BERLIN, Germany First training seminar training seminar1.pdf · RTN PRATSOLIS (HPRN-CT-1999-0050)...

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

Content~ Introduction~ Models~ Numerics~ Results~ Conclusions


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


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

~ Kinetic models for* nucleation* crystal growth* agglomeration* others

~ Numerical handling of increase in dimensionality


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


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


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


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


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


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


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


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


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


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


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


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


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



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).



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.



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



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



What we want to measure is

• Mean temperature distribution in each pipe section

• Mixing length

?Mixing temperature fluctuations

Summary



• 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.

• One particle size

• Particle size distribution

• Validation of the model by experiments.

• Application to large scale reactors.

Prospect

Kramers Laboratorium voor Fysische Technologie

Large-eddy simulations on the sample flow case

pitched blade turbine in baffled tank at Re=7,300

Jos Derksen


Department of Applied Physics

Delft University of Technology

The Netherlands

email: [email protected]


Outline

• Introduction– why large-eddy simulations (LES) in stirred tanks?

• Subgrid-scale modeling– Smagorinsky model– structure function model

– wall damping

• The sample case – pitched blade turbine case (experiments by Schäfer et al.

1998)

• Summary


Why LES in stirred tanks?

• Intrinsically unsteady flow

• Applicationse.g. agglomeration

micro-mixing

0 1 2 3 4 5t·N

Vel

ocity

tim

e se

ries

vtip


Example: agglomeration in crystallizers

• Particle-particle collisions• Contact time (to grow a bond)

agglomeration rate (#/m3s) : 2aggl mJ β=

2.2/12.22.23

collision /33.1

29.1d

νεγ+

νε≈β

&

sheared turbulenceγ&

simple shear

γ&

β

0 10 (s-1)0

2

(10-14 m3/s)

βcollision

β

Elco Hollander

averaged instantaneous

β


Agglomeration (2)

Evolution of the particle number concentration during 10 impeller revolutions


Example: micro-mixing

LES combined with (particle based) PDF methodsModeling scalar transport with competitive chemical reactions

forced turbulenceηK≈0.1∆

0

0.5

10-2

yie

ld o

f the

slo

w p

rod

uct

Damkohler: t /tturb chem

10 2100

(slow) product Q

consumed reactant A

Da=8

t=0.5tturb

t=2tturb

t=0

QCA

PBA

2

1k

k→+

→+k1>>k2

Eelco van Vliet


temporal evolution of species concentrations Da=8


Single-phase, turbulent flow: ηK/L∝Re-3/4

Computational grid: spatial low-pass filterfilter width: λfilter=2∆

Subgrid-scale motion: diffusive → νe

Subgrid-scale modeling

Smagorinsky model Structure function model*

( ) ijijsSme SSc 22∆=ν

∂∂

+∂∂

=i

j

j

iij

xu

xu

S21

( ) ( )[ ] 212

231050 //K

SFe ,xFC.,x ∆∆=∆ν −

( ) ( ) ( )∆=

+−=∆r

t,rxut,xu,xF2

2

F2: structure function

*Métais&Lesieur, JFM 239, 1992


Subgrid-scale modeling (2)

Isotropic, equilibrium turbulence, ∆ in the inertial subrange

cs=0.165 CK=1.4

Smagorinsky model versus structure function model

( ) iiijijsSF

e SSc. ωω+∆≈ν 2770 2

Smeν

Stirred tanks: anisotropic, off-equilibrium turbulence

is there an inertial subrange? → how to chose ∆?

cs=0.1 CK=2.7


Subgrid-scale modeling (3)Fully developed turbulence at Re=7,300?

possibly partly turbulent, partly transitional flow

Selective subgrid-scale modeling*:

*Voke, Theoret. Comput. Fl. Dyn. 8 (1996)

92

1 =β

βν

ν−−βν−ν=ν withexp

SmeSm

esele

ννe

( ) ν∆= /Sr/ 2122 2

0 50 1000

1

2

Smag.

