Modelling of the motion of a mixture of particles and a Newtonian fluid

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a Newtonian uid

by

Josene Maryna Wilms

Dissertation approved for the degree of Do tor of Philosophy

in the Natural S ien es at Stellenbos h University

Promoters:

Dr G.J.F. Smit

Fa ulty of Natural S ien es

Department of Applied Mathemati s

Dr G.P.J.Diederi ks

CSIR

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By submitting this dissertation ele troni ally, I de lare that the entirety of the work

ontained therein is my own, original work, that I am the sole author thereof (save

to the extent expli itly otherwise stated), that reprodu tion and publi ation thereof

by Stellenbos h University willnot infringe any third party rights and that I have not

previouslyin itsentirety or inpart submitted it for obtainingany quali ation.

Signature: ...

J.M. Wilms

2011/11/25

Date: ...

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A theoreti al model for the predi tion of parti le motion through a traversing

New-tonian uid is proposed. The model is derived by treating the uid as a ontinuum

and modelling its motion with the Navier-Stokes momentum- and mass onservation

equations. Appli ationof a Representative Elementary Volume (REV) yields

expres-sions for the onservation equations in terms of averages. The parti les are assumed

rigid andmomentum-and mass onservation equationsareinitiallyderived from

New-tonian prin iples for a single solid, spheri al parti le. A summation-based averaging

pro edure is appliedto obtain onservation expressions in terms of averaged variables

for the parti le phase.

Using the prin iple of momentum onservation, a ollision-sphere model is applied to

modelthe transfer of momentum between parti les. The momentum transfer between

the parti lesand the ontinuum ismodelledusing a modi ationof an existing

repre-sentative unit ell modelfor two-phase motion, mat hed with an REV-averaged form

of the Stokes drag law. In addition, an asymptoti mat hing pro edure is applied

between low- and high Reynolds number ows. The mat hing pro edures render the

modelappli ableto awide rangeof parti le volume fra tions and Reynolds numbers.

The theoreti almodelisimplemented intoanumeri al ode andthe numeri alresults,

yieldedfromthesesimulations,aretestedagainstresultsobtainedthroughsettlingtube

experiments done by the author at the Coun il for S ienti and Industrial Resear h

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'n Teoretiese model vir die voorspelling van partikelbeweging deur 'n omringende

dinamiese Newtoniese vloeistof word voorgestel. Die vloeistof momentum- en

mas-sabehoudwordmetdieNavier-Stokesmomentum-enmassabehoudsvergelykings

gemo-delleer. Hierdievergelykingswordintermevangemiddeldevloeistofeienskappe

voorge-stel deur 'n verteenwoordigende eenheidsvolume toe te pas. Dit word aanvaar dat

die deeltjies solied en bolvormig is. Momentum- en massabehoudsvergelykings vir

die deeltjies word afgeleideur, aanvanklik, behoudsvergelykings vir 'n enkele partikel,

op grond van Newton se wette, daar te stel. Volume gemiddeldes van bogenoemde

deeltjievergelykings word verkry deur dietoepassingvan 'nsommasietegniek.

Momentumoordragtussenindividueledeeltjiesisgemodelleerdeurdiebeginselvan

mo-mentumbehouden'nbotsing-sfeermodeltegebruik. 'nBestaandeverteenwoordigende

eenheidsselmodelisgewysigomditvantoepassingoptwee-fasevloeitemaak. 'n

Kom-binasievandielaasgenoemdemodelen dieStokesvergelykingvir diewrywingskrag op

'n sfeer, is gebruik om momentumoordrag tussen die deeltjies en die vloeistof te

mo-delleer. Daarbenewens is'nasimptotiese passingstegniekgebruikom'npassingtussen

lae- en hoë Reynolds getal vloeie te bewerkstellig. Die passingsprosedures het tot die

gevolg dat die model geskik is vir modellering oor 'n wye spektrum konsentrasie- en

Reynoldsgetalwaardes.

Die vergelykings is geïmplementeer deur 'nrekenaar programin Fortran teontwikkel.

Die afvoer van hierdie simulasies is vergelyk met eksperimentele resultate, afkomstig

van valbuis-eksperimente uitgevoer vir hierdie studie by die Wetenskaplike Navorsing

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I wish toa knowledge the following fortheir ontributions towards this study:

My supervisors, Dr G.J.F. Smit and Dr G.P.J.Diederi ks

Friends, family and pets for un onditionalsupport

The NRF for nan ialsupport duringthe rst fouryears of this study

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De laration

Opsomming ii

Contents iv

List of Figures viii

List of Tables xiii

Nomen lature xv

1 Introdu tion 1

1.1 Motivation . . . 1

1.2 Ba kground . . . 1

1.3 Obje tivesof this study . . . 3

1.4 Contributions and publi ations . . . 3

1.5 Overview of this work . . . 4

2 Literature review 6 2.1 Introdu tion . . . 6

2.2 ComputationalFluid Dynami s (CFD) development . . . 6

2.3 Classi ation of multi-phase ows . . . 9

2.4 Classi ation of modellingpro edures . . . 12

2.5 Parti le-phase methodologies . . . 12

2.6 Modelling pro edures for two-uidmodels . . . 15

3 Conservation equations 29 3.1 Introdu tion . . . 29

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3.2 Continuum mass onservation . . . 29

3.3 Continuum momentum onservation. . . 30

3.4 Parti le mass onservation . . . 32

3.5 Parti le momentum onservation . . . 33

4 Averaging 36 4.1 Introdu tion . . . 36

4.2 Arbitrary and Representative Elementary Volumes . . . 36

4.3 Averaging rules forthe ontinuum phase . . . 39

4.4 Averaging of the ontinuum mass onservation equation . . . 40

4.5 Averaging of the ontinuum momentum onservation equation . . . 42

4.6 Averaging rules forthe parti lephase . . . 44

4.7 Averaging of the parti lemass onservation equation . . . 45

4.8 Averaging of the parti lemomentum onservation equation . . . 50

4.9 Summary and on lusions . . . 52

5 Constitutive modelling 53 5.1 Introdu tion . . . 53

5.2 Continuum stress . . . 53

5.3 Parti le stress . . . 54

5.4 Appli ationof onstitutive laws . . . 56

5.5 Parti le intera tion . . . 59

5.6 Summary and on lusions . . . 73

6 Transfer laws: The Representative Unit Cell 74 6.1 Introdu tion . . . 74

6.2 1997 RUCmodel . . . 76

6.3 Adaptation tothe 1997 RUC. . . 83

6.4 Two-phase vis ous ow atthe lowReynolds numberlimit . . . 83

6.5 High Reynolds numberow . . . 88

6.6 Asymptoti mat hing . . . 88

7 Numeri al al ulation of the ow eld 92 7.1 Prin ipleof dis retisation. . . 92

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7.4 Dis retisation of the mass onservation equation . . . 104

