simulation of gas–solid particle flows by boundary domain integral method

Simulation of gas–solid particle flows by boundary

domain integral method

M. Pozarnik*, L. Skerget

Faculty of Mechanical Engineering, University of Maribor, Smetanova ul. 17, SI-2000 Maribor, Slovenia

Received 16 September 2000; revised 28 May 2002; accepted 2 June 2002

Abstract

A novel numerical scheme based on the boundary domain integral method (BDIM) for the numerical simulation of gas–solid particle

flows is presented. A program is being developed to model the hydrodynamics of fluidized bed systems by using the Eulerian approach in

terms of velocity–vorticity variable formulation. Both phases are treated as separated, incompressible, continuous and fully interpenetrating

fluids. Each phase is described by a modified Navier–Stokes equation including interphase momentum exchange. With the vorticity vector

vpi representing the curl of the velocity field vpi; computation scheme of both phases motion is partitioned into its kinematic and kinetic

aspects. Therefore, the additional equation from the drift flux theory is necessary to compute volume fraction. Main advantage of the

proposed BDIM scheme is the reduced number of additional physical models derived from the kinetic theory of granular flows for the

description of the solid phase. Numerical scheme has been tested first on a single-phase test examples. Two-phase two-component results are

studied on two-phase gas–solid particles vertical channel flow. q 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Gas–solid particle flow; Two-fluid model; Modified Navier–Stokes equations; Velocity–vorticity formulation; Boundary element method

1. Introduction

Computational fluid dynamics (CFD) is becoming more

and more an engineering tool to predict flows in various

types of apparatus on industrial scale. Although the tools for

applying single phase flow CFD are widely available,

application of multiphase CFD is, however, still more

complicated from both physical and numerical points of

view. The research efforts of most groups working in this

field are aimed at development of still more detailed CFD

models for two-phase flow, while little attention is paid to

the evaluation of the simulation results from an engineering

point of view. An additional problem is the scale-up from

laboratory towards industrial equipment. For example,

equations describing the bubble behaviour in gas–solid

fluidized beds are (semi)empirical and often determined

under laboratory conditions. For this reason there is little

unifying theory describing the flow behaviour in fluidized

beds.

Computer simulations of gas–solid flows can be done

by using either a Lagrangian or Eulerian description for

the solid phase. A Eulerian description, typically based

on a volume-averaged continuum formulation, is used to

model the fluid phase. When the Lagrangian description

of the solid phase is used, the effects of the solid phase

appear as implied sources of mass, momentum, and

energy in the continuum description of the fluid phase.

Numerical models of gas–solid particle flows that are

based on Eulerian descriptions of both particles and the

fluid employ volume-averaged formulations to model

both phases. Such models are often referred to as two-

fluid models (TFM). Examples of their use can be found

in the works of Boemer et al. [4], Ding and Gidaspow

[6], Balzer and Simonin [2], and Sinclair and Jackson

[11]. Practical models of gas–solid particle flows are

obtained by introducing the notion of volume concen-

tration in the context of superimposed continua: each

phase is treated as continuum, simultaneously occupying

the same region in space. Rigorous derivations of such

models are based on averaging procedures. Example of

their use can be found in the work of Anderson and

Jackson [1].

Numerical simulations of gas–solid particle flows are

mostly based on finite difference methods (FDMs) or finite

volume methods (FVMs). None of the simulations based on

BEM regarding hydrodynamics of fluidized bed systems are

known to the authors. In the present work, the BDIM

0955-7997/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.

PII: S0 95 5 -7 99 7 (0 2) 00 0 54 -1

Engineering Analysis with Boundary Elements 26 (2002) 939–949

www.elsevier.com/locate/enganabound

* Corresponding author.

E-mail address: [email protected] (M. Pozarnik).

http://www.elsevier.com/locate/enganabound

(Skerget [12], Skerget et al. [13], Skerget and Rek [14]) for

simulation of two-phase two-component flow is presented.

It computes both phases flow variables in the velocity–

vorticity formulation with significant assumptions regarding

volume fraction of both fluid and solid particles. The

advantages of this approach lie with the numerical

separation of kinematic and kinetic aspects of the two

phases motion from the pressure computation, which could

be determined afterwards by the solution of a linear system

of equations for known velocity and vorticity fields. Most of

the two-phase two-component models used in combination

with FDMs or FVMs are derived from the kinetic theory of

granular flows. Then the solid-phase needs a lot of

additional models to calculate the so-called virtual proper-

ties from the momentum balance of the solid particles. The

emphasis of the proposed BDIM numerical scheme is given

to the relative motion between the phases governed by the

suitable drag function. Additional equation to compute

volume fraction of the fluid phase is derived from the drift

flux theory of the two-phase flow.

The paper is organized as follows. In Section 2 governing

equations and their transformed form in the BDIM are

presented. Section 3 gives an insight into the integral

representation of a parabolic diffusion-convective partial

differential equation, as all of the basic equations in BDIM

can be cast in this form. Special attention is given to

formulations with different fundamental solutions, as this is

one of the key features of the BDIM. While parabolic

diffusion fundamental solution has been used for the

kinematics, the elliptic diffusion-convective one has been

used for the kinetics of both fluid and solid particles motion.

