population balance methods for multiphase flows

© 2015 ANSYS, Inc. 8-1 Release 16.0

16.0 Release

Multiphase Flow Modelingin ANSYS CFX

Population Balance Methodsfor Multiphase Flows


Overview

• Introduction to Population Balance Methods

• MUSIG Model– Homogeneous

– Inhomogeneous

• DQMOM (Method of Moments)– Beta feature, provides capability similar to MUSIG


Introduction

• In ANSYS CFX, normally the diameter of a dispersed Eulerian phase is

assigned

• Multiple sizes for dispersed Eulerian droplets, bubbles, or droplets must

be modeled using individual phases each with a characteristic diameter -

this is memory and CPU intensive

• Fluid droplets and bubbles can break-up and coalesce in response to fluid

forces - the mean bubble or droplet diameter may not be known a priori

• Population balance models in ANSYS CFX offer a way to model these

phenomena

– MUSIG

– DQMOM (beta)


MUSIG Model – General

• MUSIG

Multiple Size Group model

Accounts for coalescence and breakup in polydispersed phases using a population balance approach


• Population balance approach

Population balance equation for group i:

Equivalently (conservation form):

In the homogeneous implementation of the MUSIG model, all size groups are assumed to move at the same velocity(valid for ellipsoidal bubbles and droplets where the interphase drag does not depend strongly on size):

CiCiBiBii

i

i

ii DBDBx

)nU(

t

n

ii

i

i

idiidi Sx

fUr

t

fr

)()(

ii

i

i

dddidd Sx

fUr

t

fr

)()(

Population Balance Approach


Justification for Homogeneous MUSIG


• Sources

• Bij = breakup rate from group i into group j

• Bij’ = breakup rate from diameter i to diameter j

fraction) (breakup

group)in sizesover on (integrati '

i

j

BV

f

BVijij

ij ij

ijijjiddBi

m

mf

dfBB

BffBrS

BV

Sources Due to Break-up


• Models for breakup rate (Bij’)

– Luo and Svendsen

• based on theory of isotropic turbulence

• See Solver Theory documentation for details

– User-defined

• Breakup rate may be a CEL expression involving standard variables and the diameters and/or masses) or groups i and j

Models for Break-up Rate


• Coalescence:

Cij = coalescence rate from group i into group j

Xjki = mass matrix (fraction of mass due to coalescence between j and k which goes into i)

– Linear profile is the default

– Lumped mass assumption may be set using a CCL parameter

Sources Due to Coalescence

ij j j

jiijjki

kj

kj

kj

ik

jkddCim

ffCXmm

mmffCrS

1

2

12


MUSIG – Coalescence & Breakup

• Coalescence– Prince & Blanch model

• accounts for collisions due to turbulence and buoyancy

• Turbulence buoyancy coefficient is typically 0.25-1.0

– User-defined model also available

• Breakup

– Luo & Svendsen model

• Accounts for breakup due to turbulence

• Coefficient is typically 1.0

– User-defined model also available

• User-defined models

– User-defined expression/routine will be called for all size group pairs

– Rate can be a function of the diameter and mass of the two groups in the group pair by using the following variables: diami, diamj, massi, massj


MUSIG – Size Group Discretization

• When setting the size groups, specify the following:

– Minimum Diameter (eg, 0 cm)

– Maximum Diameter (eg, 1 cm)

– Number of Size Groups

– Size Group Discretization Option

• Equal mass (skewed toward large diameters)

• Equal diameter (equal weighting)

• Geometric (skewed toward small diameters)

– Reference density (of polydispersed phase density varies)

• Used only for creating group masses from specified diameters


MUSIG – BCs, ICs, Sources• At inlets/openings, need to set size distribution

– ‘Automatic’ option assumes a flat distribution

• Also need to set size distribution for initial conditions

– ‘Automatic’ options assumes a flat distribution

• Also need to set size distribution for mass sources

– No ‘Automatic’ option

– Used only when mass source is positive

– Can also be applied when mass source is negative by setting ‘MUSIG Sink Option = Specified Size Fractions’

