parameter for bubble breakage and coalescence

7292019 Parameter for Bubble Breakage and Coalescence

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THE DETERMINATION OF PARAMETERS FOR BUBBLE BREAKAGE AND COALESCENCE

FUNCTIONS FOR GAS-LIQUID SYSTEMS IN A MIXED TANK

Marko Laakkonen1 Ville Alopaeus2 Juhani Aittamaa1

1Laboratory of Chemical Engineering and Plant Design

Helsinki University of TechnologyPOB 6100 FIN-02015 HUT

Espoo Finland

2 Neste Engineering OyPOB 310 FIN-06101

Porvoo Finland

Key words population balances mixing gas-liquid dispersion inhomogeneity

Prepared for presentation at AIChE 2002 Annual Meeting Indianapolis Nov 3-11 Gas-Liquid and Solid-Liquid mixing

Copyright copy M Laakkonen V Alopaeus and J Aittamaa

Helsinki University of Technology Espoo Finland

Unpublished

AIChE shall not be responsible for statements or opinions contained in papers or printed in its publications



Abstract

Local time-averaged bubble size distributions (BSD) have been measured for air-water and CO2-n-butanol dispersions in

a baffled 138 dm3

mixed tank by Particle Image Velocimetry (PIV) technique BSD measurements were performed at six

locations in the tank at various gas feeds and mixing speeds The fundamental bubble breakage and coalescence

phenomena have been studied using the measured BSDs and a multi block stirred tank model which consists of a limited

amount of ideally mixed connected units in the tank

The multi block stirred tank model including the population balance equations for bubbles has been created It has been

used to fit the parameters of bubble breakage and coalescence functions because it is simple enough for the fitting but

capable in predicting inhomogeneities occurring in the stirred tank Computational fluid dynamics (CFD) has been used to

determine the flow patterns and local dissipations of turbulent energy in the mixed tank The flow pattern is needed to

estimate the flow rates between the blocks PIV technique has been applied for the verification of the flow pattern Local

dissipations of turbulent energy are required for the bubble breakage and the coalescence models The applicability of

various breakage and coalescence models found from the literature has been tested

The BSDs at the gas inlet are important for the fitting process They have been estimated from the PIV bubble size

distributions and from the pictures of high-speed camera The arrangement of the blocks in the tank and the selection of

the bubble velocity model which describes the relative velocities between the bubbles and the liquid are other important

issues in the fitting

The fitted bubble population balance model predicts the dependence of the bubble populations on the local turbulence

energy dissipation If local dissipation of turbulent energy is known all over the vessel or predicted by CFD then local

bubble sizes and mass transfer areas can be estimated and used to scale up the mixed tanks

1



Introduction

Gas-liquid contacting devices have been used widely in chemical biochemical petroleum and mining industries It has

been estimated that 25 percent of all chemical reactions occur between a gas and liquid Mass transfer across the gas-

liquid interface controls a considerable number of these processes (Tatterson 1991) Stirred vessels are favorable mass

transfer devices since they provide a large gas-liquid interfacial area and high shear stress to enhance the mass transfer

On the other hand the modeling of gas-liquid mixing is a very challenging topic The complexity of gas-liquid mixing

arises from the difficulties to describe the turbulent continuous phase flow and its interactions with the dispersed phase

There is not general agreement about the formulation of mass and momentum transfer equations or the turbulence

properties at the gas-liquid interface (Derksen et al 2002)

Traditionally empirical correlations of dimensionless numbers have been used to model the gas-liquid mass transfer and

vessel hydrodynamics These correlations constitute mostly on integral quantities such as gassed power input or overall

gas hold-up in the certain vessel geometry They are incapable in predicting vessel hydrodynamics in a wide range of

operational conditions or various geometries In addition they do not give information about the inhomogeneities

occurring in a vessel Mathematical tools and computational capacity allow at the moment a more fundamental approach

Computational Fluid Dynamics (CFD) has already been used in many studies to predict flow patterns and local gas

volume fractions in the stirred gas-liquid vessels (eg Gosman et al 1992 Bakker 1992 Bakker and Van Den Akker

1994 Morud and Hjertager 1996 Lane et al 2000 Derksen et al 2002) The CFD as connected to the reaction kinetic

heat and mass transfer models has become an interesting tool for the scale-up and the design of gas-liquid stirred tank

reactors and other gas-liquid contacting devices These computational tools include many modeling options assumptions

and a set of parameters Therefore their validation against experiments is needed (Schaumlfer et al 2000 Alves et al 2002

Derksen et al 2002)

The determination of heat mass and momentum transfer between a gas and liquid requires a model for the gas-liquid

interfacial area The use of population balances is a mechanistic approach that has been used to describe local particle size

distributions and hence the interfacial area between the continuous and the dispersed phase in gas-liquid liquid-liquid and

solid-liquid systems The most challenging task in the utilization of population balances is to find the generalized kinetic

equations for the breakage and coalescence Considerable efforts have been put for the mechanistic modeling of breakage

and coalescence of bubbles drops and particles (eg Hinze 1955 Coulaloglou and Tavlarides 1977 Narsimhan and Gupta

