V T T P U B L I C A T I O N S

TECHNICAL RESEARCH CENTRE OF FINLAND ESPOO 2002

Hannu Karema

Numerical treatment of inter-phase coupling and phasicpressures in multi-fluid modelling

4 5 8

VTT PU

BLICATIO

NS 458

Numerical treatm

ent of inter-phase coupling and phasic pressures in multi-fluid m

odellingKarem

a

Tätä julkaisua myy Denna publikation säljs av This publication is available from

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This thesis work concentrates on the area of dispersed multi-phase flows and,especially,  to  the  problems  encountered  while  solving  their  governingequations numerically with a collocated Control Volume Method (CVM). Alltreatment  is  expressed  in  a  form suitable  for  local Body Fitted Coordinates(BFC) in a multi-block structure. All work is related to conditions found in asimplified fluidized bed reactor. The problems covered are the treatment andefficiency of inter-phase coupling terms in sequential solution, the exceedingof  the  bounds  of  validity  of  the  shared  pressure  concept  in  cases  of  highdispersed  phase  pressure  and  the  conservation  of  mass  in  momentuminterpolation for rapidly changing source terms.It is shown that advanced methods to treat interphase coupling terms result

in  a  faster  convergence  of  momentum  equations  despite  of  the  increasednumber of computational operations required by the algorithms. When solvingthe entire equation set, however, this improved solution efficiency is mostlylost due  to  the poorly performing pressure correction step  in which volumefractions  are  assumed  constant  and  the  global  mass  balancing  is  based  onshared  pressure.  Improved  pressure  correction  algorithms  utilizing  separatefluid  and  dispersed  phase  pressures  are  therefore  introduced.  Further,  anexpanded  Rhie-Chow  momentum  interpolation  scheme  is  derived  whichallows equal treatment for all pressures. All the computations are carried outin  the  context  of  a  collocated  multi-block  control  volume  solver  CFDS-FLOW3D.

ISBN 951–38–5969–X (soft back ed.) ISBN 951–38–5970–3 (URL: http://www.inf.vtt.fi/pdf/)ISSN 1235–0621 (soft back ed.) ISSN 1455–0849 (URL: http://www.inf.vtt.fi/pdf/)

VTT PUBLICATIONS 458

TECHNICAL RESEARCH CENTRE OF FINLANDESPOO 2002

Numerical treatment of inter-phasecoupling and phasic pressures in

multi-fluid modelling

Hannu KaremaVTT Processes

Thesis for the degree of Doctor of Technology to be presented with duepermission for public examination and criticism in Auditorium K1702 at Tampere

University of Technology on the 1st of March, 2002, at 12 o�clock noon.

ISBN 951�38�5969�X (soft back ed.)ISSN 1235�0621 (soft back ed.)

ISBN 951�38�5970�3 (URL:http://www.inf.vtt.fi/pdf/)ISSN 1455�0849 (URL:http://www.inf.vtt.fi/pdf/ )

Copyright © Valtion teknillinen tutkimuskeskus (VTT) 2002

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Otamedia Oy, Espoo 2002

3

Karema, Hannu. Numerical treatment of inter-phase coupling and phasic pressures in multi-fluidmodelling. Espoo 2002. Technical Research Centre of Finland, VTT Publications 458. 62 p. + app. 51 p.

Keywords multi-phase flow, multi-fluid modelling, inter-phase coupling, phasic pressures,numerical methods, Control Volume Method, Body Fitted Coordinates, fluidizedbeds, chemical reactors

AbstractThis thesis work concentrates on the area of dispersed multi-phase flowsand, especially, to the problems encountered while solving theirgoverning equations numerically with a collocated Control VolumeMethod (CVM). To allow flexible description of geometry all treatment isexpressed in a form suitable for local Body Fitted Coordinates (BFC) in amulti-block structure. All work is related to conditions found in asimplified fluidized bed reactor. The problems covered are the treatmentand efficiency of inter-phase coupling terms in sequential solution, theexceeding of the bounds of validity of the shared pressure concept incases of high dispersed phase pressure and the conservation of mass inmomentum interpolation for rapidly changing source terms.The efficiency of different inter-phase coupling algorithms is studied intypical fluidized bed conditions, where the coupling of momentumequations is moderate in most sections of the bed and where severalalternatives of different complexity exist. The interphase couplingalgorithms studied are the partially implicit treatment, the PartialElimination Algorithm (PEA) and the SImultaneous solution of Non-linearly Coupled Equations (SINCE). In addition to these specialtreatments of linearized coupling terms, the fundamental ideas of theSINCE are applied also to the SIMPLE(C) type pressure correctionequation in the framework of the Inter- Phase Slip Algorithm (IPSA). Theresulting solution algorithm referred to as the InterPhase Slip Algorithm �Coupled (IPSA�C) then incorporates interface couplings also into themass balancing shared pressure correction step of the solution.It is shown that these advanced methods to treat interphase coupling termsresult in a faster convergence of momentum equations despite of theincreased number of computational operations required by the algorithms.When solving the entire equation set, however, this improved solutionefficiency is mostly lost due to the poorly performing pressure correctionstep in which volume fractions are assumed constant and the global massbalancing is based on shared pressure. Improved pressure correctionalgorithms utilizing separate fluid and dispersed phase pressures, theFluid Pressure in Source term (FPS) and the Equivalent Approximation ofPressures (EAP), are then introduced. Further, an expanded Rhie-Chowmomentum interpolation scheme is derived which allows equal treatmentfor all pressures. All the computations are carried out in the context of acollocated multi-block control volume solver CFDS-FLOW3D.

4

PrefaceMost of this work was carried out at the Institute of Energy and ProcessEngineering at Tampere University of Technology while the author had aposition of junior research fellow in the Academy of Finland. Researchpossibilities obtained by this position are gratefully acknowledged. Thework was later completed at VTT Energy in the field of Forest Industryunder the favourable attitude of Seppo Viinikainen, the ResearchManager.

To my thesis supervisor Professor Reijo Karvinen I owe deep gratitudefor the research opportunity and facilities he provided. My sincere thanksbelong to AEA Technology for the help and co-operation they offered. Inparticular, my co-author Dr. Simon Lo faithfully and kindly provided allavailable time.

I express my thanks to my colleagues at the Institute of Energy andProcess Engineering as well as at VTT Energy for their encouragingattitude towards my work.

Finally, I warmly thank my wife Marianne for her patience through all thelate evenings spent and for her endless encouragement to complete thework.

Tampere, January 2002

Hannu Karema

ted in

zed

LIST OF PUBLICATIONS

This thesis work contains as an essential part the following manuscript locaAppendix E:

Karema, H. and Lo, S. Efficiency of interphase coupling algorithms in fluidibed conditions. Computers & Fluids, 1999. Vol. 28, pp. 323–360.

5

39

424445

47474950

53

CONTENTS

ABSTRACT 3PREFACE 4LIST OF PUBLICATIONS 5SYMBOLS 7

1 INTRODUCTION 11

1.1 Problem considered 111.2 Background 121.3 Outline of this work 14

2 MULTI-FLUID MODEL 16

2.1 Constitutive equations 172.1.1 Stress tensor 172.1.2 Interfacial momentum source 18

2.2 Momentum balances 192.3 Generalized forms 202.4 Discretized equations 23

2.4.1 Profile assumptions 262.4.2 Values on cell faces 272.4.3 Gradients at cell centres 282.4.4 Gradients on cell faces 28

3 INTER-PHASE COUPLING ALGORITHMS 32

4 MULTI-PRESSURE ALGORITHMS 38

4.1 Phase-sequential solution method and FPS4.1.1 Treatment of fluid phase 394.1.2 Treatment of dispersed phases in FPS

4.1.2.1 Incompressible model for dispersed phases4.1.2.2 Compressible model for dispersed phases

4.2 Equation-sequential solution method and EAP4.2.1 Treatment of dispersed phases in EAP

4.2.1.1 Incompressible model for dispersed phases4.2.1.2 Compressible model for dispersed phases

5 MOMENTUM INTERPOLATION 53

5.1 Extended momentum interpolation scheme

6 CONCLUSIONS 56

REFERENCES 58

APPENDICES

Appendix E of this publication is not included in the PDF version.

Please order the printed version to get the complete publication

(http://otatrip.hut.fi/vtt/jure/index.html)

6

SYMBOLS

adjugate Jacobian matrix, area vector of cell facem2

cell face area in computational space m2

, cell centre coefficients of discretized equations

interface transfer coefficient of momentum kg/m3s

convection coefficient kg/s

diffusion tensor kgm/s

covariant basis vectors

interfacial force density kg/m2s2

interfacial source of momentum kg/m2s2

total force per unit volume due to interface kg/m2s2

momentum source other than gravity kg/m2s2

total source term (used to collect contributions) kg/m2s2

geometric diffusion tensor m

g gravitational vector m/s2

, pressure correction coefficients

the total normal phasic mass flux kg/s

the total normal phasic momentum flux kgm/s2

Jacobian determinant

interfacial source of mass kg/m3s

source term in extended Rhie-Chow method

interfacial mass transfer from phase to kg/m3s

number of phases

number of dispersed phases

coefficient in extended Rhie-Chow method

n normal vector of surface

P pressure N/m2

, apparent pressure N/m2

modified and effective pressure N/m2

interfacial mean pressure N/m2

mass residual term kg/s

S source term in discretized equation

A i( )

AC

aUα

k aΦα

Bαβ

Ci

Dij

e i( )

FD

F I

FTI

FS

FT

Gij

Hαki hα

k

Imα

i

IUαk

i

J

M I

Mα

m· βα β αNP

Nd

Nα

Pa εP

e

Pe

PI

R mα

7

source term from deferred correction

(residual) source term kgm/s

T stress tensor N/m2

collision/contact stress of dispersed phase N/m2

kinetic stress of dispersed phase N/m2

particle-presence stress N/m2

turbulent momentum transfer (Reynolds stresses)N/m2

viscous stresses of fluid N/m2

t time sU velocity vector m/s

interfacial velocity m/s

cell volume in computational space m3

weight factor for cell face interpolation

arc weight factor for cell centre gradients

weight factor for pressure correction method

phase functionx special coordinate mx, y, z i, j and k coordinates in physical space m

diffusivity of a scalar property kg/ms

diffusivity tensor of a scalar property kgm/s

volume fraction

material viscosity kg/ms

eddy viscosity kg/ms

average mean curvature of interfaces m

density kg/m3

surface tension coefficient N/m

shear stress N/m2

general dependent variable

, , i, j and k coordinates in computational space m

generic integration operator

scalar product of vectors

vector product

dyadic product

SD

sUα

k

R

Tdc

Tdk

Tdp

Tft

Tfv

UI

VC

Wc

WH

Wpc

X

Γ

Γi j

εη

ηt

κρστΦξ ς ζ

⟨ ⟩•×⊗

8

, difference of operand at specified locations

difference of cell-based averages at respective times

cell-based average at time t

weighted linear interpolation to H between P and C

arc lengths between cell centres H and P m

deviations from mass-weighted phase average

contravariant component of a vector

constant part of a linear model for variable F

first order coefficient of a linear model for variable F

transpose of matrix F

macroscopic ‘effective’ constitutive property

value of dependent variable from previous outer iteration

intermediate value at current iteration

new value at current iteration

subscriptsI interface of phasesf fluid phased dispersed phase

, phase indexes

superscripts– phase average~ mass-weighted phase average^ normal flux component

F[ ]du

Fu

Fd

–

F[ ]tt ∆∆t+

F{ }t

H{ }

C

P

HP∩u″u′

F0

F1

F( )T

ηeff

Φ*

Φ*

Φ**

α β

9

10

ws often

tics ofs and, col-ingn ofe withet asulti-e theods in.

kingdent

energyphasehase con-it was tar-t sec-tionse usu-ently,cou-ved

ntialjectorithmng isidized i.e., thetart-up,he vol-unt inns that

re con-, the

Pres-

ollo-

1 INTRODUCTION

1.1 PROBLEM CONSIDERED

The field of multi-phase flow problems is a very wide field from stratified floto highly dispersed flows such as systems of aerosols. This diverseness quiterequires modelling and solution approaches exploiting the special characteriseach sub-type. This thesis work concentrates on the area of dispersed flowespecially, to the problems encountered while solving them numerically with alocated Control Volume Method (CVM). A requirement set from the beginnwas that all treatment is expressed in a form suitable for flexible descriptiogeometry. Thereby, an approach was selected based on multi-block structurBody Fitted Coordinates (BFC) at each block. A simplified fluidized bed was sthe special application to concentrate on. Although having this sub-type of mphase flow problems in mind from the beginning, the selected method to derivconservation equations and the numerical method studied are general methnature and they are also available to other types of multi-phase flow problems

When considering the multi-fluid equations two properties of them are striand typical of multi-phase problems. The first is the large number of depenvariables and the second is the inter-phase transfer of mass, momentum andbetween the phases. It is just the latter property that actually makes multi-problems more difficult to solve than single-phase problems. If the inter-ptransfer is negligible the multi-phase problem reduces almost to solving manyservation equations of the same type than in single-phase flow. Therefore, natural to start the work from the study of inter-phase coupling algorithms. Theget application, a fluidized bed, is especially interesting in this aspect as mostions of both gas-particle and liquid-particle beds normally operate in condiwhere the inter-phase coupling can be stated to be ‘moderate’. However, therally exist at least small spots in the bed area where the coupling is tight. Currthere is no information available on the required implicitness of inter-phase pling algorithms or what benefits in overall solution efficiency could be achieby a proper selection of them.

Deficiencies related to the customary solution algorithm applied in sequeCVM methods for multi-fluid problems were also expected. A principal subhere is the mass balancing pressure correction step of the interphase slip alg(IPSA) in which the volume fraction is held constant and the mass balancibased on the adjustment of global mass by shared pressure gradient. In flubeds the average of the normal component stresses of the dispersed phase,dispersed phase pressure, e.g., in connection of compaction of the bed at scorners of the bed and at heat transfer surfaces, is strongly dependent on tume fraction and, consequently, this dependency should be taken into accothe pressure correction step. The rising dispersed phase pressure also meathe performance of a pressure correction algorithm based on shared pressucept will be poor if sufficient at all. Simple methods to reduce these problemsFluid Pressure in Source term (FPS) and the Equivalent Approximation of sures (EAP), are therefore presented.

In addition, the selected geometric description for the CVM method, the c

11

lity,t. Thetum

ethodatived ashie-

dis-e lat-itsn of

Reekshavegingtimehesarityoryhasesrag-l flowo the].localssoci-tativethods founds. Bywn as

thelocal, e.g.,es, arepend

neralheseulti-st ofs to

lowsf theal par-

cated local body fitted coordinate (BFC) system with multi-block capabirequires some interpolation scheme to mimic a staggered grid arrangemencustomary solution for collocated methods is to utilize the Rhie-Chow momeninterpolation scheme. For rapid changes in source terms this interpolation mfails to conserve mass, a severe deficiency for a CVM method utilizing itersolution. An improved Rhie-Chow interpolation method is therefore presentewell as an extension to multiple pressures which is applicable both for the RChow interpolation method and its improved version.