7.5 Pressure and velo ity orre tions . . . 106

7.6 Relaxation . . . 109

7.7 Solutionof the dis retised equations. . . 110

7.8 Assembly of a omplete method . . . 114

7.9 Implementation of boundary onditions . . . 117

7.10 Con lusions . . . 118

8 Numeri al simulations 119 8.1 Introdu tion . . . 119

8.2 Basi owsimulations . . . 119

8.3 Two-phase ow . . . 123

8.4 Verti almotion . . . 130

8.5 Con lusions . . . 142

9 Physi al experiments 143 9.1 Settlingtube omponents . . . 143

9.2 Camerasetup . . . 147

9.3 Sample hara teristi s . . . 148

9.4 Experimentalresults and pro essing . . . 149

9.5 Con lusions . . . 155

10 Dis ussion and re ommendations 157 10.1 Empiri alwork by Ri hardson and Zaki (1954). . . 157

10.2 Comparisons toempiri almodels . . . 159

10.3 Con lusions . . . 161

11 Con luding Remarks 171 Appendi es 173 A For es a ting on a sphere 174 A.1 Introdu tion . . . 174

A.2 Volumefor es . . . 174

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B.1 Introdu tion . . . 186 B.2 Volumeaveraging . . . 186 B.3 Time averaging . . . 189 B.4 Ensemble averaging . . . 189 B.5 Averaging prin iples. . . 189 B.6 Averaging theorems . . . 190

B.7 Averaging of the onservation equations . . . 194

C Evaluation of the shear stress 195 C.1 Introdu tion . . . 195

C.2 Evaluation of the stress deviationterm . . . 195

D Momentum theorem 198 D.1 Introdu tion . . . 198

D.2 Derivation of the momentum theorem . . . 198

E Extension of ollisional-kineti for e to two dimensions 200 E.1 Introdu tion . . . 200

E.2 Newton's law . . . 200

F Dire tional omponents of the momentum equations 203 F.1 Introdu tion . . . 203

F.2 De ompositionof ve tor equations into omponent form. . . 203

G Experimental amera data 206

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2.1 Dilute,dispersed, and dense ow onditions (Loth (2006)). . . 11

2.2 Dierent representations for parti letreatment where the shadedarea rep-resentstheparti leandthegrid representsthe omputationalresolutionfor the ontinuous phase solution(Loth (2006)). . . 14

(a) Point-for e treatment. . . 14

(b) Resolved-surfa e treatment. . . 14

3.1 Solid parti le. . . 32

4.1 Variation of porosity in the neighbourhood of a point as a fun tion of the average volume. . . 38

4.2 The Representative Elementary Volume (REV). . . 39

4.3 Solid parti leat the REV boundary. . . 47

5.1 The three main forms of vis ous dissipation within granular ow: kineti , kineti - ollisional,and fri tional. . . 55

5.2 Elasti two-dimensional ollisionwith spe ular ree tion. . . 62

5.3 Two-dimensionalview of a ollisionsphere formed aroundParti le1. . . . 66

5.4 Proje ted Surfa eelement,

S

⊥

. . . 68

5.5 Sphere of Type 1subje ted toshear ow of loud of Type 2 parti les. . . . 69

6.1 Representative UnitCell (RUC). . . 75

6.2 Two-dimensionalRUC s hemati . . . 75

6.3 Plane Poiseuille ow. . . 79

6.4 Plane Poiseuille owfor the adapted model. . . 84

6.5 Flowby and owthrough momentum transfer given by Equations (6.4.16) (6.4.18). . . 87

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6.6 Flowby and owthrough momentum transfer given by Equations (6.4.16)

(6.4.18) for

ǫ

c

≥ 0.95

. . . 88

6.7 Inuen e of shifting parameter,

s

, onthe momentum transfer oe ient,

β

. 91 7.1 Grid arrangement. . . 93

7.2 Representation of the line-by-line method. . . 114

7.3 Adapted SIMPLE algorithmfor two-phase ow. . . 116

7.4 Grid arrangement. . . 117

8.1 Setupfor plane Poiseuille ow simulation. . . 120

8.2 Setupfor ow simulationthrough astationary porous medium. . . 120

8.3 Setupfor ow simulationpast and through a porous medium. . . 122

8.4 Simulatedandanalyti alvelo ityprolesforowinbetweenparallelplates, owthrough aporousbed and ow overand through aporous bed. . . 122

8.5 Setupfor horizontal two-phase ow. . . 123

8.6 Change in parti levolume fra tionwith time. . . 125

(a) Parti le volume fra tionsat t=0.01s and t=0.24 s. . . 125

(b) Parti le volume fra tionsat t=0.50s and t=1.85 s. . . 125

8.7 Change in parti levelo ity prolewith time. . . 126

(a) Parti le velo ities att=0.01 s and t=0.24s. . . 126

(b) Parti le velo ities att=0.50 s and t=1.85s. . . 126

8.8 Change in ontinuum velo ity prole with time. . . 127

(a) Continuum velo ities at t=0.01s and t=0.24 s. . . 127

(b) Continuum velo ities at t=0.50s and t=1.85 s. . . 127

8.9 Changesintheorderofmagnitudeofparti le-parti leintera tionfor es per unit volumeas a deposit ollapseswithina ontinuum. . . 128

(a) Parti le-parti le for es att=0.01 s and t=0.24s.. . . 128

(b) Parti le-parti le for es att=0.50 s and t=1.85s.. . . 128

8.10 Changes inthe order of magnitudeof parti le- ontinuumintera tion for es per unitvolume as adeposit ollapses withina ontinuum. . . 129

(a) Parti le- ontinuum for es at t=0.01s and t=0.24 s. . . 129

(b) Parti le- ontinuum for es at t=0.50s and t=1.85 s. . . 129

8.11 Setupfor verti al settlingsimulation. . . 130

8.12 Convergen e withina time step. . . 132

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8.15 Parti levolumefra tionof a3.6gsampleof1mmparti lesovera