Section 4 presents the discrete model and solution procedure

of the presented method. In Section 5, the method is verified

on the single-phase symmetric sudden expansion flow and

the two-phase two-component vertical channel flow. The

paper ends with the conclusions.

2. Gas–solid multiphase model

In spite of the increasing computational power, the

number of particles in gas–solid flow in large scale

equipment is still much too large to handle each particle

separately. Simulating each particle separately is called

Lagrangian method, which can be used to study microscopic

properties of fluidized beds. The CFD model used in this

work is based on a TFM extended with the drift flux theory

of two-phase flow. In a TFM both phases are considered to

be continuous and fully interpenetrating. The TFM has first

been proposed by Anderson and Jackson [1].

2.1. Primitive variables formulation

The continuity equation or mass balance for phase p (f for

gas and s for solid) reads

›

›tð1p@pÞ þ

›

›xj

ð1p@pvpjÞ ¼ 0 andX

p¼f;s

1p ¼ 1; ð1Þ

where 1p is the volume fraction of the phase, vpi; the ith

instantaneous phase velocity component, and @p; the

density. Mass exchange between the phases, e.g. due to

reaction or combustion, is not considered.

The momentum balance for the gas phase is given by the

Navier–Stokes equation, modified to include an interphase

momentum transfer term

Dvfi

Dt¼

1

@f1f

›Sfij

›xj

þ gi 21

@f

›p

›xi

2b

@f1f

ðvfi 2 vsiÞ; ð2Þ

where xi is the ith coordinate, D=Dt represents the

substantial or Stokes derivative, Spij is the viscous stress

tensor, gi is the gravity acceleration, p is the thermodynamic

pressure, and b is the interphase momentum transfer

coefficient. The solid phase momentum balance is given by

Dvsi

Dt¼

1

@s1s

›Ssij

›xj

þ gi 21

@s1s

›pps

›xi

21

@s

›p

›xi

þb

@s1s

ðvfi 2 vsiÞ; ð3Þ

where pps is the solids pressure originally obtained from the

kinetic theory of the granular flow. Bulk viscosity, which

describes the resistance of a fluid against compression

should be used together with the shear one in viscous strain

rate tensor in general what is discussed by Bird et al. [3].

The bulk viscosity is identically zero for low density

monatomic gases and is probably not too important in dense

gases and liquids. In the case of fluidized beds shear and

bulk viscosities are in the same order of magnitude, and

therefore, bulk viscosity should not be neglected.

With 1f ¼ 1 and b ¼ 0 Eq. (2) becomes the classical

Navier–Stokes equation. Mass and momentum balances are

discussed in detail in Boemer et al. [4].

2.2. Velocity–vorticity variables formulation

In BDIM, the original sets of Navier–Stokes equations

for gas phase and solid particles are further transformed

with the use of the velocity–vorticity variable formulation.

Within this approach flow field computation is decoupled

into flow kinematics and flow kinetics. Main advantages of

this scheme in the case of single phase flow lie with the

numerical separation of kinematic and kinetic aspects of the

flow from the pressure computation. This leads to a simple

way to enforce the proper boundary conditions compared

with the primitive variables approach, whenever the

pressure is not specified on the boundary as a known

quantity. The developed algorithm can still be written in the

most general form both for two or three dimensions.

With the vorticity vector vpi representing the curl of the

M. Pozarnik, L. Skerget / Engineering Analysis with Boundary Elements 26 (2002) 939–949940

velocity field, e.g. written in symbolic notation

vpi ¼ eijk

›vpk

›xj

;›vpj

›xj

¼ 0; ð4Þ

where eijk ði; j; k ¼ 1; 2; 3Þ is the permutation unit tensor,

that equals 1, if the subscripts ijk are in cyclic order or

equals 21, when they are in anticyclic order and zero

otherwise, the two components motion computation scheme

is partitioned into its kinematic and kinetic aspects.

By taking the curl to Eq. (4) and applying the reformed

continuity equation (1) with @p ¼ const: as supposed in the

work of Jackson [9]

›vpj

›xj

¼ 21

1p

›1p

›tþ vpj

›1p

›xj

!; ð5Þ

the kinematics of both components motion is carried out

›2vpi

›xj ›xj

þ eijk

›vpk

›xj

¼›

›xi

21

1p

›1p

›tþ vpj

›1p

›xj

!" #: ð6Þ

Eq. (6) represents the kinematics of an incompressible fluid

and solid particles motion or the compatibility of the

velocity and vorticity fields at a given point in space and

time.

