RTN PRATSOLIS (HPRN-CT-1999-0050) 7. DECEMBER 2001 BERLIN, Germany First training seminar Programme : Paschedag (TU Berlin) : “Simulation of precipitation reactors using commercial CFD software” (paschedag1.pdf) M. Signorino (TU Berlin) : “Programme of investigation on mixing turbulent processes in non reactive solid-liquid systems” (signorino.ppt) J. Derksen (TU Delft) : “Large eddy simulations on the sample flow case (pitched-blade turbine in baffled tank at Re=7,300)” (derksen1.ppt) H. Saint-Raymond (IRSID) and A. Alexiadis (IRSID) : “Inclusion removal from liquid steel” (saint-raymond1.ppt) M. Vanni , D. Marchisio, G. Baldi, A. Barresi (POLITO) : “Precipitation in turbulent fluids” (vanni1.ppt)
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RTN PRATSOLIS (HPRN-CT-1999-0050)
7. DECEMBER 2001
BERLIN, Germany
First training seminar Programme :
Paschedag (TU Berlin) : “Simulation of precipitation reactors using commercial CFD software” (paschedag1.pdf)
M. Signorino (TU Berlin) : “Programme of investigation on mixing turbulent processes in non reactive solid-liquid systems” (signorino.ppt)
J. Derksen (TU Delft) : “Large eddy simulations on the sample flow case (pitched-blade turbine in baffled tank at Re=7,300)” (derksen1.ppt)
H. Saint-Raymond (IRSID) and A. Alexiadis (IRSID) : “Inclusion removal from liquid steel” (saint-raymond1.ppt)
M. Vanni, D. Marchisio, G. Baldi, A. Barresi (POLITO) : “Precipitation in turbulent fluids” (vanni1.ppt)
Simulation of 1 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Einfuhrung
Simulation of Precipitation Reactors using Commercial CFD Software
Anja PaschedagTechnical University Berlin, Department of Chemical Engineering
Simulation of 2 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Introduction
Goal
~ Development of a code for numerical support of design of precipitation reactors
~ Modelling of the interaction between mixing and crystallization in a two−phase system
Basis
~ CFD codes (commercial or academic) with large number of models included (e.g. for turbulence) and stable numerical solution algorithms
~ Experimentally determined precipitation kinetics~ Codes for solving the population balance without transport phenomena~ Experimental setup for verification of test computations
Simulation of 3 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Introduction
Requirements
Extention of models in CFD codes by
~ Improved turbulence models for* turbulent mass transfer / turbulent mixing* influence of a second phase on turbulent mixing
~ Numerical handling of increase in dimensionality
Simulation of 4 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Mass Balance EquationsMass balance for solved species
Population balance with spacial dependencies
with and
formation by agglomeration
consumption by agglomeration
For each of that equations appropriate boundary conditions and initial conditions have to be defined.