340 × 60

grid. . . 135

8.16 Parti levolumefra tionof a3.6gsampleof1mmparti lesovera

170 × 30

grid. . . 136

8.17 Parti le volume fra tion of a3.6 gsampleof 1 mmparti lesover a

85 × 15

grid. . . 137

8.18 Average groupvelo ities for verti al parti le motion.. . . 140

8.19 Comparison of Matlab's fzero terminal velo ity solution to results for ter-minal velo ity obtained with 2PMS for dierent values of the asymptoti ttingparameter,

s

. . . 142

9.1 Strainoutput. . . 145

9.2 Settlingtube. . . 146

(a) S hemati of the settlingtube . . . 146

(b) Settlingtube . . . 146

9.3 Me hanismsof settling tube. . . 147

(a) Upper me hanism of settlingtube . . . 147

(b) Lower me hanismof settling tube . . . 147

9.4 Lightand amera positions. . . 147

9.5 Sili onbeads used for the experiments. . . 149

(a)

0.25 − 0.50

mm . . . 149

(b)

0.50 − 0.75

mm . . . 149

( )

0.75 − 1.00

mm . . . 149

p

→ 0

. . . 165

(b) Correlation for

ǫ

A.1 Simplied onservation of mass. . . 177

A.2 Dire tion of the pressure for e ona parti le. . . 179

A.3 Dire tion of the Saman for eona parti le. . . 181

A.4 Dire tion of the Magnusfor e ona parti le. . . 182

A.5 Drag oe ient,

C

D

, for asmooth sphere. . . 184

B.1 One-dimensional distributionfun tion

X

α

and itsderivatives. . . 188

(a) Unit fun tion. . . 188

= 0.585

. . . 213

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2.1 Dierent regimesfor two-phase dispersed ows a ording to Ishii(1975). . 9

2.2 Examples of single- and multi- omponent, multi-phase ows (Crowe et al. (1998)). . . 10

2.3 Forms of the in ompressible unsteady Navier-Stokes momentum equations (Loth (2006)). . . 13

2.4 Mixture vis osities proposed by various authors. . . 20

2.5 Dragfun tions by various authors. . . 24

4.1 Averagingrules for the ontinuum phase . . . 41

4.2 Averagingrules for the dis rete phase. . . 45

6.1 Geometri oe ients for agranular medium. . . 81

7.1 Dis retised expressions for the

x

-dire ted ontinuum momentum onserva-tion equation. . . 98

7.2 Conve tion and diusion oe ientsfor the ontinuum momentum onser-vation equation. . . 99

7.3 Dis retised expressions for the

x

-dire ted parti le momentum onservation equation. . . 101

7.4 Conve tion oe ients for the parti ulate momentum equation. . . 101

7.5 Conve tion anddiusion oe ientsfor

y

- ontinuummomentum onserva-tion equation. . . 103

7.6 Dis retised expressions for ontinuum mass onservation equation over

P

- ontrolvolume. . . 105

7.7 Conve tion oe ients for the ontinuum mass onservation equation. . . . 105

9.1 Parti le sizes. . . 148

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9.3 Comparisonbetween amera and settlingtube data.. . . 154

10.1 Physi al properties of material used for experiments done by Ri hardson

and Zaki(1954). . . 162

10.2 Empiri al group velo ities for various solid volume fra tions and parti le

sizes (Ri hardson and Zaki (1954)). . . 163

Chara teristi dimension of physi alsystem . . . [

m

℄

D

V

Chara teristi dimension of averagingvolume . . . [

m

th

parti le . . . [

−

℄

g

0

Radial distributionfun tion . . . [

−

℄

I

Unittensor,

I = i i + j j + k k

. . . [

Vis ous ontributionto

I

pc

. . . [

kg·m

−2

_·s

−2

℄

I

pc

m

∗

_{= m}

1 m

2 /(m

1 + m

−3

℄

P

Linearmomentum . . . [

kg·m·s

−1

℄

p

coll

Collisionalpressure . . . [

Pa

℄

p

S

x

Sour etermfor

x

- omponentofmomentum onservation equation[

kg·m

−3

_·s

−1

℄

S

y

S

y

for the parti les. . . [

kg·m

−3

t

Time . . . [

s

℄

u

pore

c

Velo ity prolewithin RUC hannel for single-phaseow . . . . [

m·s

−1

℄

℄

v

_r

Relativevelo ity,

v

r

= v

c

− v

℄

β

owby

o

Vis ous momentum transfer oe ient for high porosities . . . [

kg·m

−3

_·s

−1

℄

β

owthrough

o

Vis ous momentum transfer oe ientfor lowporosities . [

kg·m

−3

_·s

−1

℄

β

_∞

Momentum transfer oe ient for two-phase inertialow . . . . [

kg·m

−3

_·s

µ

mix

Mixture dynami vis osity . . . [

kg·m

−1

_·s

−1

℄

µ

p

Parti le dynami vis osity . . . [

kg·m

−1

_·s

−1

℄

ν

_k|x

Parti levolume ontainedwithinREVwhen entroidofparti leisatposition

x

i

. . . [

m

3

℄

ν

p(i)

Volume of parti le

ρ

ξ

α

Bulk vis osity of the alpha phase . . . [

N·m

−2

τ

pc

Parti le- ontinuum shear stress . . . [

N·m

−2

℄

τ

−

℄

Abbreviations

Re

Reynoldsnumberfor single-phase ow . . . [

−

℄

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Introdu tion

1.1 Motivation

Two-phaseowis be omingin reasinglysigni antin engineeringdesign and

te hnol-ogy. In addition to its pertinent appli ations in engineering and prevailing s ienti

problems it is alsorelevant tothe interpretation of natural phenomena and thus

war-rants further investigation.

Empiri al methodsare required to emulate anumberof diverse fa tors,su h as

appa-ratus geometry and physi al uid properties. It is therefore vital that engineers and

s ientists graspthe underlyingphysi sand theoreti almodellingfundamentaltothese

appli ations in orderto design equipmenta urately.

Currently, various Computational Fluid Dynami s (CFD) pa kages (e.g. FLUENT,

CFX) employ two-uid models to predi t the behaviour of parti les immersed in a

uid. The expressions that these two-uid systems use to model the drag, due to

the relative velo ity between the two phases, are often based on empiri al models,

derived from pressure-drop experiments in uidised beds. This presents the need for

an alternative model, basedpurely onthe physi s of the intera tions.

1.2 Ba kground

Thefollowingse tionsgiveabriefoverview ofthe ne essary ba kgroundtheoryforthis

study anddenethe on eptsthatwillbeusedinlaterstagesofthiswork. A omplete

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1.2.1 Classi ation of modelling pro edures

Ishii (1975), Enwald et al. (1997) and Loth (2006) divided the modelling pro edures

for two-phase ows into three ategories: Boltzmann -, Lagrangian -, and Eulerian

methods. Sin e the Boltzmann methods are not dire tly applied to this study, they

willonlybedis ussed briey inChapter2. Forthetime beingitsu es todistinguish

between the Lagrangian and Eulerian strategies.

