Fluent Multiphase 16.0 L04 Gas Liquid Flows

1 2015 ANSYS, Inc. April 24, 2015 ANSYS Confidential

16.0 Release

Lecture 4:

Gas-Liquid Flows

Multiphase Modeling using

ANSYS Fluent


Outline Introduction

Conservation equations

Modelling strategies : Euler-Lagrangian and Eulerian

Interfacial Forces

Drag Non-Drag Forces Turbulence Interaction

Mixture Model

Validation example


Gasliquid flows occur in many applications. The motion of bubbles in a liquid as well as droplets in a conveying gas stream are examples of gasliquid flows

Bubble columns are commonly used in several process industries

Atomization to generate small droplets for combustion is important in power generation systems

Introduction

Bubble Column

Rain/Hail Stones

Spray Drying

Distillation Process

Absorption

Process

Boiling Process

Combustion


Why Study Gas-Liquid Flows

The main interests in studying gas-liquid flows, in devices like bubble columns or stirred tank reactors, are:

Design and scale-up

Fluid dynamics and regime analysis

Hydrodynamic parameters


Bubble Columns

To design bubble column reactors, the following hydrodynamic parameters are required:

Specic gasliquid interfacial area ()

Sauter mean bubble diameter, ()

Axial and radial dispersion coefcients of the gas and liquid, ()

Heat and mass transfer coefcients, (, )

Gas holdup, ()

Physicochemical properties of the liquid medium, (, )


Two types of ow regimes are commonly observed in bubble columns:

The bubbly flow regime, Gas velocity < 5cm/s

Bubbles are of relatively uniform small sizes (db =2 to 6 mm)

Rise velocity does exceed 0.025m/s

Holdup shows linear dependence with the flow

Regime Analysis < . < . /


The churn turbulent flow regime Gas velocity > 5cm/s

Bubble are Large bubbles ( > ) and show wide size distribution

Rise velocity is in the range of 1-2m/s

Regime Analysis > . > . /

Most frequently observed flow regime in industrial-size, large diameter columns


Photographic Representation of Bubbly and Churn-Turbulent Flow Regimes

Bubbly Flow Regime Churn Turbulent Flow Regime


Design and Scale-up of Bubble Column Reactors

Bubble have significant effect on hydrodynamics well as heat and mass transfer coefcients in a bubble columns

The average bubble size and rise velocity in a bubble column is found to be affected by:


In this approach, a single set of conservation equations is solved for a continuous phase

The dispersed phase is explicitly tracked by solving an appropriate equation of motion in the lagrangian frame of reference through the continuous phase flow field

The interaction between the continuous and the dispersed phase is taken into account with separate models for drag, and non-drag forces

Euler-Lagrangian Method

Eulerian Cell

Gravity

Buoyancy

Liquid Flow


Eulerian Approach

In the Eulerian approach, both the continuous and dispersed phases are considered to be interpenetrating continua

The Eulerian model describes the motion for each phase in a macroscopic sense

The flow description therefore consists of differential equations describing the conservation of mass, momentum and energy for each phase separately


Conservation Equations

Continuity equation:

Momentum equation:

sDrag ForceNon

ForceDispersionTurbulent

td,q

ForcessVirtual Ma

vm,q

ForcecationWall Lubri

wl,q

ForceLift

lift,q

Forceexternal

q

n

p

sDrag Force

fermass trans

qpqppqpq

l ForceInterfacia

qppq

Bouyancy

qq

Friction

q

essure

qqqqqqq

FFFFF

vmvmvvK gpvvt 1

Pr

2

source

q

transfermass

n

p

qppqqqqqq Smmvt

1


A key question is how to model the inter-phase momentum exchange

This is the force that acts on the bubble and takes into account:

Effect of multi-bubble interaction

Gas holdup

Turbulent modulation

Turbulent Dispersion

Turbulent Interaction

Interphase Momentum Exchange

Interphase Momentum

Exchange

Drag

Lift

Turbulent Dispersion


Virtual Mass


We can think of drag as a hydrodynamic friction between the liquid phase and the dispersed phase

We can also think of drag as a hydrodynamic resistance to the motion of the particle through the water. The source of this drag is shape of particle

Drag Force


Drag Force

For a single spherical bubble, rising at steady state, the drag force is given by:

For a swarm of bubbles the drag, in absence of bubble-bubble interaction, is given by:

4

3

2

63,

qpqp

p

q

D

p

qpqpq

pD

p

p

DswarmD

vvvvd

C

vvvvACd

NFF

qpqpq

pDD vvvvACF velocitysliptcoefficien drag

2


Drag Force

In order to ensure that the interfacial force vanishes in absence any dispersed phase, the drag force needs to multiplied by as shown:

In Fluent

4

3, qpqp

p

q

D

qp

swarmD vvvvd

CF

24

ReC

6

18

6

18

D

2

2,

qpi

p

pp

q

pp

qpi

p

pp

q

ppqppqswarmD

vvAd

d

vvfAd

dvvKF

= Interfacial Area Density, m2/m3


Drag Force

To estimate the drag force bubble diameter, ,is needed

The is often taken as the mean bubble size

For bubble columns operating at low gas superficial velocities (< 5 cm/s) works reasonably well

For bubble columns operating at higher gas superficial velocities (> 5 cm/s), bubble breakup and coalesce dominate and bubble size is no longer uniform and mean bubble size approach may not be adequate


The drag coefficient is likely to be different for a single bubble and a bubble swarm. This is because the shape and size of a bubble in a swarm is different than that of an isolated bubble

When the bubble size is small ( < 1mm in water): bubble is approximately spherical

When the bubble size is large ( > 18mm in water): bubble is approximately a spherical cap

When the bubble of intermediate size: bubbles exhibit complex shapes

Drag Coefficient Water Glycerol /


We can use the Eotvos number () together with the Morton number () to characterize the shape of bubbles or drops moving in a surrounding fluid or continuous phase

Number Ratio of bouncy force and surface tension force and

essentially gives a measure of the volume of the bubble

Number Ratio of physical properties

Constant for a given incompressible two-phase system. Water has a Morton number of .

Bubble Shape

2pgdEo

32

4

q

qgMo

Lorond Eotvos

3mm air bubble rising in tap water

Bubble Regime Map


Drag Laws for Small and Constant Bubble Sizes

At low flow rates bubbles assume an approximately spherical shape while they rise in a rectilinear path

Schiller and Naumann (1978)

Morsi and Alexander (1972)

Symmetric Drag Model: The density and the viscosity are calculated from volume averaged properties and is given by

Schiller Naumann model

1000 Re :for 44.0C

1000 Re :for Re15.01Re

24C

D

687.0

D

2

321D

ReReC

aaa

q

pqpq dvv

Re

When Reynolds number is small ( < 1) these correlations essentially reduce to the well known Stokes drag law =

24


For all other flow rate, bubble size and shapes varies with the flow

Consequently, different drag correlations are needed

Several drag correlation are found in literature Grace drag law

Tomiyama drag law

Universal drag law

Drag Laws for Variable Bubble Sizes Larger bubbles - ellipsoidal

As bubble size increases, spherical caps may be formed


Terminal Rise Velocity for Bubbles The drag correlations for

large bubbles are very different from those for spherical particles

Grace Correlation

Spherical Bubble Correlation


Bubble Regimes

Viscous and inertial forces are important

the function is given by an empirical correlation e.g. SN

Viscous Regime

Bubbles follow zig-zag paths

is proportional to the size of bubble

is independent of viscosity

Distorted Bubble Regime

Drag coefficient Reaches a constant value Cap Regime

.C .D

44.0,Re1501

Re

24max 6870

gdC pD

3

2,

3

8DC

The drag coefficient on the Reynolds number decreases with increasing values of the Reynolds number


Flow regime automatically determined from continuity of drag coefficient

Automatic Regime Detection

The determined by choosing minimum of vicious regime and capped regime

CCCCCCCC

distortedDviscousDDdistortedDviscousD

viscousDdistortedDviscousD

,,,,

,,,

,min

3cm/s

35cm/s


Drag Laws for Variable Bubble Sizes

Universal Drag Law (for Bubbly Flow) Viscous regime

Distorted regime

Capped regime

As the bubble size increases the bubble become spherical caped shaped

)1( 67.18

67.171

3

2 1.52

7/6

ppD ff

fgdC

-13

8C

2

pD

1

ReRe101Re

24 750

;

dvv; .C

p

q

e

e

ppqq.

D



Grace Drag Law The flow regime transitions between the viscous and distorted particle flow and can

expressed as follows.

