VTT TECHNICAL RESEARCH CENTRE OF FINLAND Multiphase …users.abo.fi/rzevenho/Abo_Akademi_2013_slides...

VTT TECHNICAL RESEARCH CENTRE OF FINLAND1

Multiphase flow dynamicsMultiphase flow dynamicsSirpa Kallio, VTT

B ili tBoiling water

Turbulent fluidized bed


O tliOutline

Cl ifi ti f lti h fl Classification of multi-phase flows

Modeling approaches Modeling approaches

– Homogeneous flow model

– Lagrangian particle tracking

– General description

M th d f d i (DPM MP PIC)– Methods for dense suspensions (DPM, MP-PIC)

– Eulerian-Eulerian two-fluid models

– Algebraic slip mixture modelAlgebraic slip mixture model

– Other approaches

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


Multiphase flows can be classified according to

• the combination of phases:

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


M lti h fl b l ifi d di tMultiphase flows can be classified according to

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

)1(Re3

4)1(Re

18

2

pfpDf

pppp

f

ppp C

dt

dt

u

• 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 flowsd spe sed o s


• Multiphase flows can be treated as single phase flows when f ( )• 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 multiphase flow, e.g., g

• in mixing of fluid streams of different temperature and/or concentration• in the description of the different scales of turbulencein the description of the different scales of turbulence

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

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


TYPES OF PHASE INTERACTION

O liOne-way coupling:

Fluid flow affects particles/dispersed phaseFluid flow affects particles/dispersed phase

Two-way coupling:

Above + dispersed particles modify fluid flow

Four-way coupling:

Above + particle-particle interaction


M lti h fl d t b lMultiphase flow and turbulence

lti h fl t i ll b t ’t b l t’ ( h multiphase flows are typically by nature ’turbulent’ (phase interfaces distort the flow field which leads to fluctuations in velocities and other properties)

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

carrier phase turbulence enhances dispersion and clustering of the dispersed phase

in dense suspensions the turbulent fluctuations in the different phases are strongly coupled and often of a different character than p g y pin single-phase flows


Approach 1: Homogeneous flow model

Simplest multiphase CFD model- Simplest multiphase CFD model

- Assumes that dispersed phase travels at the same velocity th i has 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 distributionsFast; 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 particleg

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


Equation of motionq

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

rdmdd pffp

22

)(vv

rd

dtdtmmm

dtm f

pfp

ffffp

pp

2

2

102)( vvvg

d

r

d

d

rr

rt f

pfp

ffpfp

fpfp

2

222 6

66

6

vvv

vvvtt

ffpffpfp

0

6

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


Validity of Maxey & Riley equationy y y

For laminar flow (and mean field of turbulent flow)

2UrWrr1,1,1

L

UrWr

L

r

f

p

f

pp

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

For turbulent flow

2kppp vrWrr

1,1,1 kf

kp

f

p

k

p

d v th K l i l f l th d l itk and kv are the Kolmogorow microscales of length and velocity:

4/1

4/13

)(

fkk vf

)(

fkk

is the dissipation rate of turbulent kinetic energy


E ti f ti i St k iEquation of motion in Stokes regime (Rep<1)

d

Usually simplified:

fpfpfpp

p rmmdt

dm vvg

v 6)(

For rapidly accelerating or oscillating flows this approximation can be poor.

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


Drag models

a large number of empirical correlations exist

d l th t t ti l h d d f ti models that account particle shape and deformation

for dilute fluid particle flows, the drag is given byp , g g y

)(4

3fpfp

fD d

CF vvvv

)(4 fpfp

pD d

Schiller & Naumann correlationSchiller & Naumann correlation

1000Re44.0 pDC

1000ReRe15.01Re

24 687.0 ppDC

pD

Re p


Di i f ti lDispersion of particles

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

The interaction time is the smaller of the eddy life time te and the eddy crossing time tcc

epce

t

Ltt

kt ln6.0

The velocity fluctuation during an eddy is obtained by sampling a normally di t ib t d d f ti

pfp

puut

distributed random function:

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


Numerical methods

One-way coupling Two-way coupling


L i h dil t lLagrangian approach, dilute example:

Pulverized coal boiler:Particle temperature along

ti l t kparticle tracks


L i th d f d lid flLagrangian methods for dense gas-solids flow

Lagrangian particle tracking methods have been applied e.g. to fluidized bedsbeds

