sirpa kallio, vtt - Åbo akademiusers.abo.fi/rzevenho/icfd sk abo_akademi2019_slides.pdf · vtt...

VTT TECHNICAL RESEARCH CENTRE OF FINLAND1

Multiphase flow dynamicsSirpa Kallio, VTT

Boiling water

Turbulent fluidized bed


Outline

t Classification of multi-phase flows

t Modeling approaches

– Homogeneous flow model– Lagrangian particle tracking

– General description– Methods for dense suspensions (DPM, MP-PIC)

– Eulerian-Eulerian two-fluid models– Algebraic slip mixture model– Other approaches

t Example: Eulerian-Eulerian modeling of dense gas-solid flow


Multiphase flows can be classified according to

• the combination of phases:

• gas-solid (fluidized beds, conveying, combustors, etc.)• gas-liquid (sprays, nuclear reactors)• liquid-solid (crystallisators, sedimentation, fiber suspensions)• liquid-liquid (oil-water separation)• gas-liquid-solid (catalytic cracking)• etc.


Multiphase flows can be classified according to

• The volume (or mass) fraction of the dispersed phase• dilute (relaxation time tp < collision time tc)• dense

• The type of distribution of the dispersed phase• homogeneous• heterogeneous (e.g. clusters, sprays, flocks)

• The geometry of the phase interfaces• separated flows• mixed flows• dispersed flows

)1(Re3

4)1(Re

18

2

>=<= pfpDf

pppp

f

ppp C

dt

dt

urr

mr


• Multiphase flows can be treated as single phase flows when• the amount of the dispersed phase(s) is very small• the carrier phase is not affected• the dispersed phase(s) are of no interest

• In special cases single phase flow can be treated as multiphaseflow, e.g.

• in mixing of fluid streams of different temperature and/orconcentration• in the description of the different scales of turbulence

• A computational phase in multiphase CFD is not same as aphase in the physical sense

• computational phase represents a mass moving at a ‘single’velocity


TYPES OF PHASE INTERACTION

One-way coupling:

Fluid flow affects particles/dispersed phase

Two-way coupling:

Above + dispersed particles modify fluid flow

Four-way coupling:

Above + particle-particle interaction


Multiphase flow and turbulence

t multiphase flows are typically by nature ’turbulent’ (phaseinterfaces distort the flow field which leads to fluctuations invelocities and other properties)

t dispersed phase modifies carrier phase turbulence– big particles enhance turbulence– small particles damp turbulence– carrier phase turbulence becomes unisotropic

t carrier phase turbulence enhances dispersion and clustering of thedispersed phase

t in dense suspensions the turbulent fluctuations in the differentphases are strongly coupled and often of a different character thanin single-phase flows


Approach 1: Homogeneous flow model

- Simplest multiphase CFD model

- Assumes that dispersed phase travels at the same velocityas the carrier phase

- Mixture momentum equation

- Mixture continuity equation

- Phase continuity equations: transport of dispersed phase through ‘diffusion’

- Fast; describes well particle size and other distributions

- Describes well effects of turbulence on mixing of the dispersed phase


Approach 2: Particle tracking methods

• also called Lagrangian approach

• utilizes the equation of motion of a single particle

• initially used for dilute suspensions (one-way and two-way coupling)

• natural description for mixtures of particles with different properties

• usable for steady state simulation only in cases with distinct inlets(eg. sprays, burners)

• can be used for dense suspensions if collisions are accounted for(the so-called micro-scale models; complicated and slow)


Equation of motion

tt

trmpmp d

t

rdd

rr

r

rdtdm

dtd

mmmdt

dm

t

t

fp

fp

ffpfp

fpfp

fp

fpff

ffpp

p

ò -

÷÷ø

öççè

æÑ--

-÷÷ø

öççè

æÑ---

÷÷ø

öççè

æÑ---+-=

0

22

222

22

66

66

102)(

vvv

vvv

vvvv

gv

according to Maxey & Riley (Phys. Fluids, 26:883-889, 1983)Note: modified from the older BBO-equation

Forces acting on the particle: gravity + added mass +viscous drag + Basset history effects


Validity of Maxey & Riley equationt For laminar flow (and mean field of turbulent flow)

L is characteristic length scale, W characteristic relative velocity, U/Lthe scale of fluid velocity gradient for undisturbed flow.

t For turbulent flow

1,1,12

<<<<<<L

UrWrLr

f

p

f

pp

nn

1,1,12

<<<<<<kf

kp

f

p

k

p vrWrrhnnh

kh and kv are the Kolmogorow microscales of length and velocity:

4/1

4/13

)( ene

nh fkk vf =

÷÷ø

ö

ççè

æ=

e is the dissipation rate of turbulent kinetic energy


Equation of motion in Stokes regime (Rep<1)

( )fpfpfpp

p rmmdt

dm vvg

v---= mp6)(

Usually simplified:

For rapidly accelerating or oscillating flows this approximationcan be poor.