1.2.1.1 Parti le phase methodologies

Basedontheframeofreferen e,modellingpro eduresfortheparti lephasearedivided

intotwo ategories namely Lagrangian orEulerian.

Lagrangianmodelstreattheuidphaseasa ontinuumand al ulateparti le

traje to-ries. Thisisdonebyeithertra kingea hindividualparti le(i.e. traje tory al ulation)

or by tra kinggroups of similar parti les(i.e. simultaneous parti le tra king).

The Euleriandes ription, when applied tothe dispersed phase, generallyassumes the

hara teristi sof the parti les (e.g. velo ity) an bedes ribed as a ontinuum.

Eulerian methods may be further subdivided into mixed- and separated-uid

ap-proa hes. The former assumes a negligible relative velo ity between phases and

de-s ribes the motion with a single set of onservation equations, whereas the latter

assumes that phase velo ities dier and the motion is modelled with two sets of

momentum-and mass onservation expressions: one set forea h phase.

1.2.2 Interphase oupling

Both Lagrangian and Eulerian treatments require a des ription for the intera tion

between the phases. The interphase oupling for e,

F

pc

, is a for e a ting on a single

parti leduetopressureandvis ousstresseswhi haretheresultofdisturban es aused

in the ow due tothe presen e of the parti le.

Su h a for e is equal in magnitude and opposite in dire tion to the hydrodynami

parti lefor ea tingonthe ontinuousphase. Itamountstothe hydrodynami surfa e

for es,

F

surf

, minus the ontributions from the undisturbed ow stresses,

F

c

. The undisturbed stresses,

F

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expressed by

F

pc

=

F

surf

− F

c

= F

D

+ F

T R

+ F

AM

+ F

HI

,

(1.2.1) where

F

D

,

F

T R

,

F

AM

,and

F

HI

,denotethedrag-,transverse orliftfor es, the added

massfor eandthehistoryfor e,respe tively(Kleinstreuer(2003),Croweetal.(1998)).

For heavy parti les (

ρ

p

≫ ρ

c

), the interphase for e is often simplied to in lude only the parti le drag (negle ting lift, added mass, and history ee ts, sin e they are

pro-portionalto

ρ

c

)i.e.

F

pc

= F

D

. For lightparti les(

ρ

p

≪ ρ

c

)with negligible ollisions, the parti lea eleration and body for e an be negle ted.

Asthenumberofparti lesin reases, ollisionsbe omemoreimportant,leadingtodense

ows. Thekeyaspe tfortheseows isthe properin orporationofthe parti le-parti le

ee ts on the parti lephase uiddynami s. Inparti ular, the parti le ollisions ause

ee tive stresses, whi h should be in orporated intothe parti le transportequation.

1.3 Obje tives of this study

Themainobje tiveofthisstudy isto reateamathemati almodelthat anpredi tthe

motion of parti le mixtures in a Newtonian uid with the potential to be modied in

futureworktoin orporateadditionalowregimes(e.g. anon-Newtonian ontinuumor

multiplephases). The integration of su h amodelintoan existing ode ould in rease

predi tion apabilities for industrial appli ations, while the pro ess of its derivation

ontributestoanimproved omprehensionoftheunderlyingphysi sthatgovern them.

It is also the obje tive of this work to provide a model that is apable of predi ting

a parti le vis osity and stress based on rst prin iples, thus eliminating the need for

estimating these parameters.

1.4 Contributions and publi ations

A novel method is used to average the parti le phase and the existing Representative

UnitCell(RUC)modelhasbeenmodiedtoin ludethe aseofvariableparti levolume

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A simulation ode was developed in Fortran and the two-phase ow equations were

solved numeri ally. Theseresults omparedwellwithdata obtainedfromsettlingtube

experiments atthe Coun il forS ienti and Industrial Resear h (CSIR).

Theappli ationofthesenewmodellingmethods,asappliedtolowparti levolume

fra -tions (

ǫ

p

≪ 1

), was presented at the International Conferen e of Numeri al Analysis andAppliedMathemati s (ICNAAM)duringSeptember2009(Wilmset al.(2009)). It

was expanded into a full arti le and published in Applied mathemati s and

omputa-tion (Smitetal.(2010)). Extensionofthedragtermtoin ludethe parti leintera tion

ee ts waspresented during September2010 at ICNAAM (Wilmset al. (2010)).

1.5 Overview of this work

Theoreti allythemotionofsolidparti lessuspendedinaNewtonianuidis ompletely

determined by requiring the Navier-Stokes equations to be satised at ea h point of

the uid, and equating ea h parti le's rate of hange of linear and angular momenta

to the resultant for e and the resultant torque applied to it. Termed a Lagrangian

des ription, the extensive pro essing power required by su h an approa h has proved

viableonlyforlowReynoldsnumbers enarios omprisingofarelativelysmallnumbers

of parti les. Hen e, the need for equationsbased onaveraged ow properties.

Averaged expressions, whi h are validfor allpointsin the ow domain, are developed

inChapter 3. Althoughtoo omplexforadire tsolution,they provideagoodstarting

pointfor thedevelopmentof mu hneeded averaging pro edures whi haredis ussed in

Chapter 4.

Following Ba hmat and Bear (1986), the mi ros opi Navier-Stokes expressions, as

derived inChapter3,areaveragedoveraRepresentativeElementaryVolume(REV)in

Chapter 4,yieldingequationsinvolume averaged form. Asummation-basedaveraging

method for the dis rete phase is used to ope with the dis ontinuous nature of the

parti les toprovide ma ros opi expressions forthe dispersed phase.

A oupling me hanism exists between the parti les for instan es of in reased parti le

volumefra tionswhi hresultinparti le-parti le ollisions. FollowingtheworkofClark

(2009), Bird et al. (2002), and Soo (1990), the losure of su h an intera tion term is

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However, the pro ess of averaging leavesa number of termsindeterminate. The

prob-lem of losure for the parti le- ontinuum intera tion is dis ussed in Chapter 6 and

yields an expression in terms of averaged variables by employing an extension of the

Representative Unitary Cell (RUC) model. The adaptation to the RUC is required

sin e itis asimpli ationof the REVand was introdu edby Du Plessis and Masliyah

(1988)fortheaveragingofsinglephaseowthroughstationary porousmedia. Chapter

6 on ludes the development of the dispersed two-phaseow model.

Chapter 7 is dedi ated to a dis ussion of the development of a simulation ode whi h

numeri allysolvestheexpressionsderived inChapter6. Theresultsobtainedfromthis

program are illustrated in Chapter 8 and ompared to experimental work ondu ted

at the CSIR inChapter 9.