∂cA B
∂t
v∇cA B
∇
D∇cA B
B fA B lmin3
∞
0
∂∂l
Gn
fA B l3dl
∂n∂t
v∇n G∂n∂l
Bagg
Dagg lmin
BG
Bagg
l
l2
2
l
0
β
l3 λ3 1
3 λ
l3 λ3
2
3n
l3 λ3 1
3 n
λ
dλ
Dagg
l
n
l
∞
0
β
l λ
n
λ
dλ
Simulation of 5 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Turbulence Models
RANS (URANS) LES DNS~ Averaging in time ~ Resolving structures ~ Direct resolution of all
over fluctuations above grid size, avera− scales in space and timeging small structures
~ No requirements concer− ~ Sufficient resolution of ~ Kolmogorov scale has toning grid and time step large structures in space be resolved by grid,from the model and time required appropriate time step
~ Only resolution of macro− ~ Direct resolution of micro−mixing (implementation of mixingmicromixing models possible)
numerical effort
accuracy
Simulation of 6 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
k − ε − Model
1) Representation of all values Φ by Φ = Φ + Φ’2) Inserting into balance and averaging of the equations
−> Balance in terms of averaged values, but with an additional term containing fluctuation values
3) Application of closure model for that term requiredmodels available at different state of complexity, high Re standard k−ε−model most common and best tested
Momentum balance:
Mass balance for species:
ρu
iu
j
µt∂ui
∂x j
∂u j
∂xi
δi j23
ρk µt
Cµρk2
ε
ρc
u
i
Dt∂c∂xi
Dt
µt
ρSct
Simulation of 7 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Probability Density Function Approach (PDF)
p
A B
Idea: define statistical measure for micromixing degree in the frame of RANS model
Realization: ~ For each cell probability of all possible
mixing states computed~ Computation of reaction rate based on
probabilities~ Additional transport equations for PDF~ Presentation of PDF
* full PDF* moments of preassumed function* finite mode
A B
Simulation of 8 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Kinetics
Approaches from literature available, but questions left about:
~ Accuracy of experimental determination (influence of mixing on the measured values)
~ Influence of activity coefficients~ Influence of surface effects
(’asymmetric’ behaviour for surplus components)
~ Influence of anisotropic growth
Mixing of inlet flows
fast chemical reaction
Supersaturation
Nucleation
Number of crystals
Crystal Growth / Agglomeration
Crystal size distribution
Simulation of 9 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Two−phase Models
Pseudo−one−phase Real multiphase
~ Particles handled like a solved ~ Particles desribed as an own phasechemical species (’concentration’considered)
~ No relative velocity between ~ Seperate transport equations forparticles and continuous phase both phases
~ Influence of particles on fluid flow ~ Interaction between phases included(esp. turbulence) has to be in balance equations and multiphasedescribed by empirical relations turbulence model
~ Concentration of a species is a ~ In present models dispersed phasesscalar and not a distribution constist of particles of unique sizefunction −> additional effort −> additional effort to simulate sizeto simulate changes in size distribution
~ Numerical effort relatively high ~ Numerical effort very high
Simulation of 1 0 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Numerics
Discretization of balance equations in most commercial codes by finite volume method (FVM)Problem: discretization of population balance terms
Method of moments Method of classes~ Transport property: moments of a ~ Transport property: mass of particles
distribution function in different size classes~ Small number of equations ~ Large number of equations~ Result: approximate continuous ~ Result: approximate discrete
distribution function distribution function~ Special derivation for the transport ~ Transport equations for classes result
equations of moments from application of FVM on size coord.~ Easy to derivate mean values of distr. ~ Some effort to derivate mean values~ Some effort to reconstruct distribution ~ Easy to reconstruct distribution
Simulation of 1 1 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Method of Classes
Particle size distribution varies along an additional ’internal’ coordinate −> Additional discretization along that direction using FVM−> Number of PDE in ’usual’ coordinates coupled by source terms containing growth
and agglomeration terms + boundary condition for lmin containing nucleation
Implementation into commercial CFD code~ Coupling terms computed explicitely, even if equations solved implicitely
−> numerical stability reduced−> inaccuracies in mass balance
~ Special description of agglomeration (discontinuous process)~ Size distribution changes strongly in space and time, but adaptive discretization
impossible
Simulation of 1 2 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Configuration of the Tubular ReactorChemical systemBa2+ + SO42− BaSO4 Na+ und Cl− as counterions
Geometry Inlet Concentrations
Rtot = 0.005 m cin,SO4 = 100.0 mol/m3
Rnozzle = 0.0005 m cin,Ba = 34.1 mol/m3
Ltot = 2.1 m
Lnozzle = 0.1 m
Na2SO4
BaCl2
Simulation of 1 3 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Numerical Setup for the Tubular Reactor Model
size classes: 45
transient ~ 1.5 residence times~ ∆t = 0.0005 s
2d simulation:rotational symmetry,in angular reaction section of 4 , 1 cellnumber of cells: 6580
(29 x 227)
geometry mesh density section of the mesh
Simulation of 1 4 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Comparison of ResultsRelative integral curves of the size distribution at the outlet averaged over the
cross section
Simulation TorinoCFD code FLUENTmethod of moments
Experiments Torinoturbidity measurements
Simulation BerlinCFD code Star−CDmethod of classessame kinetics as Torino
0 1e−06 2e−06 3e−06particle diameter (in m)
0
0.2
0.4
0.6
0.8
1
mas
s fr
acti
on
simulation Torinoexperiments Torinosimulation Berlin
Simulation of 1 5 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Comparison of ResultsDifferential size distribution at the outlet averaged over the cross section
0 1e−06 2e−06 3e−06particle diameter (in m)
0
2e+06
4e+06
6e+06
8e+06
1e+07p
arti
cle
mas
s d
ensi
ty (
in k
g/m
3 m
) simulation Torinosimulation Berlin
Simulation of 1 6 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Conclusions
~ Results of our simulation in same order of magnitude like simulations and experiments from Torino
~ Slope to steep, average size lo large − reasons?