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 treatedy p p

– Discrete Particle Method (DPM):

– no collisions (used for dilute suspensions)

soft sphere approach (the discrete element method DEM)– 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:Lagrangian methods for dense gas solids flow: soft- and hard-sphere models

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

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

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

H d h d l f i l i l lli i Hard-sphere model of particle-particle collisions:– momentum-conserving binary collision

– interactions between particles pair-wise and instantaneousp p

– faster than soft-sphere method due to longer time steps


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

Eulerian gas phase

Lagrangian tracking of parcels of particlesg g g p p– Gas-particle and particle-particle interaction forces calculated in the mesh of

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

Also called / similar models:

– Dense DPM (Dense Discrete Particle Model; Ansys Fluent)

– CPFD (Computational Particle Fluid Dynamic; Barracuda)

Description of particle size distribution easy

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 ( p , p p ; ,

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


Lagrangian approach, summaryLagrangian 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 collisions are accounted for

• description of dispersion of particles in turbulent flow requires large number of particles

• fast for laminar flow; describes well particle size and other distributions


A h 3 E l i E l i hApproach 3: Eulerian-Eulerian approach

single phase equations (momentum, continuity) are averaged over time and/or volume and/or a particle assembly (Newtons equation) t bt i lti h fl tito obtain multiphase flow equations

distribution of phases given by volume fraction p g y

models for phase interaction required

all phases are described as interpenetrating continua


Volume averagingg g

Averaging volumePhase interface

Averaging volume


Starting point in volume averaging: Single phase equations + jump g p g g g p q j pconditions

0 0 qqqt

v

gτvvv p gτvvv qqqqqqqq pt

0 nvvnvv 0 nvvnvv AqAqq

Aq

AAA pp Rσ

σnτInτInvvvnvvv

2)()( A

AqAqqqAqAqqq pp R

RσnτInτInvvvnvvv )()(

n Unit outward normal vector of phase q AR

AA RR /=qn Unit outward normal vector of phase q

Av Velocity of the interface

AR AA RR /=

R interface curvature vector

qσ interface surface tension

AR interface curvature vector

A AR

= = surface gradient operator I unit tensor


Averaging

Volume averaging:

dVV qV qq 1

Volume average

qq

V qq

q dVV q

11~ Intrinsic volume averageqq

qqV qq dV qq

qq

V q

V qq

qdV

q

q

~

Mass-weighted (Favre) average

1VVqq / Volume fraction, 1q

q


E l i E l i b l iEulerian-Eulerian balance equations

Continuity:

qqqqqq

v ~~

Continuity:

qqqqqqt

Momentum:

vvv ~~

Momentum:

qqqqqqq

qqqqqqq

p

t

τMgτ

vvv

~)~( qqqqqqq p g)(

M is the interaction forceMq is the interaction force.


Turbulent stress term:Turbulent stress term:

))((~ vvvvτ ))(( qqqqqqq vvvvτ

0Conservation of mass:

0q

q

21 σ

Surface tension:

,

2

2

1

q A

AA

qqA

qq dA

Vq

RR

σσM


Alt ti i hAlternative averaging approaches

volume averaging

time averaging

ensemble averaging ensemble averaging

volume-time averaging

two-step volume-volume or volume-time averaging (first over the local scale, then over a larger scale), g )

three-step averaging


Closures for turbulent stress terms

no generally accepted turbulence models exist; parameters in any models are probably suspension dependent

several alternatives suggested

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

dispersed turbulence model

f– appropriate for dilute suspensions

– turbulence quantities of the dispersed phase are obtained using the theory of dispersion of particles by homogeneous g y p p y gturbulence

t b l d l f h h turbulence model for each phase


C f fClosures for drag force

large number of models exist

drag force is generally written in the form:

3)(

4

3fqfq

q

ffqDq d

CM vvvv

Note: in dense flows, CD depends on αf

parameters in drag models should be functions of the averaging scale in inhomogeneous flows!