Problems can also be expected when density differencebetween the phases is small.


Drag models

t a large number of empirical correlations exist

t models that account particle shape and deformation

t for dilute fluid particle flows, the drag is given by

)(43

fpfpp

fD d

CF vvvv --=r

Schiller & Naumann correlation

[ ] 1000ReRe15.01Re24 687.0 £+= pp

pDC

1000Re44.0 >= pDC

• for dense fluid particle flows, the drag force dependsalso on volume fraction of dispersed phase


Dispersion of particles

t Eddy interaction model/discrete random walk model: a particle is assumed tointeract with an eddy over an interaction time

t The interaction time is the smaller of the eddy life time te and the eddycrossing time tc

t The velocity fluctuation during an eddy is obtained by sampling a normallydistributed random function:

úúû

ù

êêë

é

--=»

pfp

epce uut

Lttkt ln6.0e

)3/2()(' 2' kfufu ==


Numerical methods

One-way coupling Two-way coupling


Lagrangian approach, dilute example:

Pulverized coal boiler:Particle temperature alongparticle tracks


Lagrangian methods for dense gas-solids flow

t Lagrangian particle tracking methods have been applied e.g. to fluidizedbeds

t Several approaches that differ– in the way a computed particle corresponds to real particles: one particle can

present either one or a group of particles– in the way particle-particle collisions are treated

– Discrete Particle Method (DPM):– no collisions (used for dilute suspensions)– soft-sphere approach (the discrete element method, DEM)– hard-sphere approach

– Dense DPM, CPFD (computation particle fluid dynamic):– statistical approach


Lagrangian methods for dense gas-solids flow:soft- and hard-sphere models

t Soft-sphere model of particle-particle collisions:– the instantaneous inter-particle contact forces, namely, the normal,

damping and sliding forces are computed using equivalent simplemechanical elements, such as springs, dashpots and sliders.

– a very small time step is required (< 1e-6 s)

t Hard-sphere model of particle-particle collisions:– momentum-conserving binary collision– interactions between particles pair-wise and instantaneous– faster than soft-sphere method due to longer time steps


Lagrangian methods for dense gas-solids flow:multiphase particle-in-cell method (MP-PIC)

t Eulerian gas phase

t Lagrangian tracking of parcels of particles– Gas-particle and particle-particle interaction forces calculated in the mesh of

the gas phase from closures written for Eulerian-Eulerian models for densegas-particle suspensions (kinetic theory of granular flow, see later in thepresentation)

t Also called / similar models:– Dense DPM (Dense Discrete Particle Model; Ansys Fluent)– CPFD (Computational Particle Fluid Dynamic; Barracuda)

t Description of particle size distribution easy

t Coarse gas phase mesh leads to inaccuracies


Lagrangian simulation, dense example:(LES for carrier phase, collision models for particle-particle interaction; Helland et al., Powder

Technology, 110, pp. 210-221 (2000))


Coupling with direct numerical simulation (DNS) forcontinuous phase

t Lagrangian simulation of particles with hard or soft sphere models canbe coupled with direct numerical simulation of the continuous phase

– Fine mesh with several mesh elements per particle diameter– Short time steps, time consuming!– Produces accurately fluid-particle interactions– Used for submodel development

t Body-fitted grid: mesh is regenerated at each time step to fitparticle surfaces

t Non-body fitted grid: e.g. immersed boundary method in whichthe fluid flow calculations are performed by assuming that the fluidoccupies the entire flow field and the effect of particles is expressed bya body force into the momentum equation of fluid to constrain the no slipboundary condition at the nodes inside the particles.


Lagrangian approach, summary

• momentum and continuity equation for the carrier phase

• Newton equation for dispersed phase

• good for dilute suspensions, usable also for dense suspensions if collisionsare accounted for

• description of dispersion of particles in turbulent flow requireslarge number of particles

• fast for laminar flow; describes well particle size andother distributions


Approach 3: Eulerian-Eulerian approach

t single phase equations (momentum, continuity) are averaged overtime and/or volume and/or a particle assembly (Newtons equation)to obtain multiphase flow equations

t distribution of phases given by volume fraction

t models for phase interaction required

t all phases are described as interpenetrating continua


Volume averaging

Averaging volumePhase interface


Starting point in volume averaging: Single phase equations + jumpconditions

( ) ( ) 0=×Ñ+¶¶

qqqtvrr

( ) ( ) gτvvv qqqqqqqq pt

rrr +×Ñ+-Ñ=×Ñ+¶¶

( ) ( ) 0=×-+×- gggrr nvvnvv AqAqq

( ) ( ) AA

qqAqqqAqAqqq pp R

Rσ

σnτInτInvvvnvvv)g

gggggggg rr2

)()( +Ñ-×+-+×+-=×-+×-

qn Unit outward normal vector of phase q

Av Velocity of the interfaceAR

)AA RR /=

AR interface curvature vectorgqσ interface surface tension

AD Ñ×-Ñ AR)

