioreactor geometry on the flow pattern in the outer CZ comparedto the apparently minor effect of water throughput rate. The modelclearly showed that greater isolation of the outer CZ protects itfrom perpetuating eddies from the cone below. However, thevery slow speeds of fluid flow through the clarifier compared tothe mesh fineness and convergence criteria used in the model pre-vent a quantified evaluation of the effects of variable wastewaterthroughput. Still, this suggests that the potential smoothing effectof higher wastewater throughput is far less important than changesto geometry.

Conclusion

This research has applied the methods of CFD to study the principalhydrodynamic characteristics of a new type of wastewater treat-ment system, the novelty of which lies primarily in the integrationof an external clarifier zone (Fig. 1) isolated from the air-drivenflow in the aerobic and microaerophilic zones. This study hassought to establish a CFD model of the physical prototype to testthe effects of varying geometries and operational parameters onthe hydrodynamic characteristics in the reactor, including flowpatterns. A comparison of hydrodynamic characteristics (liquidcirculation time and flow patterns in the clarifier) between theCFD model and the physical prototype has confirmed that thedeveloped model provides a good representation of the physicalprototype. After confirming the validity of the model, operatingconditions and geometries were varied and the effects on circula-tion time, mean vertical velocities, and the qualitative characteris-tics of flow patterns were recorded. The model suggests that thehydrodynamic characteristics are essentially independent of waste-water flow rate, even in the clarifier zone where the wastewaterthroughput is responsible for the mean flow. Perhaps most impor-tantly, the geometry of the bioreactor around the base of the clarifierhas a direct effect on the smoothness of flow towards the exit andmust be chosen to be conducive to the settling of solids. Specifi-cally, the length of the skirt between the microaerophilic zone andthe clarifier must be extended by at least 50% (compare Runs 10and 22) beyond the dimensions of the physical prototype to ensuresmooth upward flow in the clarifier. Hydrodynamic characteristicssuch as circulation and flow pattern, which in this research havebeen described only qualitatively, and characteristics such as tur-bulence, which are not examined in this research, can be quantifiedin future study of the bioreactor configurations of interest. Pri-marily, the bioreactor configurations that suggest smooth risingin the clarifier should be explored in greater detail to more quanti-tatively model the hydrodynamic characteristics that might have animpact on the treatment potential of this technology. This researchfound that adjusting bioreactor geometry is far more likely to havean impact on flow pattern in the clarifier than modifying operatingconditions. The expansion of this model to account for a third phase(i.e., solids) and/or a chemical removal component would be a goodcontinuation of this research.

Appendix. Calculation of the Prandtl Number for theDispersed Gas Phase Model

Estimates of σPr depend on the particle relaxation time (TR) of thedispersed gas phase, relative to the turbulent dissipation timescale (TD).

If TR ≪ TD, it is permissible to set the kinematic eddy viscosity(νg) of the gas phase equal to that of the liquid phase (νL), i.e., toassume that σPr ¼ 1. If TR → TD, it is preferable to assumeσPr > 1, hence decreasing the estimate of νg relative to νL. Accord-ing to Ranade (2002), the two time scales can be expressed as

TR ¼ d2bρg18μL

ð11Þ

TD ∼ 1

juj ∼Dc

jug;rj¼ Dcεg

jug;sjð12Þ

where εg = gas holdup. The bubble diameter (db) is calculated usingEq. (1); TD is approximated by the ratio between the diameter ofthe aerobic riser zone and the rising gas velocity, ug;r. This velocitywas calculated by dividing superficial gas velocity (ug;s) by εg,where the order of ug;s is estimated as for the db calculation basedon Eq. (1). This provides an estimate for the time scales of thelargest and most energy-carrying turbulent eddies; namely, thosewith a length scale on the order of the diameter of the riser columnand a velocity scale on the order of the speed of the rising gas.

To calculate εg as a function of ug;s, several equations are inwide use for different geometries and conditions (Tabib et al.2008). Hughmark (1967) provides one such equation applicableto the dimensions of the riser column:

εg ¼�2þ 0.35

jug;sj�ρLσ72

�1=3

�−1ð13Þ

The relaxation time, TR (Eq. 11), is a function of the slip veloc-ity between air and liquid phases, which varies in space in a waythat is largely unknown a priori. To produce a conservative (high)estimate for TR, a minimum slip velocity is considered. This isassumed to be equal to the difference between the maximum liquidvelocity and minimum gas velocity, under the assumption that thegas phase has a higher velocity than the liquid phase, on the basisthat the liquid phase is agitated by inflowing air. The maximumliquid velocity is equal to the point estimate for ug;r, higher thanthe velocity actually acquired by the liquid phase via the mecha-nism of momentum exchange, but a satisfactory approximationfor this purpose. The minimum gas velocity is equal to the highestvelocity attained under acceleration due to buoyancy with an initialvelocity equal to ug;r at the sparger. The significant effects ofmomentum exchange between the gas and liquid phases, whichsignificantly slows the gas, is neglected to produce a conservativeestimate for maximum gas velocity.

Calculations, using Eqs. (1) and (11)–(13) for wastewater flowrates of Q ¼ 10–70 Lmin−1, show that TR is roughly an order ofmagnitude smaller than TD (Table 2). Therefore, the decision isjustified to set σPr equal to 1 in the dispersed-phase model usedfor the gas phase. This analysis is only relevant to the riser column,because the gas holdup, and hence, the effects of momentumtransfer and buoyancy-induced turbulence, are very small in theother regions of the bioreactor.

Acknowledgments

This study has received financial support from the Natural Sciencesand Engineering Research Council of Canada through DiscoveryGrants held by S. Samuel Li.

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Notation

The following symbols are used in this paper:A = effective bubble diameter (m);

CD = drag coefficient (−);CTD = turbulent dispersion coefficient (−);Cε1 = turbulence closure coefficient (−);Cε2 = turbulence closure coefficient (−);Cμ = turbulence closure coefficient (−);Dc = column diameter (m);db = bubble diameter (m);FD = drag force (N);FL = lift force (N);

FME = sum of momentum exchange forces (N);FTD = turbulent dispersion force (N);

g = gravity (m s−2);k = specific turbulent kinetic energy (m2 s−2);

mb = bubble mass (kg);Pk = turbulence production (kgm−1 s−3);Q = wastewater flow rate (L day−1 or Lmin−1);Qa = air flow rate (m3 s−1);r = radial distance from the reactor centerline (m);

TD = turbulent dissipation timescale (s);TR = particle relaxation timescale (s);u = velocity vector (m s−1);uj = the xj-component of u (m s−1);ur = rise velocity (m s−1);us = superficial velocity (m s−1);uT = terminal velocity (m s−1);xj = coordinates in tensor notation (m);

ðx; y; zÞ = coordinates (m);ε = turbulence dissipation rate (m2 s−3);εg = gas holdup (−);μ = dynamic viscosity of water (m2 s−1);μt = dynamic eddy viscosity (m2 s−1);ν = kinematic viscosity of water (m2 s−1);ρ = density (kgm−3);σ = interfacial tension (Nm−1);σk = turbulence closure coefficient (−);σPr = turbulent Prandtl number (−);σε = turbulence closure coefficient (−);τ = Reynolds stress tensor (m4 s−2); and

τ ij = Reynolds stress tensor in i − j plane (m4 s−2).

Subscripts

g = gas phase;j = jth component of the velocity vector; andL = liquid phase.

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