Description of particle size distribution in the Eulerian-Eulerian method

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

– Continuity and momentum equation for each size classComputationally slow– Computationally slow

– Numerical difficulties: at least earlier versions than Ansys Fluent 14 produced non-physical results

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 of the moments

– Momentum equations can in some cases be replaced by algebraic equations


Numerical methods for the Eulerian-Eulerian approach

several algorithms; usually extensions of single phase ones several algorithms; usually extensions of single phase ones

e.g. IPSA (Inter Phase Slip Algorithm) performs the following steps in it ti lone iteration loop:

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

continuity equation)– update pressure

correct velocities– correct velocities – calculate volume fractions from phasic continuity equations


Example: simulated solids volume fractions in aExample: simulated solids volume fractions in a circulating fluidized bed

Large Small

Two-phase flow

particles particles

’Three’-phase flow


E l i E l i d lEulerian-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 velocity differences/acceleration

• often very slow

• particle size and other distributions difficult to account for• particle size and other distributions difficult to account for


A h 4 Mi t d lApproach 4: Mixture model

A l l ilib i b t th ti d di d h i A local equilibrium between the continuous and dispersed phases, i.e.

at every point, the particles move with the terminal slip velocity relative to the continuous phase

velocity components for dispersed phases can be calculated from algebraic formulas

significantly less equations to be solved

Also called / similar models:

Algebraic slip (mixture) model– Algebraic slip (mixture) model

– Diffusion model

– Suspension model

– Local-equilibrium model

– Drift-flux model


M ti ti f i lifi d lti h d lMotivation for simplified multiphase models

Full Eulerian models - problemsFull Eulerian models problems

Numerical problems - long computing times - small time steps

Difficulties in convergenceg

Only a few secondary phases possible

More difficulties, if mass transfer and chemical reactions are id dconsidered

Simplified models are recommended if applicable Simplified models are recommended, if applicable

By Mikko Manninen, VTT


I t iti d i ti f th i t d l (1)Intuitive derivation of the mixture model (1)

Consider a mixture of a continuous phase (liquid) and a dispersed phase (small particles)

Write single phase equations for the mixture of phases (mass and momentum balances)

– Use mixture density and velocity

0m 0 mmm

tu

gττuuu mTmmmmmmm pt



I t iti d i ti f th i t d l (2)Intuitive derivation of the mixture model (2)

Now bubbles or particles have a slip with respect to the liquid Now, bubbles or particles have a slip with respect to the liquid

Assume a local equilibrium (fast acceleration of particles) and solve the slip velocity (velocity of particles with respect to the liquid) by balancing the drag and body forces

In gravitational field:

01

2p CAVdu

Take the phase slip into account in the balance (continuity)

021

2pDpfp

pp uCAgV

dtm

( y)equation for the dispersed phase volume fraction:

0

slipmpppptuu



D t il d d i ti f th i t d l (1)Detailed derivation of the mixture model (1)

Starting point: Eulerian model equationsg p q

Continuity equation for phase k:

Momentum balance for phase k:

kkkkkkt u

Momentum balance for phase k:

kkkTkkkkkkkkkkkk pt

Mgττuuu

Mixture equations are derived by taking a sum over all phases

t




Continuityy

n

kk

n

kkkk

n

kkkt 111

u

Define mixture density and velocity as

kkkt 111

mk

n

k k

1

n

kkk

kkkk

mm c

1

1uuu

Resulting equation

0m u mkkk 0jju 0 mmm

tu

kmk

kkk jj

If all phases are incompressible




Momentum balance

n

kkkkk

n

kkkkt 11

uuu

n

kk

n

kkk

n

kTkkk

n

kkk p

1111

Mgττ

Diffusion velocity (not slip velocity!)u u uMk k m

n

Definitions mk

k k

1

n

Viscous stress

T b l t tTm k Ik Fk Fkk

u u

1

n

u u

Turbulent stress

Diffusion stress Dmk

k k Mk Mk

1

u u Diffusion stress




Assumption: all phases share the same pressure,

Using the definitions of mixture density and velocity and the

ppk g y y

stresses above, we obtain the resulting momentum balance for the mixture

mmDmTmmmmmmm pt

Mgτττuuu

This differs from the intuitive equation by the diffusion stress d b h li d th t d t f t iMcaused by phase slip and the term due to surface tension

forces.mM




Eli i t th h l it i th ti it ti f h Eliminate the phase velocity in the continuity equation for a phase

Mkkkkmkkkk uu

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

Mkkkkmkkkktuu

Mkkmkktuu

which is a balance equation for phase k volume fraction and includes the effect of slip Note that this the same as the intuitive

t

includes the effect of slip. Note that this the same as the intuitive result except for the diffusion velocity instead of slip velocity.