= = surface gradient operator I unit tensor


Averaging

Volume averaging:

dVV qV qq ò= yy 1

qq

V qq

q dVV q

ya

yy11~ == ò

qq

qq

V q

V qq

q dV

dV

q

q

ra

yr

r

yry ~==

òò

Volume average

Intrinsic volume average

Mass-weighted (Favre) average

VVqq /=a Volume fraction, 1=åq

qa


Eulerian-Eulerian balance equations

( ) ( ) qqqqqqtG=×Ñ+

¶¶ vrara ~~

( ) ( )

qqqqqqq

qqqqqqq

pt

draa

rara

τMgτ

vvv

×Ñ+++×Ñ+-Ñ=

×Ñ+¶¶

~)~(

~~

Mq is the interaction force.

Continuity:

Momentum:


0=Gåq

q

å òå ÷÷ø

öççè

æ+Ñ--=

g

gg

g,

221

q AA

A

qqA

qq dA

Vq

RRσ

σM)

Turbulent stress term:

Conservation of mass:

Surface tension:

))((~qqqqqqq vvvvτ ---= rad


Alternative averaging approaches

t volume averaging

t time averaging

t ensemble averaging

t volume-time averaging

t two-step volume-volume or volume-time averaging (first over thelocal scale, then over a larger scale)

t three-step averaging


Closures for turbulent stress termst no generally accepted turbulence models exist; parameters in any

models are probably suspension dependent

t several alternatives suggested

t mixture turbulence model:uses mixture properties to calculate viscous stress; applicable forstratified flows and when densities of the phases are of same order

t dispersed turbulence model– appropriate for dilute suspensions– turbulence quantities of the dispersed phase are obtained

using the theory of dispersion of particles by homogeneousturbulence

t turbulence model for each phase


Closures for drag force

t large number of models exist

t drag force is generally written in the form:

t Note: in dense flows, CD depends on αf

t parameters in drag models should be functions of theaveraging scale in inhomogeneous flows!

)(43

fqfqq

ffqDq d

CM vvvv ---=raa


Description of particle size distribution in the Eulerian-Eulerianmethod

t Simplest way would be to treat each size class as a separatecomputational phase (method of classes)

– Continuity and momentum equation for each size class– Computationally slow– Numerical difficulties: at least earlier versions than Ansys Fluent 14

produced non-physical results

t Particle size distribution described by methods of moments– QMOM (Quadrature Method Of Moments)– DQMOM (Direct QMOM)

– In principle requires solution of momentum equations for velocities ofthe moments

– Momentum equations can in some cases be replaced by algebraicequations


Numerical methods for the Eulerian-Eulerian approach

t several algorithms; usually extensions of single phase ones

t e.g. IPSA (Inter Phase Slip Algorithm) performs the following steps inone iteration loop:

– solve velocities of all phases from momentum equations– solve pressure correction equation (which is based on the joint

continuity equation)– update pressure– correct velocities– calculate volume fractions from phasic continuity equations


Example: simulated solids volume fractions in acirculating fluidized bed

Two-phase flow

Largeparticles

Smallparticles

’Three’-phase flow


Eulerian-Eulerian models, summary

• momentum and continuity equations for each phase

• several averaging methods used to derive the equations

• best alternative for dense suspensions with large velocitydifferences/acceleration

• often very slow

• particle size and other distributions difficult to account for


Approach 4: Mixture model

t A local equilibrium between the continuous and dispersed phases, i.e.at every point, the particles move with the terminal slip velocity relative tothe continuous phase

Þ velocity components for dispersed phases can be calculated fromalgebraic formulas

Þ significantly less equations to be solved

t Also called / similar models:– Algebraic slip (mixture) model– Diffusion model– Suspension model– Local-equilibrium model– Drift-flux model