selective


Subgrid-scale modeling (4)

Wall damping functions (Van Driest, 1956)

21

−ν=ν

++− A/yedamped,e e

A+=26

y+

anisotropyvanishing sgs stresses at the wall

No-slip walls


LDA data at Re=7,300*

•angle-averaged•angle-resolved

•mean velocity values•RMS values (Reynolds normal stresses)

*Schäfer et al., AIChE J 44, 1998

The sample flow

Pitched blade turbine revolving in a baffled tank

ν=

2NDRe with D the impeller diameter


Experimental validation (1)(angle-averaged velocity field midway between baffles)

0.5vtip

experiment

simulations interpolated to the experimental grid

1203 2403 3603

Smagorinskymodel

structurefunction

model

Smagorinskymodel withwall damping

0.5vtip


Intermezzo: Re-number effects*

LES on a 2403

grid(Smagorinsky

model with wall damping)

0.5vtip

*see also the experiments by Bittorf&Kresta (European Mixing 10, 2000)

Re=7,300 35,000 70,000 140,000


Experimental validation (2)(angle-resolved velocity field)

30o0o 60o

simulations interpolated to

the experimental grid

vtip

experiment

LES, Smag.2403 mesh

LES, SF2403 mesh


0.0

0.012

0.024

0.036

0.048

k/vtip220o

experiment

40o 60o

LES, Smag.2403 mesh

LES, SF2403 mesh

Experimental

validation (3)(angle-resolved tke field)


Summary

• LES for stirred tanks: detailed flow information– time dependence– micro-scale physics and chemistry

• Subgrid-scale models – equilibrium turbulence / inertial subrange– wall effectswhat resolution to chose? → experimental validation

• Smagorinsky versus Structure Function model– no significant differences for the average flow– with respect to tke: no clear conclusion

07/12/2001 1PRATSOLIS Meeting HSR - AA / CP / IRSID

Inclusions Removalfrom Liquid Steel

Cooperative work between :SPIN Laboratory : Pr. M. Cournil, F. Gruy, P. Cugniet

IRSID : H. Saint-Raymond, P. Gardin, A. Alexiadis


Steelmaking route

n Flat carbon steels for automotive and packaging application


A major Challenge :Clean Steel Elaboration

n Deoxidizing process l One of the last stage before Continuous Casting

l addition of deoxidizing agent (Mn, Si, Al, Ca, …) in liquid steel

• Formation of oxide particles in bath : Inclusion

– solid (Al2O3)

– liquid (CaO-Al2O3)

• Formation of clusters by aggregation

Defects in steel productsProcess perturbations


Alumina cluster in steel


Inclusion aggregation

Objectives

n Development of knowledge about elementary mechanisms on solid inclusion elimination :

flottationV→

Cluster Flotation

bullebubble

cluster

Bubble - clusterinteraction

SLAG

Slag entrapment

WALL

Wall deposit


Successive steps

n Particle-particle interactionl non wetting effect

n Formation and removal of inclusion clustersl aggregation - fragmentation - flotation

n Experimental validationl representative cold model : turbidimetric study of SiO2

aggregation in water - ethanol mixture

n Simulation of industrial treatmentl Simulation of inclusions removal in steel

l Fluid flow calculations in industrial reactor


Particle-particle interactionin non wetting conditions

n Experimental observation (Yaminski et al. 1983)l They observed a cavity between a glass

sphere immerged in mercury and a glass wall

Gas bridge

Hgglass

Gas bridge formation

Liquid

SolidParticle

Gas

l Propagation of the cavities during a collision

θ

α

Solid particleLiquid

Gas

n Thermodynamic Analysis (Kozakevitch et al. 1968)l Gaseous cavities could exist in particle

porosity


ii

i

kkki

kikki

kikjij

i

jjji

i FnBnBnnKnnKdtdn

−−+−= ∑∑∑∑−

=+

∞

=

∞

=−

−

=−

1

1k-i

11

1

121

Population balanceSmoluchowski equation

n Kij : aggregation kernel

n Bij : fragmentation kernel

n Fi : aggregate removal term by flotation


df

clusterft C

R3

4gU

2ρ

ρρ )−(=

RNumber of particles in cluster 1000 10000 100000Fractal dimension df 2.9 2 2.9 2 2.9 2Terminal velocity Ut (m/s) 2.8 10-5 8.6 10-6 1.3 10-4 2.6 10-5 5.7 10-4 7.8 10-5

