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1

Modelling the bubble size distribution ingas-liquid reactors

with QMOM implemented with a newcorrection algorithm

Miriam Petitti

A.Nasuti, D.L.Marchisio, M.Vanni, G. Baldi,N. Mancini, F. Podenzani, A. Bennardo

Dept. Materials Science and Chemical Engineering, Politecnico di Torino(Italy)ENI R&M, S.DonatoMilanese (Italy)

[email protected]



2

Outline

oGas-liquid systems modeling: challenges and issues

oSimulation of a gas-liquid stirred tank (air-water):oApproach

oDrag force evaluation

oPopulation Balance Model

oQMOM and correction algorithm

oResults:oGas volume fraction

oBubble Size Distribution (BSD)

oConclusion and possible next steps



3

Gas-liquid systems modeling: challenges and issues

o The simulation of poly-dispersed multiphase systems (i.e.,constituted by droplets, particles and bubbles) is a complexproblem

o Gas-liquid systems are the hardest to describe because fluidparticles have no fixed shape or size:

o Small bubblesà sphericalo Medium bubblesà ellipsoidal

o Big bubblesà spherical cup

o This has a tremendous impact on the interaction between theliquid (continuous phase) and the bubbles (disperse phase)

for:o Momentum phase transfer

o Mass phase transfer

o Enthalpy phase transfer



4

Phase coupling

Liquid Bubble – BubbleBubble

One – Way Coupling

Two – Way Coupling

Four – Way Coupling

Global gas hold-up

Drag forceVirtual mass force

Lift forceBubble breakup Collisions between bubbles:

Coalescence andmomentum exchange



5

BREAK-UP COALESCENCEThree conceptual steps

Bubbles come in close proximity because of

turbulent eddies

The controlling mechanism is deformationsinduced by turbulence

For gas bubbles binary breakageis the most common event

Approach: Four-way coupling

o The fluid exerts a drag force on the bubbles that is related tothe slip velocity, bubble size, turbulence intensity and bubbleconcentration (details in back-up slides)

o In addition there are the lift and virtual mass forces

LIQUID-BUBBLE FORCES



6

Approach

o The phases are simulated as interpenetrating continua thatobey transport equation and exchange momentum. Everyphase has its own mass and momentum conservationequation.

o Phase interaction occurs via interfacial terms (especiallydrag). Closeness of the model used for this terms to thereality determines accuracy of the model.

o Most of the interfacial terms depend on particle size andmost dispersed flows are not monodispersed, so

knowledge of representative bubble diameter is essential.

Eulerian Multifluid Model (Fluent)



7

Approach

o Expressions of the drag term tailored specifically forbubbles have been introduced as UDF

o A solver for the population balanceequation has beenincluded in the code to evaluate the bubble sizedistribution in every point of the system. The solver is

based on the QMOM method, which allows accurateestimation of the integral properties of the BSD withmoderate computational load (6 additional transportequations)

o Proper expressions for bubble breakage and coalescence

rates have been tested and fitted to experimental data sets



8

Drag force evaluation

Usually in stirred reactors BSD is included in the range of

ellipsoidal bubbles and is not so wide to cause a significantchange in the bubble terminal velocity, according toMendelsons’s law.



9

Drag force evaluation

The drag coefficient is evaluated from bubble terminal velocity: the d32

of the local BSD is considered and a unique value for bubble terminal

velocity is used in the calculation.

The drag force is evaluated on the basis of the local d32 and the local slipvelocity between the continuous and dispersed phases.

