jd_2

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An Introduction to Population Balance Modeling

Joel DucosteAssociate Professor

Department of Civil, Construction, and Environmental Engineering

MBR Training SeminarGhent UniversityJuly 15-17, 2008

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Outline

• PBM? What’s that?• Flocculation theory

– Equations– Numerical methods– Solution techniques

• Examples– Mixing Tank– Secondary clarifier

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Particle Collision and Coagulation Kinetics

• If particles in air or water are unstable, collisions can result in agglomeration or “Flocculation”

• Examples :Coagulation of aerosolsGrowth of rain droplets in cloudsPrecipitation Kinetics (Inorganics)Flocculation – Both double layer compression and charge neutralization

How do we describe the collision rate between particles, aerosols, droplets etc. ?

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•DOUBLE LAYER COMPRESSION

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•CHARGE NEUTRALIZATION

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• Lets consider a box full of lot of high school sharks and one little fish (moving randomly)

The rate that seniors collide with fish is classically given by :N ∝ ns

• Now, if there are actually nF fishes, the overall rate of collision between fishes and sharks is:

N ∝ nFns

Collision rate of freshman

Concentration of seniors

ss s

ss

f

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• Simplest expression for collision rate between two particle classes i and j

• This assumes fast flocculation (no resistance to collisions)• Now consider a discrete distribution of particle sizes :

Lets assume V2 = 2* V1, V3 = 3* V1, V4 = 4* V1 ,,Vi = i* V1

{( ) (3.30) nnr,rβN ji

function frequency Collision

ji

lume)/(time)(vocollisions

ij 321=

V1r1

V2r2

V3r3

V4r4

V5r5

V6r6

V7r6

V8r8

# / vol

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• Assumption : Coalescing drop ⇒ Vk = Vi + Vj

• We can now define a rate equation for particle collision (Smoluchowski)

i j

ji

k

(3.31) n)nr,β(r n)nr,β(r21

dtdn

particlesother allwith particlesk ofcollision by causedknin decrease

particlessmaller 2 ofcollision by

causedknin increase

1ikiki

kjijiji

k

44 344 2144 344 21∑∑∞

==+

−=

GENERAL DYNAMIC EQUATION

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• Laminar Shear (2D):

Collision can only occur for particles entering the sphere⇒ Mass Transfer Rate of rj particles into collision sphere ri + rj


t = 0 t = later0

00

rirj ri

rj

r i+ r j

Collision Sphere

0

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• This is a mass balance problem :

•α

•Φ

•Φ n

u

∫∫∫∫∫ •−=∂∂

C.S.j

C.V.j dAunndVn

t

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( )

( )

( ){

( )( )dxducosrrsinun

dxdu cosrru

sinu(1)un

sin90-αsin - cosα

cosαunun

ji

ji

SlopeLocation

φ+φ−=•

φ+=

φ−=•

φ−==

=•

43421

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( ) ( )( )

{ ( ){

φφφ+=

•=

φ+φ+=

∫

∫∫π

dcossindxdurr)(2)2(

dAn un- rate transfer Mass

drrsinrr)2(dA

2

strip aldifferenti

2/

0

3ji

collisonfor quadrants 2

C.S.j

jiji

• Need to determine dA :

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3/1cossin

1 0-1 sincossin3

dcossin2sin cos sin

vdu-uvudv

: Partsby n Integratio Use

sin v dcos2sin du

dcos dv sin u

dcossin

2/

0

2

2/

0

32/

0

2

232

2

2/

0

2

=φφ

==φ=φφ

φφφ−φ=φφ

=

φ=φφφ=

φφ=φ=

φφφ

∫

∫

∫∫∫ ∫

∫

π

ππ

π

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( )

( )

( )dxdurr3/4),( Shear,Laminar For

),(dxdurr4/3

transferMass particles central n have weSince

dxdurr4/3 rate transfer

3ji

3ji

i

3ji

+=

=

+=∴

×=

+=

ji

jijiij

jiij

iij

j

rr

nnrrN

nnN

nN

nMass

β

β

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• Isotropic Turbulence :

• G is Camp and Stein characteristic velocity gradient• G is used to describe :

Power in mixing vesselsPressure drop in pipesVelocity gradient in atmospheres

Other Collision Mechanisms

( )