Motivation for simplified multiphase models

Full Eulerian models - problemst Numerical problems - long computing times - small time steps

t Difficulties in convergence

t Only a few secondary phases possible

t More difficulties, if mass transfer and chemical reactions areconsidered

t Simplified models are recommended, if applicable

By Mikko Manninen, VTT


Detailed derivation of the mixture model (1)

t Starting point: Eulerian model equationst Continuity equation for phase k:

t Momentum balance for phase k:

t Mixture equations are derived by taking a sum over all phases

( ) ( ) kkkkkktG=×Ñ+ urara

¶¶

( ) ( ) ( )[ ] kkkTkkkkkkkkkkkk pt

Mgττuuu +++×Ñ+Ñ-=×Ñ+ raaarara¶¶




t Continuity

t Define mixture density and velocity as

t Resulting equation

( ) ( ) ååå===

G=×Ñ+n

kk

n

kkkk

n

kkkt 111

urara¶¶

r a rmk

n

k k==

å1

åå=

==n

kkk

kkkk

mm c

1

1 uuu rar

( ) 0=×Ñ+ mmm

tur

¶¶r åå =×Ñº×Ñº×Ñ

kmk

kkk 0jjua

If all phases are incompressible




t Momentum balance

t Diffusion velocity (not slip velocity!)

t Definitions

( ) åååå

åå

====

==

+++×Ñ+Ñ-=

×Ñ+

n

kk

n

kkk

n

kTkkk

n

kkk

n

kkkkk

n

kkkk

p

t

1111

11

Mgττ

uuu

raaa

rara¶¶

u u uMk k m= -

t tmk

n

k k==

å1

a

tTm k Ik Fk Fkk

n

= -=

å a r u u1

t Dmk

n

k k Mk Mk= -=

å1

a r u u

Viscous stress

Turbulent stress

Diffusion stress




t Assumption: all phases share the same pressure,t Using the definitions of mixture density and velocity and the

stresses above, we obtain the resulting momentum balance for themixture

( ) ( ) mmDmTmmmmmmm pt

Mgτττuuu ++×Ñ++×Ñ+Ñ-=×Ñ+ rrr¶¶

ppk =




t Eliminate the phase velocity in the continuity equation for a phase

t Or if phase densities are constants and mass transfer is zero:

which is a balance equation for phase k volume fraction andincludes the effect of slip. Note that this the same as the intuitiveresult except for the diffusion velocity instead of slip velocity.

( ) ( ) ( )Mkkkkmkkkktuu rarara

¶¶

×Ñ-G=×Ñ+

( ) ( )Mkkmkktuu aaa

¶¶

×-Ñ=×Ñ+




t Relation between the slip velocity and the diffusion velocity

t For only one dispersed phase

t Note: so far no additional assumptions or approximations has beenmade (except the shared pressure, commonly used also in fullEulerian models)

u u uCk k c= -

å-=-=l

CllCkmkMk c uuuuu

( )u uMd d Cdc= -1

slip velocity (c denotes the continuous phase)



Slip velocity (1)

t The momentum equation for the dispersed phase is

t Momentum equation for the mixture (with Mm=0)

t Eliminate pressure gradient and solve for Mp

( ) ( ) ( )[ ]

( )[ ] ( ) ( )gτττττ

uuuuuuM

mppDmTmmpTppp

mmmppppm

mpMp

ppp tt

rraaa

rra¶

¶rr¶

¶ra

--++×Ñ++×Ñ-

Ñ×-Ñ×+úû

ùêë

é-+=

( ) ( )[ ] pppTppppppppp

pp pt

Mguuu

+++×Ñ+Ñ-=Ñ×+ rattaara¶

¶ra

( ) ( ) gτττuuu mDmTmmmmmmm pt

rrr¶¶

+×Ñ++×Ñ+-Ñ=×Ñ+



Slip velocity (2)

t Now, the following approximations are made:– All turbulent terms are omitted for the moment– Viscous and diffusion stress terms are small and are neglected– The local equilibrium assumption implies

t The momentum transfer term is given by

mmpp uuuu )()( Ñ×»Ñ×0»¶

¶tMpu

CpCpcDp

ppp C

VA

M uura

21=



Slip velocity (3)

t Finally:

t For small particles (Stokes drag) in gravitational and centrifugalfield

t This is the slip velocity. The diffusion velocity used in the modelequations must be calculated as shown above.

( ) ( ) úû

ùêë

é-+Ñ×-= guuuuu

tVCA m

mmmppCpCpDpc ¶¶rrr

21

úû

ùêë

é-

-= geu r

m

m

mppCd r

ud 22

18)( j

mrr



Closures for viscous stress in mixture models

[ ]Tmmmm )( vv Ñ+Ñ= mt

*

÷÷ø

öççè

æ-=

ma

aa

mmmax,5.2

max,

1p

p

pfm

max,pa

Viscous stress:

1=*mfp

fp

mmmm

m+

+=* 4.0

Mixture viscosity (Ishii & Zuver, 1979):

where is the maximum packing and

for bubbles or drops

for solid particlesBy Mikko Manninen, VTT


Treatment of turbulence in mixture models

t in dilute suspensions, mixture turbulent stress is assumed sameas carrier phase turbulent stress

t at higher concentrations, modeling required

t turbulence enhances spreading of dispersed phases -> takeninto account in continuity equation:

( ) ( ) ( ) ( )mpppppmpmpppp Dt

vv raarara ~~~ ×Ñ-G+Ñ×Ñ=×Ñ+¶¶

÷÷ø

öççè

æ+=

3/285.01

2

kD cp

Tcmp

vn

where Dmq is a dispersion (diffusion) coefficient:



Turbulent dispersion

t Turbulence is coupled in the mixture model in a straightforwardway, because the treatment has a single phase nature. A standardmodel, e.g. k-e model, can be coupled to the mixture equations.

t The turbulent stress and the fluctuating part of Mq are modeledusing a turbulent dispersion (diffusion) coefficient DT

t A suitable model is needed for the dispersion coefficient, forexample (Picart et al., 1986)

qq

Tmqmq

Da

aÑ+®uu

DukT TCd= +

æ

èç

ö

ø÷

-

n 1 0 852 3

2 1 2

.



Mixture model - validity

t The essential approximation is the local equilibrium assumption:the particles are accelerated instantaneously to the terminalvelocity

t Consider again the simplified one-particle equation

t Solution of particle velocity as afunction of distance (Stokes regime)

t The characteristic length ofacceleration iswith

2pDpfp

pp uCAgV

dtdu

m rr21

-D=

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

px l

tpp ut=l

f

ppp

dt

mr

18

2

= pp

t gtur

rD=



Mixture model - validity

0.0001

0.001

0.01

0.1

1

1.0 2.0 3.0 4.0 5.0 6.0rp/rc

l p(c

m)

0.1 mm

0.2 mm

Virtual mass and history effects

Stokes drag only

u If the density ratio is small, the virtual mass and history terms cannot be ignored (botheffects increase the response time)

u A suitable requirement for the local equilibrium is lp<< typical system dimension



Validity range of mixture models, cont.

- Mixture model is usually not suitable for gas-particle flows,because the characteristic length can be large

- Not suitable for clustering flow, where average slip velocityis greater than single particle slip

- Typically used for liquid-solid flows and bubbly liquid-gasflows (not for very big bubbles)


Validating simulationsSolid particles in an agitated vessel

Glass beads of three different sizes in water

Experiments: Barresi and Baldi (1987)

D = 0.390 m, H = 0.464 m4 baffles (0.039 m)45° pitched 4-blade turbine(Dt = 0.130 m, Htb = 0.046 m).Impeller power number: 1.2

Average mass fraction: 0.5 %

dp,exp.(mm) 100-177 208-250 417-500dp,sim.(mm) 140 230 460r (kg/m3) 2670 2600 2600N (rev/s) 4.83 5.67 6.50

d = 230 mm



Validating simulations

Solid particles in an agitated vessel

0.0

0.5

1.0

0.0 0.5 1.0 1.5 2.0 2.5

c/cav e

z/z m

ax

ExperimentalMixture modelMultiphase model

dp = 140 mm

0.0

0.5

1.0

0.0 0.5 1.0 1.5 2.0 2.5

c/cav e

z/z m

ax


dp = 230 mm

0.0

0.5

1.0

0.0 0.5 1.0 1.5 2.0 2.5

c/cav e

z/z m

ax


dp = 460 mm



Bubble drag correlations

Tomiyama 1

Tomiyama 2

Ishii-Zuber

( )s

rr 2pqp d

Eo-

=a



Bubble terminal velocity

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.002 0.004 0.006 0.008 0.01d (m)

Usl

ip

I-Z, gI-Z, 10gI-Z, 20gTomiyama 2, 20gSl

ipve

loci

tyd (m)

In gravitational field Effect of acceleration



Mixture model example


Mixture models (FLUENT): (N+3)t Continuity equation for the mixture and

all bubble phases (N)t Momentum equation for the mixture (3)t Bubbles: slip velocities from algebraic

balance equationst Numerically well behaved, a large

number of bubble classes possible

General Eulerian models(4xN balance equations)t Continuity equation for all phases (N)t Momentum equation for all phases (3N)t Numerically difficult and slow

Comparison of models for bubbly flows



Mixture model summary

- Length scale of particle relaxation must be small comparedto the mesh spacing

- Local equilibrium assumption yields the slip velocity

- Transport of the dispersed phase calculated from thephase continuity equation

- Fast; describes well particle size and other distributions

- Describes well effects of turbulence


Other approaches

t VOF (volume of fluid) –method for tracking phase interfaces inseparated flows or e.g. droplet formation in small scale

t Eulerian-Eulerian + Lagrangian

t Eulerian-Eulerian + mixture model

t Mesoscopic simulation methods (e.g. Lattice-Bolzmann method)

t MP-PIC (multi-phase particle in cell; Lagrangian transport ofparcels of particles + Eulerian kinetic theory description for solids)