1.2 BACKGROUND

Numerous different approaches to formulate conservation equations forpersed multi-phase flows exist. Only the most recent method in the field, thtice-Boltzmann method ([2], [3]), avoids this complication by having foundation in the description at the molecular level and in the mutual interactiothese entities. In other approaches, some kind of averaging is required. ([47], [48]) applied kinetic theory in the phase space while most of the others utilized regular space-time coordinates. The first derivation with volume averawas presented by Drew [19]. Another corner stone in the field, introducing averaging, is the more general publication of Ishii [31]. After first approac([11], [12]), the application of ensemble averaging has gained wide popul([20], [29], [30], [33]). This is in part a consequence of applying the kinetic theof dense gases [15] to derive the constitutive equations of the dispersed p([17], [6]). Especially in the context of fluidized beds, the special volume aveing applied by Anderson & Jackson [4] has functioned as a basis for severamodels, e.g, [45, 50]. The equations in this work are most closely related tmathematically formal treatment based on generalized functions by Drew [20

Although the conservation equations can be derived by integrating the conservation equations of mass, momentum and energy together with their aated interface conservation equations, i.e., jump conditions, over represendimensions of either space, time or realizations of the system, all these meproduce conservation equations of the same structure. The main difference isin the interpretation of the terms and in the content of the constitutive relationa rigorous mathematical treatment these averaged equations, customarily knothe multi-fluid equations, are an exact representation of the system. In reality,multi-fluid equations include several terms that depend on the microscopic variables or on the integrals over the inter-phase boundaries. These termsstresses, diffusional fluxes, inter-phase transfers and pseudo-turbulent stresscalled constitutive terms and they have to be replaced by models which deonly on averaged dynamic quantities or their derivatives. Discussions of geconstitutive requirements can be found, e.g., from [31] and [21]. It is just tconstitutive terms which bring in the typical problems related to solving the mfluid equation set. As no generally valid constitutive models exist and as mothem are quite crude approximations of reality, a multitude of different formwrite multi-fluid equations have evolved.

Two fundamentally different approaches to solve dispersed multi-phase fhave evolved in the past. Their conceptual difference lies in the treatment odispersed phases, where one is based on Lagrangian tracking of computation

12

con-

itstionslica-ann alsorriere dis-ds, i.e.,al-lts in aratived to hugeodsns inory

con-s.

corre- cur-

otherphased an

plicitcur-morea-e-over

innde work.re cor-t andressurese, i.e.,e frac- pres-idizedbed and itmeth- of the

Pres-

ticles ([25], [23], [56]) and the other is based on the idea of interpenetrating tinua and the Eulerian frame of reference, i.e., multi-fluid model ([19], [54], [31],[12], [20]). The former is naturally suited to dilute flows and the latter meetstheoretical justification better in denser flows [16]. However, numerous excepto this simple classification can be found, e.g., the PALAS concept [8], the apption of kinetic theory in the phase space ([47], [48]) and the lattice Boltzmmethod ([2], [3]). Efforts to combine the advantages of both approaches haveappeared [5]. Further, a variety of methods exist for the solution of the caphase balance equations and, in the case of the multi-fluid approach, for thpersed phase balance equations as well. Methods based on unstructured griFEM [28] and CVFEM [40], usually exploit the simultaneous solution of all bance equations. This is a consequence of the unstructured mesh which resusparse coefficient matrix and, consequently, prevents the use of efficient itesolvers tailored for band matrixes. This approach avoids the difficulties relatethe numerical implementation of the inter-phase transfer terms but requires aamount of run time computer memory. In contrast, Control Volume Meth(CVM) built on structured grids use the sequential solution of balance equatiocombination with iterative solvers, which results in large savings in memrequirement. The major disadvantage, as far as the multi-fluid approach iscerned, is the difficulty to implicitly consider the couplings of balance equation

The easiest and most natural way to treat the interphase coupling term sponds to considering it a linear source term, where the coefficient times therent phase variable forms the first-order term and the coefficient times the phase variable acts as a constant term [55]. This method to treat the inter-coupling terms corresponds to an implicit scheme for the first-order term anexplicit scheme for the constant term, and it is referred as an partially imtreatment. A fully explicit scheme would exploit an existing value also for the rent phase variable. To enhance convergence in conditions of tight coupling implicit methods are needed [42]. In this work, the fully implicit Partial Elimintion Algorithm (PEA) [52] and the semi-implicit Simultaneus solution of non-linarly coupled equations (SINCE) [37, 38], described in detail, are meant to cthis requirement.

Two well-known sequential iterative solution algorithms for CVM approachmulti-fluid conditions exist, i.e., the Interphase Slip Algorithm (IPSA) [51–53] athe Implicit Multifield (IMF) method [26]. As having more implicit nature, thIPSA method has been selected as the basis of the solution algorithm in thisThe distinctive features of the IPSA are related to the mass balancing pressurection step of the algorithm, in which the volume fractions are held constanthe mass balancing is based on the adjustment of global mass by shared pgradient. The average of the normal component stresses of the dispersed phathe dispersed phase pressure, is sometimes strongly dependent on the volumtion and, consequently, this dependency should be taken into a count in thesure correction step. The special case here in mind is the compaction of a flubed below the state of minimum fluidization. As the porosity of the fluidized reduces, the dispersed phase pressure steeply rises with the volume fractionis insufficient to use only the shared pressure for mass balancing. Two new ods to reduce these problems, the compressible and incompressible variantsFluid Pressure in Source term (FPS) and the Equivalent Approximation of sures (EAP), are introduced in this work.

13

e cus-sureolu-ou-

ofcitly

, the withsameesh forling ofcated thatourcehemeept isure the inde-k.

ed in-phaseer to

fluidis is

n, thend,ellrids. the

veredter 3hich, theo-

pi-ion,nter- man-lution

cus-ncy

Because the constant parts of the linearized interphase coupling terms artomarily treated as source terms, their information in a SIMPLE(C) type prescorrection equation [18], a part of the IPSA method, is lost. To improve this stion step, a solution algorithm, referred to as the Interphase Slip Algorithm-Cpled (IPSA-C), is given in this work. This method utilizes the same kindformalism as the SINCE to include the interphase coupling terms semi-impliin the pressure correction equation.

In order to achieve adequate geometric flexibility in constructing the meshCVM used is based on collocated local body fitted coordinate (BFC) systemmulti-block capability. The essential feature of collocated methods is that the mesh is used for all dependent variables. As the staggered arrangement of mmomentum equations is abandoned, some other means to avoid the decouppressure and velocity fields has to be used. The customary solution for collomethods is to utilize the Rhie-Chow momentum interpolation scheme [49]actually mimics the staggered grid arrangement. For rapid changes in the sterms this interpolation method fails to conserve mass. A better behaving schas been described in Appendix D. In addition, if the shared pressure concabandoned and a proper treatment is given for the dispersed phase pressRhie-Chow interpolation method has to be expanded. An approach availablependently or with the improved Rhie-Chow scheme is given in this thesis wor

1.3 OUTLINE OF THIS WORK

By following the theme of the work, the material presented has been dividthree conceptual parts: the theoretical and the mathematical basis, the intercoupling algorithms and further improvements of the solution method in ordgain better overall efficiency.

The first part comprises chapter 2 in which the mathematically exact multi-equations are given in together with the related interfacial jump conditions. Thmade to lay out solid basis for the later discussions of the numerics. In additiogeneralized form of multi-fluid equations for numerical solution are given afinally, discretized for collocated CVM solver. All the treatment in this part, as was in the other two parts, are carried out in a form applicable for local BFC gThis may complicate the following of the general idea but reveals betterrequirements and complications related to this geometric description.

The subject of the second part, the inter-phase coupling algorithms, are coin chapter 3 and in the related publication attached to Appendix E. The chapitself is only a short summary of the material discussed in the publication, wincludes a detailed description of the inter-phase coupling algorithms, i.e.explicit treatment, the partially implicit treatment, the semi-implicit SINCE algrithm and the fully implicit PEA algorithm. Their efficiency were tested in a tycal conditions found in gas-particle and liquid-particle fluidized beds. In additthe IPSA-C solution method is introduced. This solution method includes the iphase coupling terms also into the pressure correction step in a semi-implicitner, and consequently, enhances the accuracy of the traditional IPSA somethod.

The third part, i.e., chapter 4 and chapter 5, include improvements to thetomary IPSA solution method in order to achieve better overall solution efficie

14

on-re in, areo theluidw the wayima-

and to avoid numerical problems typical for collocated CVM in fluidized bed cditions. In chapter 4 the new multi-pressure algorithms, i.e., the Fluid PressuSource term (FPS) and the Equivalent Approximation of Pressures (EAP)explained. The relation of these methods to the multi-fluid equations and tmodel equation for discretization is given in chapter 2 in context of the multi-fbalance equations and constitutive models. Finally, chapter 5 describes homulti-pressure concept can be included in the momentum interpolation in awhich is consistent with the requirement of second-order accuracy for approxtion of pressure gradient.

15

tingphasee., the1] orovern-nd theically [20].

and

ate-n indi-of

the

m can

ed byresent

2 MULTI-FLUID MODEL

The governing equations of the multi-fluid model are derived by integralocal conservation equations of mass, momentum and energy, valid in each separately, together with their associated interface conservation equations, i.jump conditions, over representative dimensions of space [19, 54], time [3realizations of the system [12]. Each of these averaging methods produce ging equations of the same structure though the interpretation of the terms aoperations required for the interfacial source terms are unique. A mathematformal treatment based on the generalized functions has been given by DrewThis approach, followed below, has its origin in ensemble averaging.

Utilizing the notion for a generic integration process the phase averagethe mass-weighted phase average of a variable can be defined as

and . (1)

In equation (1) and are the detailed piece-wise continuous functions of mrial density and general dependent variable, whereas is the phase functiocating the phase present at a specified location and time. Thus, the definition is

(2)

It follows, that the average of phase function itself is the volume fraction ofphase in question

. (3)

Based on these definitions, the governing equations of mass and momentube written in the form:

, (4)

(5)

On the right side of equation (5), stands for the momentum source causother body forces than gravity. The two terms inside the angle brackets repthe interfacial source terms of mass and momentum denoted by

⟨ ⟩Φ

ΦαXαΦ⟨ ⟩εα

-----------------= ΦαXαρΦ⟨ ⟩

εαρα--------------------=

ρ ΦXα

Xα

Xα x t,( ) 1 if x is in phase α at time t

0 otherwise.

=

εα Xα⟨ ⟩=

t∂∂ εα ρα( ) ∇ εα ρα Uα( )•+ ρ U UI–( )[ ]α Xα∇•⟨ ⟩=

t∂∂ εα ρα Uα( ) ∇ εα ρα Uα Uα⊗( )•+ ∇ εαTα• εα ρα g Fα

S+ +=

ρU U UI–( ) T–[ ]α Xα∇•⟨ ⟩ .+

FαS

16

s where

ial

the sur- and

single

mod- beenn, ins offaceetti & Shen

into

and (6)

. (7)

The square brackets in these equations are used to express the limit procesthe interface is approached from the side of phase

. (8)

In addition, the interfacial sources and have to fulfil the interfacjump conditions of mass and momentum

and (9)

. (10)

In equation (10), is the contribution to the total force per unit volume onmixture due to the interface, mainly originating from the spatial changes in theface tension, and stands for the number of phases present. Without

, the equations (4) and (5) closely resemble the balance equations of phase flow.

2.1 CONSTITUTIVE EQUATIONS

A large number of studies and intuitive speculations about the constitutive els of the stress tensor and about the interfacial sources and haveconducted, e.g., [7, 11, 19, 20, 21, 29, 30, 31, 46]. In the following discussioaddition of the conventional treatments of Ishii [31] and Drew [20], the resultthe two consistent and mathematically rigorous control volume/control surapproaches are referred, namely the volume averaging method of ProsperJones [46] and the micromechanical ensemble averaging method of Hwang &[29, 30].

2.1.1 Stress tensor

In analogy with fluids, the stress tensor of each phase is normally dividedisotropic and deviatoric components

, (11)

MαI ρ U UI–( )[ ]α Xα∇•⟨ ⟩=

FαI ρU U UI–( ) T–[ ]α Xα∇•⟨ ⟩=

α

Φ[ ]α Φ xα t,( )xα xI→

lim=

MαI

FαI

MαI

α 1=

NP

∑ 0=

FαI

α 1=

NP

∑ FTI

=

FTI

NP MαI

FαI

Tα MαI Fα

I

Tα Pα I– τα+=

17

ation fromlised,entumbutionsersed dis-

by thestress

udesuced on themac-

ssed

Pros-tions

eanvan-

where the isotropic part , defined to be the mean of the trace of deformrate, is called the pressure. The stress tensor itself includes contributionsseveral different mechanisms. For the fluid phase two contributions are rea

and , corresponding to the viscous stresses and to the turbulent momtransfer (analogous to Reynolds stresses). For the dispersed phase, contriarise by the hydrodynamic forces at the control volume surface cutting dispentities, the particle-presence stress [7, 29], by the random motion of thepersed phase, the kinetic stress (analogous to Reynolds stresses), andcollisions and contacts between the dispersed entities, the collision/contact

[32, 39]. These contributions are noted in short as

and (12)

. (13)

2.1.2 Interfacial momentum source

The interfacial source of momentum , represented by equation (7), incltwo different participating mechanisms, namely the momentum transfer prodby the phase change process and the force resulting from the existing stressinterfaces. It is customary to describe the former mechanism by defining the roscopic interfacial velocity as [20, 31]

. (14)

Commonly, the latter portion of the interfacial momentum source is expreas [20, 31, 34]

, (15)

where is the macroscopic interfacial mean pressure [1, 31] defined by

, (16)

and denotes the interfacial force density

. (17)

Equation (15) corresponds to the ‘current engineering practice’ as stated byperetti & Jones [46]. Most of the published treatments and numerical simulaare based on this definition which is referred as the Interfacial Mean Pressuremodel, IMP, in this text.

As stated by Prosperetti & Jones [46], the definition of the interfacial mpressure given in equation (16) is restrictive in a way that the term

PαTα

Tfv

Tft

Tdp

Tdk

Tdc

Tf Tfv

Tft

+=

Td Tdp

Tdk

Tdc

+ +=

FαI

UαI

MαI Uα

I ρU U UI–( )[ ]α Xα∇•⟨ ⟩=

FαI

T v[ ]– α Xα∇•⟨ ⟩ PαI εα∇ Fα

D+=

PαI

PαI εα∇ P[ ]α– Xα∇⟨ ⟩=

FαD

FαD

P PαI

–[ ]α Xα∇ τfv[ ]α Xα∇•–⟨ ⟩=

PαI

PαI εα∇

18

pres-ntumIn theShenods refer-

can

ima- phase.rigidom8) is flowdragresultse dis-stress

e twoefini-picalal dis- This istings arersed

ishes in the case of a non-existent volume fraction gradient even for a varyingsure distribution on the interfaces. This inconsistency in the phasic momebalance may lead to non-physical results in cases where is important. mathematically rigorous treatments of Prosperetti & Jones [46] and Hwang & [29, 30], the definition (16) is avoided. Although using different averaging methand notions, both of these treatments come to parallel results, which are hereenced with a common name Direct Interfacial Force model, DIF.

According to DIF, the latter mechanism of the interfacial momentum sourcebe expressed as

. (18)

In equation (18), the first term on the right side originates from a linear approxtion of the continuous phase stress components in the scale of the dispersedThe existence of this term is supported by the equation of motion for small particles derived by Maxey & Riley [41]. In their treatment this term arises frthe contribution of the undisturbed fluid flow. The second term in equation (1the interfacial force density , a consequence of the disturbance to the fluidcaused by the particle, including the contributions from the hydrodynamic force, the apparent mass force and the Basset history force. The last term from the stress field on locations of the control surface where it intersects thpersed phase. A mathematically rigorous definition of the particle-presence

can be found from Batchelor [7] and Hwang & Shen [29].