1979 Prince and Blanch 1990 Tsouris and Tavlarides 1994 Luo and Svendsen 1996 and Martinez-Bazaacuten et al 1999)

2



Recently Lasheras et al (2002) have presented an extensive review and tested various models for bubble breakage and

daughter bubble probabilities

In spite of extensive theoretical efforts the models for the breakage and coalescence are not completely predictive or

applicable for the CFD calculations as such but require validation They also include experimental parameters that have

to be fitted against experiments Very few thorough studies have been published on the spatial variation of bubble size

distributions in the stirred gas-liquid vessels (Barigou and Greaves 1992a and 1992b Takahashi et al 1992 Machon et al

1997 Schaumlfer et al 2000 Alves et al 2002) This is partly due to difficulties to measure gas-liquid flow in turbulent

conditions Recent progress in noninvasive monitoring techniques of multiphase flows (Chaouki et al 1997 Mavros

2001) is encouraging in this sense

The main objective of this work is to fit the population balance model parameters for gas-liquid dispersions against the

local time-averaged bubble size distributions measured using the Particle Image Velocimetry technique (PIV) in the

stirred gas-liquid vessel Local inhomogeneities have been considered in the fitting process using a multi block stirred

tank model

Population balances

The bubble size distributions are generated by various phenomena the breakage coalescence growth nucleation and

shrinkage of bubbles the relative velocities (slip) between the dispersed and the continuous phase and the transportation

of bubbles in and out of the balance region with the convection Shear forces turbulent energy dissipation reactions mass

transfer and physical properties such as density and viscosity affect to these processes In this work gas-liquid dispersion

is assumed to chemically equilibrated Therefore growth shrinkage and nucleation of bubbles due to mass transfer or

reactions are neglected The discretized population balance equation for a unit volume can be written as (Alopaeus et al

1999)

( ) ( ) ( )( )( )

( ) ( )( )

1

2

1

3133

1

j

V V

j

ji jiiout i

ji

V

j

j ji j

NC

i j

j jiinii

Y aa F Y Y a g Y

Y Y aaa F aY a g aaY dt

dY

i NC

i

sum

sumsumminus

=

=+=

minusminusminus

minus+∆+= β

(1)

The first and the fourth term in the right hand side of eq (1) are the transportation of bubbles in and out of the balance

region and include the effect of relative velocities between the continuous and the dispersed phase The second and the

third term are the birth of bubbles by breaking and coalescing The fifth and the sixth term are the death of bubbles by

3



breaking and coalescing In order to work with equation (1) models for the breakage and the coalescence rates the

daughter bubble size distribution resulting from a breakage the model for the relative velocities (slip) of bubbles and a

procedure to discretize the bubble size range into categories are needed

Stirred tank model

Fitting of population balance models against the local time-averaged experiments requires a model that considers the local

dissipation of turbulent energy and the flow pattern in the stirred gas-liquid vessel Bubbles break mainly near the

impeller where turbulence and shear forces are highest and coalesce near the walls of the vessel at the surface of the

dispersion and at the bottom of the tank Spatial variations of bubble size distributions are evolved in the vessel if the

breakage and the coalescence rates of bubbles are faster compared to the circulation rates of dispersion A multi block

stirred tank approach has been used in the fitting process The multi block stirred tank model is a very suitable one since

it accounts the inhomogeneities occurring in the vessel and is simple enough in contrast to the full-scale CFD models with

thousands to millions of unit volumes

Stirred tank has been divided into a few ideally mixed unit volumes which are connected to each other Simplified flow

pattern ie the pumping numbers between the unit volumes and the local relative turbulence energy dissipations have been

determined by averaging from the full-scale single-phase CFD simulations made with FLUENT program Flow pattern is

needed to calculate the transportation of bubbles between the subregions The spatial variation in bubble size distributions

results mainly from the spatial variation of the turbulence energy dissipation Physical properties of water and n-butanol

have been used in the CFD simulations to determine the flow patterns It is well known that ungassed and gassed liquid

velocities deviate since the power consumption and the impeller pumping capacity decrease on gassing (Tatterson 1991)

Therefore Bakker and Van Den Akker (1994) corrected pumping capacity and turbulent properties when they used the

single-phase flow pattern to simulate lean (0-5 vol-) dispersions with CFD In this work overall gas hold-up in the

vessel was less than 1 vol- due to the limitations of the PIV method We have therefore neglected the differences

between the gassed and the ungassed flow pattern and used the flow fields of the single-phase CFD simulation It was also

noticed that the volume averaged dissipations of turbulence energy or pumping numbers do not deviate significantly

when physical properties of water or n-butanol are used in the CFD simulation Therefore the same flow pattern and

turbulence properties have been used for both systems in the fitting of bubble models Stirred tank has been divided into

14 subregions Subregions and their connections are shown in Figure 1 where one half of the tank is presented Vessel is

assumed to have 90deg symmetry around the impeller axis

4



Average dissipations of turbulent energy in the vessel is obtained from

35 V N D N i P ave =ε (2)

It has been assumed here that overall energy input from the impeller dissipates through the turbulence mechanism

Volume averaged relative turbulent dissipations are defined as

aveε ε ϕ = (3)