Summary of multiphase CFD:

• Several ways to derive equations

• Simplifications in all models

• Jump conditions at phase interfaces complicated (drag force )

• Multitude of time and length scales -> difficult to choosesuitable averaging and simulation scales

• Flows often time-dependent also in macroscopic length scales(e.g. bubbling fluidized beds)


Summary of multiphase CFD, cont.:

• Never fully ‘laminar’

• Turbulence modelling difficult; no general models exist;unisotropy causes problems

• Dense suspensions dominated by particle-particle collisions

• Dilute suspensions dominated by particle-turbulence interaction

• Measurements still necessary for derivation of equation closuresand for validation; often not available


Summary of multiphase CFD, cont. :

• Choose the simplest possible model

• Follow the literature

• Compare with measurement data if available

• Results for dilute suspensions already good

• Even dense suspensions can be simulated

• Increasing computer speed will facilitate simulationof more demanding cases in the future


CFB (circulating) TFB (turbulent) BFB (bubbling)

- inhomogeneous flow pattern- flow characteristics change as a function of fluidization velocity

Ref.: U. Ojaniemi, S. Kallio, A. Hermanson, M. Manninen, M. Seppälä, V. Taivassalo, Comparison of simulated and measuredflow patterns: solid and gas mixing in a 2D turbulent fluidized bed, Fluidization XII, Harrison Hot Springs, BC, Canada, 2007

Example of Eulerian-Eulerian modeling: Fluidized beds


Eulerian-Eulerian approach based on KTGF– background

t Kinetic theory of granular flow (KTGF)

t A special model developed for dense gas-solid suspensions, seee.g. (Gidaspow D., 1994, Multiphase flow and fluidization:Continuum and kinetic theory descriptions, Academic press,Boston)

t Theory similar to gas kinetic theoryt Solid phase equations are derived by ensemble averaging over the

single particle velocity distribution functiont This averaging is done in a local scale assuming locally random

distribution of particles


Eulerian-Eulerian approach based on KTGF– limitations and simplifications

t Because of the randomness assumption, the theory is basicallyonly suitable for transient simulations in fairly fine meshes

t In derivation of the models, only binary collisions between particlesare accounted for, and thus KTGF is not valid in densesuspensions, like in bubbling beds

t In practise it is used from packing limit to very dilute conditions

t The theory was originally written for a single particle sizet Later rigorous extension of the theory to mixtures has been

developed

t Here we focus on models for a single particle size


Eulerian-Eulerian approach based on KTGF– software inplementations

t Kinetic theory of granular flow is today found in– commercial codes– open source codes– researchers’ and universities own codes

t Alternative closures based on different assumptions have beensuggested in the literature

t Here we mainly present models available in the commercial Fluentsoftware


Eulerian-Eulerian approach based on KTGF– balance equations

0)()( =×Ñ+¶¶

qqqqqtvrara

)()()()( gssggggggggggggg Kpt

vvgvvv -++×Ñ+Ñ-=×Ñ+¶¶ rataarara

)()()()( sgsgsssssssssssss Kppt

vvgvvv -++×Ñ+Ñ-Ñ-=×Ñ+¶¶ rataarara

Continuity equation for phase q:

Momentum equation for gas phase g:

Momentum equation for solid phase s:


Eulerian-Eulerian approach based on KTGF– balance equation for granular temperature

2'31

ss v=Q

Granular temperature is defined as an ensemble average

where vs´ is the fluctuating solids velocity.

First term on the right is the generation term, the second term is the diffusion term, the thirdthe dissipation term and the fourth expresses the energy exchange between gas and solids.

Alternatively, the granular temperature is computed from an algebraic equation obtained fromthe original transport equation by neglecting convection and diffusion terms.

Balance equation for Granular temperature:

( ) ( ) ( ) gssssssssssss sskp

tfgrara +-QÑ×Ñ+Ñ+-=úû

ùêëé Q×Ñ+Q¶¶

QQvτIv :)(23


Eulerian-Eulerian approach based on KTGF– closures for granular temperature

1

31

max,,0 1

-

úúú

û

ù

êêê

ë

é

÷÷ø

öççè

æ-=

s

sssg

aa

Radial distribution function related to the probability of collisions (Lun et al):

where αs,max expresses the solids volume fraction of a packed bed.