2.2 MOMENTUM BALANCES

The momentum balances of the continuous and dispersed phases for thtreatments of the interfacial transfer term are now obtained by inserting the dtions (12)–(15) and (12)–(14), (18) in equation (5), respectively. Since a tydispersed multi-phase flow problem involves one continuous phase and severpersed phases, these balance equations are written in a corresponding form.achieved simply by including the phase interaction terms from all participaphases. Details of the rearrangement and simplification of the stress termgiven in Appendix A. Thus, the momentum balances for fluid and for dispephases , for the two different treatments, can be written as:

Interfacial Mean Pressure model, IMP;

(19)

PαI

Tv[ ]– α Xα∇•⟨ ⟩ εα∇ Tf

v• FαD ∇ εαTα

p( )•–+=

FαD

Tαp

fd

t∂∂ εfρf Uf( ) ∇ εfρf Uf Uf⊗( )•+ εf Pf∇– ∇ εfτf

v( )• ∇ εfTft( )•+ +=

PfI

Pf–( ) εf∇ MfIUf

I FTI Fβ

D

β 1=

Nd

∑– FfS εfρfg+ and+ + + +

19

ts nohe pha-0) are

c pres-ept ofh phase

hen

and theersedove,

.ls is

flowsthanodels

ted inener-

(20)

Direct Interfacial Force model, DIF;

(21)

(22)

If the relative velocities between the phases are not high, and if there exisappreciable dispersed phase expansion/contraction, it can be assumed that tsic pressures and interfacial mean pressures in equations (19) and (2equal in each phase [46]. Also, in the absence of surface tension the phasisures and are almost equal [22]. These simplifications lead to the concshared pressure, i.e., the pressure force realized in balance equation of eacis the fluid pressure multiplied by the volume fraction of the respective phase.

By reverting to the treatments of Prosperetti & Jones [46] and Hwang & S[29], it is realised that the term

(23)

represents the forces created by the difference of the mean interfacial stress mean fluid stress on those parts of the control surface intersecting the dispphase. Accordingly, with the same limitations as when considering abthe term (23) can be neglected. The same is true also for the approximation

Under these limitations the only difference between the IMP and DIF modefound in the viscous stress terms. On the other hand, in turbulent particulatethe contribution from the turbulent momentum transfer is often much larger from the viscous stresses and thus the difference between the mbecomes immaterial.

2.3 GENERALIZED FORMS

For numerical treatment, the balance equations (4), (19)–(22) are represena generalized form in which all problem specific models are adapted. In this g

t∂∂ εdρdUd( ) ∇ εdρdUd Ud⊗( )•+ εd Pd∇– ∇ εdτd

p( )•+=

∇ εd Tdk

Tdc

+( )( )• PdI

Pd–( ) εd∇ MdI Ud

I FdD Fd

S εdρdg+ ,+ + + + +

t∂∂ εfρf Uf( ) ∇ εfρf Uf Uf⊗( )•+ εf∇ Tf

v• ∇ εfTft( )•+=

MfIUf

I FTI Fβ

D ∇ εβ Tβp

Tfv

–( )( )•–( )β 1=

Nd

∑ FfS εfρf g+ and+–+ +

t∂∂ εdρdUd( ) ∇ εdρdUd Ud⊗( )•+ εd∇ Tf

v• ∇ εd Tdk

Tdc

+( )( )•+=

MdIUd

IFd

DFd

S εdρdg+ .+ + +

Pα PαI

Pf Pd

∇ εβ Tβp

Tfv

–( )( )•

PαI

Pα≈Pf Pd≈

Tft

Tfv

»

20

35]

terfa-ly, theulate coun-

s the

eneral

neral-

alized form, the interfacial source terms are customarily linearized as [14, 24,

and (24)

, where (25)

. (26)

In equation (26) is used to denote the constant term of the linearized incial force density and stands for the total number of phases. Consequentmulti-fluid balance equations of mass and momentum for a turbulent particflow are often expressed as a straightforward extension of their single phaseterparts as

and (27)

(28)

where the source term is used to collect different terms of that character

. (29)

Here, the treatment of viscous stresses in the momentum equation (28) followIMP-model as usual.

The generalized momentum balance has been written on the basis of a glinear viscous fluid model, which is expressed as

. (30)

In single phase flows, the constitutive model (30) is easily adapted to the geized form of the momentum balance by defining the modified pressure

MαI

m· βα m· αβ–( )β 1=

NP

∑=

FαI

m· βαUβ m·αβ Uα–( )β 1=

NP

∑ FαD

+=

FαD

Bαβ Uβ Uα–( )β 1=

NP

∑ F0 D

α+=

F0 D

αNP

t∂∂ εαρα( ) ∇ εαραUα( )•+ m· βα m· αβ–( )

β 1=

NP

∑=

t∂∂ εαραUα( ) ∇ εα ραUα Uα⊗ ηα

eff Uα∇–( )( )•+

εα Pf∇– ∇ εαηαeff Uα∇( )

T( )• εαραg+ +=

m· βαUβ m·αβ Uα–( )β 1=

NP

∑ Bαβ Uβ Uα–( )β 1=

NP

∑ FαT ,+ + +

FαT

FαT

FαS

F0 D

α+=

Tα ηα Uα∇ Uα∇( )T

+( ) Pα I– ηαb 2

3---ηα–

∇ Uα I•+=

21

thens dif-tionus, a

1) isssureequa-

es, ulent term

utiversed appli-

calaraptingle

, (31)

which is used in place of . This is in contrast with multi-phase flows, whereappearance of the hydrodynamic pressure terms and other stress contributiofer from each other. In this case, the definition (31) provides no valid simplificaof the balance equations with either of the approaches, the IMP or the DIF. Thmore suitable template for the momentum balance (28) is provided by

(32)

where the phasic pressure in the definition of the modified pressure (3associated with the other pressure like contributions than the mean fluid pre

. For the fluid phase, this modified pressure arises from the last term of the tion (31) and from the eddy viscosity hypothesis

(33)

where represents the deviations from the mass-weighted phase averagdenotes the eddy viscosity and stands for the kinetic energy of the turbfluctuations, respectively. The ‘turbulent pressure’ is identified as the secondon the right side.

In the same way, additional ‘pressure’ contributions arise from the constitequations for the kinetic and collisional momentum transfer of the dispephases [27, 32, 39]. Since these contributions become considerable in somecations, e.g., in fluidized beds [13], they can not generally be neglected.

In numerical solution the momentum balance (32) is treated like three svariables, each representing one Cartesian component of momentum, by adit to the corresponding balance equation of a generic dependent scalar variab

(34)

Pαe

Pα ηαb 2

3---ηα–

∇ Uα•–=

Pα

t∂∂ εαραUα( ) ∇ εα ραUα Uα⊗ ηα

eff Uα∇–( )( )•+

εα Pf∇– ∇ εαηαeff Uα∇( )

T( )• εαPα

e( )∇– εαραg+ +=

m· βαUβ m·αβ Uα–( )β 1=

NP

∑ Bαβ Uβ Uα–( )β 1=

NP

∑ FαT ,+ + +

Pα

Pf

εfTft

Xαρ u″α u″α⊗⟨ ⟩α f=

=

εf ηft Uf∇ Uf∇( )

T+( ) 2

3---ρf kf I

23---ηf

t ∇ Uf I•–– ,=

u″α ηft

kf

Φα

t∂∂ εα ραΦα( ) ∇ εα ρα UαΦα Γα Φα∇–( )( )•+

m· βαΦβ m· αβ Φα–( )β 1=

NP

∑ BαβI Φβ Φα–( )

β 1=

NP

∑ Sα .+ +=

22

qua-atesin theis theionalysicalls of

phase been

equa-re

entspo-

d as

acro- beenre thetized.ted

2.4 DISCRETIZED EQUATIONS

In order to facilitate the numerical solution of the macroscopic balance etions (27), (32) and (34) in a general 3D geometry using Body-Fitted Coordin(BFC) combined with the multi-block method, these equations are expressed covariant tensor form [14, 36]. The coordinate system used in this context local non-orthogonal coordinate system , referred to as the computatspace, obtained from the Cartesian coordinate system , i.e., the phspace, by the numerical curvilinear coordinate transformation . Detaithe derivation are given in Appendix B.

Because there is no danger of confusion, the phase and mass-weightedaverage notations above the dependent variables, e.g., and , haveomitted in the subsequent treatment in order to enhance the readability of thetions. Thus, the transformed balance equations in the computational space a

, (35)

(36)

(37)

where is the Jacobian determinant (B3), (Figure 1(a)) represthe adjugate Jacobian matrix (B4), stands for the normal flux velocity comnent (B21), and the total normal phasic fluxes , and are definespecified in Appendix B.

In the process of the coordinate transformation, all the derivatives of the mscopic balance equations (27), (32) and (34) in the physical space haveexpressed with the corresponding terms in the computational space, whetransformed macroscopic balance equations (35), (36) and (37) are discreThis is done following the conservative finite-volume approach with collocadependent variables.

ξ ς ζ, ,( )x y z, ,( )

ξ ix

j( )

Φα Φα

t∂∂ J εαρα( )

ξi∂∂

Imα

i( )+ J m· βα m·αβ–( )

β 1=

NP

∑=

t∂∂ J εα ραUα

k( )ξi∂∂

IUαk

i( )+ εαAk

i Pf∂

ξi∂-------–

ξi∂∂ εα ηα

eff AmiAk

j

J-------------

Uαm∂

ξj∂----------

+=

Aki

ξi∂∂ εα Pα

e( )– J εα ραg+

J m· βαUβk

m· αβUαk

–( )β 1=

NP

∑ J Bαβ Uβk

Uαk

–( )β 1=

NP

∑ J FαT

and+ + +

t∂∂ J εα ρα Φα( )

ξi∂∂

IΦα

i( )+

J m·βαΦβ m· αβΦα–( )β 1=

NP

∑ J BαβI Φβ Φα–( )

β 1=

NP

∑ J Sα .+ +=

J Ami A i( )=

Uαi

Imα

iIUα

ki

IΦα

i

23

cells in

anda rec-sical

pu-olumes

sicalmpu-

Integrating the generic balance equation (37) over a single finite-volume(Figure 1(b)) in the computational space with the help of the Gauss law result

(38)

where is the volume of the finite-volume cell in the computational space is the surface of this cell. Up to the second-order accuracy, the edges of

tangular volume in the computational space are transformed into the physpace as

, (39)

where are the covariant basis vectors (Appendix B). Accordingly, the comtational space volumes and cell face areas are related to the physical space v

and areas as

, (40)

, and . (41)

ξ

ς

ζ

xC

xP A(1)

A(3)

A(2)

ξ

ς

ζ

UuD d

N

n

S

s

Ee

Ww

P

ξi(xj)

1

1

1

Figure 1. (a) Area vectors on the faces of the finite-volume cell in the physpace. (b) The notion of the neighbouring cell centres and cell faces in the cotational space.

a) b)

t∂∂

J εα ρα Φα( ) VCdVC

∫ IΦα

i n ACd•AC

∫+

J m·βαΦβ m· αβΦα–( ) VCdβ 1=

NP

∑V

C

∫ J BαβI Φβ Φα–( ) VCd

β 1=

NP

∑V

C

∫+=

J Sα VC ,dV

C

∫+

VCAC

VC

ξδ ςδ ζδ, ,( ) ξδ e 1( ) ςδ e 2( ) ζδ e 3( ), ,( )→

e i( )

VP

VC ξ ς ζδδδ= J ξ ς ζδδδ→ VP ξ ς ζδδδ=

ς ζδδ A 1( ) ς ζδδ→ ξ ζδδ A 2( ) ξ ζδδ→ ξ ςδδ A 3( ) ξ ςδδ→

24

to beinto

lations, (36)

nd iscom-ectly a dif-), e.g.,

spec-the

Setting the lengths of the finite-volume cell edges in the computational spaceunity , all the cells in the physical space are transformed cubes of unit volume and unit face areas in the computational space.

Based on the same type of integral balances as equation (38) and on re(40) and (41), the discretization of the macroscopic balance equations (35)and (37) can be formally represented as

(42)

(43)

(44)

In equations (42), (43) and (44), the notation indicates that the operaintegrated over the volume of the finite-volume cell in question. Because the putational space cells are all of unit volume, the resulting mean value is direxpressed on the unit-volume basis. Further, the notation expressesference between the values of the operand at specified locations (Figure 1(b)

. (45)

The right-hand side notation implies that the operand is integrated over the retive face of the finite-volume cell. The result of this integration is directly on

ξδ ςδ ζδ 1= = =

J εα ρα{ }t∆

--------------------------t

t ∆∆t+

Imα

1[ ]d

uImα

2[ ]s

nImα

3[ ]w

e+ + +

J m· βα m·αβ–( ){ } ,β 1=

NP

∑=

J εα ραUαk{ }

t∆---------------------------------

t

t ∆∆ t+

IUαk

1[ ]d

uIUα

k2

[ ]sn

IUαk

3[ ]w

e+ + + εαAk

i Pα∂

ξi∂---------

–=

εαηα

Am1

Akj

J-------------

Uαm∂

ξj∂----------

d

u

εαηα

Am2

Akj

J-------------

Uαm∂

ξ j∂----------

s

n

εαηα

Am3

Akj

J-------------

Uαm∂

ξj∂----------

w

e

+ + +

Aki

ξi∂∂ εα Pα

e( )

– J εα ραg{ } J FαT{ }+ +

J m· βαUβk

m· αβUαk

–( ){ }β 1=

NP

∑ J Bαβ Uβk

Uαk

–( ){ } andβ 1=

NP

∑+ +

J εα ρα Φα{ }t∆

---------------------------------t

t ∆∆t+IΦα

1[ ]du

IΦα

2[ ]sn

IΦα

3[ ]we

+ + +

J m· βαΦβ m· αβΦα–( ){ }β 1=

NP

∑ J BαβI Φβ Φα–( ){ }

β 1=

NP

∑ J Sα{ } .+ +=

{ }

[ ]d s w, ,u n e, ,

IΦα

1[ ]du

IΦα

1

uIΦα

1

d–=

25

) andients

ng an

unit-area basis. For the time derivative term, the corresponding notation expresses the difference of cell-based volume averages at respective times

. (46)

Thus, the total normal phasic fluxes , and are represented as

, (47)

, (48)

, (49)

where

indicates the pair of faces in question. Comparison between equations (B19(47), (48) and (49) reveals that the phasic convection and diffusion coeffic

, and are specified by the relations

, (50)

, (51)

, (52)

where .

2.4.1 Profile assumptions

Because the discretization is done in the computational space, integratioperand over the finite-volume cell with a constant value results in

[ ]tt ∆∆ t+

Sα[ ]tt ∆∆t+

Sα{ }t ∆∆t+ Sα{ }t–=

Imα

iIUα

ki

IΦα

i

Imα

i[ ]lh

Cαi

hCα

i

l–=

IUαk

i[ ]l

hCα

i

hUα

k

hCα

i

lUα

k

lDα

i j

h

Uαk∂

ξ j∂----------

h

Dαij

l

Uαk∂

ξ j∂----------

l

+––=

IΦα

i[ ]lh

Cαi

hΦα h

Cαi

lΦα l

DΦα

ij

h

Φα∂

ξ j∂-----------

h

DΦα

i j

l

Φα∂

ξ j∂-----------

l

+––=

i

1 ; h u l d=,=

2 ; h n l s=,=

3 ; h e l w=,=

=

Cαi

Dαi j

DΦα

i j

Cαi

cεα c

ρα cA i( ) Uα•

cεα c

ρα cUα

i

c= =

Dαi j

cεα c

ηα c

Ami

Amj

J--------------

c

εα cηα c

Gij

c= =

DΦα

i j

cεα c

Γα c

AmiAm

j

J--------------

c

εα cΓα c

Gij

c= =

i1 ; c u d,=

2 ; c n s,=

3 ; c e w,=

=

26

lux ofs the

cell as over face.ord-

f theues are. Theentres

on the

inse

, (53)

where is the value of the operand at the centre of the cell.Since the rest of the terms in equations (42), (43) and (44) express the f

the operand through the faces of the finite-volume cell, they are calculated adifference of the average value of the operand at the opposite faces of the indicated by equation (45). For the averaging, the real profile of the operandthe face is approximated with a constant value located at the centre of theThis approximation is already reflected in the chosen form of the notation. Accingly, integration over the face leads to

, (54)

where is the value of the operand at the centre of the cell face.