2

32

3

4

∞

−=

U

dC

l

gl

D ρ

ρ ρ

hold-up < 5%: about 13 cm/s (turbulence)

hold-up > 5% :about 8.5 cm/s (turbulence + effect near bubbles)

The damping effect of turbulence and the effect of high gas hold-upare considered for the terminal velocity evaluation:



10

Population BalanceEquation

Its solution describes the evolution of bubbles size distribution:

NumberDensityFunction

( )( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( )( ) ( ) ( ) ( ) ( )1 3 1 3

3 3 3 3

0 0

, ,, , , , , , ,

1, , , , , , , , , ,

2

d

d

d

d d d d d d d d d

L

L

d d d d d d d d d d d d

n L x tU n L x t g b L n x t d g L n L x tt

h L n L x t n x t d n L x t h L n x t d

λ λ λ λ

λ λ λ λ λ λ λ λ

∞

∞

∂ + ∇ ⋅ = − + ∂

+ − − −

∫

∫ ∫

Bb Db

Bc Dc

birth due to breakage death due to breakage

birth due to coalescence death due to coalescence



11

Kernels (or frequencies) quantifies the rate with which bubblesCOALESCENCE and BREAK.

Several expressions are available in the literature but in this work thefollowing have been used (Laakkonen et al., 2006):

Coalescence and breakup rates

( ) 1/3

1 2 32 /3 5 /3 1/ 3 4 /3erfc c

d

c d c d d

g L C C CL L

µ σ ε

ρ ε ρ ρ ε = +

BREAKUP KERNEL

( ) ( ) ( ) ( )1/ 221/ 3 2 / 3 2 / 3

7, ,d d d d d d d dh L C L L Lλ ε λ λ η λ = + +

( )

4

8 2, exp c c d d

d d

d d

LL C

L

µ ρ ε λ η λ

σ λ

= − +

C7 =0.88(from theory)

COALESCENCE KERNEL

Binary breakage, C1 from fitting



12

The PBE is solved in terms of the first k = 2N moments of the Number Density

Function (NDF) n(L):

0,1,2,3,4,5k=( ) ( )ˆkd k d d k d km u m S

t ρ ρ ρ ∂ +∇ =∂

The closure on the moments source terms Sk , concerning bubble coalescence and

breakup, is obtained by resorting to a quadratureapproximation :

QuadratureMethod of Moments

( )∫ +∞

≡0

dLLLnm kkwhere the moments mk refer to bubble size L:

L2

n(L)

LL1

w1

w2

L3

w3

N =3( ) ( ) ( )i

N

ii Lf wdLLf Ln ∑∫

=

+∞

≈10

weights nodes

the first 6 momentsare tracked

( ) ( ) ( ) ( ) ( )/3

( ) 3 3

1 1 1 1 1 1

1, ,

2

d d d d d dN N N N N Nk

k k kkk i i i i i i i j i j i j i i i j j

i i i j i j

dmS g L wb L g L w w w L L h L L L w h L L w

dt = = = = = =

= = − + + −∑ ∑ ∑ ∑ ∑ ∑

Bbk Bck DckDbk



13

QMOM and Moments corruption

The population balance is coupled with the multi-fluid model through the

moment transport equations. The moments can be considered as scalars that are

convected, mixed and diffused. However they are linked by mathematicalrelationships that assure the physical existence of the underlying distribution.

The independent transport of the moments does not preserve these relationships,

that are altered by the discretization schemes used by the CFD code, and can

therefore create a set of corrupted (and not valid) moments.

Non - valid moment set

Residuals

Instability problems

Negative moments, relationships altered



14

QMOM and Moments validity testThe validity of the moment set is checked by asserting the positivity of thedeterminants of Hankel-Hadamard (Shohat e Tamarkin, 1943):

1

1 2 1

,

1 2

...