2/32,02 )1(12

ssss

ssss

dge

sQ

+=Q ra

pg

The collisional dissipation of energy can be computed from (Lun et al.)

and the energy exchange between gas and solids from (Gidaspow et al.)

sgsgs K Q-= 3f


Eulerian-Eulerian approach based on KTGF– closures for granular temperature, continued

The diffusion coefficient is computed, according to Syamlal et al., from

úûù

êëé -+-+

-Q

=Q ssssssssss gg

dk

s ,0,02 )3341(

1516)34(

5121

)3341(415

hahp

ahhh

par

where

)1(21

sse+=h

Gidaspow suggests instead:

praapr

/)1(2)1(561

)1(384150

,02

2

,0,0

sssssssssssssssss

sss gedegge

dk

sQ++úû

ùêëé ++

+Q

=Q


Eulerian-Eulerian approach based on KTGF– granular pressure

Lun et al.:

Syamlal & O’Brien:

Ahmadi & Ma:

sssssssssss gep Q++Q= ,02)1(2 arra

ssssssss gep Q+= ,02)1(2 ar

( ) úûù

êëé +-+++Q= )21)(1(

2141 ,0 fricsssssssssss eegp mara


Eulerian-Eulerian approach based on KTGF– solids shear stress based on KTGF

Ivvv qqqqTqqqqq ×Ñ-+Ñ+Ñ= )

32()( mlamatPhase q stress-strain tensor

Solids shear viscosity is given as a sum of the one predicted by the kinetic theoryand a frictional shear viscosity

frskinscolsfrsktgfss ,,,,, mmmmmm ++=+=

The collisional viscosity (Gidaspow et al) :

21

,0, )1(54

÷ø

öçè

æ Q+=

pram s

ssssssscols egd

Kinetic viscosity (Syamlal) (Gidaspow)

úûù

êëé -++

-Q

= sssssssss

sssskins gee

ed

apra

m ,0, )13)(1(521

)3(6

The granular bulk viscosity (Lun et al)

21

,0 )1(34

÷øö

çèæ Q

+=p

ral sssssssss egd

2

,0,0

, )1(541

)3(9610

úûù

êëé ++

-Q

= ssssssssss

ssskins ge

ged

aa

prm


Eulerian-Eulerian approach based on KTGF– frictional stress

( )Tssfrsfrfr p )(, vvI Ñ+Ñ+-= mt

frktgfs ppp +=

Johnson and Jackson (1987)

Shear stress:

Solids pressure:

Frictional pressure: , Fr=0.05, n=2, p=5.In FLUENT Fr=0.1αs

Frictional viscosity:

Schaeffer (1987)

Frictional viscosity:

pss

nss

fr Frp)()(

max,

min,

aaaa-

-=

fm sin, frfrs p=

D

frfrs I

p

2, 2

sinfm =


Eulerian-Eulerian approach based on KTGF– frictional stress

Syamlal et al. (1993):

Shear stress:

Solids pressure:

Frictional pressure:


Eulerian-Eulerian approach based on KTGF– drag force in dense gas-particle suspension


Eulerian-Eulerian approach based on KTGF– implentation in the software

t The models are often not implemented as such in numerical solvers.t Simplifications are introduced in the implementations to improve the

numerical stability.

t For example, the radial basis function needs to be restricted close tothe packing limit to avoid numerical problems

t The manuals of commercial codes don’t reveal details of thesemodifications.

t Sometimes it is possible to track down the actual equations used by atedious comparison of the values given by the commercial code anddifferent versions of the equations.


Validation of the KTGF model in a 2D CFBSimulation in a fine mesh (0.625 mm spacing)

Comparison of measured solids volume fraction and velocities with thesimulation results at 0.8 m and 1.2 m heights in the 2D CFB at ÅboAkademi. Particle size 0.44 mm and fluidization velocity 3.75 m/s.


Results without drag correction in a coarse mesh

1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9 s 10 s 11 s 12 s

-> the assumption of local homogeneity in the closure equations leads to problems-> subgrid scale closures required


Modified drag force model used in CFB and turbulentbed simulations

- larger particle size → smaller cluster correction

- finer mesh → smaller correction necessary

Voidage function used at slip velocity 1 m/s.The drag model by Gidaspow et al., recommendedin Fluent manuals, is shown for comparison.

Ref.: U. Ojaniemi, S. Kallio, A. Hermanson, M. Manninen, M. Seppälä, V. Taivassalo, Comparison of simulated and measuredflow patterns: solid and gas mixing in a 2D turbulent fluidized bed, Fluidization XII, Harrison Hot Springs, BC, Canada, 2007


Simulations of a 1 m X 7.3 m riser in 2D with macroscopic drag model

8000 cells sized 2.5 cm X 3.65 cm 4400, 17600 and70400 elements

Effect of the mesh: the finer the mesh, the finer structures are produced. Average voidage patterns are unaffected.