~ Agglomeration has no significant influence on results in the tubular reactor
Simulation of 1 7 Technical University BerlinPrecipitation Reactors Department of Chemical Engineering
Prospect
~ Validation for different operating parameters (inlet concentrations with different concentration ratio, flow rate − restricted by reasonable Re and residence time)
~ Application of improved kinetics if available~ Implementation of PDF model~ Implementation of model for influence of solid phase on turbulence when
available~ Simulation of stirred tank
− longer residence times− relevant agglomeration− more complicated turbulent structuresdata for validation have to be available
~ Use of a non−commercial code
Technische Universität BerlinInstitut für Verfahrenstechnik1
Ing. Manfredi Signorino
Ing. Manfredi Signorino, PhD student TU-Berlin
Precipitation and Agglomeration in Turbulent Solid Liquid Systems
Technische Universität BerlinInstitut für Verfahrenstechnik2
Ing. Manfredi Signorino
Status
Aim: Investigation of mixing turbulent processes in non reactive solid-liquid systems.
First step: Experimental investigation in a pipe reactor.
Experimental Technique: Characterisation of mixing by temperature measurements (analogy between heat and mass-transfer).
Technische Universität BerlinInstitut für Verfahrenstechnik3
Ing. Manfredi Signorino
Experimental Setup
L
D
d
AL = 1500 mm; 1000 mm
D = 7 mm
d = 4.6mm
Fluid: Water + Suspended particles (glass particles, volume concentration up to 10% , diameter less then 0,1mm).
Experimental technique: Temperature measurements
• Five measurements point per cross-section.
• Radial and Axial temperature profile reconstruction.
Technische Universität BerlinInstitut für Verfahrenstechnik4
Ing. Manfredi Signorino
83.06
≅=m
f ahBi
λ
Biot Number (glass particles a = 0.1mm, λλm= 0.8 W/mK)
Heat-balance: The particles are reactive from the heat-exchange point of view.
Particles considerations
Particles influence expected :
• Turbulent temperature fluctuation (Kolmogorov scale of the problem ~ 17.5 µm, for Re = 15000).
• Mixing length (variation in the radial and axial temperature profile).
; Not negligible thermal-inertia
Technische Universität BerlinInstitut für Verfahrenstechnik5
Ing. Manfredi Signorino
L
D
d
A
50 cm
100 cm
0 cm
Thermocouples test
31,4
31,6
31,8
32
32,2
32,4
32,6
32,8
50 60 70 80 90 100
cm
°C
Global Accuracy (Thermocouples+Data Acquisition System) ±0,1°C.
Maximum Sample Rate = 10 kHz (per thermocouple).
Thermocouples Test: Re=18.000, U=2,5 m/s
Technische Universität BerlinInstitut für Verfahrenstechnik6
Ing. Manfredi Signorino
What we want to measure is
• Mean temperature distribution in each pipe section
• Mixing length
?Mixing temperature fluctuations
Summary
Technische Universität BerlinInstitut für Verfahrenstechnik7
Ing. Manfredi Signorino
• Better understanding of the particles influence on the flow field (turbulence, mixing,…).