"G"vvv

1vv

Powerv

1 time-Mass Fluid

nDissipatioEnergy

Raten DissipatioEnergy Mass Average

vrr29.1)r,r(

2/12/12/12/1

2/13

jiji

=⎟⎟⎠

⎞⎜⎜⎝

⎛μρ

=⎟⎟⎠

⎞⎜⎜⎝

⎛μρ

×ρρ

=⎟⎟⎠

⎞⎜⎜⎝

⎛×

ρρ

=⎟⎟⎠

⎞⎜⎜⎝

⎛ ε

×ρ

==

=ε

⎟⎟⎠

⎞⎜⎜⎝

⎛ ε+=β

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• Differential Sedimentation :

• where v = Stokes terminal settling velocity• Brownian Motion


( ) rr),( 2ji jiji vvrr −+=πβ

( )

∑∑∞

==+

−=

=μ

=β⇒=

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

μ=β

1ii

'k

kjiji

'k

'jiji

jiji

ji

nKn nn2K

dtdn

GDE into ngSubstituti

K3kT8)r,r(rr

MotionBrownian on Focus

rrr1

r1

3kT2)r,r(

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constt2K

N1

dt2KNdN

N2KNKN

2K

dtNd

-:equation previous into ngSubstituti

any timeat classes size particle all ofion concentrat of no. Total nNLet

nnK nn2K

dtdn

sideeach ummingS

'

'2-

2'

2'2'

1ii

1k 1iik

'

1k kjiji

'

1k

k

+−=−

−=

−=−=

==

−=

∞

∞∞

∞∞∞∞

∞

=∞

∞

=

∞

=

∞

= =+

∞

=

∫∫

∑

∑ ∑∑∑∑

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)0(N2

2)0(N1

)0(N)(N

)0(N1

2)(N1

)0(N)(N 0, t

'

'

'

∞

∞

∞∞

∞∞

∞∞

=

+=

+=

==

K

tKt

tKt

tAt

τ

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Exercise•In the previous figure, we noted that simple expressions for n1, n2, and so on could be developed. Imagine that at the beginning of coagulation, all particle are of size class 1(sometimes called the “primary particles”), that is, n1(t=0)=N∞(t=0). Therefore, it follows that at t=0, the concentration of all larger-size-class particles is zero.

•Examine GDE on slide 17 and show that the differential equation for n1 is

μ3kt8K

where

NKndt

dn1

1

=

−= ∞

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Exercise

)0(KN2

bygiven is where

)/t1()0(Nn

tointegratesequation aldifferenti that thisshow Next

21

∞

∞

=τ

τ

τ+=

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Solution

•Contd.

t2

)0(NK1

)0(N)t(N

: as solved wasand t offunction a is N that Note

NKndt

dn

then,nN

: that know weand 3kT8Klet weIf

1/τ beit Let 1/time are Units

'

11

1ii

43421

∞

∞∞

∞

∞

∞

=

∞

+=

−=

=

μ=

∑

•Solution :

•a) For k=1, there no smaller particles that will collide to from larger k=1 particles therefore the gain term is zero

44344214434421Term Loss

1iik

TermGain

kjiji

k n3kT8n nn

3kT4

dtdn ∑∑

∞

==+μ

−μ

=

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Solution

( )

2)0(NK hence )0(NK

2 Since

/t1ln)0(NK)0(N

nln

dt/t1

1/1

)0(NK)0(N

nln

dt/t1

1)0(NKn

dn

,separationby Solve/t1

)0(NKndt

dn

:equation aldifferenti into ngSubstituti/t1

)0(N)t(N

1

t

0

1

t

0

n

)0(N 1

1

11

1

=τ=τ

τ+τ−=⎟⎟⎠

⎞⎜⎜⎝

⎛

τ+τ−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

τ+−=

τ+−=

τ+=

∞∞

∞∞

∞

∞

∞

∞

∞∞

∫

∫∫∞

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Solution( )

( )

( )

( )21

/t1ln1

/t1ln21

1

/t11

)0(Nn

e)0(N

n

e)0(N

n

/t1ln2)0(N

nln

2

τ+=∴

=

=

τ+−=⎟⎟⎠

⎞⎜⎜⎝

⎛

∞

τ+

∞

τ+−

∞

∞

−

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• Breakage of flat particles in impeller

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breakage

birth death

+ +

aggregation

birth death

+ +

Floc of size x breakagg txhtxht

txn ),(),(),(+=

∂∂

• Birth and death concept

∫= )),'(),,(),',(,(),( txntxnxxftxh agg βα )),(,),'(),((),( txnxxxSgtxh break ∫ Γ=