The kinetics are governed by the vorticity transport

equations obtained as a curl of the momentum balances

(Eqs. (2) and (3)). In the case of low solid concentrations the

approach of Chapman and Cowling [5] with constant

viscosities is applied. Vorticity transport equations can be

written in the following form

Dvfi

Dt¼ nf0

›2vfi

›xj ›xj

2 vfi

›vfj

›xj

þ vfj

›vfi

›xj

2b

@f1f

ðvfi 2 vsiÞ2nf0

1f

›1f

›xj

eijkeklm

›vfl

›xm

� �

þ4

3

1

1f

nf0eijk

›2vfl

›xj ›xj

›1f

›xk

2b

@f12f

eijkðvfj 2 vsjÞ›1f

›xk

;

ð7Þ

Dvsi

Dt¼ ns0

›2vsi

›xj ›xj

2 vsi

›vsj

›xj

þ vsj

›vsi

›xj

þb

@s1s

ðvfi 2 vsiÞ2ns0

1s

›1s

›xj

eijkeklm

›vsl

›xm

� �

þ1

1s

4

3ns0 þ ks0

� �eijk

›2vsl

›xj ›xj

›1s

›xk

21

@s12s

eijk

›pps

›xj

›1s

›xk

þb

@s12s

eijkðvfj 2 vsjÞ›1s

›xk

; ð8Þ

describing the redistribution of the vorticity vector in the

fluid and solid particles flow field. In Eqs. (7) and (8) np0 ¼

hp0=@p0 is the phase shear viscosity and kp0 ¼ lp0=@p0 is the

phase bulk viscosity. While thermodynamic pressure p is

out of the computation, Eq. (8) is still dealing with the solids

pressure pps : Therefore, the proposed numerical scheme is

based on the subdomain technique in its limit version. Each

internal cell represents one subdomain called macroelement

bounded by four boundary elements. Discrete model is

discussed in detail in Section 4.1. Volume fraction of a

phase 1p is assumed to be constant in each macroelement

within one iteration of the numerical algorithm. Gradient

›1p=›xj equals zero all over the macroelement. Then Eqs.

(6)–(8) can be rewritten in a simple manner, but a lot of

physics is moved to the macroelement interface boundary

conditions (Section 2.5).

After the assumption regarding volume fraction 1p the

kinematics of both phases motion is written in the sense of

the parabolic equation where false transient approach is

implemented afterwards (ap is the relaxation parameter)

›2vpi

›xj ›xj

21

ap

›vpi

›tþ eijk

›vpk

›xj

¼ 0: ð9Þ

Using the same principle the kinetics given by Eqs. (7) and

(8) is rewritten as

Dvfi

Dt¼ nf0

›2vfi

›xj ›xj

þvfi

1f

›1f

›tþ vfj

›vfi

›xj

2b

@f1f

ðvfi 2 vsiÞ ð10Þ

for the fluid phase and

Dvsi

Dt¼ ns0

›2vsi

›xj ›xj

þvsi

1s

›1s

›tþ vsj

›vsi

›xj

þb

@s1s

ðvfi 2 vsiÞ ð11Þ

for the solid particles.

For the two-dimensional flow case the vorticity vector

vpi has just one component perpendicular to the plane of the

motion, and it can be treated as a scalar quantity. The vortex

deformation term is identically zero, reducing the vector

vorticity equations (10) and (11) to scalar ones for the fluid

vorticity transport

Dvf

Dt¼ nf0

›2vf

›xj ›xj

þvf

1f

›1f

›t2

b

@f1f

ðvf 2 vsÞ ð12Þ

and solid particles vorticity transport

Dvs

Dt¼ ns0

›2vs

›xj ›xj

þvs

1s

›1s

›tþ

b

@s1s

ðvf 2 vsÞ; ð13Þ

where eij ði; j ¼ 1; 2Þ is the permutation unit symbol ðe12 ¼

þ1; e21 ¼ 21; e11 ¼ e22 ¼ 0Þ: Eqs. (12) and (13) show that

the rate of change of the vorticity in the case of plane flow as

one follows a fluid or solid particle, given by the Stokes

derivation on the left-hand side of the equations, is due to

the viscous diffusion, bubble formation and interphase

M. Pozarnik, L. Skerget / Engineering Analysis with Boundary Elements 26 (2002) 939–949 941

momentum transfer, represented by the terms on the right-

hand side.

2.3. Volume fraction

To close the system of equations the additional equation

to compute volume fraction of the fluid phase is derived

from the drift flux theory [15] of the two-phase flow. This

model treats the general case of modelling each phase or

component as a separate fluid with its own set of governing

balance equations. In general each phase has its own

velocity, vorticity, and temperature. Drift flux theory has

widespread application to the bubbly, slug, and drop

regimes of gas–liquid flow as well as to fluid-particle

systems such as fluidized beds. It provides a starting point

for extension of the theory to flows in which two and three-

dimensional effects are significant.

Volume fraction of the component is determined by the

following equation

1f ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffilvfi 2 vsil

lv1ln21

s; ð14Þ

where v1 is the terminal speed of a single particle in an

infinite stationary liquid. The evaluation of index n has been

shown by Richardson and Zaki [10]. The complete

correlation of Richardson and Zaki over the whole range

of Reynolds numbers is giving the value of index n between

4,56 and 2,39 bearing in mind that particles are rigid spheres

and small compared to the diameter of the channel. The

value of n can be enlarged when particles flocculate. In our

study only low Reynolds number flow is encountered;

therefore, an intermediate value of index n for fluid-particle

systems is taken into account ðn ¼ 3Þ: A correction factor

can also be introduced in terms of the ratio of the particle

diameter to the tube diameter.