• Formulation of a model for the influence of solid particles on turbulent flow and mixing.
• Implementation of the model in the CFD code.
• Simulation of different reactors, with different particle size.
n collision frequency between two aggregates of size i and j (no interaction)l local velocity gradient
( ) jjiij naaJ 30
34
+γ= &γ&
n collision efficiency to build an cluster of size i+j
Hydrodynamic interaction no interaction G(r)=1
V : interaction potential Van der Waals attractive force
Turbulent flow
+π= rGrJt drdV
6πaµ2n
)(4 2
∂rn∂
ββ r2νε )(
12
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PRATSOLIS Meeting HSR - AA / CP / IRSID
Aggregation - Fragmentation
n Current aggregation-fragmentation modelsl Van der Waals attractive force between particles
l hydrodynamic interactions (turbulent flow)
l cluster morphology : fractal description
l gas cavities : low breakage probability
slipping condition (parameter b)
n Adaptation to a non wetting system
l liquid - gas - particle interface :
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n Wetting system: liquid-solid interfacel Non slipping condition
Liquid-gas-particle interface
Gas layerGas layer ?
liquid
z
b solid
n Non wetting system: liquid-gas-solid interfacel Slipping condition with parameter b
z
liquid
solid
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Fluid
Particleb
v Hydrodynamic force acting on the fixed particle :Fh=fh vh
When h >> a :baba
a fh 32
6++
→ πµ
)(h f
ha
f 12
46 µπ
→ with ( ) ( )
−++
=h/b
h/blnb/hf )(
3161611When h<<a :
(based on Vinogradova’s work)
Rep <<1
with
a
Slipping parameter
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Steelmaking application
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
0,00001 0,0001 0,001 0,01 0,1 1 10
Col
lisio
n ef
ficie
ncy
γµπ=
&3 36 a
ACA
Calculated collision efficiency
b/a = 10-4b/a = 10-2
b/a = 101
b/a = 1
X
XXXX X XXX X X X X X X XX
X
X b/a = 106
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Non wetting effect
l Similar collision trajectories in both cases
l Calculated collision efficiencies of the same order of magnitude
l Non wetting conditions affect:
• cluster cohesion
• breakage probability (negligible in the case of steel)
• cluster morphology (reorganization ?)
Small effect of non wetting conditions on aggregation
07/12/2001 16
PRATSOLIS Meeting HSR - AA / CP / IRSID
( )
⋅=λτ
suspension
blank
IILn
L1
( ) ( ) dDDfmDCsca ⋅⋅= ∫∞
°)(,,λλτ
Experimental methodthe Turbidimetry
n A good knowledge of the optical properties of fractal clusters is required
n In-situ measurement of light scattered by particles or clusters
n Turbidity depends on the particle size distribution f(D)
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Experimental apparatus
0.5µm 1.5µm
1 µm1 µm
n Mono dispersed silica particlesl naturally hydrophilic
l hydrophobic by means of a surface treatment (silanation)
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PRATSOLIS Meeting HSR - AA / CP / IRSID
Experimental apparatus
TankXe-Hg Lamp
Light detector
Monochromator (grating)
Spectrophotometer
Optical fiber
