jd_2
TRANSCRIPT
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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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Particle Collision and Coagulation Kinetics
•DOUBLE LAYER COMPRESSION
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Particle Collision and Coagulation Kinetics
•CHARGE NEUTRALIZATION
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Particle Collision and Coagulation Kinetics
• 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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Particle Collision and Coagulation Kinetics
• 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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Particle Collision and Coagulation Kinetics
• 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
Particle Collision and Coagulation Kinetics
t = 0 t = later0
00
rirj ri
rj
r i+ r j
Collision Sphere
0
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Particle Collision and Coagulation Kinetics
• This is a mass balance problem :
•α
•Φ
•Φ n
u
∫∫∫∫∫ •−=∂∂
C.S.j
C.V.j dAunndVn
t
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Particle Collision and Coagulation Kinetics
( )
( )
( ){
( )( )dxducosrrsinun
dxdu cosrru
sinu(1)un
sin90-αsin - cosα
cosαunun
ji
ji
SlopeLocation
φ+φ−=•
φ+=
φ−=•
φ−==
=•
43421
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Particle Collision and Coagulation Kinetics
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Particle Collision and Coagulation Kinetics
( ) ( )( )
{ ( ){
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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Particle Collision and Coagulation Kinetics
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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Particle Collision and Coagulation Kinetics
( )
( )
( )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
Other Collision Mechanisms
( ) 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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Other Collision Mechanisms
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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Other Collision Mechanisms
)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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Other Collision Mechanisms
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Other Collision Mechanisms
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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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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
AggregationBreakupΔm4
Gm = 40s-1 Gm = 150s-1 Δm
4
AggregationBreakupΔm4
• (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)
D43=550 microns (evolved)
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Results (Secondary Clarifier)• Sludge Mass Contour
D43=200 microns (constant)
D43=550 microns (evolved)
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Results (Secondary Clarifier)• d43 Contour
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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=550 microns (evolved)
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