Inadditiontotheaforementionedexperimentalveri ation,themodelistestedagainst

experimentaldataobtained byRi hardsonand Zaki(1954)inChapter 10. Theworkis

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Literature review

2.1 Introdu tion

Therehavebeenmany ontributorstotheadvan ementoftwo-phaseow. This hapter

attempts to provide a ba kground of the history of this resear h area and to get the

reader a quainted withterminology, enablingthemto distinguish between the various

lassi ations s hemes used in two-phaseow.

Detailed derivations of existing two-phase ow averaging identities, presented in this

part of the work, are done inpreparationfor envisaging ideas presented insubsequent

hapters.

2.2 Computational Fluid Dynami s (CFD)

development

An a ount of the history of multi-phase Computational Fluid Dynami s (CFD) is

given by Ly kowski (2010) in whi h the initiation and development of multi-phase

CFD from1970 to2010are dis ussed. A synopsis ofthe key ontributorsisgiven here

and the reader is referred to Enwaldet al. (1997)for a detailed summaryon uidised

bed simulationsup until1997.

Up untilthe 1970's,nu lear rea torli ensingsoftware appliedthe Homogeneous

Equi-librium Model (HEM), whi h meant that both phases were modelled as one. This

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Gidaspow, setout todevelopanewset ofequationsfortwo-phaseowwhi hwouldbe

equivalent to those developed for single-phase ow by Bird and his team (Bird et al.

(2002)). In1971Solbrigsu eededandthederivation,publishedinSolbrigandHughes

(1971), was in orporated intothe Seriated Loop(SLOOP) software.

In paralleltothesedevelopments,LosAlamosS ienti Laboratory(LASL)developed

a similar ode alled KACHINA. KACHINA was the rst software to provide

sta-ble numeri alsolutionsfor multidimensionaltwo-phaseuid dynami s (Amsdenet al.

(1999)).

During the mid1970's, Spalding(Spalding(1980) and Runshal(2009))who onsulted

with both LASL and Gidaspow, developed the Inter Phase Slip Algorithm (IPSA)

(Spalding (1976)): Apro edure tosolvePartialDierentialEquations(PDE's) similar

to that published by Solbrig and Hughes (1971). The methodwas embedded intothe

PHOENICS sour e ode in 1978.

Systems, S ien e andSoftware (

S

3

)startedwork in1975onageneral omputermodel

of uidised bed oal gasi ation alled CHEMFLUB, and the ompany, JAYCOR,

startedonasimilarsour e ode inthe early1980's alledFLAG.Thesewere transient,

two-dimensional programs whi h ontained PDE's similar to those in SLOOP (later,

STUBE(Solbrigetal.(1976)))andKACHINAsour e odesandin ludedvis ousstress

terms and an expression for the solids pressure. Work terminated onthe

S

3

software

beforeit wasdo umented.

KFIX was sour e ode used by LASL for modelling two-dimensional ow in

Loss-of-Fluid Tests (LOFT). Gidaspow had an idea to develop KFIX for the simulation of a

uidised bed and a quired the sour e ode in 1977 from LASL. It was subsequently

modiedby Gidaspow, Ly kowski and Galloway,and installed atthe Illinois Institute

of Te hnology (IIT).

Modi ations to KFIX involved the addition of a stabilising solids pressure term to

prevent over ompa tion. The addition of this term is dis ussed in Bouillard et al.

(1992). KFIX would later be known as FLUFIX whi h in turn was oupledwith the

EROSION/MOD1 software and was designated FLUFIX/MOD2. This was followed

by FORCE2, developed by Bab o k and Wil ox (Ding et al. (1993)). These sour e

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modellingof dense suspension (i.e. slurry) ows.

In 1985developmentontheCFDLIB software startedatLASLunderKashiwa(1987).

Itwasonlyin1991thattherstInternationalConferen eonMulti-phaseFlow(ICMF)

was held in Tsukuba, Japan. It was the rst of many with the 2010 ICMF held in

Tampa,USA.

In 1991, O'Brien and Syamlal started development on the open sour e ode alled

MFIX (Multi-phase Flow model with Interphase Ex hanges). Their obje tive being

the development of a ode that ould yield a reliable modelof uidised bed rea tors.

The rst version of MFIX applied numeri al te hniques found in early versions of

the previously mentioned IIT ode. MFIX was ompleted in 1993 and is maintained

by Oak Ridge National Laboratory (ORNL) inpartnership with the National Energy

Te hnologyLaboratory (NETL) inthe United States. Itis available atwww.mfix.org

and the latest version was released in 2007.

After ompletinghisPh.D.underGidaspowin1985,SyamlaljoinedFluent,In . where

he tookpartinfurtheringthedevelopment oftheFLUENT pa kagewhi hwasstarted

in 1983 by a small group at Creare In near Fluent In .'s present headquarters in

Lebanon, New Hampshire, USA.It wasoriginally reated by Swithenbank atSheeld

University inthe U.K.

Work on the ode has ontinued and is presently known as ANSYS FLUENT 12.0.

ANSYS also a quired the CFX ode, formerly FLOW3D, whi h was developed at

Harwell inthe U.K.It is nownamed ANSYS CFX.

TheOpenSour eFieldOperationandManipulation(OpenFOAM),C++based,sour e

ode is another appli ation that may be used to model multiple-phase ows. It is

produ ed by the UK ompany, OpenCFD Ltd., and is based on equations similar to

those usedinitsANSYS CFX ounterpart. Mostuiddynami ssolverappli ationsin

OpenFOAM use the pressure-impli it split-operator (PISO) or semi-impli it method

for pressure-linked equations (SIMPLE) algorithms. These algorithms are iterative

pro eduresforsolvingequationsforvelo ityandpressure,PISObeingusedfortransient

problems and SIMPLE forsteady-state (Barton (1998)).

Themajorityofsour e odesmentionedaboveutiliseatwo-uidapproa hasmodelling

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2.3 Classi ation of multi-phase ows

The numeri als hemes, appliedinthesour e odes dis ussedinSe tion2.2,havebeen

do umented and ategorised by Enwald et al. (1997) in a ordan e with Ishii (1975)

as a guideline.

Ishii set up a lassi ation whi h depended on the topology of the ow and

distin-guished between three lasses: separated, mixed and dispersed ows. For the purpose

ofunderstandingthe urrentworkonthemotionofparti lesinaNewtonianuid,only

the sub ategories of dispersed ows are listed inTable 2.1.

Table 2.1: Dierent regimes for two-phase dispersed ows a ordingto Ishii (1975).