Figure 1 Subregions for the stirred tank simulation model

Relative dissipations of turbulence energy and corresponding volumes of subregions are given in Table 1 They are scaled

so that

1=

sum

NB

iiV (4)

and

(5)1=sum i

NB

i

iV ϕ

Dimensionless areas between the subregions are needed to calculate the effect of relative velocities between the gas and

liquid (slip) to the population balances They are defined as

42

i

ij

ij D A A

π sdot= (6)

5



Only the acceleration of bubbles due to gravity is considered Therefore only axial slip velocities have been included to

the multi block model Dimensionless areas that are needed in the horizontal direction are defined so that area is positive

from a subregion to next below They are presented in Table 2

Table 1 Volumes and relative turbulent energy dissipations in the subregions of multi block model

subregioniV iϕ

1 0113581 0025158

2 0201753 0015869

3 0166078 009479

4 0093496 0175606

5 0063023 0515762

6 0068275 0264278

7 0007222 2121584

8 0100999 15867569 0086571 0821006

10 0028857 0683946

11 0009096 4904083

12 0014878 0931224

13 0012505 1491787

14 0033666 7780494

Table 2 Dimensionless horizontal areas between the subregions

1 2 3 3 4 5 6Betweensubregions 4 3 12 6 5 8 8

A 03600 06400 02500 03900 03600 03600 03900


A 01109 07500 01391 01109 01391 02500 02500

Dimensionless pumping numbers between the subregions are defined as

3

iijij D N QQ sdot= (7)

They are given in table 3

Transportation of bubbles in equation (1) in and out of the subregions are calculated as

sumsum==

++= NB

k j

k ik ikj NB

k j

k ikj

feed jiin jiV

Y U A

V

Y QY Y

1

1

(8)

and

1

1

sumsum == ++=

NB

k j

ji ji jk NB

k j

ji jk

prod jiout ji V

Y U A

V

Y Q

Y Y (9)

6



They include 1) feed and product flows in a particular region of vessel 2) convections between the vessel subregions and

3) the flow due to relative axial velocity between the continuous and the dispersed phase Y is the number concentration

of bubble size i in block j and U is the slip velocity of bubble size i in subregion j

ji

ji

Table 3 Dimensionless pumping numbers ( ) between the subregionsQ

from to 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 0396

2 0396

3 0124 0815

4 0396 0543

5 0939 0149

6 0059 0102 0158

7 0810

8 1088 0004

9 0004

10 0004

11 0123 0265

12 0687 0286

13 0046 1029

14 1033

Experimental

Experimental information about the inhomogeneities in bubble size distributions is needed for the fitting of bubble

models In this work Particle Image Velocimetry technique (PIV) has been used PIV is a noninvasive method which can

be used to determine particle size distributions flow patterns and relative velocities between the dispersed and the

continuous phase simultaneously from transparent lean dispersions It is based on imaging a cross section of a flow which

is illuminated with a pulsing laser light plane Flowing fluid is seeded with small particles that follow the continuous

phase illuminated by laser light CCD camera is used to record the pictures from the illuminated sheet High concentration

of bubbles hampers the visibility on the measurement plane and attenuates the intensity of light Therefore the

applicability of PIV technique for the gas-liquid flows is restricted to low at most 4 gas volume fractions (Deen et al

2002) Local fluid and bubble velocities are calculated from the time delay and the displacement of bubbles and the

seeded particles between subsequent exposures The displacement of bubbles and seeded particles is calculated through

correlation analysis The discrimination between the seeded particles and the bubbles and the recognition of the actual size

and shape of bubbles are demanding tasks Honkanen and Saarenrinne (2002) have discussed more about the data

acquisition and the digital image-processing systems that were needed to perform the analysis

7



Experiments were carried out in a flat-bottomed cylindrical glass vessel (00138 m3) which was equipped with four-

bladed radial impeller and four baffles Gas was fed through a 066 mm (inner diameter) single tube nozzle which was

located in the middle of the vessel 30 mm from the bottom of the tank Particle imaging system consisted of two CCD

cameras and Nd-YAG-laser (400 mJ) The dimensions of the vessel and the experimental setup are presented in Figure 2

Experimental setup and processing of PIV results to bubble size distributions are discussed more profoundly by

Laakkonen et al

Air-tap water and CO2-n-butanol systems were measured from six locations A-F (Figure 2) of the vessel Gassing rate and

stirring speed were varied to study their effect on local bubble size distributions Experiments were performed at

atmospheric pressure and room temperature 22 Cdeg The surface tensions of tap water mmN 69=σ and n-butanol

mmN 23=σ against the air were measured with Sigma 70 Tensiometer Other physical properties in the fitting were

density =)( 2O H C ρ 997 kgm3 =)(air D ρ 129 kgm

3 (C ρ n-butanol =) 804 kgm

3 =)( 2CO D ρ 183 kgm

3and

viscosity =)(water C 098 cP (C n-butanol =) 28 cP

Gassing rates were chosen so that local gas hold-ups in the vessel were suitable for the PIV technique Stirring speeds

were adjusted so that the vessel operated as close as possible to turbulent flow regime At the same time it was required

that gas was dispersed properly by the impeller but was not sucked from the liquid surface to the dispersion

Gas feed and stirring speed were adjusted and the flow pattern was allowed to settle into stationary state Approximately