• General one-dimensional PBM (Ramkrishna, 2000) without growth

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• Net aggregation rate expression– Aggregation frequency β

• Describes transport of flocs towards one another

Temperature driven

‘Perikinetic flocculation’

Velocity gradient driven

‘Orthokinetic flocculation’

DifferentialSedimentation

∫ )),'(),,(),',(,( txntxnxxf βα

33

13

1)'()',( ⎟

⎠⎞

⎜⎝⎛ −+=− xxxGxxx

πβ Spicer & Pratsinis (1996)

31131

0 )'()',( ⎟⎟⎠

⎞⎜⎜⎝

⎛−+=−

−DfDfDf xxxvGxxx

πβ Lee et al. (2000)

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• Net aggregation rate expression– Aggregation efficiency α

• Expresses success of collision– Constant {0,1}– Adler (1981)

– Kusters et al. (1997)• Shell-core model• Similar, but using RH,i instead of di

∫ )),'(),,(),',(,( txntxnxxf βα

( )

18.0

0

3

0,11

132

3

−

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +=

ji

ham

jiji

ddd

Addπμ

αα

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• Net breakage rate expression• Breakage frequency S

– Power law – empirical approach

– Shear dependent - Ducoste (2000)

)),(,),'(),(( txnxxxSg∫ Γ

⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

ευε

π i

i

xKCK

xS 12

1

exp15

4)(

axAxS =)(

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• PBM = Integro-differential equation

• Analytical solutions are rarely found• Several numerical solutions in literature• Discretisation of property x

– Acceptable calculation times– Ease of implementation

ttxn

∂∂ ),(

∫∫∞

−−−=00

'),'()',(),('),'(),'()','(21 dxtxnxxtxndxtxntxxnxxx

x

αβαβ

)(),('),'()'(),'( xStxndxxxxxStxnx

−−Γ+ ∫∞

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• Discretisation

continuousx

n(x,t)

discretex

N(x,t)

Integrals become summationsM ordinary differential equationsOnly pivots or representative x for a class

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• Hounslow and fixed pivot

Vi+1ViVi-1 Vi+2

class i-1 class i class i+1

xi-1 xi xi+1

One equationper class (Ni)

M ordinary

differentialequations

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• moving pivot

Vi+1ViVi-1 Vi+2

class i-1 class i class i+1

xi-1 xi xi+1xi

Two equationsper class (Ni and xi)

2*M ordinary

differentialequations

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• Combined aggregation/breakage – steady state (Nopens et al, 2005)

moving

improvedaccuracy

fixed 25 cl.31 cl.46 cl.

3146

Hounslow

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Introduction : Population Balance Modeling

and the QuadratureMethod of Moment

Part I

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Introduction : General Background

• PBM for simulating particle aggregation-breakup typically use average turbulent quantities to account for fluid flow characteristics

• Experimental studies have revealed the influence of the turbulence spatial heterogeneity on the behavior of flocculation dynamics (Hopkins and Ducoste 2003)

• The local influence of the turbulence could be investigated by combining a CFD model with PBM equations (Marchisio et al. 2003c, Prat and Ducoste 2006, 2007)

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Introduction : PBM for aggregation‐breakup problems

• General transport equation of particles in presence of aggregation-breakup

∂n(x,t)/∂t + <ui> ∂n(x,t)/∂xi – ∂[ (ρ 0.09 kε2/εp) ∂n/∂xi]/∂xi(I) (II) (III)

= 1/2 ∫ α[l,(d3 – l3)1/3] β[l,(d3 – l3)1/3] n(l) n((d3 – l3)1/3) dl Aggregation

– n(d) ∫ α[l,(d3 – l3)1/3] β[l,(d3 – l3)1/3] n(l) dl(IV)

+ ∫ kb(l) F(d│l) n(l) dlBreakup

– kb(d) n(d)

(I : Transient term) + (II : Convective term) – (III : Diffusion term) = (IV : Source term)

► Traditional Class Size methods => one equation to be solved for each particle Class Size

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Introduction : Advantage of the QMOM formulation

• Method of moments appear to be an alternative approach to the PBM class size method

• QMOM (McGraw 1997) : – Provide statistical information about the evolution of the floc size distribution– Does not completely capture the shape of the floc size evolution (equivalent to a 3 (or