– Similar to treatment of components for multicomponent fluids

• Can also set a source for an individual size group

– Does not contribute to net mass source for the phase

– User’s responsibility to ensure that the sources sum to zero when summed over all size groups

– Similar to treatment of component sources for multicomponent fluids (except in that case, if the sources do not sum to zero, there is an implied nonzero source of the ballast component)


Solver Details

• Interphase transfer processes (eg, drag) use interfacial area density based on Sauter mean diameter, d32

• Solver for size groups implemented carefully such that size groups remain bounded and sum to 1 every iteration


MUSIG – Post-processing

• Following variables are available:

– Size Fraction

– Cumulative Size Fraction

– Interfacial Area Density

– Mean Particle Diameter (ie, Sauter Mean Diameter)


Accessing MUSIG in CFX

• To access the MUSIG population balance model in CFX-Pre, set the Morphology Option for the dispersed phase to Polydispersed Fluid on the Basic settings tab for the domain

• A Polydispersed Fluids tab will appear on the Domain Details form. The properties of the Polydispersed Fluid can be set by clicking on this tab, which include MUSIG size group settings, MUSIG fluid type, coalescence and breakup models, etc.


Poydispersed Fluids: MUSIG Settings

•On the Polydispersed Fluids tab, click on the New icon to define a new polydispersed fluid

•There are four different optionsfor polydispersed fluid types:

– Homogeneous and Inhomogeneous MUSIG (which divide the bubbles into discrete size groups)

– Homogeneous and Inhomogeneous DQMOM which are based on the quadrature method of moments (beta capability)

– In order to see the options other than Homogeneous MUSIG, you will need to turn on beta options under Case Options/General


MUSIG Fluid Details

• The details of the MUSIG fluid(i.e. the number of size groups, minimum and maximum diameter, etc., can then be specified)


Inhomogeneous MUSIG

•Homogeneous MUSIG

– Assumes single velocity field for all dispersed phases

– Dispersed phases may interact with continuous phases

• Interfacial area density calculated from Sauter mean diameter

– Valid for bubbly flows in elliptic regime and when lift force can be neglected

• Inhomogeneous MUSIG

– Allows multiple velocity fields for dispersed phases

– But several dispersed phases can belong to the same ‘velocity group’

– Example: 21 size groups and three velocity groups would allow different velocity

groups to be partitioned according to their mean size (i.e. could predict radial

distribution of different sized bubbles for bubbly flow in vertical pipes)


Inhomogeneous MUSIG

• Momentum equations are solved for N gas phases (vel. groups)

• Size fraction equations for M bubble size classes in each vel. group

• Bubble coalescence and break-up over all NxM MUSIG groups

dP1 dPa

V1 V2

dPa+1 dPb dPx+1 dPM

VN

size classes (M)

velocity groups (N)

dPdP,krit

N(dP)

coalescence

breakup


Inhomogeneous MUSIG

Inhomogeneous MUSIG model solves for:

• N volume fraction equations

• N+1 momentum equations

• (>) 1 turbulence model equations

• NxM size fraction equations

1 1 1

, , 1 , 0N M N M N M

dg

d dg dg dg g

g g gd

rr r f f S

r

1, ,

1, ,

g N M

j N


Inhomogeneous MUSIG

coalescence birth

1

1

1

1

coalescence dea

breakup birth

br

th

eakup death

1

1

1

2

N M

g d dh

h g

g

d dg

h

g g

h id dh di g hi

h i h i

N Mdh

d dg

h h

gh

g

h

gh

h

i

S r

r

mC

C

B

B

mr r X

m m

rr

m

g

g

g

g


• TOPFLOW facility at FZR studies different types of bubbly flows:

– Finely dispersed (121)

– Bubbly flow

• Void maximum near the wall(039)

• Transition region (083)

• Centred void fraction maximum(118)

• Centred void fraction maximum bimodal (129)

– Slug flow (140)

– Annular flow (215)