Rcluster (microns) 5.4 20.1 12.0 63.7 26.5 201

radius of elementary particle a0 : 0.5 µm

Flotation

fdcluster NaR

1

0 ⋅=

Fractal description of clusters

07/12/2001 10

PRATSOLIS Meeting HSR - AA / CP / IRSID

0ijij JJ ααij=

Aggregation kernel: Kij

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

07/12/2001 11


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 :

07/12/2001 12


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

07/12/2001 13


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

07/12/2001 14


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

07/12/2001 15


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


( )

⋅=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)

07/12/2001 17


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)

07/12/2001 18


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


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

07/12/2001 20


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

07/12/2001 21


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


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


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


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


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


Why study bubble size distribution?

n Physical proprieties related with the interfacial surface

n Drag coefficient for the velocity pattern

07/12/2001 27


Bubble interactions

n Coalescencen Break-up

07/12/2001 28


Coalescence mechanism

n Turbulent Collisionn Buoyancy collisionn Coalescence efficiency

07/12/2001 29


Mechanism of collision

07/12/2001 30


Rate of turbulentcollision (s-1 m-3)

( ) ( ) 213232312, 2796.0 bjbibjbijiT

ji ddddnn ++= εθ

•Prince et al.,1990

07/12/2001 31


Rate of buoyancy collision (s-1 m-3)

( ) ( )rjribjbijiB

ji uuddnn ++= 2, 1963.0θ

•Friedlander,1977

07/12/2001 32


Coalescence efficiency

)exp( ,,, jijiji t τλ −=

•Coulaloglou and Tavlarides; 1977

07/12/2001 33


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

07/12/2001 34


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()(

07/12/2001 35


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]

07/12/2001 36


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]

07/12/2001 37


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

Berlin, Berlin, Pratsolis Pratsolis MeetingMeeting

December 7th, 2001December 7th, 2001

Precipitation and reactive crystallization

• 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

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)

Reactor geometry and operating conditions

• Internal diameter (main flow) = 10 mm• Length = 1500 mm• Internal diameter = 1 mm• Outer diameter = 1.5 mm• Velocity = 1 m/s• Re = 10000

• The two reactants were fed alternatively in the main flow and in the small coaxial tube

• The inlet concentrations were varied in order to study the effect on the CSD

Effect of ion excess on crystal size

Re = 10000; VR = 1; cA0 = 34.101 mol/m3

Exp. data with BaCl2 in nozzle Exp. data with Na2SO4 in nozzle

Model predictions w/o aggregation Model predictions with aggregation

Crystal morphology at high sulfate concentration

ØAt higher concentrations aggregation becomes important

cBO = 34.101 mol/m3

Aggregates morphology

Conclusions

• A model for investigating turbulent precipitation has been presented

• The model developed is CFD based• Micro-mixing has been included with a

presumed PDF model (finite-mode PDF)• The population balance has been modelled by

using the Standard Moment Method but a new approach (QMOM) has been presented

• The model has been validated through comparison with experimental data

Conclusions

• The finite-mode PDF model has been shown to describe with sufficient accuracy mixing and reaction in liquid turbulent media

• The role of the micro-mixing in CFD modeling has been investigated and cleared

• Kinetics expressions for barium sulfate nucleation and growth have been shown to be inadequate

• The SMM has been shown to be inadequate for high aggregation rate

• An alternative is constituted by the QMOM which has been partially validated

BERLIN, Germany First training seminar training seminar1.pdf · RTN PRATSOLIS (HPRN-CT-1999-0050)...

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