Class Typi alregimes Geometry Conguration Examples

Dispersedow Bubblyow

b

c

b

c

b

c

b

c

_b

_c

b

c

b

c

Gasbubblesinliquid Chemi alrea tors

Dropletow bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bcbc bc

Liquiddropletsingas Spray ooling

Parti ulate ow bc bc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc

Solidparti lesingas

orliquid

Sedimentation

A ording to this lassi ation s heme, dispersed media are divided into bubbly-,

droplet-, and parti ulate ows: Bubbly ow physi ally manifests as gas bubbles in

liquid, whi h in ludes the everyday soda drink or the physi al pro esses in hemi al

rea tors. Flows in whi h liquid droplets oin ide within a gas is lassied as droplet

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Following Kleinstreuer (2003), Crowe et al. (1998), and Ishii (1975), the ow of

par-ti les and droplets in uids an be seen as a subset of multi- omponent, multi-phase

ows. Crowe et al. (1998), denes a omponent as a hemi al spe ies su h as

nitro-gen, oxygen or water whereas phase refers to the solid, liquid or vapour state of the

matter. Examples of single-phase, single- omponent ows in lude water- and

nitro-gen ows, whereas multi-phase single- omponent examples in lude steam-water ow.

Multi- omponent examples of single- and multi-phase ows are given by air ow and

air-water ow, respe tively. Theseexamples are listed in Table 2.2.

Table2.2: Examplesofsingle-andmulti- omponent,multi-phaseows(Croweetal.(1998)).

Single- omponent Multi- omponent Single-phase Waterow Nitrogenow Airow Flowofemulsions

Multi-phase Steam-waterow

Air-waterow

Slurryow

Thestudyof parti lesinwater maythereforebequaliedasamulti- omponent

exam-ple, sin e there are two separate hemi al spe ies involved (i.e. sili on (

Si

) parti les in water (

H

2 O

)). Moreover it may be qualied as multi-phase ow due to the sili on parti les being in a solid state and the water being in a liquid state. It follows that

the fo usinthisworkispla edonmulti-phase,multi- omponentregimesand on erns

itself with the motion of dispersed matter (i.e. a parti ulate phase) in a arrier uid

(i.e. a ontinuum phase).

Multi-phase,multi- omponentowsmayfurtherbedividedintosub lassesonthebasis

of how the omponents intera t with the arrier phase and with ea h other. These

intera tions are termed oupling me hanisms by various authors (e.g. Loth (2006),

Croweetal.(1998)andKleinstreuer(2003))andthe lassi ationoftheparti lephase

is most aptlydes ribed, followingLoth (2006), inFigure 2.1.

The broadest division isbetween dispersed and denseows and isbased onwhether it

isthe ontinuumordispersedphase thatdominatestheoverall motionof theparti les.

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aforementionedwithinstan eswheretheparti leee tsontheuidbe omesigni ant

through interphase oupling(e.g. drag for e).

DISPERSED











DILUTE











sparse







One-way oupling:

Continuous-uidae tsparti lemotion

(e.g. parti lerotated byvortex)

In reasing

volume

fra tion Two-way oupling:

Above plus parti le motion ae ts

ontinuous-uidmotion

(e.g. parti lewakein reasesdissipation)

Three-way oupling:

Above plus parti ledisturban e of the uid

lo allyae tsanotherparti le'smotion

(e.g. draftingofatrailingparti le)

DENSE











Collision-dominated ow







High-frequen yof ollisions

(e.g. energeti uidized

beds) Conta t-dominated ow

High-frequen yof onta t

(e.g. nearlysettledbeds)

Figure 2.1: Dilute, dispersed,anddense ow onditions (Loth(2006)).

Astheparti levolumefra tionin reases,dispersedowissubje ttothree-way oupling

where the parti le wakes and other lo al ontinuum disturban es ae t the motion of

nearby parti les. A further in rease in parti le volume fra tion indu es the last level

of the dispersed regimewhere four-way ouplingdominatesas parti le ollisionso ur

in ombinationwith allof the aforementionedintera tions.

When the parti le-parti leintera tions dominate, the ow is onsidered dense. These

intera tions an refer to two separate me hanisms: parti le-parti le ollisions and

parti le-parti le dynami intera tions. The former refers to intera tions where

par-ti les an rebound, shatter or oales e, whereas the latter refers to ases where the

parti les glide uponea h other, ausing fri tion.

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shouldbe appliedto adequately represent its motion. The various types of modelling

methodsavailablein literature are subsequently des ribed.

2.4 Classi ation of modelling pro edures

Ishii(1975)dividedthe modellingpro edures fortwo-phaseowsintothree ategories,

namely Boltzmann, Lagrangian,and Eulerianmethods.

Boltzmann theory uses a method analogous to dilute gas kineti theory to des ribe

the intera tions present in gas-parti le systems (Ahmadi and Ma (1990), Ding and

Gidaspow(1990)). This method denes a mole ulardistribution fun tionfor the

on-tinuum phase and another for the parti ulate phase. However, a ounting for size

distribution and the ollisionpro esses of the solid parti les with ea hother and with

the gas mole ules, proves hallenging.

The motionof asuspension anbeviewed intwoways: Inthe eldsofuidisationand

gas-parti le transport, separate equationsof motionare sought forea h of the phases,

whereas thoseinterested inthe rheology ofsuspensionsoften viewthe suspensionas a

whole. The two viewpoints should however be equivalent (Gidaspow (1986), Ja kson

(1997)). These diverse modelling approa hes mainly involve the parti le phase and a

on ise dis ussion follows inthe next se tion.

2.5 Parti le-phase methodologies

Based on the frame of referen e, the parti le phase is divided into two lassi ation

s hemesasEulerianorLagrangian. TheEulerianapproa h anbefurther lassiedinto

mixed orpoint-for e approa hes, while the Lagrangian methodis grouped into

point-for eorresolved-surfa e approa hes. Table2.3showsthevariousmodellingapproa hes

for the two-phasemedium.

2.5.1 Lagrangian method

The Lagrangian method, often referred to as the dis rete method, assumes that ea h

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ve-Table 2.3: Forms ofthein ompressible unsteadyNavier-Stokesmomentumequations (Loth

(2006)).