2.4. Boundary conditions

To solve the equations of gas–solid particles flow we

need appropriate boundary conditions not only for the fluid

phase but also for the solid particles. Classical boundary

conditions in the case of primitive variable formulation are

velocities of the gas phase and the solid particles, boundary

condition for gas phase pressure and boundary granular

temperature derived from the kinetic theory of granular flow

to compute the solids pressure. In velocity–vorticity

variables approach only appropriate boundary conditions

for velocities or velocity fluxes of the gas and the solids are

necessary.

The boundary conditions assigned to kinematic velocity

equation (9) are in general of the first and second kind, e.g.

vpi ¼ �vpi on G1 and›vpi

›xj

nj ¼›vpi

›non G2: ð15Þ

The most physical boundary conditions arise when the

velocity is prescribed over the whole surface. In this case,

normal derivatives of the velocity components are the

unknown boundary values in the set of kinematic equations,

assuming known vorticity distribution in the solution

domain. More difficulties arise when the velocity vectors

are not known a priori over part of the surface, i.e. outflow

regions. In such cases reasonable choice is to assume zero

velocity normal flux values through the specific part of the

boundary. The most important computation part of the

kinematics is the determination of the new boundary

vorticity values, which are the proper boundary conditions

associated with parabolic kinetic equation (9) written for the

whole boundary, e.g.

eijk

›vpk

›xj

¼ �vpi on G; ð16Þ

while the vorticity normal flux values

›vpi

›xj

nj ¼›vpi

›non G ð17Þ

are the only unknown boundary values in the vorticity

kinetics.

At an impenetrable solid wall, the gas phase velocities

are generally set to zero. This no-slip condition cannot

always be applied to solid motion. Since the particle

diameter is usually larger than the length scale of surface

roughness of the rigid wall, the particles may partially slip

the wall. But it is also important to note that for small

particle diameters the boundary condition is close to the no-

slip condition.

Initially, in gas phase, solid particles mixture volume

fraction of the two phases can be set on any physically

acceptable value.

2.5. Macroelement interface conditions

When dealing with the modified Navier–Stokes system

of equations written for the constant volume fraction of the

component over each macroelement within one iteration the

most critical part of the numerical scheme are macroelement

interface conditions. Therefore, particular attention has

been given to the analysis of a jump discontinuity in flow

properties. Velocity vectors at the macroelement interface

boundaries are split to the normal and tangential component.

The major interface characteristics are a jump in the volume

fraction and continuity of the tangential velocity (no-slip).

Due to the volume fraction jump also the jump in normal

velocity is necessary. For the interface macroelement

boundary conditions the relation relating the vorticity

values with normal and tangential velocity component

fluxes is derived. It reads

vp ¼›vpt

›n2

›vpn

›tð18Þ

for the two-dimensional motion in x–y plane.

Appropriate macroelement interface boundary conditions


derived from the conservation laws are for the kinematics

vptl2I ¼ 2vptl

1I ; ð19Þ

vpn1pl2I ¼ 2vpn1pl

1I ; ð20Þ

›vpt

›n

2I¼

›vpt

›n

1I; ð21Þ

1p

›vpn

›n

2I¼ 1p

›vpn

›n

1I; ð22Þ

and for the kinetics

vpl2I ¼ vpl

1I þ 1 2

11p

12p

!›vpn

›t

1I; ð23Þ

›vp

›n

2I¼ 2

›vp

›n

1I2 1 2

11p

12p

!›2vpn

›n ›t

1

I

; ð24Þ

where subscript I denotes the interface boundary GI between

the macroelements V1 and V2: With 1p ¼ 1 for both sides of

the boundary GI; Eqs. (19)–(24) become the classical form

for the single phase fluid motion.

3. Integral representation of velocity–vorticity

formulation

3.1. Gas and solid phase kinematics

Considering first the kinematics of both phases motion in

the boundary domain integral representation one has to take

into account that each component of the velocity vector vpi

(Eq. (9)) obeys the non-homogeneous parabolic equation

ap

›2vpi

›xj ›xj

2›vpi

›tþ bpi ¼ 0: ð25Þ

Equating the body force term bpi to the vortical fluid or solid

particles flow term from Eq. (9)

bpi ¼ apeijk

›vpk

›xj

ð26Þ

renders an integral statement

cðjÞvpiðj; tFÞ þ ap

ðG

ðtF

tF21

vpi

›upp

›ndt dG

¼ ap

ðG

ðtF

tF21

›vpi

›n2 eijkvpjnk

� �up

p dt dG

þ ap

ðV

ðtF

tF21

eijkvpj

›upp

›xk

dt dVþðV

vpi;F21upp;F21 dV;

ð27Þ

where upp is the parabolic diffusion fundamental solution, i.e.