500 pictures including 4000-70000 bubbles depending on the system location and the operation conditions were

recorded from all investigated locations (Figure 2) This is assumed to be a statistically relevant sample since most

authors (Barigou and Greaves 1992a 1992b Takahashi et al 1993 Machon et al 1997) have measured only 500-1000

bubbles to determine the bubble size distribution The smallest detectable bubble size was 010 mm due to spatial

resolution of CCD camera The largest observed bubbles were approximately 85 mm

The solution of equation (1) requires the discretization of bubble size range An uniform geometrical or other

discretization could be used Eg Litster et al (1995) have presented an adjustable geometrical discretization of the form

where q is an integer equal to or greater than one Also in this work adjustable geometrical

discretization was observed to be very applicable since bubbles were concentrated to the lower end of size range in PIV

experiments Geometrical discretization gives narrow size categories (and hence higher accuracy) at the lower end of size

range where sharp peak of bubble density is located The categories are wider for larger bubbles which have low density

q

ii aa 133

1 2 =+

8



in the distribution If the number of bubble size categories the smallest detectable bubble size (gt 0) and the

largest observed bubble size are fixed equations (10) and (11) can be easily derived to describe the average size of

each category and the width of each category

NC mina

maxa

ia

ia∆

1a

a

+ min

a

a

min1

minus

=

2

1

max

min

min

max aa

NC NC

i

i

= (10)

1

max

min

min

max aa

a

a

aa

NC NC

i

i

∆ (11)

Figure 2 Dimensions of the stirred tank (left) (in millimeters) and top view of the experimental setup (right)

Local bubble concentrations ie the gas hold-up is in relation to the ability of bubbles to coalesce Therefore local gas

hold-ups were needed for the fitting process They can be determined straightforwardly from the PIV results if the depth

width and height of PIV pictures are known The width and the height of PIV picture were determined by optical settings

of camera The depth of illuminated laser light sheet 65 mm was obtained from the calibration experiments with a bubble

gel Sensitivity analysis denoted that local gas hold-up determined from the PIV results is relatively insensitive to the

depth of laser light sheet Therefore we have confidence in the determined local gas hold-ups

Some of the bubbles are only partially in the laser light sheet in the cross sectional direction of sheet and are observed

smaller than their actual size in the PIV pictures Therefore a correction method based on the principles presented by

Tadayyon and Rohani (1998) was developed (Laakkonen et al )

9



Flow patterns were measured simultaneously with the bubble size distributions in the PIV experiments Simulated flow

patterns for water and n-butanol and measured flow patterns for lean dispersions are compared in figure 3 As can be

noticed the directions of flow from the CFD and PIV experiments are well comparable for the air-water system For the

CO2-n-butanol system the directions of flow are quite comparable elsewhere but below the impeller where they are

opposite Deviation in the simulated and measured flow pattern for CO2-n-butanol system results probably from the

differences between the ungassed and gassed flow

Figure 3 Comparison of flow patterns in the stirred tank 1) Single-phase CFD simulations (continuous vector field over the vessels) water (left) and n-butanol (right) (simulations made by Manninen and Taivassalo VTT processes 2002) 2)

Evaluated from the PIV experiments (arrows in the boxes) air-water gassing rate 025 dm3min stirring speed 400 rpm

(left) CO2-n-butanol gassing rate 0185 dm3min stirring speed 400 rpm (right) (Honkanen and Saarenrinne 2002)

Bubble size distribution of gas inlet

Bubble size distribution of gas inlet affects to the local size distributions everywhere in the vessel It is therefore necessary

to characterize it for the fitting of bubble models Bubble size distribution injected from the submerged gas nozzle

depends complicatedly on the geometry of the nozzle physical properties of gas and liquid gassing rate and

hydrodynamic conditions surrounding the nozzle Models for the formation of bubbles from the submerged nozzles are

based generally on the balance of hydrodynamic forces (Tsuke 1977 Geary and Rice 1991) Especially turbulent flow

conditions around the nozzle make the modeling of initial bubble size difficult Therefore the formation of bubbles from

the gas nozzle was studied visually using the high-speed video imaging technique It was observed that with low gassing

rates bubbles came out from the nozzle one by one and formed a trail At high gassing rates a gas jet was evolved

10



Evolved bubble trails or gas jets fluctuated periodically Video imaging revealed also that trailing vortices of bubbles were

formed in the impeller discharge flow at high gassing rates (Figure 4)

Bubble size distributions of gas inlet have been evaluated for the fitting by using the PIV results The distributions that

were determined below the impeller were bimodal It has been assumed that the density peaks of largest bubbles represent

the effect of bubble trail or gas jet evolved by the nozzle This is quite evident when Figure 4 and the locations of

measured areas in Figure 2 are compared The bubble size that corresponds the maximum bubble density of the peak of

largest bubbles in the PIV distribution has been chosen to the average size of bubbles from the gas feed The size

distribution from the gas feed is assumed to gaussian with the estimated average bubble size and 10 mm standard

deviation The standard deviation was obtained by estimating from the experimental PIV distributions and high-speed

camera pictures Estimated average bubble sizes from the gas feed were compared to those calculated from the model of

Rice and Geary (1991) The average sizes of bubbles from that model were slightly higher than those evaluated from the