2.4.2 Values on cell faces

To calculate the flux terms , the material properties and the values odependent variables at the centres of the cell faces are needed. These valfound by a weighted linear interpolation scheme from the cell centre valuesweight factors used in this context are based on the distances between cell cand the corresponding cell faces in the physical space (Figure 2). The value cell face is then given by

Φ{ } ΦP

VC≈ ΦP

ΦP= =

ΦP

Φ ACi

dAC

i∫ Φ

cAC

i≈ Φc

Φc= =

Φc

Φ[ ]du

Figure 2. Notations used in the context of discretization approximations. The tof the physical space reveals the weight factors.

P

PD

D

U

U

N

N

S

S

UN

UN

US

US

DN DN

DSDS

d

d

u

u

n

n

s

s

ξi(xj)

P

Uu∆Pu

∆uU

∩PU

27

s. In a, e.g.,

nts arene forse areg thethodces,lineard thee cell

ntum the

theseation

denter-ation

termsusing

, (55)

where and .

This scheme is second-order accurate in rectangular non-uniform meshes.

2.4.3 Gradients at cell centres

The source terms include the pressure gradients calculated at cell centregeneral case, other source terms involving gradients of dependent variablesthe model used in the test case of this article, may also exist. These gradieapproximated by the same type of weighted interpolation scheme as the ovalues on cell faces, but to enhance accuracy, the weight factors in this cabased on arc lengths between the cell centres (Figure 2). Although calculatinarc lengths is computationally expensive, the Rhie-Chow interpolation me[49], used to provide the normal flux velocity components on the cell fanecessitates the second-order accuracy also on the non-uniform curvimeshes. Thus, by defining two weight factors, one on the positive side another on the negative side of the cell centre in consideration, the gradient at thcentre is calculated as

, (56)

in which the weight factors are defined to be

,

.

Accordingly, the pressure gradient source term in the discretized momeequation (43) is not calculated on the computational space but directly fromphysical space gradient, as specified by equation (56). The relation betweentwo forms of the pressure gradient source term is given in Appendix B by equ(B25).

2.4.4 Gradients on cell faces

The total normal phasic fluxes (48) and (49) include gradients of the depenvariable on the cell faces in their terms involved with diffusion fluxes. Furthmore, in the discretized momentum balance (43) those terms of the deformrate tensor not analogous with the viscous diffusion are classified as sourceand also involve gradients of this type. These gradients are all discretized

Φc

Φc 1 Wc–( )ΦP WcΦC+= = Wc∆Pc

∆Pc ∆Cc+----------------------------=

c u d n s e w, , , , ,( )= C U D N S E W, , , , ,( )=

Uαi

Φ∂x

i∂-------

P

WL ΦH WH WL–( )ΦP WH ΦL–+=

WH1

HP∩ PL∩+--------------------------------------------

HP∩PL∩

-------------------=

WL1

HP∩ PL∩+--------------------------------------------

PL∩HP∩

-------------------=

28

e cell

e, theted at

cross-

analo-

con-llowduced

l faceating

ion forto the), therms

ation.

ancee theey areious

central differences in the computational space. For the gradients normal to thfaces, the central difference method is applied as

with . (57)

In the case of cross-derivatives, i.e., derivatives on the plane of the cell facgradients are approximated as the mean of the two central differences calculathe cell centres on both sides of the face in question. As an example, the derivatives on the cell face u are (Figure 2)

, (58)

. (59)

The other ten cross-derivatives on the remaining five faces are defined in an gous way.

The approximation of the cross-derivatives as in equations (58) and (59)nects 18 neighbouring cells to the treatment of every finite-volume cell. To athe use of the standard linear equation system solvers, the treatment is reback to the usual one involving only the eight neighbouring cells sharing a celwith the cell under consideration by the deferred correction approach, i.e., trethe cross-derivatives as source terms by using values from the previous iteratthe dependent variables in question. Thus, in addition to the terms related gradients normal to the cell faces (the diagonal terms of the diffusion tensordeferred correction approach results in the following additional source teexpressing the effect of non-orthogonality

(60)

where denotes the value of the dependent variable from the previous iter

Following the details given above, the formally discretized macroscopic balequations (42), (43) and (44) can now be written in their final form. To enhancreadability and to keep more physical nature in the discretized equations, thnext given in a partly discretized form retaining most of the terms in their prev

Φ∂ξi∂

-------

c

ΦC ΦP–= i

1 ; C U c u=,=

2 ; C N c n=,=

3 ; C E c e=,=

=

Φ∂ς∂

-------u

14--- ΦN ΦS– ΦUN ΦUS–+( )=

Φ∂ζ∂

-------u

14--- ΦE ΦW– ΦUE ΦUW–+( )=

SΦα

DDΦα

12 Φα*∂ς∂

------------ DΦα

13 Φα*∂ζ∂

------------+d

u

DΦα

21 Φα*∂ξ∂

------------ DΦα

23 Φα*∂ζ∂

------------+

s

n

+=

DΦα

31 Φα*∂ξ∂

------------ DΦα

32 Φα*∂ς∂

------------+

w

e

,+

Φα*

29

s toalanceriables

thectionentralow

form. This practice should also aid in transferring the presented methodapproaches using different discretization schemes. Thus the macroscopic bequations of the phasic mass, momentum and generic dependent scalar vaare

(61)

(62)

(63)

In equations (62) and (63), , are the coefficient part and , constant part of the linearized total source term (29). Furthermore, the adveterms have been treated using the hybrid differencing scheme, in which the cdifferencing is utilized when the cell Peclet number ( ) is bel

Cα uCα d

– Cα nCα s

– Cα eCα w

–+ +

MαI

PVP

εα Pρα P

VP

t∆----------------------------t

t ∆∆t+

,–=

aUα

k

cUα

k

Pc

u d n s e w, , , , ,

∑ aUα

k

cUα

k

Cc

C

u d n s e w, , , , ,U D N S E W, , , , ,

∑=

εα PAk

i

P

Pf∂

ξi∂-------

P

Aki

P ξi∂∂ εαPα

e( )P

–– εα Pρα P

gVP+

εαηαeff Am

1Ak

j

J-------------

Uαm∂

ξ j∂----------

d

u

εαηαeff Am

2Ak

j

J-------------

Uαm∂

ξ j∂----------

s

n

εαηαeff Am

3Ak

j

J-------------

Uαm∂

ξ j∂----------

w

e

+ + +

m· βα PUβ

k

Pm· αβ P

Uαk

P–

VPβ 1=

NP

∑ Bαβ PUβ

k

PUα

k

P–

VPβ 1=

NP

∑+ +

FαT1

PUα

k

PVP Fα

T0

PVP S

Uαk

D εα Pρα P

Uαk

PVP

t

t ∆∆t+ and–+ + +

aΦα cΦα P

c

u d n s e w, , , , ,

∑ aΦα cΦα C

c

C

u d n s e w, , , , ,U D N S E W, , , , ,

∑=

m· βα PΦβ P

m· αβ PΦα P

–( )VPβ 1=

NP

∑ BαβI

PΦβ P

Φα P–( ) VP

β 1=

NP

∑+ +

Sα1

PΦα P

VP Sα0

PVP SΦα

Dεα P

ρα PΦα P

VP

t∆------------------------------------------t

t ∆∆t+

.–+ + +

FαT1

Sα1

FαT0

Sα0

Peα Cα DΦα⁄=

30

is asso-

inanceemeilt upontrix

g cells

mma-

two and the upwind differencing, ignoring diffusion, is performed when the greater than two. This scheme together with the deferred correction approachciated with the gradients on the cell faces (60) guarantees the diagonal domof the resulting coefficient matrix. For that reason hybrid differencing schserves as the base method for which other more accurate schemes can be buby the deferred correction approach [14, 36]. With hybrid differencing the macoefficients can be written as

, and (64)

, . (65)

To save some space, the summations over the cells and the neighbourinare simplified as

and (66)

(67)

in the following treatment. The latter notation should be understood as a sution of a product term with two simultaneously changing indices.

Peα

aΦα hmax

12--- Cα h

DΦα

ii

h,

12--- Cα h

–= i

1 ; h u=

2 ; h n=

3 ; h e=

=

aΦα lmax

12--- Cα l

DΦα

i i

l,

12--- Cα l

+= i

1 ; l d=

2 ; l s=

3 ; l w=

=

aΦα cc

u d n s e w, , , , ,

∑ aΦα cc

∑→

aΦα cΦα C

c

C

u d n s e w, , , , ,U D N S E W, , , , ,

∑ aΦα cΦα C

c C,∑→

31

bal-f other. For

ations (61),por-ciated

meth-rithm

ling

esti-te of aat the thec bal- A trues at the

n the

n theionsis at

3 INTER-PHASE COUPLING ALGORITHMS

The characteristic feature of multi-fluid model equations is that the phasicance equations (4) and (5) are coupled to the phasic balance equations ophases through the interfacial source terms of mass and momentum numerical treatment, these terms are customarily linearized as given by equ(24)–(26) and, correspondingly, appear in the discretized balance equations(62) and (63) in that form. As the coupling of phases becomes tighter, the imtance of these interfacial sources grows and, accordingly, the time scale assowith the interfacial transport decreases. Then, in sequential iterative solution ods, the treatment of the inter-phase coupling terms in the solution algobecomes critical for both the efficiency and stability of the solver.

Four different possibilities exist to handle the linearized inter-phase coupterms, namely,

the explicit treatment

, (68)

the partially implicit treatment

, (69)

the semi-implicit treatment

and (70)

the fully implicit treatment

. (71)

In equations (68)–(71) the notations , and denote the current mate, an intermediate estimate during an inner iteration and the new estimadependent variable of the phase . However, it should be remembered thfully implicit treatment can never be fully attained in sequential solvers, butterm is understood as meaning a treatment where all information of a phasiance equation, except source terms, is taken at the level of a new estimate.fully implicit treatment would be approached by calculating new source termevery inner iteration cycle. Inter-phase coupling algorithms intended to mimicclassification (68)–(71) have been studied by Karema & Lo [34], presented iAppendix E of this publication.

Customarily, the sequential iterative control volume solvers are based oSIMPLE (Semi-Implicit Method for Pressure-Linked Equations) type [44] solutalgorithm utilizing a pressure correction step to conserve mass on cell ba

MαI Fα

I

Bαβ Φβ Φα–( )β 1=

NP

∑ Bαβ Φ* β Φ* α–( )β 1=

NP

∑→

Bαβ Φβ Φα–( )β 1=

NP

∑ Bαβ Φ* β Φα**–( )

β 1=

NP

∑→

Bαβ Φβ Φα–( )β 1=

NP

∑ Bαβ Φβ* Φα

*–( )β 1=

NP

∑→

Bαβ Φβ Φα–( )β 1=

NP

∑ Bαβ Φβ** Φα

**–( )β 1=

NP

∑→

Φ* α Φα* Φα

**

α

32

ws,d bySAr toolution10],ter-

were bedr and

s, thewith af the

ces a thisody-ly, thecu--fluideth-

source

ted rater the

hen ato thest few rate.le ofs in

ent oper-ired ortimeu-

ciencyntumtiallynly

tumit istionon-

puta-ions

every outer iteration. An extension of the SIMPLE method to multi-phase flocalled the IPSA (InterPhase Slip Algorithm) algorithm [51, 52, 53], was useKarema & Lo [34] in their study of inter-phase coupling algorithms. As the IPalgorithm neglects inter-phase coupling terms in its formulation, in ordeachieve a more accurate pressure correction step, an improved IPSA based smethod, called the IPSA–C (InterPhase Slip Algorithm – Coupled) algorithm [was also studied. With this solution algorithm, a semi-implicit treatment of inphase coupling terms can be included into the solver.

The test cases, the gas-particle bed (A) and the liquid-particle bed (B), selected to represent typical but simplified problems in the area of fluidizedhydrodynamics and the constitutive models for inter-phase momentum transfedispersed phase stresses were set as given by Bouillard et al. [9]. There, to preventthe dispersed phase from compacting to unreasonably high volume fractionnormal component of the dispersed phase stress is retained and modelled simple Coulombic approach (see Eq. 69 in Appendix E). When the porosity obed decreases below the state of minimum fluidization this model produstrong repulsive force to prevent bed compaction. In hydrodynamic termsreflects a condition in which the collision time scale reduces below the aernamic time scale related to momentum transfer by phase interaction. Naturalrepulsive force is a strong function of fluid volume fraction . All tests domented in Appendix E were based on the customary representation of multiequations (27) and (28) and they were solved by IPSA or IPSA–C solution mods. Accordingly, the dispersed phase normal stress was implemented as a term with a zero first order term.

In both flow configurations A and B the semi-implicit SINCE method resulin about 5% and the fully implicit PEA method about 35% higher convergenceof phasic momentum equations, respectively. The most important reason foimprovement can be found in the correctness of the few first approximates wnew time step is entered. The effect of including interphase coupling terms inpressure correction scheme, the essence of the IPSA–C, is limited to the firtime steps, and there is practically no improvement of the overall convergence

When the time step of simulation is kept below the characteristic time scainterfacial transfer processes, the partially implicit treatment of coupling termcombination with the IPSA solution algorithm provides the most efficiapproach. This result is totally based on the smaller number of computationalations required by these simpler schemes. When a longer time step is requthere is a spot in the solution domain involved with very small characteristic scales of interfacial coupling, the semi-implicit or fully implicit treatment of copling terms becomes necessary. Then the PEA method provides the best effifor two-phase flows, providing a considerably better convergence with momeequations but being only slightly more laborious computationally than the parimplicit treatment. In multi-phase flows the SINCE method provides the ooption for the partially implicit treatment. Its convergence rate with momenequations is only slightly better than with the partially implicit method though computationally more involved. The IPSA–C is arguably the preferred solumethod for conditions where the interfacial coupling is tighter than in the flow cfigurations studied, e.g., in bubbly air-water flows.

Studying mass residuals reveals that in practice the improvement in comtional speed attained with implicit-like algorithms in solving momentum equat

εmf

Tdn εf

33

oulde cou- withd byrticle somectsis it

espe-where flu-ssure to an

lutionp hin- IPSAection fluid-es areequa-of the

the pres-ncy ofs thessureblem-ove the thee andhysi- thirdorrec-

the dis-articlesigherurcerate real-, i.e.,asedstill

this in non-

us-arent

is almost completely lost in the pressure correction step of the solution. This whave been acceptable for the IPSA solution method as it neglects inter-phaspling terms in correcting the pressures but the existence of this problem alsothe IPSA–C method indicate that the convergence rate of solution is limiteother factors. In addition, this problem is much more pronounced in gas-pabed than in liquid-particle bed and in both test cases it becames worse afterinitial period of integration time starting from the initial field. All these aspepoint towards a poorly performing pressure correction algorithm. After analyswas realized that the limiting factor in predicting the total mass balance and cially the mass balance of the dispersed phase was threefold. First, in cells the bed compaction was high, i.e., the porosity of the bed below the minimumidisation porosity , the shared pressure, for which the mass balancing precorrection step is based on, is not adequate to prevent the bed compactingunreasonably low porosity. Second, the characteristic feature of the IPSA somethod to keep the volume fractions constant during pressure correction steders the too high compaction to be realized and prevented. Third, as in thethe volume fractions are solved after momentum equations and pressure correquation, the large value of static pressure compared to dynamic pressure inized beds is not effectively countered, i.e., as long as cell-based imbalancgoverned by the velocity field, the enhanced convergence rate of momentum tions enters the mass balances. Apparently, the pressure correction step multi-fluid solution method should be further improved by incorporating alsopressure-like contributions of the dispersed phases, in addition to the sharedsure, and by including a mechanism to take into account the strong dependethe solid pressure on volume fractions. Further, in the momentum equationpressure-like contributions should be treated equivalently with the shared pregradient. These topics are covered in detail in chapter 4. In Figure 3 the proatic areas of the gas-particle bed are seen to be located beneath and abobstacle at the centre line and below the bubble. Evidently, the flow field invicinity of the obstacle has a dramatic effect on the development of the bubblits accurate prediction is necessary. In liquid-particle bed the problem of unpcally high compaction does not exist (Figure 4) and the main problem is in theitem, namely considering volume fractions as constant during the pressure ction step.