the solution of the equation

ap

›2upp

›xj ›xj

2›up

p

›tþ dðj; sÞdðtF ; tÞ ¼ 0 ð28Þ

and given by expression

uppðj; s; tF; tÞ ¼

1

ð4papTÞd=2e2r2=4apT ; ð29Þ

where ðj; tFÞ and ðs; tÞ are the source and reference field

points, respectively, d stands for the dimensionality of

the problem, T ¼ tF 2 t and riðj; sÞ is the vector from

the source point j to reference point s, i.e. ri ¼

xiðjÞ2 xiðsÞ for i ¼ 1; 2 or i ¼ 1; 2; 3; while r is its

magnitude r ¼ lril:Assuming constant variation of all field variables

within the individual time increment, the time integrals

in Eq. (27) may be evaluated analytically, as shown in

the work of Skerget and Rek [14] for the plane case and

for the three-dimensional geometry. Integral statement

(Eq. (27)) can be finally rewritten in the following

form:

cðjÞvpiðj; tFÞ þðG

vpi

›Upp

›ndG

¼ðG

›vpi

›n2 eijkvpjnk

� �Up

p dGþðV

eijkvpj

›Upp

›xk

dV

þðV

vpi;F21upp;F21 dV:

ð30Þ

3.2. Gas and solid phase kinetics

Integral representation describing kinetics of both

components motion can be formulated by using the

fundamental solution of steady diffusion-convective PDE

with reaction term. Since it exists only for the case of

constant coefficients, the velocity field vpjðrk; tÞ has to be

decomposed into an average constant vector �vpjðtÞ and

perturbated one ~vpjðrk; tÞ; such that vpj ¼ �vpj þ ~vpj: In the

case of kinetics a non-symmetric finite difference

approximation of the field function time derivative for

the time increment Dt ¼ tF 2 tF21 is applied, e.g. for

the vorticity

›vpi

›t¼

vpi;F 2 vpi;F21

Dt; ð31Þ

and for the volume fraction

›1p

›t¼

1p;F 2 1p;F21

Dt: ð32Þ


Rewriting Eqs. (10) and (11) we obtain

np0

›2vpi

›xj ›xj

2›�vpjvpi

›xj

21

Dtvpi þ bpi ¼ 0: ð33Þ

The above differential formulation can be transformed into

an equivalent integral statement, e.g.

cðjÞvpiðjÞ þðGvpi

›Upp

›ndG

¼1

np0

ðG

np0

›vpi

›n2 vpi �vpn

!Up

p dG

þ1

np0

ðV

bpiUpp dV ð34Þ

with Upp ¼ np0up

p and �vpn ¼ �vpjnj: upp is the fundamental

solution of the diffusion-convective equation with first order

reaction term, e.g. the solution of the equation

np0

›2upp

›xj ›xj

þ›�vpju

pp

›xj

21

Dtup

p þ dðj; sÞ ¼ 0 ð35Þ

given for the plane case as

Upp ¼

1

2pK0ðmprÞexp

�vpjrjðj; sÞ

2np0

" #; ð36Þ

where the parameter mp is defined as

m2p ¼

�vp

np0

!2

þ1

np0Dtð37Þ

being �v2p ¼ �vpj �vpj: K0 and K1 are the modified Bessel

function of the second kind. Formulation for elliptic

diffusion-convective fundamental solution is discussed in

detail in the work of Skerget et al. [13] and Hribersek and

Skerget [8].

The pseudo-body term bpi in Eq. (34) includes the

convective flux for the perturbated velocity field �vpj;deformation, initial conditions, vorticity change on account

of the bubble formation and interphase momentum

exchange term, e.g.

bpi ¼ 2›~vpjvpi

›xj

þ›vpjvpi

›xj

þvpi;F21

Dt

þvpi

1pDtð1p;F 2 1p;F21Þ7 bðvfi 2 vsiÞ ð38Þ

with sign 2 in front of the interphase momentum

exchange term for the fluid phase and sign þ for the

solid particles rendering the following final integral

representation

cðjÞvpiðjÞ þðGvpi

›Upp

›ndG

¼1

np0

ðG

np0

›vpi

›n2 vpivpn þ vpnvpi

� �Up

p dG

þ1

np0

ðVðvpi ~vpj 2 vpivpjÞ

£›Up

p

›xj

dV7b

@p1p

ðVðvfi 2 vsiÞU

pp dV

þ1

np0Dt

ðVvpi;F21Up

p dV

þ1

1pnp0Dt

ðVvpið1p;F 2 1p;F21ÞU

pp dV: ð39Þ

4. Numerical solution

4.1. Discrete model

The discrete model in the proposed numerical scheme

is based on subdomain technique derived to its limit

version following the concept of finite volume with

supposed constant volume fraction of the component

within one iteration of the algorithm. One subdomain

called macroelement consists of one internal cell and

four boundary elements. The quadratic interpolation

functions were adopted since a good approximation of

boundary values of velocity gradients ensures an accurate

evaluation of boundary vorticity values, which influence

the stability of the proposed method. Discontinuous

three-noded quadratic boundary elements and continuous

nine-noded quadratic internal cells are used.