Dispersedphaseapproa h Dispersedphasemomentum Continuousphasemomentum

Eulerianwith

mixeduidtreatment

∂(ρ

m

v

m

)/∂t + ∇ · (ρ

m

v

m

v

m

) = ρ

m

g

− ∇p + µ

m

∇

2 v

m

where

ρ

m

= ǫ

p

ρ

p

+ ǫ

c

ρ

c

Appliedthroughoutdomain

Eulerianwith

point-for etreatment

ρ

p

_∂t

∂

(ǫ

p

v

p

) + ρ

p

∇ · (ǫ

p

v

p

v

p

) =

ǫ

p

ρ

p

g − ǫ

p

∇(p + p

coll

) +

ǫ

p

µ

c

∇

2 v

p

+ ǫ

p

F

pc

/ U

p

Appliedthroughoutthedomain

ρ

c

_∂t

∂

(ǫ

c

v

c

) + ρ

c

∇ · (ǫ

c

v

c

v

c

) =

ǫ

c

ρ

c

g

− ǫ

c

∇p + ǫ

c

µ

c

∇

2 v

c

−

ǫ

p

F

pc

/ U

p

Lagrangianwith

point-for etreatment

m

p

∂ v

p

∂t

= F

V ol

+ F

Surf

Applied along parti le tra

je to-ries

ρ

c

_∂t

∂

(ǫ

c

v

c

) + ρ

c

∇ · (ǫ

c

v

c

v

c

) =

ǫ

c

ρ

c

g

−ǫ

c

∇p+ǫ

c

µ

c

∇

2 v

c

−N

p

F

pc

Lagrangianwith resolved-surfa e treatment

m

p

∂ v

p

∂t

= F

V ol

+ F

Surf

+ F

pp

where

F

Surf

=

R

S

[−p + τ

pc

]nd S

anddoesnot ontain

F

pp

Applied along parti le tra

je to-ries

ρ

p

∂ v

_∂t

c

+ ρ

c

v

c

∇· v

c

= ρ

c

g

−∇p+

µ

c

∇

2 v

c

Applied outside of parti le

p

.

In ontrast,theEulerianmethodaveragesparti lepropertiesovera omputational

vol-ume. In brief,the Eulerianreferen eframeisastationarymeasurementof theaverage

of the system whilst the Lagrangian framemoveswith the elementit is measuring.

Forthetreatmentofsurfa efor es, thepoint-for emethodrepresentsthe owoverthe

parti lewithempiri alandtheoreti almethods(e.g. byspe ifyingadrag oe ient)to

obtain the for eonthe parti le. Forthe resolved-surfa eapproa h,the uid dynami s

(e.g. pressureand shearstressdistributions)are fullyresolvedovertheentireparti le's

surfa e and then integrated to obtain the overall hydrodynami for es. Hen e, for

the resolved-surfa etreatment, high spatial resolution of the ontinuous phase is thus

required over the parti le surfa e. Therefore, this method is sometimes alled dire t

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termsof omputationalresour es. FollowingLoth (2006),the dieren ebetween these

two approa hes is illustratedin Figures2.2a and 2.2b.

(a)Point-for etreatment. (b) Resolved-surfa e

treat-ment.

Figure2.2: Dierentrepresentationsforparti letreatmentwheretheshadedarearepresents

the parti le and the grid represents the omputational resolution for the ontinuous phase

solution (Loth(2006)).

Lagrangian models treatthe uid phaseasa ontinuum and al ulatesparti le

traje -tories. Typi al te hniques whi h may be applied to solve Lagrangian models in lude

(Wassen and Frank(2000)):

Traje tory Cal ulation (TC) A large number of parti le traje tories are

sequen-tially omputed. The average properties of the traje tory segments that ross a

omputational ell are determined in order to derive ma ros opi properties for

the dis retephase. The TC methodis however limitedtosteady ows.

Parti le-parti le ollisionshavebeen a ounted for by Oesterle and Petitjean (1993).

Simultaneous Parti le Tra king (SPT) The motions of a representative number

of parti les are al ulated simultaneously. Ea h simulated parti le represents a

ertain number of real parti les with similar hara teristi s. The ma ros opi

propertiesoftheparti ulatephasefora ertaingrid ellareobtained atanytime

by averagingoverallparti lesthatare lo atedinthat ellatthat time.

Parti le-parti le ollisionswere a ounted forby Tanaka and Tsuji (1991). In the

major-ity of appli ations, ollisions are treated sto hasti ally using Dire t Simulation

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2.5.2 Eulerian method

The Eulerian des ription, appliedto the dispersed phase, generallyassumes the

har-a teristi s of the parti les (e.g. velo ity) may be des ribed as a ontinuum. As listed

in Table 2.3, Eulerian te hniques are subdivided into mixed- and separated-uid

ap-proa hes.

2.5.2.1 Mixed-uid model

Inthemixed-uidapproa h,theassumptionismadethatthedieren esinvelo ityand

temperaturebetween the two phases are smallin omparison tovariations inthe eld

as a whole. The use of these models results in a singleset of momentum onservation

equations forthe ow mixtureas opposed toone set for the ontinuous phaseand one

set forthedispersedphase. Theapproa hisnumeri allyun ompli atedand,moreover,

is able to ope with both dispersed and dense onditions.

2.5.2.2 Separated-ow model

The separated-uid approa h for aEulerian des ription of the parti lephase with the

point for eassumption assumes that both the arrier uid and the parti les omprise

two separate, but intermixed, ontinua. Therefore, two sets of momentum equations

are required: one for the ontinuous phaseand the otherfor the dispersed phase. The

separated uid method is also alled the two-uid method. Here the relative velo ity

betweenthe phasesaretakenintoa ountand theequationswillgenerallybe oupled.

Su h an approa h will be applied in this work and the following se tion is devoted

to introdu ingthe reader to the approa hes followed by various authors in setting up

appropriate models.

2.6 Modelling pro edures for two-uid models

Generally, the ontinuum phase is modelled with the Navier-Stokes momentum- and

mass onservation equations. The onstru tion of a model for the dis rete phase is,

however, approa hed either with the Navier-Stokes expressions or, alternatively, by

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Theappli ationoftheNavier-Stokesrelationstotheparti ulatephaserequiresthe

for-mulationofdes riptionsforvariousvariableswhi hare welldened fromthemole ular

theory for uids,but are relatively unknown forsolids. Thesein lude thedenition of

the solid stress termwhi hinturn requiresexpressions for the dis retephasevis osity

and pressure.

Alternatively,thedis retephasemaybemodelledusingakineti theoryapproa h: The

momentum equation for a single sphere is onstru ted using Newton's se ond law of

motion and extended toa ount fora single parti leinsuspension(Cliftet al. (1978),

Soo(1990), and Enwaldet al.(1997)).

2.6.1 Traditional two-uid formulation

In the absen e of mass transfer, the ontinuity and momentum equations for both

phases are respe tively given by

∂ρ

α

∂t

+ ∇ · (ρ

α

v

α

) = 0,

(2.6.1) and

∂ρ

α

v

α

∂t

+ ∇ · ρ

α

v

α

v

α

− ∇ · σ

α

− ρ

α

g = 0,

(2.6.2) wherethe dis rete (or parti ulate) and ontinuum phases are respe tively denoted by

α = p

and

α = c

. Density and stress are denoted by

ρ

α

and

σ

α

respe tively. The lo al velo ity is denoted by

v

α

.

