Simulations for modelling of population balance equations of particulate processes using the discrete particle model (DPM) [Elektronische Ressource] / Nageswara Rao Narni

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• Publié par: otto-von-guericke-universitat_magdeburg
• Publié le: 01 janvier 2009
• Nombre de lectures: 114
• Langue: English
• Poids de l'ouvrage: 1 Mo

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SIMULATIONS FOR MODELLING OF
POPULATION BALANCE EQUATIONS OF
PARTICULATE PROCESSES USING
DISCRETE PARTICLE MODEL (DPM)
NARNI NAGESWARA RAO
FakultätfürMathematik
Otto von GuerickeUniversitätMagdeburgSimulationsforModellingofPopulationBalance
EquationsofParticulateProcessesusingthe
DiscreteParticleModel(DPM)
Dissertation
zurErlangungdesakademischenGrades
doctorrerumnaturalium
(Dr. rer. nat.)
genehmigtdurchdieFakultätfürMathematik
derOtto von Guericke UniversitätMagdeburg
vonNarniNageswaraRao
geb. am01. July1979inKappaladoddi,India
Gutachter:
Prof. Dr. rer. nat. habil. GeraldWarnecke
Prof. Dr. Ing. habil. StefanHeinrich
Eingereichtam: 04. Februar2009
Verteidigungam: 24. März2009Nomenclature
LatinSymbol
a,b,c Constants —
2A Areaoftheparticlei mi
C Dragcoefficient —d
d Diameteroftheparticleα mα
hdi Averagediameter m
e Coefficientofrestitution —
2g Accelerationduetogravity m/sec
G Growthrate 1/sec
h Heightofthebed m
I Unittensor —
2I Momentofinertiaofparticleα Kgmα
J Impulsevector Kgm/sec
−16 0k Boltzmannconstant 1.380×10 erg/K
3K(t,x,y) Aggregationkernel m /sec
3K(x,y) Collisionfrequencyfunction m /sec
K (t) Aggregationefficiencyfunction 1/sec0
2K (t)gationkernelamongclassesiandj m/seci,j
L Lengthoftheparticlei mi
M Averagemolecularweightofair Kg/mol
m Massofparticleα Kgα
n Normalunitvector —
3n(t,x) Numberdensityattimetofparticlepropertyx 1/m
3n (x) Initialnumberdensityofthewithpropertyx 1/m0
3n Particleconcentrationofclassi 1/mi
3n Pofclassj 1/mj
3N Numberofcollisionsbetweenclassiandclassj m /seci,j
Nofparticlesofclassi —i
N Numberofofclassifromsimulation —i,sim
Nofparticlesofclassifromexperiment —i,expt
N Numberofofclassj —j
NofEuleriangridcells —cell
2p Pressureofthegas dyne/m (Pa)
r Radiusoftheparticleα mα
7 0R Gasconstant 8.31×10 erg/K mol
Re ParticleReynoldsnumber —α
hRei AverageRe —6
2s Surfaceareasoftheparticle m
S Stokesnumber —t
∗S CriticalStokesnumber —t
t Time sec
t Tangentialunitvector —
t Simulationtime secsim
0T Temperature K
T Torque Nmα
u Velocityofthegas m/sec
U Relativevelocity
v Velocityoftheparticleα m/secα
3V Volumeoftheα mα
3V Vofthebed mbed
3V VolumeofEuleriangridcell mcell
3V Vofthefluid mfluid
3V Volumeoftheparticles mparticles
3W Effectivevolumeoftheparticle m
∗ 3W Criticalvolumeofthe m
x Positionoftheparticleα mα
3x,y Volumeofthe m
z Randomnumber —
z Heightoftheparticleα mα
GreekSymbol
3β Interphasemomentumtransfercoefficient Kg/m sec
β Coefficientoftangentialrestitution —0
μ Gasphaseshearviscosity Kg/msec
3j −3μ Thejthmoment m ·mj
δ Distance m
δt Timestepforgasphase secflow
δ(x) Dirac deltadistribution —
δ Kroneckerdelta —i,j
Voidfraction —
λ Gasphasebulkviscosity Kg/msec
λ(t) Specificaggregationratefunction 1/sec
Ω Discretizeddomain —
ω Angularvelocity 1/sec
3ρ Densityofthegas Kg/m
η Bedparameter (dimensionless)
Γ Laminarshearvelocityofgas m/sec
2τ Gasphasestresstensor Kg/msec
σ Standarddeviation m7
Subscripts
0 Initialcondition
α Particleindex
a Pa
ab Betweenparticleaandb
b Particleb
agg Aggregation
bed Fluidizedbed
break Breakage
coll Collision
cell Cellsize
expt Experiment
i,j,k Index
n Normal
step Timestep
sim Simulation
t Tangential
Superscripts
− Meanoraveragevalues
∧ Simulatedvalue
n Valueatnthlevel
Acronyms
CA CellAverage
DPM DiscreteParticleModel
DEMElementMethod
EKE Equi partitionKineitcEnergykernel
IPSD InitialParticleSizeDistribution
KTGF KineticTheoryofGranularFlow
PBE PopulationBalanceEquation
PSD ParticleSizeDistribution
RE RelativeError
SSE SumofSquareofErrorsContents
1 GeneralIntroduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fluidizedbedspraygranulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Particleformationmechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1 Agglomeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.3 Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Modellingofthespraygranulationprocesses . . . . . . . . . . . . . . . . . . . 5