Inhomogeneous MUSIG Validation:TOPFLOW

Test case FZR-074: dilute bubbly flow with near wall maximum of void fraction

Experiments by Prasser et al., FZR


Inhomogeneous MUSIG: FZR-074

• Near wall void fraction peak at inlet, spreads with increasing height

• 3 velocity groups; 21 size groups

0.0

5.0

10.0

15.0

0.0 25.0 50.0 75.0 100.0

x [mm]

Air

vo

lum

e f

rac

tio

n [

-]Exp. FZR-074, level CExp. FZR-074, level FExp. FZR-074, level IExp. FZR-074, level LExp. FZR-074, level OExp. FZR-074, level RCFX, Inlet level (z=0.0m)CFX, level CCFX, level FCFX, level ICFX, level LCFX, level OCFX, level R


MUSIG Versus DQMOM

• The MUSIG model fits in the PBM category of a Class Method.

• Attempts to directly predict the Particle Size Distribution (PSD).

• The internal coordinate of the PSD is the particle mass (M).

•Mass divided by a fixed number of intervals (or classes) over a fixed range.

• Size fraction (f) used as a measure of the mass interval weight.

• The size fraction changes in response to coalescence/breakup processes.

f

M

Range of M is

predetermined

Attempts directly predict the

PSD shape

M not necessarily uniform

Collection of size groups (or

all of them) can be

transported at their own

velocity.


MUSIG Versus DQMOM Cont’d•MUSIG model requires a large number of size groups to accurately predict a broad or bi-modal PSD shape.

• Approach is problematic if the physical mechanisms affecting the PSD shape (i.e. nucleation, growth, coalescence/breakup) lead to large changes in the PSD which are not known ahead of time.

Supersonic nozzle solution with nucleation

followed by growth of the PSD

(QMOM solution using a 1D code). 1 2

3 4

Large movement in PSD


MUSIG Versus DQMOM

• DQMOM does not predict directly the PSD shape, but only tracks a representative distribution.

• Representative distribution statistics (i.e. number, mean, area density and volume fraction) accurately reflects the statistics of the real distribution.

• Give up directly predicting the PSD shape, but gain a method where the particle size range dynamically changes with the solution.

• In DQMOM we do not predefine either the size group range or the intervals.

• DQMOM also needs significantly less size groups to maintain high accuracy (N=3 generally).

• DQMOM may be less stable than MUSIG due to the introduction of the additional non-linearity of the size scale calculation. It has remained beta capability since it was first introduced at 12.0 and offers no real advantages over MUSIG at this level of implementation


Brief History of the Method of Moments

•Method of Moments (MOM) (Hulburt and Katz,1964)

– Useful because of minimal number of scalars required to represent properties of a complex PSD

– Primary limitation being simplified rate equations (i.e. growth, coalescence/breakup) needed in order to close problem

• Quadrature Method of Moments (QMOM) (McGraw,1997)

– removed closure problems so that MOM can be applied widely

– practically limited to homogeneous flows (in terms of velocity)

• Direct Quadrature Method of Moments (DQMOM)

– addresses the need for inhomogeneous solutions in terms of velocity

– provides a systematic framework for including multivariate PSD’s

– recovers the QMOM solution for monovariate/homogeneous cases


DQMOM In Principle

E

w

E1E2

w2

w1

E3

w3

N=3

Quadrature representation of an underlying PSD. The quadrature weights (wi) and abscissas (Ei) are obtained so as to represent the aggregate properties of the PSD. Level of quadrature (N) is practically in the range of 2 to 4.

Distance between nodes in quadrature

are dynamic

Overall extent of distribution is dynamic

Weights of individual

nodes are dynamic


Homogeneous DQMOM: Example• Example CFX-DQMOM solution

with bubble coalescence in a static mixing vessel.

• Widely disparate distribution can be employed at boundaries.

• Prediction of outlet distribution.

Dsauter = 1.70 mmDsauter = 9.20

Dsauter = 12.28 mm

population balance methods for multiphase flows

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