Another type of difficulty present in the gas-particle bed but non-existent inliquid-particle bed, still related to mass balancing, is originating from the largecrepancy of the source terms across sharp interfaces between the fluid and pin circumstances when the particle density is several orders of magnitude hthan the fluid density. Then, the weighted linear interpolation applied to soterms while carrying out the Rhie-Chow momentum interpolation is not accuenough resulting in mass imbalance in adjacent cells of the interface. This isised as the small wrinkles around the interface with finer mesh in Figure 3sharper interface, and will ultimately lead to divergence of a control volume bsolver. For the liquid-particle bed the customary Rhie-Chow interpolation is adequate. A geometric interpolation procedure [43] for source terms ininstance could have been used but as it does not necessarily conserve massuniform curvilinear meshes a so-called improved Rhie-Chow momentum interpo-lation [14] was utilized. This method is briefly described in Appendix D. If the ctomary representation of multi-fluid equations (27) and (28), where the app

εmf

34

re-occ

0.0000E+001.3447E-012.6895E-014.0342E-015.3789E-016.7236E-01

> 8.0000E-01

0.0000E+001.3447E-012.6895E-014.0342E-015.3789E-016.7236E-01

> 8.0000E-01

0.0000E+001.3447E-012.6895E-014.0342E-015.3789E-016.7236E-01

> 8.0000E-01

0.0000E+001.3447E-012.6895E-014.0342E-015.3789E-016.7236E-01

> 8.0000E-01

Figure 3. Volume fraction field of particles in test case A with different repsentations of geometry: (a) multi-block rectangular mesh, (b) multi-blk

BFC-mesh, (c) multi-block rectangular mesh and (d) multi-blok BFC-mesh.

εd31 48×

31 48× 62 96×62 96×

a) b)

c) d)

35

re-occ

0.0000E+001.3333E-012.6667E-014.0000E-015.3333E-016.6667E-018.0000E-01

0.0000E+001.3333E-012.6667E-014.0000E-015.3333E-016.6667E-018.0000E-01

0.0000E+001.3333E-012.6667E-014.0000E-015.3333E-016.6667E-018.0000E-01

0.0000E+001.3333E-012.6667E-014.0000E-015.3333E-016.6667E-018.0000E-01

Figure 4. Volume fraction field of particles in test case B with different repsentations of geometry: (a) multi-block rectangular mesh, (b) multi-blk

BFC-mesh, (c) multi-block rectangular mesh and (d) multi-blok BFC-mesh.

εd31 48×

31 48× 62 96×62 96×

a) b)

c) d)

36

ed, thisction.

terms

pressure term of the dispersed phase is modelled as a source term, is applimethod should also enhance stability of the solver in areas of high bed compaIn order to reach an equal order of approximation for the apparent pressureas for the shared pressure, the extended Rhie-Chow momentum interpolationderived in section 5 can be used instead of or in combination with the improvedRhie-Chow interpolation.

37

andstric- lengthn bee tur-oximate treat-ntumd 2.4,elatedtionsntial

e ises ared the

SAtrol

thisntribu-ns aresharedure is

basis andhmse or

oned.hases,ersedlected.featureic massn forent of

aterial

ivendedS) andctionFPS is equa-

4 MULTI-PRESSURE ALGORITHMS

In section 2.2 it was shown that the momentum balances of both the IMPDIF approaches simplify to the same form under certain restrictions. These retions necessitate that the dispersed phase entities are small compared to thescales of the mean fluid flow field, that the interfacial mean pressure caapproximated with the phasic mean pressure and that the fluid phasbulent stresses exceed considerably the viscous stresses. The resulting apprmomentum balance was shown to be conveniently represented for numericalment as the generalized form (32). The associated formally discretized momebalance was then given by equation (62), as discussed in sections 2.3 anrespectively. Since in the equation (62), there are two pressure terms, one rwith the mean fluid pressure and the other with the pressure like contribu

originating from the other constitutive equations, in the frame of sequecontrol volume methods (CVM) some extended pressure correction schemrequired in order to solve this kind of equation set. These extended schemnext studied in the context of two different sequential solution strategies, callephase-sequential and the equation-sequential solution methods, respectively.

The main features of the treatment in this chapter follow closely the IPmethod [51, 52, 53] and its implementation to the collocated multi-block convolume solver, described by Karema & Lo [34]. The two distinctive features ofmethod are related to the pressure correction step. The first neglects the cotion of the volumetric changes to the pressure, i.e., the phasic volume fractioheld constant during this step. The second feature relates to the use of the pressure concept in performing the pressure correction. While only one pressdefined, it is adjusted in a way which fulfils the total mass balance on elementat every outer iteration. Though allowing simple pressure correction algorithmcontrol over the fulfilment of global mass conservation, the IPSA type algoritbecome inefficient when the pressure like contributions grow in importancwhen they become strongly dependent on the phasic volume fractions.

In the subsequent treatment, these two distinctive features are partly abandFor the fluid phase and for the incompressible treatment of the dispersed pthe first feature is still retained, but in the compressible treatment of the dispphases, the contributions of the volume fractions on the pressure are not negBecause the balance equations (62) involve multiple pressures, the second can not be applied, and the pressure corrections are then based on the phasbalances. In this context, it is important to notice that the naming conventiothe treatments of the dispersed phases originates from the analogous treatmsingle phase flows. Thus, the compressible treatment here implies that the mdensities are independent of pressure but the volume fractions are not.

There are two solution methods referred in this chapter, i.e., the phase-sequen-tial and the equation-sequential solution methods, which represent two alternatapproaches to construct a sequential CVM solver. Two versions of the extepressure correction scheme are explained, Fluid Pressure in Source term (FPEquivalent Approximation of Pressures (EAP). Both of these pressure correschemes can be applied in either of the solution methods. However, as the naturally suited to the phase-sequential solution method and the EAP to the

PαI

Pα=

PfPα

e

Pαe

38

solu-

orderdomi-eneral

f thisling

ritten

mod-his is thetreat-

tions,

tion-sequential solution method, they are discussed in connection with thesetion methods.

4.1 PHASE-SEQUENTIAL SOLUTION METHOD AND FPS

The key feature of the phase-sequential solution method is found from the of balance equations solved. As seen from Figure 5, the solution proceeds nantly from phase to phase. This approach represents a natural extension of gsolvers originally designed for single phase flows. The major disadvantage osimple structure lies in the difficulty to implement efficient inter-phase coupalgorithms, e.g., such as the PEA, the SINCE and the IPSA–C [34].

4.1.1 Treatment of fluid phase

The treatment starts from the discretized phasic momentum balance (62) wfor the fluid phase

, where (72)

(73)

. (74)

In the spirit of the pressure correction algorithm, the term associated with the ified pressure of fluid has been classified as part of the source term . Tjustified by considering the importance of this term to be small compared toactual pressure gradient term in normal conditions. With this assumption, the ment resembles the customary IPSA method.

As shown in Appendix C, the connection between the fluid pressure correc

aUf

k

PUf

k

Pa

Ufk

cUf

k

Cc C,∑ εf P

Aki

P

Pf∂

ξi∂-------

P

– sUf

k

R+=

aUf

k

PaUf

k

cc

∑ m·f β PVP

β 1=

NP

∑ Bf β PVP

β 1=

NP

∑ FfT1

PVP–+ +=

εf Pρf P

VP

t∆------------------------- and+

sUf

k

RAk

i

P ξi∂∂ εfPf

e( )P

– εfηfeff Am

1Ak

j

J-------------

Ufm∂

ξ j∂----------

d

u

εfηfeff Am

2Ak

j

J-------------

Ufm∂

ξ j∂----------

s

n

+ +=

εfηfeff Am

3Ak

j

J-------------

Ufm∂

ξ j∂----------

w

e

εf Pρf P

g VP m·βf PUβ

k

PVP

β 1=

NP

∑+ + +

Bf β PUβ

k

PVP

β 1=

NP

∑ FfT0

PVP S

Ufk

Dεf P

ρf PUf

k

PVP

t∆-------------------------------------------t

+ + + +

Pfe

sUf

k

R

39

com-tten in

which obey the mass balance of fluid (61), and the phasic Cartesian velocityponents can be obtained on the basis of equation (72). The result can be wria compact form as

Solve momentum equations of fluid f by using values for dependent variables obtained from initial guess or

from previous iteration cycle

Solve pressure correction equation for fluid f

Update fluid pressure

Correct Cartesian velocity components of fluid

Solve pressure correction equation for dispersed phase

Update dispersed phase pres-sure

Correct Cartesian velocity components of dispersed

phase

Correct normal flux velocities of dispersed phase at cell cen-

tres

Solve dispersed phase momentum equation with updated fluid pressure

Calculate normal flux velocity components of all phases at cell faces with Rhie-Chow interpolation

method

Calculate phasic convection coefficients

Solve phasic continuity equa-tions for phasic volume frac-

tions

Correct normal flux velocities of fluid at cell centres

Start

End

Phase index < NP

1

1

Yes

No

Figure 5. Computational flow chart of phase-sequential solution method.

Solve granular temperatures of dispersed phases

Equation set converged

Yes

No

40

ents

d the

alsoan be

bal-

, where (75)

and (76)

. (77)

The corresponding corrections to the phasic normal flux velocity componare then found directly from equation (75) by utilizing the associated definition

. (78)

This results in the following relation between the fluid pressure corrections anphasic normal flux velocity components of fluid

, where (79)

. (80)

The corrections to the phasic normal flux velocity components produce new estimates for the phasic convection coefficients of the fluid (50). These cexpressed as

. (81)

By inserting equation (81) to the mass balance of the fluid (61) the followingance equation for the corrections is obtained

, where (82)

. (83)

Ufkδ

PHf k

i

P

Pfδ∂

ξi∂-----------

P

–=

Hf ki

P

εf PAk

i

P

VP-------------------- h f

k

P=

h fk

P

VP

aUf

k

PWpc a

Ufk

cc C,∑–

------------------------------------------------=

Uf

iδ Ak

iUf

kδ=

Uf

i

Pδ H f

i j

P

Pfδ∂

ξ j∂-----------

P

–=

H fi j

PAk

i

PH fk

j

P

εf PAk

i

PAk

j

P

VP------------------------------- h f

k

Pk 1=

3

∑= =

Cfi *

Cfi

Cfiδ+ εf ρf Uf

iεf ρf Uf

iδ+= =

Cf1δ

uCf

1δd

– Cf2δ

nCf

2δs

– Cf3δ

eCf

3δw

– Rmf+ + + 0=

Rmf

εf Pρf P

VP[ ]t

t ∆∆t+

t∆-----------------------------------------=

Cf1

uCf

1

d– Cf

2

nCf

2

s– Cf

3

eCf

3

w– Mf

I

PVP–+ + +

41

lancefluid

po-valuesationt the

e cell equa-of the

ssureormal

vious

fluid, equa-

The substitution of the relation (81) to the mass balance (82) results in the baequation for the corrections of the phasic normal flux velocity components of

(84)

In the equation (84) the corrections of the phasic normal flux velocity comnents are introduced on the cell faces necessitating the interpolation of these from the cell centres where they are originally determined, as seen from equ(79). The adjugate Jacobian matrix is naturally defined on the cell face bucoefficient must be interpolated. If the pressure correction gradients on thfaces are approximated with central differences on the computational space,tion (79) can be substituted to the fluid mass balance (84) to give corrections fluid pressure fulfilling continuity

, where (85)

and . (86)

The term in equation (85) denotes the cross-derivatives of the fluid precorrections treated according to the deferred correction approach. With the fdiscretization notion it can be written as

(87)

. (88)

In equation (88) denotes the value of pressure correction from the preiteration.

4.1.2 Treatment of dispersed phases in FPS

In the phase-sequential solution scheme, the pressure correction of theand consequently the updated fluid pressure, is solved before the momentum

εf uρf u

Uf1

δu

εf dρf d

Uf1

δd

– εf nρf n

Uf2

δn

εf sρf s

Uf2

δs

–+

εf eρf e

Uf3

δe

εf wρf w

Uf3

δw

Rmf+–+ 0 .=

Aki

h fk

aPf PPfδ

PaPf c

PfδC

c C,∑ SPf

D Rmf–+=

aPf PaPf c

c C,∑= aPf c

εf cρf c

H fi i

c=

SPf

D

SPf

DEPf

12 Pfδ*∂ς∂

------------ EPf

13 Pfδ*∂ζ∂

------------+

d

u

EPf

21 Pfδ*∂ξ∂

------------ EPf

23 Pfδ*∂ζ∂

------------+

s

n

+=

EPf

31 Pfδ*∂ξ∂

------------ EPf

32 Pfδ*∂ς∂

------------+

w

e

, where+

EPf

ij εf cρf c

H fi j

c=

Pfδ*

42

e fluidis con-ressed

d withpdatedifiedplaced

areents

tions of the dispersed phases are entered. By following the FPS treatment, thpressure gradient term in the momentum equations of the dispersed phases sidered as a known source term. Thus, the momentum equations can be expas

, where (89)

(90)

(91)

In equation (91) the fluid pressure gradient term locater bar has been denotean asterisk superscript in order to emphasize that the fluid pressure is an uvalue fulfilling the mass balance of the fluid phase (61). In addition, the modpressure of the dispersed phase, defined in the equation (31), has been rewith an apparent pressure

. (92)

By proceeding as with the fluid phase in section 4.1.1, the details of whichgiven in Appendix C, the corrections to the phasic Cartesian velocity componof the dispersed phase can be expressed in the form

, where (93)

and (94)

aUα

k

PUα

k

Pa

Uαk

cUα

k

Cc C,∑ Ak

i

P

Pαa∂

ξi∂---------

P

– sUα

k

R+=

aUα

k

Pa

Uαk

cc

∑ m· αβ PVP

β 1=

NP

∑ Bαβ PVP

β 1=

NP

∑ FαT1

PVP–+ +=

εα Pρα P

VP

t∆----------------------------- and+

sUα

k

R εα PAk

i

P

Pf∂

ξi∂-------

P

*

– εαηαeff Am

1Ak

j

J-------------

Uαm∂

ξ j∂----------

d

u

εαηαeff Am

2Ak

j

J-------------

Uαm∂

ξ j∂----------

s

n

+ +=

εαηαeff Am

3Ak

j

J-------------

Uαm∂

ξ j∂----------

w

e

εα Pρα P

g VP m· βα PUβ

k

PVP

β 1=

NP

∑+ + +

Bαβ PUβ

k

PVP

β 1=

NP

∑ FαT0

PVP S

Uαk

Dεα P

ρα PUα

k

PVP

t∆-----------------------------------------------t

.+ + + +

Pαa εαPα

e=

Uαkδ

PHαk

i

P

Pαaδ∂

ξi∂-------------

P

–=

Hαki

P

Aki

P

VP---------- h α

k

P=

43

ions

ities,hase,th the3) and

n as

t pres-ction

. (95)