Nd) class size method)

► CFD/QMOM approach used to model the transient spatial evolution of floc size in a turbulent stirred reactor

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Introduction : QMOM approach

• The QMOM approach (McGraw 1997)

– Based on the theory of Orthogonal Polynomial and uses a Nd points Gaussian quadrature approximation for closure

mk = ∫ n(L) Lk dL ≈ ∑wi Lik

– The particle size evolution is tracked by solving a system of differential equation for lower order moments

– Abscissas (Li) and Weights (wi) are extracted from the moment sequence (mi) using Wheeler’s algorithm

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QMOM/PBM Methodology :Moment formulation

• QMOM Aggregation-breakup methodology

– Moment transformation of the PBE with aggregation/breakup

Δmk = 1/2 Σ wi Σ αij wj (Li3 + Lj3)k/3 βij – Σ Lik wi Σ αij wj βij

+ Σ kbsi Fi(k) wi – Σ Lik kbsi wi

– 3 (Nd) points QMOM to represent the PSD• Involves tracking the evolution of the 6 lower order moments

► extract 3 Abscissas (L1,L2,L3) and 3 weights (w1,w2,w3)• Determine statistical information about the PSD

► volume based average floc size (d43 = m4/m3), length based average floc size (d10 = m1/m0), average floc density (m0), …

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QMOM/PBM Methodology : Aggregation• Aggregation dynamics

– Collision frequency function (βij) ► Based on orthokinetic flocculation assuming turbulent shear (Saffman and Turner 1956)

βij = 1/6.18 (Li + Lj)3 (ε / ν)0.5

– Collision efficiency function (αij) ► Accounts for unsuccessful particle collisions due to electrostatic repulsion or hydrodynamic retardation (Adler 1981)

αij =C1 [ { ( 3 π μ (ε / ν)0.5 Li4 (Li / Lj) (Lj / Li + 1 )2 } / (32 A d0) ] -0.18

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QMOM/PBM Methodology : Breakup• Breakup dynamics

– Breakup frequency function (kbsi) ► Breakup of particles in the viscous dissipation sub-range (Kusters 1991)

kbsi = [ 4 / ( 15 π ) ] (εp / ν)0.5 exp[ – C2 / (Li εp ) ]

– Fragment distribution function (Fi (k)) ► Formation of two fragments with mass ratio 1:4

Fi (k) = Lik [ (4k/3 + 1 ) / 5k/3 ] if Li ≥ Li 51/3

else Fi (k) = 0

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QMOM/PBM Methodology :Marchisio et al. (2003)

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Numerical Methods : The Flow Field (1)

1 / Solve the flow field

► The flow field is solved for each geometrical configuration: i.e. impeller type: axial (A310 Foil), radial (Rushton turbine), Taylor-Couette flow), tank geometry and volume, and average characteristic velocity gradient (Gm = [εpmean/ν]0.5 in s-1)

► A grid sensitivity analysis is performed to insure the validity (and cell number independency) of the flow field. A compromise has to be found between grid-insensitivity and total number of cells (and near wall refinement)

Keep in mind that if the flow field is solved quickly, the CFD/QMOM (6 equations to solve i.e. 1 for each moment) is, on the contrary, CPU demanding

► The flow field is used as initial condition for the CFD/QMOM model and PT/QMOM

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► Tips : Monitor and track floc size evolution in regions of particular interest (impeller zone : high shear zone, recirculation zone, bulk zone, etc…).

2 / Example of flow fields (Gm = 40s-1)

Numerical Methods : The Flow Field (2)

Axial Impeller : A310 Fluid Foil Radial Impeller : Rushton TurbineSpatial distribution of the normalized turbulent

energy dissipation rate (εp/εpmean) for Gm = 40s-1

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Numerical Methods : Set Initial Conditions

1 / Set initial conditions for QMOM/CFD model

► Initial particle size distribution set by the choice of 3 abscissas (dimensionless size indicator) and 3 weights (dimensionless number concentration indicator)

2 / Determine empirical constants with experimental data for (Gm) = 40s-1

► C1 is determined with the slope of experimental (d43) evolution during the initial linear growth period ► C1 = 24 (A310) and 42 (Rushton)► C2 is determined with the steady-state value of (d43) at the end of the flocculation process ► C2 = 2.5 (A310) and 9.3 (Rushton)

► Constants depend on various parameters (particle type, water and coagulant chemistry, temperature, …) But not on flow field characteristics