Turbidity sensor
Data acquisition
n Wetting properties (contact angle) depend on water-ethanol mixture composition
ethanol content in
water ethanol mixture (%)Contact angle
0% 125
3.45% 118.5°
5% 116.4°
10% 140.8°
15% 89.9°
20% 79.1°
07/12/2001 19
PRATSOLIS Meeting HSR - AA / CP / IRSID
Experimental results
l Then different evolutions In non wetting conditions, clusters are bigger
with different optical properties (gas bridge)
l Different final levels aggregation - breakage competition
l At the beginning : similar evolution in both casesdominating phenomenon : primary particles aggregation
Wetting conditionsNon wetting conditions
0 400 800 1200 1600 2000time (s)
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
turb
idit
y(c
m-1
)
Silica 0.5 µm
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PRATSOLIS Meeting HSR - AA / CP / IRSID
Experimental results
0 400 800 1200 1600 2000time (s)
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
turb
idit
y(c
m-1
)
Wetting conditions Non wetting conditions
Silica 0.5 µm
Low level big aggregates formation
20 µm
High level (limit size)
small aggregates
20 µm
l Different final levels aggregation - breakage competition
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PRATSOLIS Meeting HSR - AA / CP / IRSID
0
0,2
0,4
0,6
0,8
1
1,2
1,4
0 5 10 15 20 25 30 35
Time (min)
Tu
rbid
ity
((cm
-1) λ = 501 nm
λ = 752 nm
MeasurementsModel
Wetting conditions
Silica 1,5 µm
l Good agreement between measurements and simulations (turbidity at different wavelength)
description of the time evolution of the particle size distribution
l The non wetting effect is correctly predict compare to experimental results
Simulations
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
0 5 10 15 20 25 30Time (min)
Tu
rbid
ity
(cm
-1)
Wetting system
Non wetting system
Simulation in non wetting conditions
Silica 1,5 µm
07/12/2001 22
PRATSOLIS Meeting HSR - AA / CP / IRSID
ε, m2/s3
Fluent package - Lagrangian Approach
Population balance in each reactor zone
ε5, Vup
ε3
ε4
Simulation of industrial treatment
ε1 ε2
07/12/2001 23
PRATSOLIS Meeting HSR - AA / CP / IRSID
Application
0 50 100 150 200
t (s)
Tot
al o
xyge
n of
met
al (p
pm)
100
200
300
400
Particle number in cluster
Clu
ster
nu
mb
er /
m3
10 09
1 10 100 ≥1000
10 10
10 11
10 12
10 13
10 14
t=100 s
df =3 compact aggregatedf =2 loose aggregate
Effect of industrial process parameters
Effect of model parameters
Qg =40 Nm3/h
Qg =114 Nm3/h
07/12/2001 24
PRATSOLIS Meeting HSR - AA / CP / IRSID
Particle interaction modellingPart I - Bubble size
distribution in a gas plume•Final objective : implementation in a CFD package of specific module for :
•coalescence (bubbles, droplets),
•aggregation (solid particles),
•fragmentation
07/12/2001 25
PRATSOLIS Meeting HSR - AA / CP / IRSID
Steps
n Identification of the best models (literature, otherPratsolis project)
n Selection among different modelsn Implementation in CFD packages of specific
modules for coalescence and break-upn Validation
07/12/2001 26
PRATSOLIS Meeting HSR - AA / CP / IRSID
Why study bubble size distribution?