In the discretized equations for the kinematics and

vorticity kinetics of the fluid phase and solid particles,

integrals such as hpe; gpe; dpci; bpc; etc. are involved,

representing the integration over individual boundary

elements and internal cells, respectively, i.e.

hnpe ¼

ðGe

fn ›Upp

›ndGe; ð40Þ

gnpe ¼

ðGe

fnUpp dGe; ð41Þ

dnpci ¼

ðVc

Fn ›Upp

›xi

dVc; ð42Þ


and

bnpc ¼

ðVc

Fnupp;F21 dVc ð43Þ

for the kinematics or

bnpc ¼

ðVc

FnUpp dVc ð44Þ

for the kinetics of both components motion. The index n

refers to the number of nodes in each boundary element

or internal cell that also relates to the degree of the

respective interpolation polynomials. Integrals mentioned

above are functions of the geometry, time increment,

material properties and in case of kinetics also constant

velocity vector. They must be evaluated separately for

both fluid phase and solid particles, making this

numerical scheme quite rather time consuming.

Consider a discretized equation set for both fluid and

solid particles motion case. The discretization of integral

representations for velocities and vorticities can be readily

obtained from corresponding integral equations (30) and

(39) as follows, for the kinematics

cðjÞvpiðj; tFÞ þXEe¼1

{hpe}T{vpi}n

¼XEe¼1

{gpe}T ›vpi

›nþ eijkvpknj

� n

2eijk

XCc¼1

{dpcj}T{vpk}n

þXCc¼1

{bpc}T{vpi;F21}n ð45Þ

and for the vorticity kinetics

cðjÞvpiðjÞ þXEe¼1

{Hpe}T{vpi}n

¼1

np0

XEe¼1

{Gpe}T np0

›vpi

›n2 vpivsn þ vpnvpi

� n

þ1

np0

XCc¼1

{Dpcj}T{vpi ~vpj 2 vpivpj}

n

þ1

np0Dt

XCc¼1

{Bpc}T{vpi;F21}n þ1

np0Dt

£XCc¼1

{Bpc}T{vpi}n 2

1

np01sDt

£XCc¼1

{Bpc}T{vpi1p;F21}n 7b

@p1p

XCc¼1

{Bpc}T{vfi 2 vsi}n

ð46Þ

using superscript T to denote the transposition.

The plane kinematics ði; j ¼ 1; 2Þ can be given by the

following statement based on Eq. (45)

cðjÞvpiðj; tFÞ þXEe¼1

{hpe}T{vpi}n

¼XEe¼1

{gpe}T ›vpi

›nþ eijvpnj

� n

2eij

XCc¼1

{dpcj}T{vp}n

þXCc¼1

{bpc}T{vpi;F21}n: ð47Þ

Plane vorticity kinetics reads

cðjÞvpðjÞ þXEe¼1

{Hpe}T{vp}n

¼1

np0

XEe¼1

{Gpe}T np0

›vp

›n2 vpvsn

� n

þ1

np0

XCc¼1

{Dpcj}T{vp ~vpj}

n þ1

np0Dt

XCc¼1

{Bpc}T{vp;F21}n

þ1

np0Dt

XCc¼1

{Bpc}T{vp}n 21

np01sDt

£XCc¼1

{Bpc}T{vp1p;F21}n 7b

@p1p

XCc¼1

{Bpc}T{vf 2 vs}n:

ð48Þ

4.2. Solution procedure

Kinematic relation equation (45) or (47) and vorticity

transport equation (46) or (48), for both phases of motion

are coupled in two sets of non-linear equations. These two

sets are related by Eq. (14), which is used to compute

volume fraction of the fluid phase knowing the velocity

fields of both components. In order to obtain a solution of

the fluid and solid particles motion problem, a sequential

computational algorithm was developed. Main steps in this

algorithm are

1.F Start with initial values for the fluid phase vorticity

distribution.

1.S Start with initial values for the solid phase vorticity

distribution.

2.F Fluid phase kinematic computational part:

(a) solve implicit sets for boundary fluid phase

velocity or velocity normal flux values,

(b) transform new values from element nodes to

cell nodes,

(c) determine new boundary fluid phase vorticity

values,

(d) compute new fluid phase matrices for the


kinetics if the constant fluid phase velocity

vector is perturbated more than the prescribed

tolerance.

2.S Solid phase kinematic computational part:

(a) solve implicit sets for boundary solid phase

velocity or velocity normal flux values,


cell nodes,

(c) determine new boundary solid phase vorticity

values,

(d) compute new solid phase matrices for the

kinetics if the constant solid phase velocity

vector is perturbated more than the prescribed

tolerance.

3.F þ S Determine volume fraction of the fluid and solid

phase.

4.F Fluid phase vorticity kinetic computational part:

(a) solve implicit set for unknown boundary fluid

phase vorticity flux and internal domain fluid

phase vorticity values,


cell nodes.

4.S Solid phase vorticity kinetic computational part:

(a) solve implicit set for unknown boundary solid

phase vorticity flux and internal domain solid

phase vorticity values,


cell nodes.