Viscous regime

Distorted regime

Capped regime

/10x9, 3

4H

59.3H ,42.3

59.3H2 ,94.0

)857.0( 3

4

4

0.14-

ref

149.0

441.0

757.0

149.0

q

q

2

mskgEoMo

H

HJ

JMod

vv

gdC

ref

q

p

t

tq

pD

Re15.01Re

24C 687.0D

3

8C D



Tomiyama Model (1998)

Like the Grace et al model and universal drag model the Tomiyama model is well suited to gas-liquid flows in which the bubbles can have a range of shapes

43

8,

Re

72),Re15.01(

Re

24minmax 687.0

Eo

EoC

p

D

Viscous Regime

Distorted Regime

Cap Regime


Non-Drag Forces For gasliquid flows, non-drag forces have a profound influence on the flow characteristics,

especially in dispersed flows

Bubbles rising in a liquid can be subject to a additional forces including:

Lift Force

Wall Lubrication Force

Virtual Mass Force

Turbulence Dispersion Force


Lift Force When the liquid flow is non-uniform or rotational, bubbles experience a lift force

This lift force depends on the bubble diameter, the relative velocity between the phases, and the vorticity

and is given by the following form

The lift coefficient, , often is approximately constant in inertial flow regime ( < < ), Following the recommendations Drew and Lahey, it is

set to 0.5

Lift forces are primarily responsible for inhomogeneous

radial distribution of the dispersed phase holdup and could be important to include their effects in CFD simulations

qpqqpLlift vvvCF


Saffman and Mei developed an expression for lift force constant by combining the two lift forces:

Classical aerodynamics lift force resulting from interaction between bubble and liquid shear

Lateral force resulting from interaction between bubbles and vortices shed by bubble wake

Lift Coefficients: Saffman Mei Model

100Re40 :for ;2

Re0.0524

40 Re :for ;Re

Re

2

13314.0

Re

Re

2

10.3314-1

46.6C

Re;Re2

3C

Re)1.0(

'

L

2

'

L

e

dC q

q

pq

L

Suitability Mainly spherical rigid particles Could be applied to small liquid

drops

Shear Lift Force Vorticity induced Lift Force


Lift Coefficients: Moraga et al Model

Moraga et al. (1999) proposed an al alternative expression for the lift coefficient that correlated with the product of bubble and shear Reynolds numbers

7

73

ReRe

36000

ReRe

L

105ReRe63530

105ReRe6000 20120

6000ReRe07670

C7

e

for .-

for ee..

for .

Suitability Mainly spherical rigid particles Could be applied to small liquid

drops


Lift Coefficients: Legendre and Magnaudet Model

Legendre and Magnaudet proposed an expression for the lift coefficient that is a function of bubble Reynolds number and dimensionless shear rate

This model accounts of induced circulation inside bubbles

q

q

pq

highL

lowL

highLlowL

dJ

C

JSrC

CC

2

2

32

'

1

1

Re,

'5.0

2Re,

2

Re,

2

Re,L

Re ,Re

Re

2

1 ,

Re

2 ,

1.01

255.2

Re291

Re161

2

1

Re 6

12Sr , 500Re0.1for ,C

Suitability Mainly small spherical bubbles

and liquid drops


Tomiyama et al correlated the lift coefficient for larger bubbles with a modified Etvs number and accounts for bubble deformation

Lift Coefficients Tomiyama Model

g ,163.01 ,

g

0.4740.0159-0.00105

Eo10 27.0

10Eofor

4Eofor Re,121.0tanh288.0min

C

2

q3

1757.0

2

q'

2'3''

'

''

''

L

pp

pH

Hp dEoEodd

dEo

EoEoEof

Eof

Eof

Suitability All shape and size of bubble

and drops

Dependence of lift coefficient on bubble diameter


This is a force that prevents the bubbles from touching The main effect of this force is to ensure zero void

fraction (found experimentally) near vertical walls

Wall lubrication force is normally correlated with slip velocity and can be expressed as force is defined as:

Wall Lubrication Force

wqpqpWLWL nvvCF||

gas void fraction

Slip velocity component parallel to the wall


Wall Lubrication Coefficient: Antal et al Model

Antal et al. (1991) proposed a wall lubrication force coefficient according to:

Only active in thin region near wall where:

As a result, the Antal model will only be active on a sufficiently fine mesh

llnearest wa todistance

05.0

01.0

,0max

2

1

21

w

W

W

w

W

p

WWL

y

C

C

y

C

d

CC

bb

W

Ww dd

C

Cy 5

1

2

Suitability Mainly small bubbles Requires Fine Mesh


Wall Lubrication Coefficient: Tomiyama Model

Modified the Antal model for special case of pipe flow and accordingly:

Coefficients were developed on a single air bubble in a glycerol solution but results have been extrapolated to air-water system

Depends on Eotvos number, hence accounts for dependence of wall lubrication force on bubble shape

eter Pipe DiamD

Eo for .

Eofor .Eo.

Eo for e

o for E .

C

yDy

dCC

.Eo.