PIV distributions This is reasonable since the model of Rice and Geary (1991) was developed for the formation of

bubbles to a stagnant liquid In this work bubbles were formed to the flowing liquid at turbulent conditions

Figure 4 Bubble trail from the submerged nozzle (left) trailing vortices of bubbles in the discharge flow of impeller

(right) for the air-water system

Bubble breakage and coalescence models

The breakage frequencies of bubbles have been calculated from the modified model of Narsimhan et al (1979) where

viscous forces have been included to the energy balance for the breakage of bubbles from the work by Wang et al (1986)

Since the viscosities of gases are very low relative to those of liquid we have assumed that the viscous stresses that resist

the breakup of bubbles are not proportional to the viscosity of gas but to the viscosity of liquid surrounding a bubble

Therefore the viscosity of dispersed phase in the viscous term of bubble breakup balance is replaced with the viscosity of

continuous phase From this we end up to the following model for the breakage frequencies

11



( ) 3431335322

31

1

+=

j DC

C

jC

ja

C a

C erfcC a g ε ρ ρ

micro

ε ρ

σ ε (12)

The collision rates of bubbles have been calculated from the model of Coalaloglou and Tavlarides (1977) which stands as

( ) ( ) ( ) 312132322

4 ε ji ji ji aaaaC aah ++= (13)

The coalescence efficiencies of bubbles were assumed to unity since their consideration did not give better fittings but

would have required an extra parameter for the fitting This means that all collisions between the bubbles result as

coalescence and the coalescence term ji aa F in equation (1) becomes equal to ji aah

In addition a distribution of the formed bubbles when a breakage occurs is needed The following presented by Bapat et

al (1983) has been used

( )2

3

32

3

3

3

2

190

minus

=

j

i

j

i

j

i ji

a

a

a

a

a

aaa β (14)

The probability distribution function has been scaled in the simulation model so that volume is conserved during the

break-up of the bubbles The chosen bubble breakage and coalescence equations were found to be reasonable and present

data relatively well

Bubble slip velocity model

Local bubble size distributions and gas hold-up are sensitive to the specification of drag force between a gas and liquid

phase Therefore the model for the drag forces must be specified when bubble breakage and coalescence models are

fitted Under steady state conditions drag and buoyancy forces are in balance and the bubble attains the terminal velocity

Terminal velocities have been correlated widely against the experiments at stagnant liquid (Clift et al 1978)

However it is well known (Magelli et al 1990 Brucato et al 1998 Lane et al 2000) that the turbulence reduces

significantly the settling and the rising velocities of particles bubbles and drops Magelli et al (1989) measured settling

velocities for the small particles and noticed that the ratio of settling velocity at turbulent conditions U and the terminal

velocity U in a stagnant liquid varied in the range 04-12 Brucato et al (1998) found that U was as low as 015

for under 05 mm sized particles Spelt et al (1997) made simulations on the motion of 1 mm sized gas bubble in

turbulent isotropic flow and observed that rise velocities reduced down to 50 of the value in the stagnant liquid

t U

t t U

12



Several mechanisms have been proposed to decrease the particle slip velocities in turbulent flows particle lsquoinertiarsquo as it

responds to turbulent fluctuations changes in the instantaneous drag coefficient or the effects of added resistance of

lsquovirtual massrsquo acceleration (Brucato et al 1998) The reasons for the reduction of bubble rise velocities are still unclear

since reliable experimental data is not available The effect has been shown to increase with increasing turbulence

intensity (Spelt et al 1997) and decrease with Kolmogoroff timescale (Maxey et al 1994) Bakker (1992) proposed that

the drag coefficients could be calculated from the standard drag curve by using a modified Reynolds number where the

viscosity is the sum of the liquid viscosity and a term proportional to turbulent viscosity Brucato et al (1998) correlated

the reduction of drag coefficient to the ratio of particle size and Kolmogoroff length scale A simple cube law was

obtained through the fitting against the experiments with small particles

In this work a model of Brucato et al (1998) has been taken as a basis but the linear dependence between the increase of

drag coefficient and the ratio of particle size to the Kolmogoroff length scale is preferred since it gives a more reasonable

dependence between the slip velocities and bubble size for large bubbles After a short manipulation the ratio of slip

velocity at turbulent conditions U to terminal velocity U in a stagnant liquid can be written ast

1

21

minus

+=

λ

i

it

i a K

U

U (15)

Where K is left as adjustable parameter and λ is the Kolmogoroff scale of dissipative eddies which is obtained from

the equation (16)

413

=

ε

ν λ (16)

Terminal velocities of the bubbles in the stagnant liquid have been determined from the model proposed by Clift et al

(1978 s 114 Table 53)

Re = ND24 ndash 17569sdot10-4 ND2 + 69252sdot10-7 ND

3 ndash 23027sdot10-10 ND4 for ND le 73

log Re = -17095 + 133438sdotW ndash 011591sdotW2for 73 lt ND le 580

log Re = -181391 + 134671sdotW ndash 012427sdotW2 + 0006344sdotW3 for 580 lt ND le 155sdot107

log Re = 533283 - 121728sdotW + 019007sdotW2 - 0007005sdotW3 for 155sdot107 lt ND le 5sdot1010