5.F Relax fluid phase vorticity values and check the

fluid phase convergence. If convergence criterion

is satisfied, then stop; otherwise, go to step 2.F.

5.S Relax solid phase vorticity values and check the

solid phase convergence. If convergence criterion

is satisfied, then stop; otherwise, go to step 2.S.

5. Test examples

5.1. Single-phase symmetric sudden expansion flow

Because of the recirculation zones at larger values of

Reynolds number flow downstream of the expansion is

similar to the backward facing step flow. The flow is

symmetric at sufficiently low values of the Reynolds

number based on the step height and maximal inlet velocity.

It becomes asymmetric as the Reynolds number is increased

beyond the critical value. Our results were obtained at a

Reynolds number of 56 what ensures the symmetric flow

Fig. 1. Single-phase symmetric sudden expansion flow—geometry and

boundary conditions.

Fig. 2. Single-phase symmetric sudden expansion flow. Comparison with experimental data at different distances downflow the expansion.

Fig. 3. Single-phase symmetric sudden expansion flow. Comparison with

experimental data along the horizontal line through the centre of the

channel.


and enables us to compare the numerical results with

experimental values of Durst [7]. Geometry is simple with

the inlet region constituting 1/3 of the channel height based

at the middle of the channel. Geometry and boundary

conditions are shown at Fig. 1.

Prescribed inlet velocity profile is exactly equal to the

measured one [7] and differs a little from the parabolic

profile of fully developed laminar flow. At the outlet normal

velocity profiles are prescribed. The discretization is

consisting of 20 £ 18 subdomains with the ratio of two

between the longest and the shortest element. The calculated

separation regions behind the expansion are as reported by

Ref. [7] of equal length leading to the fully developed,

parabolic profile far downstream. Numerical results are

compared to the experimental values of Durst, Melling and

Whitelaw [7]. Fig. 2 is showing the comparison at different

distances downflow the expansion X=H ¼ 0; 1.5, 2.5, 3.5, 5

and 10, where H denotes the height of the step. In Fig. 3

maximum velocity values are shown along the centreline of

the channel.

5.2. Two-phase two-component flow in vertical channel

The aim of the research was to establish with the

proposed BDIM numerical scheme the influence of

different drag coefficients combined with different terminal

velocities of solid phase on the velocity fields of the two

components. Geometry and boundary conditions for

Fig. 4. Two-phase two-component vertical channel flow—geometry and

boundary conditions.

Fig. 5. Two-phase two-component vertical channel flow. Influence of interphase momentum exchange coefficient b on vertical component of phase 1 (upper

line) and phase 2 (lower line) along the vertical line through the centre of the channel at discretization 6 £ 24 subdomains ðvinlet max1 ¼ 1 m=s; vinlet max2 ¼

0:5 m=s (left), vinlet max2 ¼ 0:4 m=s (right), v1 ¼ 10 m=sÞ:


investigated two-phase two-component vertical channel

flow are presented in Fig. 4. Different parabolic velocity

profiles were prescribed at the inlet. In the first set of

calculations for phase 1 (gas) parabolic inlet velocity profile

with vfymax ¼ 1 m=s was defined. For phase 2 (solid phase)

maximum inlet velocity was set to the value of vsymax ¼

0:5 m=s: In the second part of calculations maximum inlet

velocity of phase 2 was decreased to the vsymax ¼ 0:4 m=s:Velocity profiles along the vertical line through the centre of

the channel were compared to each other. No slip conditions

on rigid walls were prescribed for both phases motion.

Normal velocity fluxes at the outlet were given as known

quantities. All over the computational domain realistic

value of the interphase momentum transfer coefficient was

given. Calculations with different values of drag coefficient

between b ¼ 10 and 30 kg/m3 s have been done. Steady

state analysis was simulated by a transient one for one very

large time step ðDt ¼ 1015Þ: Convergence criteria for all

runs has been set to 1024. Figs. 5 and 6 show the vertical

component of the velocity vectors along the vertical line

through the centre of the channel. The shape of velocity

profiles strongly depends on the drag coefficient b: The

results in Fig. 5 show the influence of drag coefficient on the

velocity fields at fixed Stokes velocity of a single solid

particle. On the other hand, Fig. 6 shows the results at fixed

drag coefficient and variable Stokes velocity. In both the

cases, one can see moving velocity profiles closer to each

other with the increasing interphase momentum exchange

coefficient. Also the decrease of maximum velocities due to

the increasing Stokes velocity was expected.