S. Kallio, The role of the gas-solid drag force in CFD modeling of fluidization,Åbo Akademi, Heat Engineering Laboratory, Report 2005-3, 2005

Solidsvolumefraction


Relationship between the number of elements and element size:

3D simulation, furnace volume 10 x 30 x 40 m3

Elements Element size20 000 0.853 m3

100 000 0.53 m3

1 000 000 0.233 m3

10 000 000 0.113 m3

100 000 000 0.053 m3

2D simulation, furnace ‘area’ 30 x 40 m2

Elements Element size20 000 0.252 m2

100 000 0.112 m2

500 000 0.052 m2

5 000 000 0.0152 m2

At least when mesh spacing > 0.5 cm, drag laws need to be adjusted.When mesh spacing > 10 cm, drag laws need further modifications andprobably also stress terms need to be modified ?!?Macroscopic/steady state/coarse mesh models should be developed!

3D simulation of large BFB’s and CFB’s


Macroscopic CFD models

t take into account large control volume sizet are written for steady statet large fluctuation terms in the equations → equation closure highly demandingt equation closure requires measurementst produces reasonable solids distributionst see e.g.

– S. Kallio, V. Poikolainen, T. Hyppänen, Report 96-4, Åbo Akademi/Värmeteknik– Taivassalo, V., Kallio, S., Peltola, J.: On time-averaged modeling of circulating

fluidized beds. 12th Int. conf. on multiphase flow in industrial plants, Ichia, Italy, 21.-23.9.2011, paper 69.

t coarse mesh models (filtered models) similar; meant for transient simulations oflarge processes

t no generally accepted macroscopic models available today!


Simulation of real fluidization processes- special features

t Complicated by a multitutude of particle sizes/types

t Complicated by differences in temperature → energy balance

t Complicated by homogeneous and heterogeneous reactions →species balances necessary

t Laboratory scale processes can be modeled in more details in thetransient mode


Additional equations/models required (examples)

t Energy balance equations for each phase– Heat capacity for each phase– Inter-phase heat transfer models, depend on particle size,

temperature, suspension density, velocities etc– Reaction enthalpies– Radiative heat transfer– Conductivity of each phase

t Species balance equations– Models for homogeneous and heterogeneous reactions– Inter-phase transfer rates– Diffusion/dispersion

t Comminution (attrition, fragmentation)t Solid-solid momentum transfer


Simulation of large industrial processes

t Complicated by a multitutude of particle sizes/types

t Complicated by differences in temperature → energy balance

t Complicated by homogeneous and heterogeneous reactions →species balances necessary

t Complicated by a large size → coarse meshes, long computations

t Simplifications/special approaches necessary


Examples of approaches for large industrial furnaces

t BFB modeling– Simplified models for the bed and 1-phase + Lagrangian particles in

the freeboard– Coarse mesh models

t CFB modeling– Empirical bulk solids distribution + balance equations for gas phase,

energy, species; fuel mixed using Lagrangian models or dispersionmodels

– Full two-phase model, transient (slow!!!) or steady-state (closuresbeing developed and validated e.g. at VTT)

– Full multi-phase model (slow!!!)– Particle-in-cell method (problems with coarse mesh)


Example: steady-state simulation of Chalmers boiler with a time-averaged model

t Gas phase: 5 componentst Solid phase: ash and sand as one Eulerian solid phaset Coal: particle tracking; drying, devolatilzation, combustion; two-way

coupled with gas phaset Solved using rigorous time-averaged equations for steady-state

hydrodynamics, with balance equs for Reynolds stressest Species balance equations solved for all the componentst Source terms calculated from homogeneous and heterogeneous

reactionst Mixing of enthalpy and chemical components described by

dispersion coefficients in energy and species balance equations;dispersion calculated from Reynolds stresses and time scales

Ref.: Taivassalo et al., FBC21, Naples 2012


Example2: steady-state simulation of Chalmers boiler with thetime-averaged model (Taivassalo et al., FBC21, Naples 2012)

a) Time-averaged solid volume fraction,. b) time-averaged mass-weightedvertical solid velocity, c) typical pathlines for a 1 mm fuel particle coloured bythe residence time , d) mole fraction of O2 and e) the time-averaged gastemperature (oC) .

a) b) c) d) e)


Conclusions on simulation of dense gas-solids flow

t Eulerian-Eulerian is a good approach; MP-PIC is also usable and good fordescription of particle size distributions

t Cluster formation is characteristic of dense gas-solid flows

t Transient simulations produce correct fluctuation patterns and averagedistributions, but as such usable only for small geometries

t Industrial processes large, large processes require coarse meshes

t Remember: closure model parameters for transient equations coarse-meshsimulations should be mesh-dependent! No such models available!

t Steady-state models now available at VTT and used for simulation of CFBboiler furnaces

sirpa kallio, vtt - Åbo akademiusers.abo.fi/rzevenho/icfd sk abo_akademi2019_slides.pdf · vtt...

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