(17)

where W = log ND and all logarithms have base 102

3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13



The model of Brucato et al (1998) has been developed for the particles and does not consider the effect of wobbling of

bubble shape on the rising velocities of bubbles We have assumed that the wobbling of bubble shape in the fully

developed turbulent dispersion is caused mainly by the turbulence and its effect is lumped to the adjustable parameter K

in equation (15) Since the reasons for the reduction of bubble rise velocities at turbulent conditions are still unclear

further experiments and modelling work would be needed for the development of more accurate models

Parameter fitting

Parameters and C in the breakage model C in the coalescence model and1C 2C 3 4 K in the turbulence slip model

were left as adjustable for the fitting Parameters were fitted by comparing the local time-averaged bubble size

distributions from the experiments to the corresponding calculated bubble size distributions in the subregions from the

multi block model Also measured and calculated local gas hold-ups and Sauter mean diameters were compared in the

fitting The locations of the subregions and the measured areas did not overlap completely Therefore in some cases it

was necessary to compare the averaged experimental distributions of several measured locations to the averaged

distributions of several adjacent subregions Nelder-Mead non-linear Simplex algorithm followed by Davidon algorithm

was used to minimise the residual function In all iterations dynamic population balances were solved to the stationary

state in the simulation model Since the gas feed was observed to fluctuate it was introduced to two subregions in the

simulation model It was estimated based on the visual observation that 70 vol- of the gas went to the subregion 14 and

30 vol- to the subregion 8 (Figure 1) The residual for the fit was calculated from the errors in the 1) relative volumetric

density 2) local gas hold-up and 3) Sauter mean diameter Residual function is defined as

)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)

where are weight functions Relative volumetric densities are defined asiw

)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



Relative volumetric densities were needed to control independently the fitting of shape of distribution and the fitting of

gas hold-up Reason was that less emphasis was put for the fitting of gas hold-up near the impeller than for the shape of

distribution This was necessary since multi block model is not capable in predicting the fluctuating gas wakes or jets

evolved by the nozzle and the trailing vortices in the impeller discharge flow which affect strongly to the gas hold-up

near the impeller It is also probable that PIV method gives worse results for the bubble size distributions and gas hold-up

near the impeller Reasons are found from the difficulties to recognize the bubbles from the trailing vortices in the PIV

images or to recognize the size and shape of bubbles when gas holdup is high The phenomena that affect to the gas hold-

up near the impeller should be included in the multi block model since the fitted parameters of bubble breakage and

coalescence models depend also on the bubble density

Four experimental sets from six locations of the vessel were available for the fitting at various gassing rates and stirring

speeds for both systems studied At first systems were fitted independently After that all experimental data for both

systems were fitted simultaneously to obtain the model that is predictive at various operational conditions and physical

properties of dispersion

Results for the air-water system

PIV results were available for air-water system at gassing rates 025 050 and 100 dm3min with stirring speed 400 rpm

and at gassing rate 100 dm3min with stirring speed 500 rpm Measured bubble size distributions for the air-water system

from all conditions and locations of the vessel were fitted simultaneously The optimised values of parameters for the

multi block model are C 58861 = 026702 =C 8033 =C and Measured and

calculated relative volumetric distributions are compared in Figure 5 In the fitting measured and calculated distributions

with the corresponding markers have been compared When the sum of several measured areas (eg A+B) or subregions

(eg 14+13) is marked as a legend an averaged distribution based on these measured areas or subregions has been used in

the fitting As can be noticed the model fits all conditions reasonably considering the complex nature of the studied

process Clearly the model has difficulties to follow the bimodal shape of experimental distributions especially at gassing

rate 050 dm3min (400 rpm)

4

4 10433 minussdot=C 1240= K

The comparison of calculated and measured local gas hold-up and Sauter mean diameter for two experiments is presented

in Figure 6 The fitted model predicts the gas hold-up and the Sauter mean diameter relatively well near the surface of the

dispersion Below the impeller gas hold-up and Sauter mean diameter from the multi block model are lower than were

observed in the experiments Reason is that the emphasis in gas hold-up residual was set to the gas hold-up near the liquid

15



surface in the fitting We assume that this is necessary since the fluctuating gas feed and the trailing vortices of bubbles

from the impeller affected significantly to the local gas hold-up at the bottom of the tank These phenomena were not

included to the multi block model Therefore the deviation between the calculated and the measured values is explicable

Figure 5 Relative volumetric bubble size distributions for air-water system from PIV experiments (locations of measureddistributions denoted as A-F in legends) and from the fitted model obtained from the simultaneous fitting of air-water experiments (locations of distributions denoted as subregions 1-14) The predicted distributions have been calculated fromthe same fitted model in all subplots

Figure 6 Comparison of measured and simulated (in brackets) local gas hold-up (vol-) and Sauter mean diameter (mm)for air-water system

16



Results for the CO2-n-butanol system

PIV experiments were available for CO2-n-butanol system at gassing rates 0185 037 and 050 dm3min with stirring

speed 400 rpm and at gassing rate 0185 dm3min with stirring speed 500 rpm Measured bubble size distributions for the