3 / CFD/QMOM Approach => Transport Equation is solved for each moment

initial (d43) = 1.83

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Numerical Methods : The Flow Field : BC

Gm (s-1) VOLUFLOW

40 -1.734980E-03

70 -2.540000E-03

90 -3.010000E-03

150 -4.258000E-03

► BC for A310 fluid foil ► BC for Rushton turbine

Gm (s-1) VTIP

40 0.2748

70 0.4000

90 0.4736

150 0.6644

► Values of the Boundary Conditions (BC) VOLUFLOW (A310) and VTIP(RUSHTON) to set in the Q1 file in order to obtain the appropriate value of the CVG (Gm). BC are valid for a given tank geometry, cell number, and impeller type. Values were adjusted empirically in order to obtain the correct CVG.

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Numerical Methods : CFD/QMOM : Results for D43n► Results for A310 fluid foil (30min) ► Results for Rushton turbine (30min)

(Gm) = 40s-1

(Gm) = 90s-1

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Results :Mean Particle Size Evolution (<d43n>)

1 / Evolution of the normalized spatially-averaged floc size

Slope = Δ <d43n> / Δ time

► Growth rate increases when (Gm) increases

► Floc size decreases when (Gm) increases due to higher breakup rate

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Results :Mean Particle Size Evolution (<d43n>)

2 / Transient evolution of <d43n>

► CFD/QMOM model confirms experimental observations for mixing speeds (Gm ≥ 70s-1) that display a peak followed by a smaller steady state value

► Growth region :aggregation rate > breakup rate

Growth region

► For (Gm) ≥ 70s-1 :peak region before decreasing to a lower steady-state mean particle size

Peak region

► Steady state region :aggregation rate ≈ breakup rate

Single steady state

Lower steady state

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Results : Spatial Distribution of d43n

► Larger flocs are found in the center of the recirculation zone

Gm = 40s-1

1 minute 5 minutes 15 minutes

Gm = 90s-1

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Results : Tank Average Moment Rate Equation (Δm4)

• (Gm) = 40s-1 :

► (Δm4) increases during the linear growth phase while breakage rate is negligible

► A breakage rate that exceeds the aggregation rate => peak region

Δm

4

AggregationBreakupΔm4

Δm

4


Gm = 40s-1 Gm = 150s-1 Δm

4


• (Gm) = 150s-1 ► (Δm4) ≤ 0 when breakage rate exceeds aggregation rate

► breakage rate increases but (Δm4) ≥ 0

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Results : Spatial Distribution of (Δm4)

5 min

2 min

► 85% of the tank volume promote aggregation while 15% promote floc breakup

Gm = 40s-1 Gm = 150s-1

15 sec

45 sec

► same ratio (85/15%) is observed but higher breakup rates with increasing (Gm)

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Questions

► Transient spatial variation of d43n

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Numerical Methods : Set Initial Conditions (1)1 / Set initial conditions for PT/QMOM model► A stand alone FORTRAN code is used to generate the initial location of flocs to be tracked. A sensitivity study was performed and showed that 1000 flocs were sufficient to insure no change in the floc size evolution with increasing number of particles.

2 / Example of path of one particle for 30min (as a function of impeller type)

A310 Fluid Foil Rushton turbine

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Numerical Methods : Set Initial Conditions (2)

3 / PT/QMOM Approach

• The path of each particle is solved using PHOENICS (GENTRA option).

• The result file (GHIS) that includes location and time for each tracked particle) is used in a stand alone QMOM FORTRAN code (TRACKINGhis.for). The stand alone QMOM model uses for the aggregation-breakup dynamics the same constants (C1, C2) than in the CFD/QMOM model.

• Tips : The CPU time to compute GENTRA particle tracking is long (simulation of 30min of particle transport in the tank). Particle tracking (PT) is typically run for 200/250 particles at a time.

• PT/QMOM vs CFD/QMOM : The main advantage of the PT/QMOM approach over the CFD/QMOM approach is that flow field conditions (Gm, impeller type, tank size, …) are separated from flocculation dynamics (QMOM).

This approach (PT/QMOM) allows to investigate only “non flow field” conditions ( particle type, water and coagulant chemistry, … i.e. reflected in the values of QMOM constants C1 and C2) and to explore alternative aggregation and breakup kernel formulations.