W

ww

p

WWL

331790

33501870005990

51

1470

11

2

17909330

22

Suitability Viscous Fluids and all bubble size and shapes Could be used for low air-water system


Wall Lubrication Coefficient: Frank Model

Generalised Tomiyama model to be geometry independent Model constants calibrated and validated for bubbly flow in vertical pipes

1.7m

6.8llnearest wa toDistance

331790

33501870005990

51

1470

11

,0max

17909330

1

WD

.Eo.

W

m

bWC

ww

bWC

w

WD

WWL

C

Eo for .

Eofor .Eo.

Eo for e

o for E .

C

dC

yy

dC

y

CCC

Suitability Viscous Fluids and all bubble size and

shapes in vertical pipe flows Could be used for low air-water system


Wall Lubrication Coefficient: Hosokawa Model

Hosokawa et al. (2002) investigated the influence of the Morton number and developed a new correlation for the coefficient:

Includes the effects of Eotvos number and bubble relative Reynolds number on the lift coefficient

EoCWL 0217.0,

Re

7max

9.1

Suitability All bubble size and shapes Could be used for low air-water system


The turbulent dispersion force accounts for an interaction between turbulent eddies and particles

Results in a turbulent dispersion and homogenization of the dispersed phase distribution

The simplest way to model turbulent dispersion is to assume gradient transport as follows:

Turbulent Dispersion forces

turb.

dispersion

force

fluid vel.

gas void fraction pqqTDTD kCF


Turbulent Dispersion Models

Lopez de Bertodano Model, Default CTD = 1 CTD = 0.1 to 0.5 good for medium sized bubbles in ellipsoidal flow regime. However, CTD up to 500

required for small bubbles

Burns et al. Model Default CTD = 1 The defaults value of CTD are appropriate for bubbly flows

Simonin Model Default CTD = 1 Same as Burns et al. Model

Diffusion in VOF Model Instead of modelling the turbulent dispersion as an interfacial momentum force in the phase

momentum equations, we can model it as a turbulent diffusion term in the phasic continuity equation


Turbulence in bubbly flows are very complex due:

Bubble-induced turbulence

Interaction between bubble-induced and shearinduced turbulences

Direct interaction between bubbles and turbulence eddies and

Turbulence Dispersion Models in Fluent Sato

Simonin

Only available when dispersed and per phase turbulence models are enabled

Troshko and Hassan

Alternative to Simonin Model



The virtual mass force represents the force due to inertia of the dispersed phase due to relative acceleration

Large continuous-dispersed phase density ratios, e.g. bubbly flows

Transient Flows can affect period of oscillating bubble plume.

Strongly Accelerating Flows e.g. bubbly flow through narrow constriction.

Virtual Mass Force

5.0;

VM

pq

qpVMvm CDt

vD

Dt

vDCf

Dip your palms into the water and slowly bring them together. Such a movement will

require small effort. Now try to clap your hands frequently. The speed of hands now is

low and will require considerable effort


Mixture Multiphase Model


Introduction

The mixture model, like the Eulerian model, allows the phases to be interpenetrating. It differs from the Eulerian model in three main respects:

Solves one set of momentum equations for the mass averaged velocity and tracks volume fraction of each fluid throughout domain

Particle relaxation times < 0.001 - 0.01 s

Local equilibrium assumption to model algebraically the relative velocity

This approach works well for flow fields where both phases generally flow in the same direction and in the absence of sedimentation


Underlying Equations of the Mixture Model

Solves one equation for continuity of mixture

Solves one equation for the momentum of the mixture

Solves for the transport of volume fraction of each secondary phase

0

mm

m ut

rkrkkn

k

km

T

mmmmmm uuFguupuut

u

1

eff

).().()( rpppmpppp uut


Constitutive Equations

Average density

Mass weighted average velocity

Drift velocity

Slip Velocity

Relation between drift and slip velocities

n

kkkm

1

m

n

k kkk

m

uu

1

mk

r

k uuu

qppq uuu

qk

n

k m

kkpq

r

k uuu

1


Relative Velocity

If we assume the particles follows the mixture flow path, then, the slip velocity between the phases is

In turbulent flows, the relative velocity should contain a diffusion term in the momentum equation for the disperse phase. FLUENT adds this dispersion to the relative velocity as follows:

q

Dp

m

p

mpp

pqf

au

drag

p

mpvpq

f

au

drag

t

uuuga mmm


16.0 Release

Validation of the Multiphase Flow in

Rectangular Bubble Column


Investigate air-water bubbly flow in a rectangular bubble column as investigated at HZDR by Krepper et al., Experimental and numerical studies of void fraction distribution in rectangular bubble columns, Nuclear Engineering and Design Vol. 237, pp. 399-408, 2007

Validation of Momentum Exchange Models for disperse bubbly flows accounting: Drag force