Relation between the slip velocity and the diffusion velocity

u u uCk k c

slip velocity (c denotes the continuous phase)

For only one dispersed phase

l

CllCkmkMk c uuuuu

u uc 1 For only one dispersed phase

Note: so far no additional assumptions or approximations has been

u uMd d Cdc 1

p ppmade (except the shared pressure, commonly used also in full Eulerian models)



Sli l it (1)Slip velocity (1)

The momentum equation for the dispersed phase is The momentum equation for the dispersed phase is

pppTppppppppp

pp pt

Mguuu

Momentum equation for the mixture (with Mm=0)

gτττuuu mDmTmmmmmmm p

Eliminate pressure gradient and solve for Mp

u

gτττuuu mDmTmmmmmmm pt

uuuuuu

M mmmppppm

mpMp

ppp tt

gτττττ mppDmTmmpTppp




Now the following approximations are made: 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

mmpp uuuu )()( 0 Mpu

The momentum transfer term is given by

mmpp )()(0t

CpCpcDpp

p CV

AM uu

21

pV2




Finally:

g

uuuuu

tVCA m

mmmppCpCpDpc

21

For small particles (Stokes drag) in gravitational and centrifugal fi ldfield

mmpp ud 22 )(

geu rm

m

mppCd r18

)(

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



Closures for viscous stress in mixture modelsClosures for viscous stress in mixture models

T Tmmmm )( vv Viscous stress:

max,5.2

1p

p

Mixture viscosity (Ishii & Zuver, 1979):

max,

1p

pfm

max,p

40

where is the maximum packing and

fp

fp

4.0

for bubbles or drops

1 for solid particlesBy Mikko Manninen, VTT


Treatment of turbulence in mixture models

in dilute suspensions, mixture turbulent stress is assumed same as carrier phase turbulent stressas carrier phase turbulent stress

at higher concentrations, modeling required

turbulence enhances spreading of dispersed phases -> taken into account in continuity equation:y q

D vv ~~~ mpppppmpmpppp Dt

vv

where Dmq is a dispersion (diffusion) coefficient:

3/285.01

2

kD cp

Tcmp

v

3/2kp



T b l t di iTurbulent dispersion

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

Th t b l t t d th fl t ti t f M d l d The turbulent stress and the fluctuating part of Mq are modeled using a turbulent dispersion (diffusion) coefficient DT

Dq

q

Tmqmq

D

uu

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

u 2 1 2

Du

kT TCd

1 0 85

2 3.



Mi t d l liditMixture model - validity

The essential approximation is the local equilibrium assumption: The essential approximation is the local equilibrium assumption: the particles are accelerated instantaneously to the terminal velocity

C id i th i lifi d ti l ti Consider again the simplified one-particle equation

2Df

p uCAgVdu

m 1

1

Solution of particle velocity as a

pDpfpp uCAgVdt

m 2

0.6

0.8

function of distance (Stokes regime)

The characteristic length ofacceleration is

0.2

0.4

utacceleration iswith

0

0 0.5 1 1.5 2 2.5 3

px

tpp ut

ppp

dt

18

2

pt gtu

f

p 18 p



Mixture model - validity y

1

0.1

m)

0.2 mm

Virtual mass and history effects

Stokes drag only

0.001

0.01

l p (

c m

0.1 mm

Stokes drag only

0.0001

1.0 2.0 3.0 4.0 5.0 6.0pc

If the density ratio is small, the virtual mass and history terms cannot be ignored (both

effects increase the response time)

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



Validity range of mixture models, cont. y g ,

Mi t d l i ll t it bl f ti l fl- Mixture model is usually not suitable for gas-particle flows, because the characteristic length can be large

N t it bl f l t i fl h li l it- 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 simulations Solid particles in an agitated vessel

Glass beads of three different sizes in water d = 230 m

Experiments: Barresi and Baldi (1987)

D 0 390 H 0 464D = 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 %

d ( ) 100 177 208 250 417 500dp,exp.(m) 100-177 208-250 417-500dp,sim.(m) 140 230 460 (kg/m3) 2670 2600 2600N (rev/s) 4 83 5 67 6 50N (rev/s) 4.83 5.67 6.50