CO2-n-butanol system from all conditions and locations of the vessel were fitted simultaneously The optimised values of

parameters for the multi block model are 62431 =C 048202 =C 9803 =C and

Measured and calculated relative volumetric distributions are compared in Figure 7 The fitted model gives

qualitatively reasonable prediction for the spatial variation of bubble size distributions but it is incapable to follow the

exact shape of bimodal distributions Probably the majority of the bubbles from the gas feed enter into a region of high

turbulence near the impeller and break to smaller ones in the simulation model Therefore more accurate description of

the effects of impeller to the bubble size distributions and a denser grid in the impeller region would be needed to get

better fittings

4

4 10628 minussdot=C

0890= K

The calculated and the measured local gas hold-ups and Sauter mean diameters have been compared in Figure 8 As can

be noticed local gas hold-ups and Sauter mean diameters calculated from the model do not vary significantly in the vessel

in contrast to those determined from the PIV experiments where they increase from the surface of the dispersion to the

impeller

17



Figure 7 Relative volumetric bubble size distributions for CO2-n-butanol system from PIV experiments (locations of

measured distributions denoted as A-F in legends) and from the fitted model obtained from the simultaneous fitting of CO2-n-butanol experiments (locations of distributions denoted as subregions 1-14) The predicted distributions have been

calculated from the same fitted model in all subplots

Figure 8 Comparison of measured and simulated (in brackets) local gas hold-up and Sauter mean diameter for CO2-n-

butanol system

18



Results from the simultaneous fitting of air-water and CO2-n-butanol systems

All experimental data of both studied systems were fitted simultaneously to get the model that predicts the local variation

of bubble size distributions the effect of operational conditions of mixing and the physical properties of dispersion to the

bubble size distribution The optimised values of parameters for the fitted multi block model are 67581 =C

and03902 =C 2113 =C 4

4 10695 minussdot=C 09770= K Relative volumetric bubble size distributions

calculated using these parameters are presented in figures (9) and (10) As can be seen the model that has been fitted

simultaneously against the both measured systems is almost equally good as the fittings for the individual systems

presented above For the air-water system the fitted model predicts slightly too strong decrease of bubble size with the

increasing stirring speed (Figure 7 100 dm3min and 500 rpm) The comparison of air-water and CO2-n-butanol results

indicates that the fitted model is capable of predicting the local bubble size distributions with the changing physical

properties

Figure 9 Relative volumetric bubble size distributions for air-water system from PIV experiments (locations of measureddistributions denoted as A-F in legends) and from the fitted model obtained from the simultaneous fitting of air-water and

CO2-n-butanol systems (locations of distributions denoted as subregions 1-14) The predicted distributions have beencalculated from the same fitted model in all subplots

19




measured distributions denoted as A-F in legends) and from the fitted model obtained from the simultaneous fitting of air-water and CO2-n-butanol systems (locations of distributions denoted as subregions 1-14) The predicted distributions

have been calculated from the same fitted model in all subplots

Results from the fitting of turbulent slip model

The parameter K in the turbulent slip model (eq 15) was fitted simultaneously with the bubble rate functions The fitted

value was for the air-water system 124 K 0= 0890= K for the CO2-n-butanol system and for the

simultaneous fitting of studied systems Results from the fitting of air-water system (

09770= K

1240= K ) indicate that U

the ratio of corrected slip velocity and the terminal velocity at stagnant liquid varies in the range 070-026 for the 10-76

mm bubbles near the impeller Similarly for the CO2-n-butanol system

t U

0890= K the ratio U varies in the range

088-046 for the 10-76 mm bubbles near the impeller

t U

Slip velocities were determined also in the PIV experiments and are reported by Miettinen et al 2002 Experimental slip

velocities and calculated slip velocities from equation (17) corrected with the fitted turbulent slip model eq (15) are of the

same order of magnitude but it seems that the slip velocities of bubbles from the PIV experiments are not completely

reasonable physically since slip do not vanish with infinitesimally small bubbles

20



Summary of the fitted parameters

The following bubble rate functions were used in the fitting of population balance equations breakage eq (12)

coalescence eq (13) daughter bubble distribution eq (14) terminal velocity eq (17) and turbulence correction for the

relative bubble velocities eq (15)

The resulting parameter values are presented in Table 4

Table 4 Resulting parameter values from the fitting procedure

Parameter number

Air-water fitting

CO2-n-butanolfitting

Simultaneous fittingof both systems

C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

The bubble rate and slip model parameters were fitted against the experimental data measured with the Particle Image

Velocimetry technique from lean dispersions in the mixed tank Two systems air-water and CO2-n-butanol were

investigated At first bubble rate functions and slip model were fitted for the systems separately After that both systems

and all experiments at various operational conditions were fitted simultaneously to obtain the model that predicts the

effect of physical properties and mixing conditions to the local bubble size distributions in the vessel We suppose that the

fitted model could be applicable for the gas-liquid CFD simulations since it is based on the mechanistic phenomena in

gas-liquid systems In addition the inhomogeneities in the bubble size distributions were considered within the fitting

process using the multi block stirred tank model

It was observed both visually and from the high-speed camera images that the gas-liquid flow had periodic nature near the

impeller Reasons were found to be the fluctuating gas trail or jet from the gas nozzle and the evolution of trailing vortices

of bubbles in the discharge flow of impeller These phenomena have significant effect on the local bubble size

distributions and the gas hold-up at the lower part of the vessel and they should not be neglected It was also noticed that

the decrease of relative velocities between the continuous and the dispersed phase in turbulent conditions is significant

and must be considered

Experiments were carried out at very low gassing rates since the applicability of measurement technique was limited to

lean dispersions In industrial gas-liquid devices dense dispersions are generally met Therefore the utilization of fitted