Simulations of decreasing velocity of phase 1 and

increasing velocity of phase 2 have been carried out also

Fig. 6. Two-phase two-component vertical channel flow. Influence of Stokes velocity of phase 2 v1 on vertical component of phase 1 (upper line) and phase 2

(lower line) along the vertical line through the centre of the channel at discretization 6 £ 24 subdomains ðvinlet max1 ¼ 1 m=s; vinlet max2 ¼ 0:5 m=s (left),

vinlet max2 ¼ 0:4 m=s (right), b ¼ 25 kg=m3 sÞ:

Table 1

Two-dimensional two-phase two-component vertical channel flow. Com-

parison of vertical component of the outlet velocity vpy regarding to the

mesh density ðvinlet max1 ¼ 1 m=s; vinlet max2 ¼ 0:5 m=s; b ¼ 25 kg=m3 s;

v1 ¼ 10 m=s)

Mesh Phase 1 Phase 2

6 £ 12 0.946 0.534

6 £ 16 0.943 0.535

6 £ 20 0.942 0.535

6 £ 24 0.942 0.536

6 £ 28 0.942 0.536

6 £ 40 0.942 0.536

12 £ 40 0.942 0.536


on different mesh densities. Convergence of the results is

given in Tables 1 and 2.

6. Conclusion

The BDIM has been used to simulate the two dimen-

sional two-phase two-component gas–solid flow. The

velocity–vorticity approach in combination with modified

Navier–Stokes equations is employed. The set of governing

equations is simplified under assumption that volume

fraction of each component is constant in each macroele-

ment within one iteration. Discontinuous velocity distri-

bution on the interfaces between the subdomains has been

overcome with the prescription of appropriate interface

macroelement conditions. Additional equation to compute

volume fraction of the fluid phase is obtained from the drift

flux theory. The advantage of the proposed scheme is the

reduced number of gas–solid physical models. Numerical

model has been validated first on single-phase test examples

such as single-phase symmetric sudden expansion flow.

Finally, the influence of the drag force and terminal velocity

of the solid phase on both the components velocity fields has

been tested in case of two-phase two-component vertical

channel flow.

Acknowledgments

The work of Dr Matej Pozarnik has been supported by

the Ministry for Science and Technology of the Republic of

Slovenia. This support is gratefully acknowledged.

References

[1] Anderson TB, Jackson R. A fluid mechanical description of fluidized

beds. Ind Engng Chem Fundam 1967;6:527–39.

[2] Balzer G, Simonin O. Extension of Eulerian gas–solid flow modelling

to dense fluidized bed prediction. Proceedings of the Fifth

International Symposium on Refined Flow Modelling and Turbulence

Measurements, Paris; 1993. p. 417–24.

[3] Bird RB, Stewart WE, Lightfoot EN. Transport phenomena. New

York: Wiley; 1960.

[4] Boemer A, Qi H, Renz U, Vasquez S, Boysan F. Eulerian computation

of fluidized bed hydrodynamics—a comparison of physical models.

Proceedings of the 13th International Conference on Fluidized Bed

Combustion, Orlando; 1995. p. 775–81.

[5] Chapman S, Cowling TG. The mathematical theory of non-uniform

gases. Cambridge: Cambridge University Press; 1970.

[6] Ding J, Gidaspow D. A bubbling fluidization model using kinetic

theory of granular flow. AIChE J 1990;36:523–38.

[7] Durst F, Melling A, Whitelaw JH. Low Reynolds number flow over a

plane symmetric sudden expansion. J Fluid Mech 1974;64(1):

111–28.

[8] Hribersek M, Skerget L. Fast boundary–domain integral algorithm for

the computation of incompressible fluid flow problems. Int J Numer

Meth Fluids 1999;31:891–907.

[9] Jackson R. Fluidization. In: Davidson JF, Clift R, Harrison D, editors.

Hydrodynamic stability of fluid-particle systems. New York: Academic

Press; 1985. p. 47–72.

[10] Richardson JF, Zaki WN. Sedimentation and fluidization. Part I. Trans

Inst Chem Engrs 1954;32:35–53.

[11] Sinclair JL, Jackson R. Gas-particle flow in a vertical pipe with

particle–particle interactions. AIChE J 1989;35(9):1473–86.

[12] Skerget L. Mixed convection cavity flows. Boundary Elements, vol.

XIX. Rome, Italy: Computational Mechanics Publications; 1997. p.

505–14.

[13] Skerget L, Hribersek M, Kuhn G. Computational fluid dynamics by

boundary–domain integral method. Int J Numer Meth Engng 1999;

46:1291–311.

[14] Skerget L, Rek Z. Boundary–domain integral method using a

velocity–vorticity formulation. Eng Anal Bound Elem 1995;15:

359–70.

[15] Wallis GB. One-dimensional two-phase flow. New York: McGraw-

Hill; 1969.

Table 2

Two-dimensional two-phase two-component vertical channel flow. Com-

parison of vertical component of the outlet velocity vpy regarding to the

mesh density ðvinlet max1 ¼ 1 m=s; vinlet max2 ¼ 0:4 m=s; b ¼ 25 kg=m3 s;

v1 ¼ 10 m=sÞ

Mesh Phase 1 Phase 2

6 £ 12 0.935 0.442

6 £ 16 0.934 0.443

6 £ 20 0.934 0.443

6 £ 24 0.934 0.444

6 £ 28 0.933 0.444

6 £ 40 0.933 0.444

12 £ 40 0.933 0.444


simulation of gas–solid particle flows by boundary domain integral method

Documents