Validating simulationsValidating simulations

Solid particles in an agitated vessel

1.0Experimental

dp = 140 m

1.0Experimental

dp = 230 m

1.0Experimental

dp = 460 m

ExperimentalMixture modelMultiphase model



0.5

z/z

max

0.5z/

zm

ax

0.5

z/z

max

0.0 0.0 0.0

0.0 0.5 1.0 1.5 2.0 2.5

c/cav e

0.0 0.5 1.0 1.5 2.0 2.5

c/cav e

0.0 0.5 1.0 1.5 2.0 2.5

c/cav e



B bb d l tiBubbe drag correlations

Tomiyama 1

Tomiyama 2y

2pqp d

Eo

a

Ishii-Zuber



B bbl t i l l itBubble terminal velocity

0.8

0.6

0.7

y

03

0.4

0.5

Usl

ip

I-Z, g

I-Z, 10g

I-Z, 20g

Tomiyama 2, 20gSlip

vel

oci

ty

0.1

0.2

0.3

0

0 0.002 0.004 0.006 0.008 0.01

d (m)d (m)

In gravitational field Effect of acceleration



MUSIG d lMUSIG-model

Developed primarily for bubble column applications– Bubble motion determined by gravitational accelerationy g

– The basic approximation of equal slip velocities of all bubble sizes is fairly good

In strongly swirling flows (stirred reactors pumps) the model may In strongly swirling flows (stirred reactors, pumps) the model may not be justified

Reduces the number of equations to be solved

Is a two-phase model and has therefore the convergence and stability problems of multiphase CFD



Mixture model example


Comparison of models for bubbly flows

General Eulerian models

Comparison of models for bubbly flows

(4xN balance equations) Continuity equation for all phases (N) Momentum equation for all phases (3N)

MUSIG-model (CFX): (N+6) Mixture models (FLUENT): (N+3)

Numerically difficult and slow

MUSIG model (CFX): (N 6) Continuity equation for all phases (N)

Momentum equation for one bubble phase (average diameter) (6)

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

all bubble phases (N)

Momentum equation for the mixture (3)phase (average diameter) (6)

All other bubble classes: same velocity as the class with average diameter

Momentum equation for the mixture (3)

Bubbles: slip velocities from algebraic balance equations

diameter

A large number of bubble classes possible

Numerically well behaved, a large number of bubble classes possible



Mixture model summary

- Length scale of particle relaxation must be small compared to the mesh spacingto the mesh spacing

- Local equilibrium assumption yields the slip velocity

- Transport of the dispersed phase calculated from the phase continuity equation

- Fast; describes well particle size and other distributions

- Describes well effects of turbulence


Oth hOther approaches

VOF (volume of fluid) –method for tracking phase interfaces in separated flows

Eulerian-Eulerian + Lagrangian

Eulerian-Eulerian + mixture model

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


Summary of multiphase CFD:

• Several ways to derive equations

• Simplifications in all models• 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)( g g )


Summary of multiphase CFD, cont.:

• Never fully ‘laminar’

T b l d lli diffi lt l d l i t• Turbulence modelling difficult; no general models exist; unisotropy causes problems

• Dense suspensions dominated by particle-particle collisions

• Dilute suspensions dominated by particle-turbulence interactionp y p

• Measurements still necessary for derivation of equation closures and for validation; often not availableand for validation; often not available


Summary of multiphase CFD, cont. :

• Choose the simplest possible model

• Follow the literature

• Compare with measurement data if availableCompare with measurement data if available

• Results for dilute suspensions already good

• Even dense suspensions can be simulated

I i t d ill f ilit t i l ti• Increasing computer speed will facilitate simulation of more demanding cases in the future


Example of Eulerian Eulerian modeling: Fluidized beds

- inhomogeneous flow pattern

Example of Eulerian-Eulerian modeling: Fluidized beds

- flow characteristics change as a function of fluidization velocity

TFB ( b l ) BFB (b bbli )CFB (circulating) TFB (turbulent) BFB (bubbling)Ref.: U. Ojaniemi, S. Kallio, A. Hermanson, M. Manninen, M. Seppälä, V. Taivassalo, Comparison of simulated and measured flow patterns: solid and gas mixing in a 2D turbulent fluidized bed, Fluidization XII, Harrison Hot Springs, BC, Canada, 2007


Eulerian-Eulerian approach based on KTGFEulerian Eulerian approach based on KTGF – background

Kinetic theory of granular flow (KTGF)

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

Theory similar to gas kinetic theory

Solid phase equations are derived by ensemble averaging over the single particle velocity distribution functionsingle particle velocity distribution function

This averaging is done in a local scale assuming locally random distribution of particles