Again, by utilizing the associated definition (78), the corresponding correctto the phasic normal flux velocity components can be written as

, where (96)

. (97)

4.1.2.1 Incompressible model for dispersed phases

As the correction equations of the phasic Cartesian and normal flux veloc(93) and (96), closely resemble to the corresponding equations of the fluid p(75) and (79), a fully analogous treatment for the dispersed phases apply. Wihelp of the dispersed phase mass balance and the correction equations (9(96), the pressure correction equation for the dispersed phases can be writte

, where (98)

and . (99)

In equation (98) denotes the mass residual of the phase

(100)

and the term is used to denote the cross-derivatives of the phasic apparensure corrections, treated as for the fluid phase, with the deferred correapproach

hαk

P

VP

aUα

k

PWpc a

Uαk

cc C,∑–

-------------------------------------------------=

Uαi

Pδ H α

i j

P

Pαaδ∂

ξ j∂-------------

P

–=

H αij

PAk

i

PHαk

j

P

Aki

PAk

j

P

VP--------------------- hα

k

Pk 1=

3

∑= =

aPα PPα

aδP

aPα cPα

aδC

c C,∑ SPα

D Rmα–+=

aPα PaPα c

c C,∑= aPα c

εα cρα c

H αi i

c=

Rmαα

Rmα

εα Pρα P

VP[ ]t

t ∆∆t+

t∆--------------------------------------------=

Cα1

uCα

1

d– Cα

2

nCα

2

s– Cα

3

eCα

3

w– Mα

I

PVP–+ + +

SPα

D

44

treat-und inaterial. Thisronglys the

algo-

luid, inress-

po-ing

s has

e (61)

(101)

. (102)

4.1.2.2 Compressible model for dispersed phases

As stated at the beginning of chapter 4, the naming convention for the ments of the dispersed phases originates from the analogous treatment fosingle phase flows. Thus, the compressible treatment here implies that the mdensities are independent on pressure but the volume fractions are notapproach is desirable in conditions where the modified pressure depends ston the phasic volume fractions, in which case the IPSA-like treatment, called aincompressible model here, would lead to poor performance of the solution rithm.

Consequently, by treating the dispersed phase as a pseudo-compressible fwhich the dependency is handled like the dependency in compible single phase flows, the corrections to the phasic normal flux velocity comnents (96), satisfying the phasic continuity equation, produce the followchanges to the phasic convection coefficients (50)

(103)

In equation (103), the second order term including changes to both variablebeen neglected.

Substitution of equation (103) to the mass balance of the dispersed phasresults in the following balance equation

SPα

DEPα

12 Pαaδ*∂

ς∂-------------- EPα

13 Pαaδ*∂

ζ∂--------------+

d

u

EPα

21 Pαaδ*∂

ξ∂-------------- EPα

23 Pαaδ*∂

ζ∂--------------+

s

n

+=

EPα

31 Pαaδ*∂

ξ∂-------------- EPα

32 Pαaδ*∂

ς∂--------------+

w

e

, where+

EPα

i j εα cρα c

H αi j

c=

εα Pαa( ) ρf Pf( )

Cαi * εα εαδ+( )ρα Uα

iUα

iδ+( )=

εαραUαi εαρα Uα

iδ εαραUαiδ εαρα Uα

iδδ+ + +=

εαραUαi εαρα Uα

iδ εαραUαi

.δ+ +≈

45

ritingsisting time

lume

flux form

ts are

(104)

where is the phasic mass residual defined in equation (100). When wequation (104), the time dependent term has been divided into two parts conof the time dependent pseudo-compressible term and the conventionaldependent term as

(105)

By denoting the dependency as , the correction of the phasic vofraction can be expressed as

. (106)

Substitution of the definition (106) and the corrections of the phasic normalvelocity components (96) to the phasic mass balance (104) changes it into the

(107)

If the coefficient terms of the apparent pressure corrections and their gradienunderstood as the pseudo convection and diffusion coefficients,

εαρα Uα1δ εαραUα

1δ+( )u

εαρα Uα1δ εαραUα

1δ+( )d

–

εαρα Uα2δ εαραUα

2δ+( )n

εαρα Uα2δ εαραUα

2δ+( )s

–+

εαρα Uα3δ εαραUα

3δ+( )e

εαρα Uα3δ εαραUα

3δ+( )w

–+

Rmα

εαδP

ρα PVP( )

t∆-------------------------------------

t ∆∆t+

+ + 0 ,=

Rmα

εα εαδ+( )Pρα P

VP[ ]t

t ∆∆t+

t∆----------------------------------------------------------------

εαδP

ρα PVP( )

t∆-------------------------------------

t ∆∆t+

εα Pρα P

VP[ ]t

t ∆∆t+

t∆-------------------------------------------- .+=

εα Pαa( ) cα

εαδ

εαδεα∂

Pαa∂

--------- Pαaδ cα Pα

aδ= =

cα Pρα P

VP Pαaδ

P

t∆-------------------------------------------------t ∆∆t+

cαραUα1

Pαaδ εαραHα

1j Pαaδ∂

ξ j∂-------------–

d

u

+

cαραUα2

Pαaδ εαραHα

2j Pαaδ∂

ξ j∂-------------–

s

n

+

cαραUα3

Pαaδ εαραHα

3 j Pαaδ∂

ξ j∂-------------–

w

e

Rmα+ + 0 .=

46

ely thelized

ions

9) ispress-

s setf ther each

atment cou-

pres-alentonse-

ods,solved knownradientgradi-ach is

and , (108)

respectively, the apparent pressure correction equation (107) resembles closform of the momentum equation (36) or, more accurately, the form of a generabalance equation (B12) for single phase flow [14, 34]. Thus, with the definit(108), the apparent pressure correction equation (107) can be written as

(109)

In addition, the resulting phasic apparent pressure correction equation (10fully analogous with the pressure correction scheme used for high-speed comible gas flows [14].

4.2 EQUATION-SEQUENTIAL SOLUTION METHOD AND EAP

A natural base for a solver originally designed for multi-phase problems iby the equation-sequential method. As shown on Figure 6, the main loop osolution proceeds in this case from one balance equation type to an other. Foof these balance equations there is an inner loop consisting of sequential treof the phases. This structure allows easy implementation of many inter-phasepling algorithms [34].

With the assumptions mentioned in section 4.1.1, the treatment of the fluid sure correction equation for the equation-sequential solution method is equivwith the treatment of the phase-sequential solution method derived there. Cquently, only the treatment of dispersed phases is shown in this section.

4.2.1 Treatment of dispersed phases in EAP

By comparing Figures 5 and 6 it is clear that in both of the solution methbecause of the sequential structure, the fluid pressure correction equation is before the pressure corrections of the dispersed phases. Despite of being aterm in the momentum balances of the dispersed phases, the fluid pressure gterm in the EAP approach is treated equivalently with the apparent pressure ent term. Thus, the dispersed phase momentum equation in the EAP approwritten as

, (110)

CPα

i

ccα c

ρα cUα

i

c= DPα

i j

cεα c

ρα cHα

i j

c=

cα Pρα P

VP Pαaδ

P

t∆-------------------------------------------------t ∆∆t+

CPα

1Pα

aδ DPα

1j Pαaδ∂

ξ j∂-------------–

d

u

+

CPα

2Pα

aδ DPα

2 j Pαaδ∂

ξ j∂-------------–

s

n

CPα

3Pα

aδ DPα

3j Pαaδ∂

ξ j∂-------------–

w

e

Rmα+ + + 0 .=

aUα

k

PUα

k

Pa

Uαk

cUα

k

Cc C,∑ Ak

i

P

Pαa∂

ξi∂---------

P

– εα PAk

i

P

Pf∂

ξi∂-------

P

– sUα

k

R+=

47

where

(111)

(112)

Solve momentum equations of all phases using val-ues for dependent variables obtained from initial

guess or from previous iteration cycle

Solve pressure correction equation for fluid f

Start

Figure 6. Computational flow chart of equation-sequential solution method.

Solve pressure correction equations for dispersed

phases

Update pressures of all phases

Correct Cartesian velocity components of all phases

Correct normal flux velocities of all phases at cell centres

Calculate normal flux velocity components of all phases at cell faces with Rhie-Chow interpolation

method

Calculate phasic convection coefficients

Solve phasic continuity equa-tions for phasic volume frac-

tions

Solve granular temperatures of dispersed phases

End

Equation set converged

Yes

No

1

1

2

2

aUα

k

Pa

Uαk

cc

∑ m· αβ PVP

β 1=

NP

∑ Bαβ PVP

β 1=

NP

∑ FαT1

PVP–+ +=

εα Pρα P

VP

t∆----------------------------- and+

sUα

k

R εαηαeff Am

1Ak

j

J-------------

Uαm∂

ξ j∂----------

d

u

εαηαeff Am

2Ak

j

J-------------

Uαm∂

ξ j∂----------

s

n

+=

εαηαeff Am

3Ak

j

J-------------

Uαm∂

ξ j∂----------

w

e

εα Pρα P

g VP m· βα PUβ

k

PVP

β 1=

NP

∑+ + +

Bαβ PUβ

k

PVP

β 1=

NP

∑ FαT0

PVP S

Uαk

Dεα P

ρα PUα

k

PVP

t∆-----------------------------------------------t

.+ + + +

48

nt ofionsuringesianof the

pec-

po- (78).

nd

S, themblesutingphaseppar-

s for

t of thee haserivingranged

The treatment proceeds now in a fully analogous manner with the treatmethe fluid phase given in section 4.1.1 and in Appendix C, including the definitof the new estimates (C2) and the approximation of values at the neighbocells (C3). As a result of this treatment, the corrections of the phasic Cartvelocity components of the dispersed phase are related to the corrections fluid pressure and apparent pressure of the dispersed phase as

, (113)

where and are coefficients defined by equations (94) and (95), res

tively. As previously, the corrections of the phasic normal flux velocity comnents of the dispersed phase are obtained by utilizing the associated definitionThis results in

, (114)

where the coefficient is defined by equation (97), in line with a.

4.2.1.1 Incompressible model for dispersed phases

As for the incompressible treatment of the dispersed phases with the FPderivation of the apparent pressure correction equations with the EAP reseclosely to the derivation of the fluid pressure correction equations. By substitthe new estimates of the phasic convection coefficients (81) of the dispersed to the corresponding mass balance (61), and by utilizing equation (114), the aent pressure correction equation of the EAP approach can be written as

(115)

where the coefficients and are given by equations (99). stand

the dispersed phase mass residual (100), as before. Similar to the treatmenfluid phase, the central difference approximation on the computational spacbeen assumed for the pressure correction gradients of both phases while dthe apparent pressure correction equation (115). This equation can be rear

Uαkδ

PHαk

i

P

Pαaδ∂

ξi∂-------------

P

εα P

Pfδ∂

ξi∂----------

P

+

–=

Hαki

Ph α

k

P

Uαi

Pδ Hα

i j

P

Pαaδ∂

ξ j∂-------------

P

εα P

Pfδ∂

ξ j∂----------

P

+

–=

Hαij

PHαk

i

Phαk

P

aPα PPα

aδP

εα caPα c

PfδP

c C,∑+ aPα c

Pαaδ

Cc C,∑=

εα caPα c

PfδCc C

∑ SPα

DSPα f

D Rmα,–+ + +

aPα PaPα c

Rmα

49

essure

nding

,

as

res-

defi-

2).

onal

small

plifi- the

essi-

ndled

ngly,

withon ispo-

orre-

into the form

, (116)

where the source term , related to the cross-derivatives of the apparent prcorrection, is given by equation (101).

The apparent pressure correction equation (116) reverts to the correspo

equation of the FPS approach with two additional source terms, and

originating from the fluid pressure gradient. These source terms are specified

and (117)

(118)

The origin of the source term is again in the cross-derivatives of the fluid p

sure corrections. There the coefficient is the same that is found from the

nition of in the FPS approach (101), and which is defined by equation (10

If the grid in the physical space is nearly orthogonal, in the computati

space, the cross-derivatives included in the source terms and are

compared to the components normal of the cell faces. Adaptation of this simcation, originally suggested by Rhie & Chow [49], the only difference betweenFPS and the EAP approaches is in the source term .

4.2.1.2 Compressible model for dispersed phases

By following the same naming convention as with the FPS, the term compr

bility here means that the dependency of the dispersed phases is ha

similar to the dependency in compressible single phase flows. Accordi

the derivation of the apparent pressure correction equation is fully analogousthe treatment given for the compressible model with FPS. The only exceptithat with the EAP model the corrections to the phasic normal flux velocity comnents are given by equation (114) instead of equation (96).

On these basis, the following apparent pressure correction equation, c

aPα PPα

aδP

aPα cPα

aδC

c C,∑ SδPα f

SPα

DSPαf

D Rmα–+ + +=

SPα

D

SδPα fSPα f

D

SδPα fεα c

aPα c

PfδC

PfδP

–( )c C,∑=

SPα f

D εαEPα

12 Pfδ*∂ς∂

------------ εαEPα

13 Pfδ*∂ζ∂

------------+

d

u

εαEPα

21 Pfδ*∂ξ∂

------------ εαEPα

23 Pfδ*∂ζ∂

------------+

s

n

+=

εαEPα

31 Pfδ*∂ξ∂

------------ εαEPα

32 Pfδ*∂ς∂

------------+

w

e

.+

SPα f

D

EPα

i j

SPα

D

SPα

DSPα f

D

SδPα f

εα Pαa( )

ρf Pf( )

50

spec-ion

onal equa-

pres-

sponding to equation (107), is obtained

(119)

In equation (119), and are defined by equations (106) and (100), retively. By utilizing the definitions of the phasic pseudo convection and diffuscoefficients (108), equation (119) can be expanded to the form

(120)

Again, by applying the central difference approximation on the computatispace to the gradients of the fluid pressure, the apparent pressure correctiontion (120) can be transformed into the form

(121)

where and are the source terms related to the gradients of the fluidsure corrections, given by equations (117) and (118), respectively.

cα Pρα P

VP Pαaδ

P

t∆-------------------------------------------------t ∆∆t+

cαραUα1

Pαaδ εαραHα

1 j Pαaδ∂

ξ j∂------------- εα

Pfδ∂

ξ j∂----------+

–

d

u

+

cαραUα2

Pαaδ εαραHα

2 j Pαaδ∂

ξ j∂------------- εα

Pfδ∂

ξ j∂----------+

–

s

n

+

cαραUα3

Pαaδ εαραHα

3 j Pαaδ∂

ξ j∂------------- εα

Pfδ∂

ξ j∂----------+

–

w

e

Rmα+ + 0 .=

cα Rmα

cα Pρα P

VP Pαaδ

P

t∆-------------------------------------------------t ∆∆t+

CPα

1Pα

aδ DPα

1j Pαaδ∂

ξj∂-------------–

d

u

+

CPα

2Pα

aδ DPα

2j Pαaδ∂

ξ j∂-------------–

s

n

CPα

3Pα

aδ DPα

3 j Pαaδ∂

ξ j∂-------------–

w

e

+ +

εαDPα

1j Pfδ∂

ξ j∂----------

d

u

εαDPα

2j Pfδ∂

ξ j∂----------

s

n

εαDPα

3j Pfδ∂

ξ j∂----------

w

e

Rmα+––– 0 .=

cα Pρα P

VP Pαaδ

P

t∆-------------------------------------------------t ∆∆t+

CPα

1Pα

aδ DPα

1j Pαaδ∂

ξ j∂-------------–

d

u

+

CPα

2Pα

aδ DPα

2 j Pαaδ∂

ξ j∂-------------–

s

n

CPα

3Pα

aδ DPα

3 j Pαaδ∂

ξ j∂-------------–

w

e

+ +

SδPα f– SPα f

D– Rmα

+ 0 ,=

SδPα fSPαf

D

51

EAPssiblerly

The same remarks about the differences between the FPS and theapproaches as with the incompressible model are also valid for the compremodel. Clearly, under the approximation of Rhie & Chow [49] for the neaorthogonal grid, the only difference is found to be the source term .SδPα f

52

ge-tesianhasic[34]onse-

ts. Totion

tro-luidescribedadientterpo-othings, the

stag-dja-

lied be

or the

5 MOMENTUM INTERPOLATION

The principal difficulty, related to the otherwise beneficial collocated arranment of computational elements, is the need to interpolate the phasic Carvelocity components from cell centres to cell faces, in order to calculate the pconvection coefficients (50). A simple weighted linear interpolation scheme would lead to the chequer-board oscillations in the phasic pressure field, a cquence of the discretization approximation used for the pressure gradienavoid this decoupling of adjacent cells, the Rhie-Chow momentum interpolascheme [49] imitating the staggered grid solution (Figure 7) is used.