1.4.1 Multiscalemodellingoffluidizedbeds . . . . . . . . . . . . . . . . . . 5
1.4.2 Micro Macroofthefluidizedbed . . . . . . . . . . . . . . . . 7
1.5 Outlineofthethesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 PopulationBalanceEquations 10
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Aggregationequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Continuousformoftheaggregationequation . . . . . . . . . . . . . . . 12
2.3 Scalingoftheequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Dimensionalanalysisfordiscreteequation . . . . . . . . . . . . . . . . 13
2.3.2 Dimensionalanalysisofthecontinuousequation . . . . . . . . . . . . . 14
2.3.3 Number density selection and dimensional analysis of the kernel in ap
plications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Propertiesofagglomerationkernels . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 Theoreticalkernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.2 Empiricalkernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.3 Experimentalkernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 DiscreteParticleModel 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Thediscreteparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Thegas phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 Initialandboundaryconditions . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Two waycoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.1 Hardspherecollisionmodel . . . . . . . . . . . . . . . . . . . . . . . . 33
ixx CONTENTS
3.5 Numericalcalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 ModellingofcollisionfrequencyfunctionsusingDPM 43
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Derivationofcollisionfrequencyfunctions . . . . . . . . . . . . . . . . . . . . 44
4.3 Correctiontotheaggregationequation . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Physicaldescriptionoftheflowpatterninsidefluidizedbeds . . . . . . . . . . . 47
5 Simulationresults 49
5.1 Initialparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1.1 Initialassumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1.2 Initialparticlesizedistributions . . . . . . . . . . . . . . . . . . . . . . 50
5.2 Evaluationofthebedparameter . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3 Simulationresultsforapseudo3Dbed . . . . . . . . . . . . . . . . . . . . . . . 55
5.3.1 Log Normaldistribution . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.4 Simulationresultsfor3Dbeds . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6 Evaluationofaggregationefficiencyrate 66
6.1 Evaluationoftheaggregationefficiencyrate . . . . . . . . . . . . . . . . . . . . 66
6.2 Simulationresultsfortheaggregationefficiencyrate . . . . . . . . . . . . . . . 68
6.2.1 Aggregationinpseudo3Dfluidizedbed . . . . . . . . . . . . . . . . . . 69
6.2.2gationin3Dfluidizedbeds . . . . . . . . . . . . . . . . . . . . . 71
6.2.3 Aggregationinpseudo3Dfluidizedbedwithlog normaldistribution . . 74
6.3 Numericalmethodsforpopulationbalanceequations . . . . . . . . . . . . . . . 74
6.4 Computationoftheparticlesizedistributions . . . . . . . . . . . . . . . . . . . 77
6.4.1 Computationoftheparticlesizedistributionforpseudo3Dfluidizedbed 77
6.4.2ofthesizeutionfor3Dfluidizedbed . . . . 79
6.4.3 of the particle size distribution for 3D fluidized bed with
lognormaldistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7 Generalconclusionsandoutlook 81
A 83
A.1 Analyticalderivationofshearkernelforfluidizedbed . . . . . . . . . . . . . . . 83
A.2solutionforshearkernelformonodisperseinitialconditions . . . . . . 85
B 87
B.1 PhysicalandNumericalparametersofthesimulations . . . . . . . . . . . . . . . 87
B.2 Initialparticlesizedistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
B.2.1 Particlesizedistributionfor5000particles . . . . . . . . . . . . . . . . . 88
B.2.2 Psizeutioncorrespondingto15000particles . . . . . . . . 90Chapter1
GeneralIntroduction
1.1 Introduction
Mathematical modelling plays a vital role in all discliplines of science and engineering. Its aim
is to describe the real world phenomena through mathematics. Due to enormous improvements
in computational speed and algorithms, simulating many real world phenomena is within reach.
Using these simulation results modelers try to apply them to the realistic problems in different
fields such as indus