Eulerian-Eulerian approach based on KTGFEulerian Eulerian approach based on KTGF – limitations and simplifications

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

In derivation of the models, only binary collisions between particles are accounted for and thus KTGF is not valid in denseare accounted for, and thus KTGF is not valid in dense suspensions, like in bubbling beds

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

The theory was originally written for a single particle size

L t i t i f th th t i t h b Later rigorous extension of the theory to mixtures has been developed

Here we focus on models for a single particle size


Eulerian-Eulerian approach based on KTGFEulerian Eulerian approach based on KTGF – software inplementations

Kinetic theory of granular flow is today found in commercial codes– commercial codes

– open source codes

– researchers’ and universities own codes

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

Here we mainly present models available in the commercial Fluentsoftware


Eulerian-Eulerian approach based on KTGFEulerian Eulerian approach based on KTGF – balance equations

Continuity equation for phase q:

0)()(

qqqqqtv

Momentum equation for gas phase g:

)()()()( gssggggggggggggg Kpt

vvgvvv

Momentum equation for gas phase g:

t

Momentum equation for solid phase s:

)()()()( sgsgsssssssssssss Kppt

vvgvvv


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

2'1

v

Granular temperature is defined as an ensemble average

3ss v

where vs´ is the fluctuating solids velocity.

Balance equation for Granular temperature:

kp

vτIv :)(3

First term on the right is the generation term, the second term is the diffusion term, the third th di i ti t d th f th th h b t d lid

gssssssssssss sskp

t

vτIv :)(

2

the dissipation term and the fourth expresses the energy exchange between gas and solids.

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


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

11

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

3

1

max,,0 1

s

sssg

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

2

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

2/32,02 )1(12

sss

s

ssss

d

ges

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

sgsgs K 3


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

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

ssssssssss gg

dk

s ,0,02 )3341(

15

16)34(

5

121

)3341(4

15

where

)1(2

1sse

2

Gidaspow suggests instead:

/)1(2)1(5

61

)1(384

150,0

22

,0,0

sssssssssssssssss

sss gedegge

dk

s


Eulerian-Eulerian approach based on KTGFEulerian Eulerian approach based on KTGF – granular pressure

Lun et al.:

sssssssssss gep ,02)1(2

Syamlal & O’Brien:

2)1(2

Ahmadi & Ma:

ssssssss gep ,02)1(2

)21)(1(

2

141 ,0 fricsssssssssss eegp

2


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

Ivvv qqqqTqqqqq )

3

2()( Phase q stress-strain tensor

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

frskinscolsfrsktgfss ,,,,,

The collisional viscosity (Gidaspow et al) :

1

The granular bulk viscosity (Lun et al)

2

1

,0, )1(5

4

sssssssscols egd 2

1

,0 )1(3

4

sssssssss egd

Kinetic viscosity (Syamlal) (Gidaspow)

d 22

410 d

sssssssss

sssskins gee

e

d

,0, )13)(1(

5

21

)3(6,0

,0, )1(

5

41

)3(96

10

ssssssssss

ssskins ge

ge

d


Eulerian-Eulerian approach based on KTGFEulerian Eulerian approach based on KTGF – frictional stress

Johnson and Jackson (1987)

Tssfrsfrfr p )(, vvI

Shear stress:

frktgfs ppp Solids pressure:

nssFrp

)( min, Frictional pressure: , Fr=0.05, n=2, p=5.

In FLUENT Fr=0.1αs

Frictional viscosity:

pss

fr Frp)( max,

sinpFrictional viscosity:

Schaeffer (1987)

sin, frfrs p

frp sinFrictional viscosity:

D

frfrs

I

p

2

,2


Eulerian-Eulerian approach based on KTGFEulerian Eulerian approach based on KTGF – frictional stress

Syamlal et al. (1993):

Shear stress:

Solids pressure:

F i ti lFrictional pressure:


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

The models are often not implemented as such in numerical solvers.

Simplifications are introduced in the implementations to improve the p p pnumerical stability.

F l th di l b i f ti d t b t i t d l t For example, the radial basis function needs to be restricted close to the packing limit to avoid numerical problems

The manuals of commercial codes don’t reveal details of these modifications.

S ti it i ibl t t k d th t l ti d b Sometimes it is possible to track down the actual equations used by a tedious comparison of the values given by the commercial code and different versions of the equations.