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Numerical Methods : PT/QMOM : Std Alone QMOM► The stand alone FORTRAN program TRACKINGHIS_EPDT is used with PHOENICS PT results files (GHIS) to compute the floc size evolution using the QMOM. The program is organized as follows (only key elements are reported) :

PROGRAM TRACKINGhis

C This program uses the Quadratic Method of Moments for agglomeration/breakup processes. We use there a quadratureC approximation with 3 nodes. This program is for stand alone QMOM and devoted to be used with results files of PHOENICSC particle TRACKING (GHIS).

C Variables definition !!! DEFINE TIME STEP (TGAP) FOR QMOM ROUTINE

C Step 1 : a ) Delete x,y,z coordinates in GHIS raw data file REMOVE UNECESSARY DATA IN (GHIS) PT FILEb ) Delete reduntant data in PT filec ) Transform files in equispaced time step files PREPARE DATA FOR QMOM

• The GHISEPDT file is used with the QMOM (1 data every TGAP for algorythm stability)• The GHISEPDT2 file is used for average Ep values (1 data every 1 sec)

C Step 2 : Mean EP calculation at each time step DT COMPUTE THE MEAN EP AT EACH TIME STEP FOR ALL (1000) PARTICLES (TANK)

Mean EP calculation for each particle all time long COMPUTE THE MEAN EP FOR EACH PARTICLE (30MIN)

C Step 3 : D43 calculation with QMOM (Define here C1 and C2) COMPUTE THE FLOC SIZE EVOLUTION (D43) FOR EACH PARTICLE (30MIN WITH DT=TGAP)

Mean D43 calculation COMPUTE THE VOLUME AVERAGED FLOC SIZE (D43) AT EACH TIME STEP FOR ALL (1000) PARTICLES (TANK)

END

C Turn key subroutines (ORTHOG and GAUCOF) from Press et al. (1992)

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Numerical Methods : PT/QMOM : Results for D43n► Results PT/QMOM ► Comparison PT/QMOM and CFD/QMOM

A310

(Gm) = 40s-1

RUSHTON

(Gm) = 90s-1

A310 vs Rushton

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Summary : FrameworkS

tep

1

Flow

Fie

ld

Ste

p 2

Floc

cula

tion

Ste

p 3

Set

tling

Particle Tracking

Std Alone QMOM (C1, C2)

GHIS File

.DAT File (D43, Mi, ABi, Wi)

IPSA/QMOM (Setlling)

(Mi_ini, R1_ini, R2_ini, …)

Result File (R1, R2)

Solve CFD flow field for tank geometry, CVG, impeller type

Flow field DINI File

(EP, KE, U, V, W)

ASM (ABi, Wi, …) (Setlling)

RST File ([PTi], FRSL, FRLQ)

CFD/QMOM

(Mi_ini, C1, C2, …)

Result File

(D43, Mi, ABi, Wi)

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Example: Secondary Clarifier

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Example (Secondary Clarifier)

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Example (Secondary Clarifier)• The Mixture Model/QMOM were used to solve this problem. Physical

properties of the two phases according to the following table:

Volume fraction of solid phase of incoming sludge

0.003φin

Outlet velocity-0.05 m/svout

Inlet velocity1.25 m/svin

Solid phase maximum packing concentration0.62φmax

z-component of gravity vector-9.82 m/s2gz

Diameter of solid particles (initial conditions)2·10-4 mdd

Solid phase density1100 kg/m3ρd

Liquid phase viscosity1·10-3 Pa·sηc

Liquid phase density1000 kg/m3ρc

DESCRIPTIONVALUEVar

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Example (Secondary Clarifier)• Initial Distribution:

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Results (Secondary Clarifier)• Velocity Contour

D43=200 microns (constant)

D43=550 microns (evolved)

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Results (Secondary Clarifier)• Sludge Mass Contour

D43=200 microns (constant)


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Results (Secondary Clarifier)• d43 Contour

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Results (Secondary Clarifier)• Sludge mass Contours (influence of floc porosity Kinnear (2002))


D43=650 microns (evolved)Floc porosity

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Thoughts on ASM/QMOM

• Current formulation is not rigidly linked to dispersed phase

• Does not completely capture the change in flocculation distribution above and below the solids blanket

• Simple approach to include particle interaction kinetics with two phase problem

jd_2

Documents

rj n i n j

n u n jda

sin n u

j n uda u n

rj ni n j dt

kni n j

nin j n

rj cos