Asmentionedearlier,theordinarydierentialequationforea hparti lemaybesolved

using a Lagrangian approa h. Sin e this is omputationallyexpensive the alternative

is to apply anaveraging operatora tion onthe lo alinstantaneousequations.

Averagingmodelsmay be dividedintovolume, time,and ensembleaveragingmethods

and are dis ussed in Appendix B. Volume averaging, whi h, due to itsphysi al

inter-pretability, is the preferred method of averaging in this work, is applied to the mass

and momentum onservation expressions in the following se tions.

Theaveragingpro eduresforthemass-andmomentum onservationequations,byway

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2.6.1.1 Averaging of the mass onservation equation

Applying the volume averaging te hnique tothe mass onservation equation,yields

∂

∂t

hǫ

α

ρ

α

i + ∇ · hρ

α

v

α

i = 0.

(2.6.3)

2.6.1.2 Averaging of the momentum onservation equation

The averaging pro ess for the momentum equation yieldsvarious terms whi hrequire

further modelling in order to a hieve losure. The rst step is to apply the denition

of volume averagingto ea hterm in the momentum onservation equation:

∂ρ

α

v

α

∂t

+ h∇ · ρ

α

v

α

v

α

i −

D

∇ · σ

_α

E

−

ρ

α

g

=

0.

(2.6.4)

This is followed by the appli ation of Rules (B.6.23), (B.6.24) and (B.6.20) to the

averages of derivatives, to give

∂

∂t

hρ

α

v

α

i + ∇ · hρ

α

v

α

v

α

i − ∇ ·

D

σ

α

E

−

ρ

α

g

=

1 U

o

Z

S

αβ

ρ

α

v

α

w

αβ

· n

α

d S −

1 U

o

Z

S

αβ

ρ

α

v

α

v

α

· n

α

d S +

1 U

o

Z

S

αβ

σ

_α

_{· n}

_α

_{d S.}

(2.6.5)

In the absen e of ombustion or ondensation (i.e. when the interfa e velo ity,

w

αβ

,

equals that of the velo ity of the phase itself,

v

α

) Equation (2.6.5)willsimplify to

∂

∂t

hρ

α

v

α

i + ∇ · hρ

α

v

α

v

α

i − ∇ ·

D

σ

_α

E

₋

ρ

α

g

=

1 U

o

Z

S

αβ

σ

_α

_{· n}

_α

_{d S.}

(2.6.6) 2.6.1.3 Reynolds de omposition

Equation (2.6.6) annot be solved dire tly as it ontains averages of produ ts of the

dependent variables. To obtain a solvable set of equations, it must rst be rewritten

into expressions ontaining produ ts of the averaged variables. This is done by

em-ployingReynoldsde omposition. Reynoldsde ompositionofvariablesistypi allyused

in the eld of single-phase turbulen e modelling in order to separate the u tuating

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how-into produ ts of averages. The pro edure will result in extra terms in the equations,

ontaining produ ts of the u tuating omponents. These extra terms are analogous

to the Reynolds stressterms inthe ase ofsingle-phase turbulen emodelling(Enwald

et al. (1997)). Administering Reynoldsde omposition toageneralvariable,

Ω

α

, yields

Ω

α

= hΩ

α

i

α

+ e

Ω

α

,

(2.6.7)

where the denition of the intrinsi phase average is given by Equation (B.5.4). The

averageofthe deviationtermisassumedtobezero,whi h orrespondswiththe notion

that the volume over whi haveragingis doneisindeedasensiblerepresentation ofthe

ma ros opi average

D

e

Ω

α

E

= 0.

(2.6.8)

When Reynolds de ompositionis appliedto Equation (2.6.5)it yields

∂

∂t

hρ

α

v

α

i + ∇ · (ρ

α

h v

α

i

α

h v

α

i

α

) + ∇ · hρ

α

˜v

α

˜v

α

i − ∇ ·

D

σ

α

E

−

ρ

α

g

=

1 U

o

Z

S

αβ

σ

_α

_{· n}

_α

_{d S.}

(2.6.9) Theterm

∇·hρ

α

˜v

α

˜v

α

i

isgenerallyreferredtoastheReynoldsstresstermanddenoted by

σ

Re

α

(Enwald et al. (1997)). The right-hand side of Equation (2.6.9) is termed the interfa ialmomentum transfer.

TheReynolds stress forthe ontinuum phaseis modelledusing astandard Boussinesq

approximation. Fora more detailed a ount of this approa h the reader is referred to

the work of Enwaldet al. (1997), Simoninand Viollet(1989) and Simonin(1995).

Turbulen e models for the parti ulate phase available in literature are based on the

kineti theory of granular ow. Su h an approa h to the modelling of the parti ulate

phase uses lassi al results from kineti theory of dense gases, f. Dartevelle (2003),

ChapmanandCowling(1970)in ombinationwithGrad'stheory, f. Grad(1949),and

,andavis ousshear stress term,

τ

α

, i.e.

σ

α

= p

α

I + τ

α

.

(2.6.10)

Inthe following two se tionsthese twoterms are dis ussed.

2.6.1.5 Vis ous shear stress

The stress tensor for both phases is often modelled using the Newtonian strain-stress

relation:

τ

α

= ξ

α

(∇ · v

α

)I + 2µ

α

(S

α

−

1

3 (∇ · v

α

)I),

(2.6.11) wherethe strain-rate tensor is dened by

S =

1

2 ∇ v

α

+ (∇ v

α

)

T

_.

(2.6.12)

Ina ordan e withStokes' assumption,the bulkvis osity,

ξ

α

,is ommonlyset tozero inbothphases(Panton(1984)). Inpra ti e, thereasonfornegle tingthebulkvis osity

is the la k of reliable measurement te hniques (Prit hett et al. (1978)). A theoreti al

expression is however possible using the kineti theory of granular ow.

Fromtheassumption thatthereisnomasstransfer between thephases,itfollowsthat

∇ · v

α

= 0

. The remainingdynami vis osity,

µ

α

,is easyto spe ify forthe ontinuum phase with mole ulartheory but proves di ultfor the dis rete phase.

The parti le vis osity may be modelled as a fun tion of the parti le volume fra tion

(Enwald et al. (1997)). However, the majority of vis osity models available are for

mixture vis osities only. Examples of su h vis osity formulaeare listed inTable 2.4.