Bubbling bed simulations using the kinetic theory of granular flow g g y g

Comparison of the measurements (top) and simulations with the model of Gidaspow et al (middle) and with the drag model of Syamlal & O’Brien (below). The figure shows (from left) examples of instantaneous bubble frames, the locations of the bubbles, bubble diametersexamples of instantaneous bubble frames, the locations of the bubbles, bubble diameters vs.height, and the bubble rise velocities vs. height.

S. Kallio, U. Ojaniemi, A. Hermanson, K. Fagerudd, M. Engblom, M. Manninen, Experimental study and CFD modeling of bubble formation in a 2D bubbling fluidized bed, Fluidization XI, Ischia, Italy, May 2004.


Validation of the KTGF model in a 2D CFBValidation 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 theComparison of measured solids volume fraction and velocities with the simulation results at 0.8 m and 1.2 m heights in the 2D CFB at Åbo Akademi. Particle size 0.44 mm and fluidization velocity 3.75 m/s.


Results without drag correction in a coarse meshResults without drag correction in a coarse mesh

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

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


Modified drag force model used in CFB and turbulentModified drag force model used in CFB and turbulent bed 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., recommended in Fluent manuals is shown for comparisonin Fluent manuals, is shown for comparison.

Ref.: U. Ojaniemi, S. Kallio, A. Hermanson, M. Manninen, M. Seppälä, V. Taivassalo, Comparison of simulated and measured flow 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

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

Solids volumefraction

8000 cells sized 2.5 cm X 3.65 cm 4400, 17600 and

70400 elementsS. Kallio, The role of the gas-solid drag force in CFD modeling of fluidization, Åbo Akademi, Heat Engineering Laboratory, Report 2005-3, 2005


Simulation results: solids volume fraction

a) the Gidaspow models and base grid, b) the denser grid and c) with the modified drag model and base grid. The modified drag model produced a lower bed. p

MFIX simulation resultMFIX-simulation resultfor comparison: the flowpattern is the same as obtained with Fluent.

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


3D simulation of large BFB’s and CFB’sRelationship between the number of elements and element size:

3

3D simulation of large BFB s and CFB s

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

Elements Element size

20 000 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 size

20 000 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 andWhen mesh spacing > 10 cm, drag laws need further modifications and probably also stress terms need to be modified ?!?Macroscopic/steady state/coarse mesh models should be developed!


M i CFD d lMacroscopic CFD models

take into account large control volume size are written for steady state large fluctuation terms in the equations → equation closure highly demanding equation closure requires measurements produces reasonable solids distributions produces reasonable solids distributions 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.

coarse mesh models (filtered models) similar; meant for transient simulations of large processes

no generally accepted macroscopic models available today!


Simulation of real fluidization processesSimulation of real fluidization processes- special features

Complicated by a multitutude of particle sizes/types

Complicated by differences in temperature → energy balance

Complicated by homogeneous and heterogeneous reactions → species balances necessary

Laboratory scale processes can be modeled in more details in the transient mode


Additi l ti / d l i d ( l )Additional equations/models required (examples)

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

Species balance equationsp q– Models for homogeneous and heterogeneous reactions– Inter-phase transfer rates– Diffusion/dispersionDiffusion/dispersion

Comminution (attrition, fragmentation) Solid-solid momentum transfer


Example of simulation of a real lab-scale process:

CFD Modeling of Combustion Experiments in a Bubbling Fl idi d B d R t

Example of simulation of a real lab scale process:

Fluidized Bed Reactor

Combustion experiments in the reactor at the Tampere University of Technology (Wallen 2005)Combustion experiments in the reactor at the Tampere University of Technology (Wallen 2005) - bottom: 0.73 m x 0.135 m; total reactor height is 3 m - air distributor is of tuyere type with six caps (fluidization velocity about 1 m/s)- fuel: sawdust (pellets and peat)

Modeling approach – main features• Kinetic theory model of Fluent (2005)• Sand and fuel particles constitute a single solid phase. • The fuel components, water, char, volatiles and ash (in 3 size categories) as species • The gas mixture comprises of seven species

O2, H2, H2O, CO, CO2, N2 and Others (hydrocarbons)• Temperature for the gas and solid phases separately + the P1 model implementedTemperature for the gas and solid phases separately + the P1 model implemented• Heterogeneous char reactions with O2, H2O and CO2• Homogeneous gas phase reactions: Jones&Lindstedt + “EDC”• ∆t=0.1 ms; on 6 processors: simulation of 1 s takes about 2 weeks