5.1 EXTENDED MOMENTUM INTERPOLATION SCHEME

The Rhie-Chow momentum interpolation scheme has originally been induced for single phase flow, but a straightforward application to the multi-fmodel equations can be achieved based on the shared pressure concept, din section 2.2. To achieve the same level of accuracy for both pressure grterms in the momentum equations of the dispersed phases, the Rhie-Chow inlation scheme has to be extended by applying an analogous fourth order smoscheme for the apparent pressure as for the hydrodynamic pressure. Thumomentum balance of the dispersed phase is written in the form

, where (122)

and (123)

. (124)

The phasic Cartesian velocity components at the cell faces, i.e., the pseudogered grid velocities, are found by interpolation of equation (122) from the acent cell centres to the staggered grid location as illustrated in Figure 7.

In the Rhie-Chow interpolation scheme, weighted linear interpolation is appto the source-like term . This allows the left side of equation (122) toapproximated with the same scheme. Accordingly, the momentum balance fimaginary staggered cell can be expressed as

Uαk

PNαk

i

P

Pαa∂

ξi∂---------

P

εα PNαk

i

P

Pf∂

ξi∂-------

P

+ + Mα P=

Mα Pa

Uαk

cUα

k

Cc C,∑ s

Uαk

R+

aUα

k

P⁄=

Nαki

P

Aki

P

aUα

k

P

-------------=

Mα

53

een

la

(125)

where the notation stands for the weighted linear interpolation betwthe cell centres . If it is assumed that

, (126)

, (127)

and , (128)

equation (125) can be simplified to the form

PD U

N

S

UN

US

DN

DS

d u

n

s

0.5 0.5

Figure 7. Staggered cell locations for x-component (DU) velocity for a regurnon-uniform mesh. Grey circles denote staggered cell centres.

Uαk

cNαk

i

c

Pαa∂

ξi∂---------

c

εα cNαk

i

c

Pf∂

ξi∂-------

c

+ +

Uαk

HNαk

i

H

Pαa∂

ξi∂---------

P

εα HNαk

i

H

Pf∂

ξi∂-------

H

+ +

C

P

= ,

H{ }

C

P

H P and C=

Nαki

H

Pαa∂

ξi∂---------

H

C

P

Nαki

H

C

P Pαa∂

ξi∂---------

H

C

P

=

εα HNαk

i

H

Pf∂

ξi∂-------

H

C

P

εα H{ }

C

PNαk

i

H

C

P Pf∂

ξi∂-------

H

C

P

=

εα cεα H

{ }C

P= Nαk

i

cNαk

i

H

C

P

=

54

y cen-ns ofd withe cellrence

asesing

of thecross-ith the

erenttermsrate

diffi--

the

(129)

As the pressure gradients in the pressure correction equation are calculated btral differences, it is essential for the conservation of mass that the interpolatiocell centre pressure gradients to the cell faces in equation (129) are calculatethe weight factor equal to one half. In contrast, the pressure gradients at thfaces are approximated, as in the staggered solution method, with the diffebetween the pressures on cell centres of adjacent control volumes.

The required phasic normal flux velocity components of the dispersed phare found with the help of the associated definition. This results in the followexpression

(130)

where the coefficients are defined as

. (131)

As seen from the interpolation equation (130), only the diagonal components pressure gradient terms are retained. This simplification is valid because the derivative terms cancel out when all the pressure gradients are calculated wcentral difference approximation.

In case of rapidly changing source terms, e.g., the buoyancy force on diffsides of a sharp interface, the weighted linear interpolation applied to source while carrying out the Rhie-Chow momentum interpolation may not be accuenough resulting in mass imbalance in cells adjacent to the interface. Thisculty can be relieved with a so-called improved Rhie-Chow momentum interpolation method described in Appendix D. The extended Rhie-Chow momentuminterpolation derived above can be used instead of or in combination withimproved Rhie-Chow interpolation.

Uαk

cUα

k

H

C

P

Nαki

H

C

P Pαa∂

ξi∂---------

H

C

P

Pαa∂

ξi∂---------

c

–

+=

εα H{ }

C

PNαk

i

H

C

P Pf∂

ξi∂-------

H

C

P Pf∂

ξi∂-------

c

–

.+

Uαi

cUα

i

H

C

P

Nαi i

H

C

PPα

a∂

ξi∂---------

H

C

P

Pαa∂

ξi∂---------

c

–

+=

εα H{ }

C

PNα

ii

H

C

PPf∂

ξi∂-------

H

C

P Pf∂

ξi∂-------

c

–

,+

Nαi i

Nαi j

PAk

i

PNαk

j

P

Aki

PAk

j

P

aUα

k

P

---------------------= =

55

dis-med effi-unds

ressureging, i.e., a

con- thed by thend tondi- char-na initions,s, pipeases iseatingitction

ted rater the

hen a intoe firstencee cor-

calermsent oper-ired ortimeu-

ciencyntumtiallynly

tumit iscan

6 CONCLUSIONS

This thesis work concentrated on the problems encountered while solvingpersed multi-phase problems numerically with a collocated Control VoluMethod (CVM). The essential aspects were considered to be the treatment anciency of inter-phase coupling terms in sequential solution, exceeding the boof validity of the shared pressure concept in cases of high dispersed phase pand the conservation of mass in momentum interpolation for rapidly chansource terms. These aspects were studied only a special application in mindfluidized gas-particle bed or a liquid-particle bed.

It would be intuitively expected that to have a sequential iterative solver verge, the inter-phase coupling terms have to be treated fully implicitly wheninterfacial transfer coefficient becomes large. This has also been confirmeOliveira and Issa [42]. The requirement of implicit treatment is connected tovery short characteristic times of the interfacial transfer in these conditions athe nature of the iterative solution method itself. In typical fluidized bed cotions, represented here by air-glass particle and water-glass particle flows, theacteristic times of interfacial momentum transport by hydrodynamic phenomemost of the bed sections are not as small. However, there always exist conde.g., start-up of the bed, and locations in the bed, e.g., heat transfer surfacebundles and corners, where the inter-phase coupling between dispersed phtight and the dispersed phase pressure is significant. Thus, the benefits of trinter-phase coupling terms in a partially implicit, semi-implicit or fully implicway and of including inter-phase coupling terms into the pressure correscheme were studied in these conditions.

In both flow configurations A and B the semi-implicit SINCE method resulin about 5% and the fully implicit PEA method about 35% higher convergenceof phasic momentum equations, respectively. The most important reason foimprovement can be found in the correctness of the few first approximates wnew time step is entered. The effect of including inter-phase coupling termsthe pressure correction scheme, the essence of the IPSA–C, is limited to thfew time steps, and there is practically no improvement of the overall convergrate. This can be attributed to the generally poor performance of the pressurrection step.

When the time step of simulation is kept well below the characteristic time sof interfacial transfer processes, the partially implicit treatment of coupling tein combination with the IPSA solution algorithm provides the most efficiapproach. This result is totally based on the smaller number of computationalations required by these simpler schemes. When a longer time step is requthere is a region in the solution domain involved with very small characteristic scales of interfacial coupling, the semi-implicit or fully implicit treatment of copling terms becomes necessary. Then the PEA method provides the best effifor two-phase flows, providing a considerably better convergence with momeequations but being only slightly more laborious computationally than the parimplicit treatment. In multi-phase flows the SINCE method provides the ooption for the partially implicit treatment. Its convergence rate with momenequations is only slightly better than with the partially implicit method though computationally more involved. However, a slight improvement of efficiency

56

A–Cterfa- air-better

ncepthouldially

of con- step

.tivepres-ted inS-ent is Rhie- as thetion

luid inter-er of dis-e pri-mallf flu-

at thes and,

refore,ten-uchver

s oflectedcon-ed.

be obtained by calculating new source terms at every inner iteration. The IPSmethod is arguably the preferred solution method for conditions where the incial coupling is tighter than in the flow configurations studied, e.g., in bubblywater flows, or if the pressure correction algorithm can be enhanced to give approximations in general.

To overcome the poor performance of the pressure correction step, the coof multi-pressure algorithms was introduced in chapter 4. Their usefulness sbe obvious but no test of their real functionality has been included. Espectempting are the compressible versions of the FPS and the EAP, as despite sidering the volume fraction also as a variable during the pressure correctionthey still reduce to the same treatment as in compressible single-phase flows

A very crucial operation for the success of a collocated CVM utilizing iterasolution is the conservation of mass in momentum interpolation, a part of the sure-correction step. The improved Rhie-Chow interpolation scheme presenAppendix D is known to be efficient in reducing the error in interpolation (CFDFLOW3D). If the shared pressure concept is abandoned and a proper treatmgiven for the dispersed phase pressure both the customary and the improvedChow interpolation method must be expanded. Thus an approach, referred toextended Rhie-Chow interpolation, applicable to both momentum interpolaschemes is presented.

Thus far the majority of the studies concerning numerical solution of multi-fequations have concentrated on the special difficulties of these flows, i.e., thephase couplings, moving interfaces, strong non-linearity and the large numbdependent variables. However, the development of constitutive models forpersed multi-phase flows has revealed that in contrary to single phase flows thmary transfer mechanism of information does not operate only from large to sflow scales but also in opposite direction. This is especially true in the case oidized beds. The consequence of the opposite transfer of information is thsmall scale phenomena are important in creation of the macro-scale structureaccordingly, they should be modelled and solved to adequate accuracy. Thereasonably accurate and predictive simulations will be computationally very insive and all progress in solution efficiency will be highly useful. Approaches sas special multi-grid solution cycles, being efficient to transport information othe spectrum of flow scales, should be utilized.

Another way to improve the transfer of information between different scaleflow can be achieved by solving separate conservation equations for a seranges of scales. In this approach, an additional difficulty in constituting the servation equations as well as describing there mutual interactions will be fac

57

f two-ux,n-

us-

us-t-

ized

IC)

lingd

les.

simu-

ns

rt 1.

ds.

ces,

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62

A1

APPENDIX A

Interfacial Mean Pressure model, IMP

Equation (5) together with the definitions (14) and (15) produce the followingbalance equation of momentum

(A1)

By extracting the pressure contribution from the first term on the right side of equa-tion (A1), on the basis of the definition (11), and combining it with the fourth termresults in

. (A2)

For problems involved with the surface tension the relation between the interfacialmean pressures can be written as

, (A3)

where is the surface tension coefficient and is the average mean curvature.Further discussion on pressure approximations can be found, e.g., in [1]. Thus, bytaking into account the definitions of stresses (12) and (13), for particulate flows,the momentum equations of fluid and dispersed phases can be written as given in(19) and (20).

Direct Interfacial Force model, DIF

By combining equations (14) and (18) with equation (5) results in the momen-tum equations for the fluid and dispersed phases of the form

(A4)

(A5)

t∂∂ εαραUα( ) ∇ εαραUα Uα⊗( )•+ ∇ εαTα( )• εαραg+=

MαIUα

IPα

I εα∇ FαD

FαS

.+ + + +

εαPα( )∇– PαI εα∇+ εα Pα∇– Pα

IPα–( ) εα∇+=

PdI

PfI

– σκ=

σ κ

t∂∂ εfρf Uf( ) ∇ εfρf Uf Uf⊗( )•+ ∇ εfTf( )• εfρf g+=

MfIUf

I FTI εβ∇ Tf

v• Fβ

D ∇ εβTβp

( )•–+( )β 1=

Nd

∑– FfS

and+ + +

t∂∂ εdρdUd( ) ∇ εdρdUd Ud⊗( )•+ ∇ εdTd( )• εdρdg+=

MdI Ud

I εd∇ Tfv

• FdD ∇ εdTd

p( )•– FdS

.+ + + +

rightd and

nces

By utilizing the definition (12) the viscous stress parts of the first term on the side and of the interfacial momentum source in equation (A4) can be expandethen subsequently simplified as

(A6)

With the result (A6) and with the definitions (12) and (13) the momentum bala(A4) and (A5) can be written into the form of (21) and (22).

1. Drew, D. A. Mathematical modeling of two-phase flow. Ann. Rev. FluidMech., 1983. Vol. 15, pp. 261–291.

∇ εfTfv

( )• εβ∇ Tfv

•( )β 1=

∑–

εf∇ Tfv• Tf

v εf∇ ∇ εβTfv( )• Tf

v εβ∇–( )β 1=

Nd

∑–+=

εf∇ Tfv• ∇ εβTf

v( )•( )β 1=

Nd

∑– Tfv εf εβ

β 1=

Nd

∑+

∇+=

εf∇ Tfv• ∇ εβTf

v( )•( )Nd

∑ .–=

A2

Car-ear the

obian

asisd thets are

,gni-

is

other

APPENDIX B

The non-singular coordinate transformation mapping from a right-handed tesian frame in the physical space to a right-handed curvilinframe in the computational space is determined by specifyingJacobian matrix of this transformation

. (B1)

Additional useful mathematical concepts in this context are the inverse Jacmatrix , the Jacobian determinant and the adjugate Jacobian matrix

, (B2)

and (B3)

. (B4)

Every curvilinear frame can be associated with two distinct frames of bvectors of which the first one is tangential to the coordinate curves another one is normal to the coordinate surfaces . Their Cartesian componengiven by

and . (B5)

The area vectors of the surfaces of an elementary grid cell (Figure 1(a)) i.e., the vectors pointing to the outward normal direction of a cell face with matude equal to the area of the face, are obtained as

, and . (B6)

With the help of these area vectors the volume of an elementary grid cell then related to the Jacobian determinant as

. (B7)

Because the two frames of basis vectors defined by (B5) are dual to each, equation (B7) can be written in the form

. (B8)

xi

x y z, ,( )=ξi ξ ς ζ, ,( )=

Jji x

i∂ξj∂

-------=

Jji J Aj

i

Jji ξi∂

xj∂

--------=

J det Jji

( )=

Aji J Jj

i=

ξix

j( )e i( )

e i( )

e i( )k J ik

= e i( )k Jk

i=

A i( )

A 1( ) e 2( ) e 3( )×= A 2( ) e 3( ) e 1( )×= A 3( ) e 1( ) e 2( )×=

VP

A i( ) e j( )• e 1( ) e 2( ) e 3( )ו δ ji VP δ j

i J δ ji

= = =

e i( ) e j( )• δ ji

=

A i( ) e j( )• J e i( ) e j( )•=

B1

vec-

jugate

rformnts ofletedix

spacependent4) inhysi-

In this way the contravariant frame of basis vectors is related to the areators , which in turn are dependent on the covariant frame of basis vectors

. (B9)

Thus the Cartesian components of area vectors are determined by the adJacobian matrix as

. (B10)

As evident in the subsequent treatment, the necessary information to pethe coordinate transformation includes only volumes and Cartesian componethe area vectors of grid cells in the physical space. This information is compby calculating the Jacobian determinant and the adjugate Jacobian matrof the transformation.