...0

...

n n n l

n n n l

n l

n l n l n l

m m mm m m

m m m

+ +

+ + + +

+ + + +

∆ = ≥M M M M

for n =0,1 and l ≥ 0

These conditions for the first few moments {m0 , m1 , m2 , m3 } are equivalent to the

convexity check of the curve ln(mk ) in function of k:

2

2 1 0k k kmm m− −− ≥

k

ln (mk)

k

ln (mk)

valid non-valid



15

QMOM and Moments correction

Adjust that moment whichafter adjustment maximizessmoothness through minimizationof a vector related to third orderdifferences

Initial moments set

Form second orderdifferences in ln(mk)

Satisfies convexity

Form third order differences inln(mk): verctor a

Determine k* for which

vector a is minimized

Adjust mk* to get newmoment sequence

check m4 and m5

Stop: momentsok

Minimum Square

Gradient Algorithm

(McGraw 2006)



16

QMOM and Moments correction1) In certain cases the iterative algorithm enters an infinite loop and

does not manage to correct the moment sequence.

In these cases the moment set is re-built through two log-normaldistributions whose parameters are obtained fromthe first 4 moments

of the original set that must be corrected (Wright 2007)

NDF(1) from m0 , m2 , m3 NDF(2) from m0 , m1 , m3

2 2

exp

2k T

km N k

σ µ

= +

( )

( )2

2

lnexp

2

2L T

L

n L N

L

µ

σ

σ π

− − = ⋅NDF: moments:

Momentsevaluated as averages of the momentsof the two distributions

2) Control on gas volume fraction, particularly useful at higher gassing

rates: if gas vof <10-3 sources =0; mk=0; d32=dbinlet

Without this change at higher gassing rates it is observed the formationof zones with m5 < 0, tending to spread in neighbouring zones



17

Effects of moments correction

Negativevalues! A

0

≅ -10 4

B103

0

103

0

m2 without correction m2 with correction

ResidualsNo instabilityproblemswhencorrectionapplied,alsowith highunder-relaxationfactors

Residuals

Instabilityproblems



18

Effects of moments correction

d32

withoutcorrection

d32

with correction

The correction algorithm does not alter the physical behaviour of the system,

the bubblesizedistribution

“Lucky” simulation with lownumerical diffusion for the moments



19

Standard Rushton turbine

Baffles and blades width: 1.03 cmShaft diameter: 3.3 cm

Disk diameter: 13.6 cm

Turbine diameter: 21cm

Disk width: 1.06 cm

Sparger position: z =-10.5 cm; d =3.3 cm

Sparger width d =15 mm

Reactor volume: 194 liter

Porous sparger

Reactor configuration - 1

rotatingfluid

baffles

shaft

impeller

spargerComputational domain on half reactorà Ncells: 226776



20

Ring sparger (12 holes 2 mm of diameter)

Computational domain on half reactorà Ncells: 226776

Standard Rushton turbine

Baffles and blades width: 1.03 cm

Shaft diameter: 3.3 cm

Disk diameter: 13.6 cm

Turbine diameter: 21cm

Disk width: 1.06 cm

Sparger position: z =-10.5 cm; d =3.3 cm

Sparger width d =15 mm

Reactor configuration - 2

Reactor volume: 194 liter

sparger

zona rotantebaffles

albero

impeller

sparger

zona rotantebafflesbaffles

albero

impeller

rotatingfluid

shaft

impeller

sparger

baffles



21

Results-fluid dynamics

The fluid dynamics predicted by the model was also tested for astandard geometry stirred reactor of 15.4 liter, agitated by a Rushtonturbine, in terms of:

-Velocity profiles

-Fluid dynamics regime transitions

-Gas cavity structure transitions

- Power dissipated

The agreement with experimental data and/orempirical correlations is good



22

Results – gas distribution

Reactorconfiguration 1

GAS VOLUME FRACTION



23

0.00

0.02

0.04

0.06

0.08

0 0.4 0.8 1.2gas flow rate, vvm

g l o b a l g a s h o l d - u p

Results – Gas hold-up

N = 390 rpmReactor configuration 2

T R A S I T I O N A L

R E G I M E Experimental

data

Simulation



24

8 different operating conditions for reactor configuration 1 and 7different conditions for reactor configuration 2 were considered

resulting in the following best fit values:C1=6 (from fitting) and C7 =0.88 (from theory)