Lift force

Turbulent dispersion

Turbulence Interaction

Turbulence models

Objectives


Duct Dimensions: Height: 1.0 m

Width: 0.1 m

Depth: 0.01 m

Bubbles are introduced at the bottom LW 0.020.01 m

Computational Geometry

Outlet: Degassing or Pressure Outlet

Inlet: Velocity or mass inlet


Fluid Materials and Phase Setup

Phases Setup

Phase Specification Primary Phase: water (Material: water) Secondary Phase: gas bubble (diameter: 3mm with Material: air)

Phase Interaction Drag: Grace Drag Force Lift: Tomiyama lift force Wall Lubrication: Antal et al (default coeff.) Turbulent Dispersion Burns et al. (cd=0.8) Turbulent Interaction Sato Model (default coeff.) Surface Tension Coeff.: 0.072

Materials Setups

Gas Bubble FLUENT Fluid Materials: air

Water FLUENT Fluid Materials: water-liquid (h2o)


Boundary Conditions

Boundary Patch Properties

Inlet

Type: Mass flow inlet Gas Bubble: 2.37E-05 kg/s Gas Volume Fraction (VF): 1.0 Turbulence Intensity 10% Viscosity Ratio 10 Water: mass flow rate: 0 kg/s Water VF: 0.0

Outlet

Type: Degassing Degassing outlet: Symmetry for water Sink for air

Walls

No Slip


Solution Methods and Control

Solution Methods

Pres.-Vel. Coupling Coupled Scheme

Spatial Discretization Gradient: Least Squared Cell Based Momentum: QUICK Volume Fraction: QUICK TKE: 1st Order Upwind

Transient Formulation Bounded 2nd Order Implicit

Solution Controls

Courant No. 200

Explicit Relax. Factors Momentum: 0.75 Pressure: 0.75

Under-Relax. Factors Density: 1 Body Forces: 0.5 Volume Fraction: 0.5 TKE: 0.8 Specific. Diss. Rate: 0.8 Turb. Viscosity: 0.5


Instantaneous Gas Volume Fraction

Gas volume fraction at 25s, 35s, 45s

k-SST-Sato k- Troshko-Hassan

Gas volume fraction at 20s, 30s


Turbulence Validation, Sato Model

Mean gas volume fraction distribution at plane y=0.63m



Turbulence Validation, Troshko-Hassan Model




Summary and Conclusions

It was found that the most appropriate drag which is in good accordance with the measurements is the Grace Drag law

The k- turbulence model combined with the Sato Model reproduced well the experiments with no fundamental differences to the k- SST plus the Sato Model. This may indicate that the bubble induced turbulence is quite significant in this bubble column

The Troshko-Hassan k- turbulence model performed well, particularly near the injection point, a region of interest as it seemed to be problematic when the validations were carried out with ANSYS CFX using k- SST plus the Sato Model


Numerical Schemes and Solution Strategies


Numerical schemes for multiphase flows

Three algorithms available for solving the pressure-velocity coupling Phase coupled SIMPLE (PC-SIMPLE)

Pressure Coupled (Volume Fraction solved in a segregated manner)

Full multiphase coupled (Volume Fraction solved along with pressure and momentum)

A possibility of solving all primary and secondary phase volume fractions directly rather than solving only the secondary phases directly

Ability to use the Non-Iterative Time Advancement (NITA)


Multiphase coupled solver Simultaneous solution of the equations of a multiphase system offers a more robust

alternative to the segregated approach

Can be extended to volume fraction correction (Full multiphase coupled)

For steady state problems the coupled based methodology is more efficient than segregated methodology

For transient problems the efficiency is not as good as for steady, particularly for small time steps. Solver efficiency increases with increase in time steps used for discretization of the transient terms.


Solution controls for PC-SIMPLE Conservative solution control settings are shown

If convergence is slow, try reducing URFs for volume fraction and turbulence

Tighten the multi-grid settings for pressure (lower it by two orders of magnitude). Default is 0.1

Use gradient stabilization (BCGSTAB)

Try using F (or W) cycle for pressure

Solution Strategies


For steady state problems using coupled multiphase solver is effective

Use lower courant numbers for steady state and higher URFs for momentum and pressure

Recommended values

Courant number = 20

URF pressure and momentum = 0.5 - 0.7

URF volume fraction = 0.2 - 0.5

For transient problems the efficiency of coupled not as good as for steady, particularly for small time steps.

Use larger time steps and high courant numbers (1E7) for coupled solvers and high URFs (> 0.7)

Solution Strategies

Documents

Fluent Multiphase 16.0 L04 Gas Liquid Flows