Since separate balance equations for velocity components in the physicalare used, these components are treated in essence like any other scalar devariable . Thus it is only necessary to transform the prototype equation (3order to describe the method. Using tensorial notation, equation (34) in the pcal space coordinates can be written as

(B11)

Equation (B11) can be simply rewritten as

(B12)

by defining the total phasic flux and the total phasic source to be

and (B13)

. (B14)

e i( )

A i( ) e i( )

e i( ) Ai( )

J---------

12---εijk

e j( ) e k( )×

J----------------------------------= =

A ki( )

Aki

A ki( ) J Jk

iAk

i= =

J Aki

Φ

t∂∂ εαραΦα( )

xi∂

∂ εα ραUαi Φα Γα

Φα∂

xi∂

----------–

+

m· βαΦβ m· αβΦα–( )β 1=

NP

∑ BαβI Φβ Φα–( )

β 1=

NP

∑ Sα .+ +=

t∂∂ εαραΦα( )

xi∂

∂IΦα

i( )+ SαT

=

IΦα

iSα

T

IΦα

i εα ραUαi Φα Γα

Φα∂

xi∂

-----------–

=

SαT

m· βαΦβ m·αβΦα–( )β 1=

NP

∑ BαβI Φβ Φα–( )

β 1=

NP

∑ Sα+ +=

B2

B3

On the basis of the Gauss law, equation (B12) is equivalent to the conservation law

, (B15)

where is the volume of any physical space domain and its surface.According to tensor calculus, the covariant divergence of a vector is defined as

, (B16)

in which is a generic vector and its contravariant component. Instead ofusing the contravariant component of a vector, it is more beneficial to utilize thenormal flux component defined as

. (B17)

With definitions (B16) and (B17), and on the basis of the information in equa-tion (B15), the balance equation (B12) can be transformed to the computationalspace coordinates

, (B18)

in which is the total normal phasic flux

(B19)

For mass balances and fluxes the dependent variable should be set equal toone.

In equation (B19) is the inverse metric tensor

, (B20)

which is used to raise the index of the covariant vector and is thenormal flux velocity component

. (B21)

εαραΦα VPd

VP

∫ IΦαn APd•

AP

∫+ SαT VPd

VP

∫=

VP AP

∇ V• 1J-----

ξi∂∂ J V'i( )=

V V'i

V i

V i J V'i J J jiV

jAj

iV

j A i( ) V•= = = =

t∂∂

J εαραΦα( )ξi∂∂

IΦα

i( )+ J SαT

=

IΦα

i

IΦα

i J I 'Φα

i J εα ραU'αi Φα Γα g

ij Φα∂

ξ j∂-----------–

= =

εα ραUαi Φα Γα J g

ij Φα∂

ξ j∂-----------–

.=

Φα

gij

gij e i( ) e j( )• A i( ) A j( )•

J 2------------------------

AmiAm

j

J 2--------------= = =

Φα∂ ξi∂⁄ Uαi

Uαi

J U'αi A

i( ) Uα•= =

as inith the

is

opic

alsospace

ate

to the

ide of

The prototype conservation equation in the computational space (B18) heffect the same structure as the one in the physical space, equation (B12), wexception that the diffusivity has been replaced by the diffusivity tensor

. (B22)

The multiplier in equation (B22) containing only geometric informationreferred to as the geometric diffusion coefficient.

Following the structure of the prototype equation (B11), the macroscmomentum balance (32) is expressed in a tensorial form as

(B23)

Accordingly, the first three terms on the right-hand side of equation (B23) aretreated as source terms. Utilizing the Gauss law on a volume of the physical

, the pressure gradient term can be put into the following form

. (B24)

Associating the volume with a differential control volume, the coordintransformation of the pressure gradient term can be given as

. (B25)

The chain rule can be used to relate the derivatives in the physical space derivatives in the computational space

. (B26)

This relation can be utilized to transform the second term on the right-hand sequation (B23) as follows

Γα

Γαi j J g

ij ΓαAm

iAm

j

J-------------- Γα G

ij Γα= = =

Gij

t∂∂ εαραUα

k( )x

i∂∂ εα ραUα

iUα

k ηαUα

k∂

xi∂

------------–

+

εαPα∂

xk∂

---------–x

i∂∂ εαηα

Uαi∂

xk∂

-----------

xk∂

∂ εαPαe

( ) εα ραg+ ++=

m· βαUβk

m· αβUαk

–( )β 1=

NP

∑ Bαβ Uβk

Uαk

–( )β 1=

NP

∑ FαT.+ + +

VP

εα Pα∇ n APd•AP

∫– εα Pα VPd∇VP

∫–=

VP

J εαPα∂

xk∂

---------– J εαJki Pα∂

ξi∂---------– εαAk

i Pα∂

ξi∂---------–= =

Φα∂

xi∂

-----------ξ j∂x

i∂--------

Φα∂

ξ j∂----------- J i

j Φα∂

ξ j∂-----------= =

B4

(B27)x

i∂∂ εαηα

Uαi∂

xk∂

----------

Jmi

ξi∂∂ εα ηαJk

j Uαm∂

ξj∂-------------

=

ξi∂∂ εα ηα

AmiAk

j

J 2-------------

Uαm∂

ξ j∂-------------

.=

B5

thef the

newe new

cityrded.

heC1)

to the

ationn of

APPENDIX C

In order to obtain the pressure correction equation for the fluid phase inframe of the phase-sequential solution method, the momentum equation ofluid is written in the form

. (C1)

Such corrections to the fluid pressure are then introduced which produceestimates of the phasic velocities obeying the fluid mass balance. Thesestimates are defined as a sum of the current value and the correction as

and . (C2)

In order to limit the interdependency of the cells in equation (C1), the velocorrections to the neighbours of the cell under consideration are discaAccordingly, the velocities in the neighbouring cells are approximated as

, (C3)

where the weight factor for the SIMPLE algorithm and for tSIMPLEC algorithm [1]. Substitution of equations (C2) and (C3) to equation (results in

(C4)

By expanding the terms on the right side, equation (C4) can be arranged inform

(C5)

The first three terms on the right side equal to the corresponding side of equ(C1) calculated with the current values of the fluid mean velocity. Substitutio

aUf

k

PUf

k

Pa

Ufk

cUf

k

Cc C,∑ εf P

Aki

P

Pf∂

ξi∂-------

P

– sUf

k

R+=

PfδUf

k *

Ufk *

Ufk

Ufkδ+= Pf

* Pf Pfδ+=

Ufk

C

*Uf

k

CWpc Uf

k

P

*Uf

k

P–

+=

Wpc 0= Wpc 1=

aUf

k

PUf

k

P

*a

Ufk

cUf

k

CWpc Uf

k

P

*Uf

k

P–

+

c C,∑=

εf PAk

i

P

Pf∂

ξi∂-------

P

– εf PAk

i

P

Pfδ∂

ξi∂----------

P

– sUf

k

R.+

aUf

k

PWpc a

Ufk

cc C,∑–

Ufk

P

*a

Ufk

cUf

k

Cc C,∑ εf P

Aki

P

Pf∂

ξi∂-------

P

–=

sUf

k

RWpc a

Ufk

cUf

k

Pc C,∑– εf P

Aki

P

Pfδ∂

ξi∂----------

P

.–+

C1

C2

that equation into equation (C5) gives

(C6)

With the help of the definition (C2) equation (C6) can be solved for the unknowncorrections to the phasic Cartesian velocity components of the fluid phase

, where (C7)

and (C8)

. (C9)

Equation (C7) provides now the desired relation between the fluid pressure correc-tions and the phasic Cartesian velocity components of the fluid.

1. Van Doormal, J. P. and Raithby, G. D. Enhancements of the SIMPLE methodfor predicting incompressible fluid flows. Numerical Heat Transfer, 1984.Vol. 7, pp. 147–163.

aUf

k

PWpc a

Ufk

cc C,∑–

Ufk

P

*aUf

k

PUf

k

PWpc a

Ufk

cUf

k

Pc C,∑–=

εf PAk

i

P

Pfδ∂

ξi∂----------

P

.–

Ufkδ

PHfk

i

P

Pfδ∂

ξi∂-----------

P

–=

Hfki

P

εf PAk

i

P

VP-------------------- h f

k

P=

h fk

P

VP

aUf

k

PWpc a

Ufk

cc C,∑–

------------------------------------------------=

emeadientehav-ssurey areurce

epa-

ining

ourceereast, all

es by aom-

tion cell theses with

pres-rsed toorm

APPENDIX DThe essential idea in the improved Rhie-Chow momentum interpolation sch

is that after proper treatment the source term is immersed in the pressure grterm, partly or completely. The advantage of this treatment is the smoother biour of this combined term as the source term alone. Although both the pregradient term and the source term are multiplied by the volume fraction, thekept in this form as the volume fraction diminishes the strong variations of soterm.

At first, the part of the source term having rapid changes in it’s value is srated from the rest

, (D1)

where and are the part for improved treatment and the rema

part of a customary source term. As in the discretized balance equation the sterm is in a linearized form, these terms include only the zero order terms whthe first order terms are combined to the current point coefficient . Nex

the extracted source terms, defined at cell centres, are interpolated to cell facweighted linear interpolation with a weight factor of one half. The covariant cponents of these interpolated source terms are then calculated

. (D2)

In equation (D2) the notation stands for the weighted linear interpolabetween the cell centres and is a unit area vector of theface between the current and neighbour cells. To obey the conservation lawscell face covariant components are used to define new cell centre source termthe help of the generalized Gauss law

, (D3)

in which is the weight factor from cells face c to current cell centre P. In theimproved procedure the cell face source terms (D2) are combined to sharedsure gradients at cell faces and the cell centre source terms (D3) are immecell centre pressure gradients. The momentum balance is now written in the f

, where (D4)

SUα

R SUα

R Imp, SUα

R Base,+=

SUα

R Imp, SUα

R Base,

aUα

k

P

SUα

R Imp,

cSUα

R Imp,

H

C

PAA------

c

•=

H{ }

C

P

H P and C= A A⁄

SUα

R Imp,

PWc

A i( )c

J------------- SUα

R Imp,

cc

∑=

Wc

Uαk

P

1aUα

k

P

------------- εα PAk

i

P

Pf∂

ξi∂-------

P

sUα

kR Imp,

P–

+ Mα P=

D1

rpo-

hie- andd as

ts is

. (D5)

By following the treatment in section 5.1 the momentum balance (D4) is intelated for the imaginary staggered cell resulting to

(D6)

By defining the coefficient of pressure gradient term as for the extended RChow method, i.e., by the equation (124), and with assumptions (127)(128) the improved Rhie-Chow momentum interpolation scheme can be state

(D7)

. (D8)

As before, the interpolation scheme for phasic normal flux velocity componenobtained by applying the definition (B21)

(D9)

. (D10)

Mα Pa

Uαk

cUα

k

Cc C,∑ s

Uαk

R Base,

P+

aUα

k

P⁄=

Uαk

c

1aUα

k

c

------------ εα cAk

i

c

Pf∂

ξi∂-------

c

sUαk

R Imp,

c–

+

Uαk

H

1aUα

k

H

------------- εα HAk

i

H

Pf∂

ξi∂-------

H

sUα

kR Imp,

H–

+

C

P

.=

Nαki

Uαk

cUα

k

H

C

P

εα H{ }

C

PNαk

i

H

C

P Pf∂

ξi∂-------

H

C

P Pf∂

ξi∂-------

c

–

+=

1 aUα

k

H⁄

C

P

sUα

kR Imp,

H

C

P

sUαk

R Imp,

c–

–

Nαki

H

Aki

H

aUα

k

H

-------------=

Uαi

cUα

i

H

C

P

εα H{ }

C

PNα

ii

H

C

PPf∂

ξi∂-------

H

C

P Pf∂

ξi∂-------

c

–

+=

Nαki

H

C

P

sUα

kR Imp,

H

C

P

sUαk

R Imp,

c–

and–

Nαi j

PAk

i

PNαk

j

P

Aki

PAk

j

P

aUα

k

P

---------------------= =

D2

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Series title, number andreport code of publication

VTT Publications 458VTT�PUBS�458

Author(s)Karema, Hannu

Title

Numerical treatment of inter-phase coupling and phasicpressures in multi-fluid modelling

AbstractThis thesis work concentrates on the area of dispersed multi-phase flows and, especially, to the problems encounteredwhile solving their governing equations numerically with a collocated Control Volume Method (CVM). To allowflexible description of geometry all treatment is expressed in a form suitable for local Body Fitted Coordinates (BFC)in a multi-block structure. All work is related to conditions found in a simplified fluidized bed reactor. The problemscovered are the treatment and efficiency of inter-phase coupling terms in sequential solution, the exceeding of thebounds of validity of the shared pressure concept in cases of high dispersed phase pressure and the conservation ofmass in momentum interpolation for rapidly changing source terms.The efficiency of different inter-phase coupling algorithms is studied in typical fluidized bed conditions, where thecoupling of momentum equations is moderate in most sections of the bed and where several alternatives of differentcomplexity exist. The interphase coupling algorithms studied are the partially implicit treatment, the PartialElimination Algorithm (PEA) and the SImultaneous solution of Non-linearly Coupled Equations (SINCE). In additionto these special treatments of linearized coupling terms, the fundamental ideas of the SINCE are applied also to theSIMPLE(C) type pressure correction equation in the framework of the Inter- Phase Slip Algorithm (IPSA). Theresulting solution algorithm referred to as the InterPhase Slip Algorithm � Coupled (IPSA�C) then incorporatesinterface couplings also into the mass balancing shared pressure correction step of the solution.It is shown that these advanced methods to treat interphase coupling terms result in a faster convergence of momentumequations despite of the increased number of computational operations required by the algorithms. When solving theentire equation set, however, this improved solution efficiency is mostly lost due to the poorly performing pressurecorrection step in which volume fractions are assumed constant and the global mass balancing is based on sharedpressure. Improved pressure correction algorithms utilizing separate fluid and dispersed phase pressures, the FluidPressure in Source term (FPS) and the Equivalent Approximation of Pressures (EAP), are then introduced. Further, anexpanded Rhie-Chow momentum interpolation scheme is derived which allows equal treatment for all pressures. Allthe computations are carried out in the context of a collocated multi-block control volume solver CFDS-FLOW3D.

Keywordsmulti-phase flow, multi-fluid modelling, inter-phase coupling, phasic pressures, numerical methods, ControlVolume Method, Body Fitted Coordinates, fluidized beds, chemical reactors

Activity unitVTT Processes, Koivurannantie 1, P.O.Box 1603, FIN�40101 JYVÄSKYLÄ, Finland

ISBN Project number951�38�5969�X (soft back ed.)951�38�5970�3 (URL:http://www.inf.vtt.fi/pdf/ )

Date Language Pages PriceFebruary 2002 English 62 p. + app. 51 p. C

Name of project Commissioned by

Series title and ISSN Sold byVTT Publications1235�0621 (soft back ed.)1455�0849 (URL: http://www.inf.vtt.fi/pdf/)

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