Results Bubble size distribution

G a s f l o w r a t e , v v m

Stirring rate, RPM





25

0

1

2

3

4

0 2 4 6 8

R2

2.90

2.77R4

3.25

3.19 R8

2.43

2.90R9

2.23

2.71 R12

3.34

3.30

Simulation resultsExperimental data

Reactor configuration 1N =250 RPM; 0.093 vvm

0

1

2

3

4

0 2 4 6 8

0

1

2

3

4

0 2 4 6 8

0

1

2

3

4

0 2 4 6 8

0

1

2

3

4

0 2 4 6 8

B S D

Mean bubblesize, mm

d, mm

R2 R4

R8

R9

R12

Results – Bubble size distribution



26

R2_A

2.24

2.86

R2_B

2.90

2.77

R2_C

2.252.74

R2_D

2.03

2.47

R2_E

2.92

3.37

Simulation results

Experimental data

0

1

2

3

4

0 2 4 6 8

0

1

2

3

4

0 2 4 6 8

0

1

2

3

4

0 2 4 6 8

0

1

2

3

4

0 2 4 6 8

0

1

2

3

4

0 2 4 6 8

R2_E R2_D

R2_C

R2_B

R2_A

B u b b l e v o l u m

e

d e n s i t y

d, mm

d32, mm

Reactor configuration 1 – N =250 RPM; 0.093 vvm


l bbl i di ib i



27


N =250 RPM; 0.052 vvmN =157 RPM; 0.052 vvm

R2

R8R9

R4

R12

R2

R8R9

R4

R12

Reactor configuration 1

N =250 RPM; 0.072 vvm

ExperimentalSimulation (45° )

Simulation (135°)

Simulation (45° )

Simulation (135°)

R lt B bbl i di t ib ti



28

Results – Bubble size distributionReactor configuration 1

Experimental

Simulation

N =157 RPM; 0.052 vvm N =250 RPM; 0.052 vvm

N =250 RPM; 0.072 vvm



29


experimental

simulation

experimental

simulation

Reactor configuration 2 – N =390 RPM; 0.7 vvm

measurement points

d 3 2 , m

m



30

Reactor configuration 2 – N =390 RPM


0.000

0.002

0.004

0.006

0.008

0 0.2 0.4 0.6 0.8 1 1.2

gas flow rate, vvm

m e a n b u b b l e

s i z e , m

Experimental data (derivedthrough emp. correlation)

Simulation



31

• Some solid guidelines for the simulation of gas-liquid stirrer tanks inrealistic conditions have been identified and validated.

•Great attention must be paid to the calculation of the drag force (thiscan be calculated via the bubble terminal velocity taking into account theeffect of turbulence and of the other bubbles for very high hold-ups)

• The algorithm for the calculation of the bubble size distribution (QMOM+correction algorithm) was found to be stable under very different

operating conditions and reactor configurations.

•The coalescence and breakage kernels and the drag law are able todescribe with satisfactory accuracy both the global hold up, the bubblesize distribution and the mean bubble size.

•The simulation settingscan be used to describe the behavior of the

stirrer tank under different operating conditions ranging from very lowgas hold-up (0.5 %) to high gas hold-up (7 % for the air-water system)

Conclusions



32

n Application to mass transfer in gas-liquid systems

Mass transfer in gas-liquid systems is very sensitive to the bubble sizedistribution, because of the effect of interfacial area and because thetransfer coefficient depends on gas-liquid slip velocity (which in turndepends on bubble size). An important point is that there may be adifference between the size range of bubbles that control fluid dynamics(usually the larger ones) and those that determine interfacial area (the

smaller ones). This point could be captured well by proper use of thepopulation balance method.

n Gas-liquid chemical reactions

Introduction of chemical reaction paths and interfacial mass transfermodels for process reactions in multicomponent two-phase systems.Probable need to consider heat generation/transfer effects as well.

Possible next steps



33

Thankyouverymuch for your

kindattention

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