n Physical proprieties related with the interfacial surface
n Drag coefficient for the velocity pattern
07/12/2001 27
PRATSOLIS Meeting HSR - AA / CP / IRSID
Bubble interactions
n Coalescencen Break-up
07/12/2001 28
PRATSOLIS Meeting HSR - AA / CP / IRSID
Coalescence mechanism
n Turbulent Collisionn Buoyancy collisionn Coalescence efficiency
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PRATSOLIS Meeting HSR - AA / CP / IRSID
Mechanism of collision
07/12/2001 30
PRATSOLIS Meeting HSR - AA / CP / IRSID
Rate of turbulentcollision (s-1 m-3)
( ) ( ) 213232312, 2796.0 bjbibjbijiT
ji ddddnn ++= εθ
•Prince et al.,1990
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PRATSOLIS Meeting HSR - AA / CP / IRSID
Rate of buoyancy collision (s-1 m-3)
( ) ( )rjribjbijiB
ji uuddnn ++= 2, 1963.0θ
•Friedlander,1977
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PRATSOLIS Meeting HSR - AA / CP / IRSID
Coalescence efficiency
)exp( ,,, jijiji t τλ −=
•Coulaloglou and Tavlarides; 1977
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PRATSOLIS Meeting HSR - AA / CP / IRSID
Break-up
n Turbulent break-up due to eddies collisionn Small eddies haven’t enough energyn Large eddies just transport the bubblesn Efficient eddy size = 0,2DB ÷ DB
( )( )
( )ξ
ξερ
σξ
ξεα ξ
dd
c
dnuv
bl
f
bb
B
+
=
−Ω
∫ 3113532
1
311
231
2 41.2
12exp
1923.0
1:
min
•Luo and Svensen, 1996
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PRATSOLIS Meeting HSR - AA / CP / IRSID
Local bubbles population balance
dtdtnnN
ki
k
i
Brik
Brik
k
i
N
i
Cik
k
ij
Cjikk ∫ ∑ ∑∫ ∑ ∑∑
−+
−+=
+=
−
=
−
= =
−
=
ττ
θθθθτ0 1
1
1,,
0
1
1 0,
1
, 21
)0()(
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PRATSOLIS Meeting HSR - AA / CP / IRSID
Example 1wg = 0,3 m s-1,ε = 0,2 m2 s-3,
τ = 1 s,NTOT= 2.000.000 bubbles
d0= 2mm
0,00E+00
1,00E+05
2,00E+05
3,00E+05
4,00E+05
5,00E+05
6,00E+05
7,00E+05
8,00E+05
9,00E+05
1,00E+06
0 0,001 0,002 0,003 0,004 0,005 0,006
dB[m]
ndt=1 [bulles m-3]
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Example 2wg = 0,3 m s-1,ε = 0,2 m2 s-3,
τ = 1 s,NTOT= 23.000.000 bubbles.
d0= 2 mm
0,00E+00
2,00E+05
4,00E+05
6,00E+05
8,00E+05
1,00E+06
1,20E+06
1,40E+06
1,60E+06
0 0,001 0,002 0,003 0,004 0,005 0,006 0,007 0,008
d [m]
ndt [bolle m-3]
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Conclusion
l Aggregation of primary particles is not very much influenced by wetting conditions.
l For high contact angle, fragmentation phenomenon appears to be less significant and big clusters can be formed.
l The aggregation model developed for wetting conditions has been adapted for non wetting system. Results are in good agreement with multi-wavelengths turbidity measurements.
l In the future, we want to improve the model with new developments concerning:
• - bubble description in reactor and interaction with inclusions
• - wall caption and behavior of cluster near the slag steel interface
Precipitation in Turbulent Fluids
D. D. MarchisioMarchisio, G. , G. BaldiBaldi, A. , A. BarresiBarresi, M., M.VanniVanniPolitecnico di TorinoPolitecnico di Torino
• Precipitation is a multi-step process:• Chemical reaction and nucleation• Crystal growth• Aggregation and breakage
Nucleation
• Nucleation is a very fast process
• Its rate is non-linear with respect to supersaturation
• Usually expressions are empirically based:
( )( )( )
1.77510
153
2.83 10,
2.53 10
A B s
A B
A B s
c c kJ c c
c c k−
× −= × −
Crystal growth
• Usually two processes control the over-all phenomenon
• In the case of diffusion and surface reaction:
( ) ( )( ) ( )( ) ( )
2,
,
,
A B r As Bs s
A B d A As
A B d B Bs
G c c k c c k
G c c k c c
G c c k c c
= −
= −
= −d
p
ShDMk
d ρ=
Aggregation and Breakage
• Crystals can aggregate (with possible further cementation: agglomeration) or break up
• Aggregation is a second-order process with respect of particle concentration
1 13 3
23
3 3 3 32
3 30
0
( ) , ( ) , ( )( )
2 ( )
( ) ( ) ( , ) ( )
L L n L n dLB L
L
D L n L L n d
β λ λ λ λ λ λ
λ
β λ λ λ∞
− − =−
=
∫
∫
Aggregation and breakage
• The Brownian mechanism has been considered for aggregation
• Aggregation kernel
• In the considered case the effect of breakage is negligible
( ) ( )22
,3
B Lk TL
L
λβ λ
µ λ
+=
L
λ
Mixing effects: CFD approach
• The CRE approach defines:– macro-mixing– meso-mixing– micro-mixing
• Using CFD macro- and meso- mixing are solved together but what about micro-mixing?