V. Taivassalo, VTT, report under preparation.


CFD Modeling of Combustion Experiments in a BubblingCFD Modeling of Combustion Experiments in a Bubbling Fluidized Bed Reactor

Solids volume fraction Conversion rate:char burning and gasification

O2 (mass fraction of gas)

Fuelfeed 10 g/s

V. Taivassalo, VTT, report under preparation, 2007.


Si l ti f l i d t i lSimulation of large industrial processes

Complicated by a multitutude of particle sizes/types

Complicated by differences in temperature → energy balance

Complicated by homogeneous and heterogeneous reactions → Complicated by homogeneous and heterogeneous reactions → species balances necessary

C li t d b l i h l t ti Complicated by a large size → coarse meshes, long computations

Simplifications/special approaches necessaryp p pp y

Two examples are illustrated at the end


E l f h f l i d t i l fExamples of approaches for large industrial furnaces

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

the freeboard

– Coarse mesh models (see example 2)

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

energy, species; fuel mixed using Lagrangian models or dispersion modelsmodels

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

– Full multi-phase model (slow!!!)

– Particle-in-cell method (problems with coarse mesh)


Example 1: Simulation of CFB combustionExample 1: Simulation of CFB combustion - fuel described as solid species

Gas phase: 13 components

Solid phase: 15 components (amount of volatiles and char in 6 fuel p p (particle size classes, moisture in fuel, ash and sand)

Solved using simplified macroscopic equations for steady-state hydrodynamicshydrodynamics

Species balance equations solved for all the components

Source terms calculated from homogeneous and heterogeneous g greactions

Mixing of enthalpy and chemical components described by effective dispersion coefficients in energy and species balanceeffective dispersion coefficients in energy and species balance equations

Ref.: VTT Tutkimusraportti Nro VTT-R-03656-06, 7.6.2006


CFB combustion simulation results from example 1p

(a)

(b)

(c)

(a) Mass fraction of oxygen in gas phase, (b) mass fraction of char in one solids size fraction and (c) solid phase temperature (oC)(c) solid phase temperature ( C).



Example 2: steady state simulation of Chalmers boiler with a timeExample 2: steady-state simulation of Chalmers boiler with a time-averaged model - fuel simulated by particle tracking

Gas phase: 5 components

Solid phase: ash and sand as one Eulerian solid phasep p

Coal: particle tracking; drying, devolatilzation, combustion; two-way coupled with gas phase

S l d i i ti d ti f t d t t Solved using rigorous time-averaged equations for steady-state hydrodynamics, with balance equs for Reynolds stresses

Species balance equations solved for all the componentsp q p

Source terms calculated from homogeneous and heterogeneous reactions

Mi i f th l d h i l t d ib d b 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 theExample2: steady state simulation of Chalmers boiler with the time-averaged model (Taivassalo et al., FBC21, Naples 2012)

a) b) c) d) e)

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


Example 3: Simulation of roasting of zink concentrate inExample 3: Simulation of roasting of zink concentrate in a BFB

2D simulation

Coarse mesh → no single bubbles simulatedg

4 phases: continuous primary gas phase, big particles, small particles, gas bubble phase

T i t E l i i l ti f l it fi ld f th 4 h Transient Eulerian simulation of velocity fields of the 4 phases

Energy equation for each phase

Species balance equations for gas phase components Species balance equations for gas phase components

Laminar flow

Reactions between particles and oxygen were taken into account by considering individual representative particles, the paths of which were calculated by Lagrangian integration



R lt f th i k ti i l tiResults from the zink roasting simulation

Volume fraction of large particles The track of a representative particle ando u e ac o o a ge pa c es(0-0.5).

The track of a representative particle and remaining amount of sulphur (0-100% of original)

Mass fraction of SO2 in primary gas(0-0.4).

Primary gas temperature (900-950 ºC)



C l i i l ti f d lid flConclusions on simulation of dense gas-solids flow

Eulerian-Eulerian best approach

Cl t f ti h t i ti f d lid fl Cluster formation characteristic of dense gas-solid flows

Transient simulations produce correct fluctuation patterns and averagep p gdistributions

Industrial processes large large processes require coarse meshes Industrial processes large, large processes require coarse meshes

Remember: closure model parameters for transient equationsmesh-dependent!

Steady-state model is a goal; first models are available Steady state model is a goal; first models are available

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