• Micromixing is a sub-grid scale phenomenon • Only introducing a SGS model micromixing is
taken into account
Governing equations
• The main challenge in turbulent reacting flow is to find a closure for the chemical source term appearing in the last equation
( )φ+φ∂∂
−
∂φ∂
∂∂
=∂φ∂
+∂φ∂
∂∂
−∂∂
ρ−
∂∂
ν∂∂
=∂
∂+
∂∂
=∂
∂
φφ kkSu
xxD
xxu
t
uuxx
px
u
xx
uu
t
u
x
u
kjjj
k
jj
kj
k
jijjj
i
jj
ij
i
i
i
'
''1
0
• Reynolds-averaged transport equations:
PDF methods: full PDF
• The problem is intractable with standard methods
• Monte-Carlo solvers
( )
'
2
j ij i
f fu u f
t x x
D f S f
φ φφ
α α φ α φα α
ψ
φ ψ ψψ ψ
∂ ∂ ∂ + + = ∂ ∂ ∂
∂ ∂ − ∇ + ∂ ∂
• Transport equation of the composition-PDF:
Presumed PDF methods
ξε−∂
ξ∂∂
ξ∂Γ+
∂
ξ∂Γ
∂∂
=∂
ξ∂+
∂
ξ∂
∂
ξ∂Γ
∂∂
=∂
ξ∂+
∂ξ∂
22''' 222
iit
it
iii
it
iii
xxxxxu
t
xxxu
t
• The functional form of the PDF can be assumed a prioriin terms of the mixture fraction, that is a non-reacting scalar
• The problem of mixing and reaction can be shifted to the problem of mixing of a conserved scalar:
Presumed PDF methods
• For two non-premixed streams the mixture fraction is defined to be zero in one feed stream and equal to unity in the other
• Non-reacting system
• Instantaneous reaction
ξ=Ao
oA
cc
ss
s
Ao
oA
sAo
oA
cc
cc
ξξξξξ
ξξ
>−−
=
<=
if 1
if 0
BoAo
Bos cc
c
+=ξ
Finite-mode PDF
• The mixture fraction PDF can be expressed in terms of a finite set of delta functions:
( )1
( ; , ) ( , )eN
n nn
f x t p x tξ ζ δ ζ ξ=
≡ −∑• Ne affects the ability to approximate the real PDF• The advantage of this method is to give an accurate
description with a small Ne
Comparison between the models
0 ζ 1
fξ(ζ)
Is=0.95
0 ζ 1
fξ(ζ)
Is=0.2
0 ζ 1
fξ(ζ)
Is=0.01
• The effect of the number of modes (Ne) has been investigated and Ne=3 was found to give an accurate description of mixing
Comparison between the models
• The finite mode PDF has been validated by comparison with Full PDF and beta PDF predictions
the Full PDF (--------), Beta PDF (_ _ _ ) and Finite Mode PDF (_______)
Finite-mode pdf transport eqs.
• Transport equations are written for volume fractions/probabilities of modes 1 and 2 and for weighted concentrations
( ) ( )
( ) ( )2232
22
1131
11
1
1
pppxp
xpu
xtp
pppx
p
xpu
xt
p
si
ti
ii
si
ti
ii
−γ−γ+
∂∂Γ
∂∂=
∂∂+
∂∂
−γ−γ+
∂∂
Γ∂∂
=∂∂
+∂∂
• The model parameters (γ, γs) are determined by forcing the variance to follow an adequate transport equation
2'Ckφ φ
εε φ=
sα
Population balance
• Population balance is a continuity statement based on the number density function
( ) ( ) ( ) ( )i ti i i
n nu n Gn B n D n
t x L x x
∂ ∂ ∂ ∂ ∂+ + = Γ + − ∂ ∂ ∂ ∂ ∂ • Different approaches can be used:• Classes methods: good accuracy, high
computational costs (30-50 scalars)• Standard Moment Method: poor accuracy
(expecially for aggregation problems) but low computational costs (4-6 scalars)
• Quadrature method of moments
Population balance
40 2 t 3 43
3, , V , dt t a v
mN m A k m k m m= = = =
• Lower-order moments are of particular interest
• The SMM solves the population balance in terms of the moments of the CSD
∫+∞
=0
)( dLLLnm jj
• With the SMM only size-independent growth and simple aggregation problems can be solved
Population balance
• In order to close the problem the QMOM can be used
• The method is based on an ad hoc quadratureformula in which abscissas and weights are obtained from the lower-order moments themselves
3
10
( ) j jj k k
k
m n L L dL w L+∞
=
= ≅ ∑∫
Validation of the QMOM
• Comparison of QMOM predictions with rigorous population balance solution (CM) in the case of perikinetic aggregation
Population balance
• The QMOM has been formulated for size-dependent growth rate
• The method has been validated for modeling aggregation by comparison with a Classes Method
• In this work the SMM has been used to model barium sulfate precipitation
Experimental setup and validation
• Precipitation of BaSO4in:
• Semi-batch Taylor-Couette reactor
• Continuous tubular reactor
Experimental setup and validation
• CFD has been used to model the flow field (Fluent)• The micromixing model and the population balance (SMM and/or QMOM) were included in the code itself by using user-defined scalars
Case Study 1: Couette reactor
• Validation of CFD predictions concerning the flow and turbulence field
• Validation of CFD predictions concerningdispersion of an inert tracer
• Validation of the model for parallel reactions and for barium sulfate precipitation
Flow fieldinvestigation
LASERMirror
Difraction lences
Glass window
Teflon rotating cylinder
Glass static cylinder
LASERMirror
Difraction lences
Glass window
Teflon rotating cylinder
Glass static cylinder
z
x y
v
w
RECEPTOR
TRANSMITOR
Transmitting lens
Receiving lens
Collimating lens
Optic fiber
Receiving fiber
z
x y
v
w
RECEPTOR
TRANSMITOR
Transmitting lens
Receiving lens
Collimating lens
Optic fiber
Receiving fiber
H/2
CFD validation: flow field• 2D and 3D simulations by using
different turbulence models and different near wall treatments with (FLUENT®) release 5.2
z
x y
z
x y
zu
uθ2
2
2
x x z x
x z z z
x z
u u u u u
u u u u u
u u u u u
θ
θ
θ θ θ
′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′
• Comparison was made in terms of number of vortices and mean velocities and Reynolds stress tensor components
CFD validation: flow field• Comparison showed that the RSM with standard wall functions
gives the best agreement
Experimental CFD predictionsM
ean
axia
l vel
ocity
, m/s
FP1
FP2
FP3
FP4
FP1
FP2
FP3
FP4
M1
M7
M10
M5
CFD validation: tracer dispersion
• Tracer dispersion was investigated at differentinjection positions (FP)
Mixing properties
• During injection, p1 enters into the reactor• Within 1 second, p2 disappears
Effect of operating conditions
• Effect of rotation speed of the inner cylinder
• Increasing the rotation speed, the degree of segregation is reduced
• This results in a slightly higher number of particles with lower dimension
Effect of operating conditions
• Effect of initial nominal supersaturation (So)
• An increase in So
results in an increase of the mean crystal size, because in these operating conditions growth is favored in respect of nucleation
Case Study 2: Tubular reactor
Ø The reactor was modeled by using a commercial CFD code (FLUENT®) release 5.2 Ø The standard k-ε model was used in a 2D axy-simmetric geometryØ Computational domain: 130×55 (8416 live cells)