21



model for real systems requires inevitably extrapolation to dense systems However since the fitted model is based on the

fundamental mechanistic ideas of bubble breakage and coalescence it has more reasonable extrapolation characteristics in

comparison with the traditional correlations of dimensionless numbers It must be noted that the fitted breakage

coalescence and slip functions should not be used independently when bubble size distributions are calculated The

reason is that these phenomena have a very complicated interrelation in the stirred gas-liquid vessel at turbulent

conditions The availability of reliable experimental bubble size distributions is one of the most significant reasons for the

difficulties to develop and fit bubble models

Results indicate that the multi block model along with the population balances is a useful tool for the fitting of bubble size

distributions against experiments Flow pattern is needed for the multi block model It can be determined from the CFD

simulations or from the experiments The comparison of measured and simulated (CFD) flow patterns revealed

differences between the single-phase CFD simulation and the CO2-n-butanol system From this we come to a conclusion

that single-phase CFD is not reliable for the prediction of flow pattern in all cases even for the lean (0-1 vol-)

dispersions Instead multiphase CFD or preferably experiments should be utilized when the multiphase flow pattern is

determined for the multi block model

Acknowledgement

Financial support from the Graduate School of Chemical Engineering (GSCE) and KaNeMa project that is a part of the

MANDI program coordinated by the National Technology Agency of Finland (TEKES) are gratefully acknowledged

Joakim Majander from Fortum Power and Heat Oy and Mikko Manninen and Veikko Taivassalo from VTT Processes are

gratefully acknowledged for carrying out the CFD simulations In addition we would like to thank Markus Honkanen

Pentti Saarenrinne and Hannu Maumlkelauml from Tampere University of Technology Laboratory of Energy and Process

Engineering for carrying out the PIV experiments and Ari Kankkunen from Helsinki University of Technology

Laboratory of Energy Engineering and Environmental Protection for the help and supplying the apparatus for the high-

speed camera imaging

Notation

)( iV index number of bubble class of characteristic volume V i

ij A area between vessel subregions i and j m2

ij A dimensionless horizontal area between vessel subregions i and j

22



a∆ width of bubble size category m

a bubble diameter m

mina minimum detectable bubble size in the experiments m

maxa maximum observed bubble size in the experiments m

32a Sauter mean diameter m23

32 ii aaa ΣΣ=

41C C empirical constants dimensionless

i D impeller diameter m

()erfc complementary error function

)( ji aa F binary coalescence rate between bubbles a and in unit volume m3s-1 i ja

)( ja g breakage frequency of bubble size s-1 ja

)( ji aah collision frequency between bubbles and a in unit volume m3s

-1 ia j

K empirical constant in turbulent slip model dimensionless

N impeller speed s-1

NB number of subregions in the simulation model

NC number of bubble size categories

P N impeller power number dimensionless

Q N impeller pumping number dimensionless

ijQ flow rate between subregions i and j m3s-1

ijQ dimensionless flow rate from subregion i to j

Re Reynolds number C iC it aU ρ Re sdotsdot= dimensionless

t time s

iU slip velocity of bubble class i ms-1

it U terminal velocity of bubble class i in stagnant liquid ms-1

)(i

av volumetric bubble density of bubble class i m3[bubbles]m-3[dispersion]m-1

23



)( irel av relative volumetric bubble density of bubble class i m-1

V total vessel volume m3

iV volume of a subregion i m3

32 aiivi www φ weight functions for the relative volume densities gas hold-ups and Sauter mean diameters in the

fitting dimensionless

iY number concentration of bubble size category i m-3

out iini Y Y flow of bubble class i per unit volume in and out from a subregion s-1m-3

Greek symbols

)( ji aa β probability that a bubble of size is formed when breaks m-1 ia ja

aveε average turbulent energy dissipation per unit mass m2s-3

ε turbulence energy dissipation in a unit volume m2s-3

C D viscosities of dispersed and continuous phase Pas

ν kinematic viscosity m2s

-1

C D ρ ρ densities of dispersed and continuous phase kgm-3

ρ ∆ absolute difference in density between dispersed and continuous phase kgm-3

λ Kolmogoroff length scale m

σ interfacial tension Nm-1

relative dissipation of turbulent energy in subregion i dimensionless

φ gas volume fraction dimensionless

References

Alopaeus V Koskinen J Keskinen K Simulation of the population balances for liquid-liquid systems in a nonideal

stirred tank Part 1 Description and qualitative validation of the model Chem Eng Sci 54 (1999) pp 5887-5899

Alves SS Maia CI Vasconcelos JMT Experimental and modeling study of gas dispersion in a double turbine

stirred tank Chem Eng Sci 57 (2002) pp 487-496

24



Bakker A Hydrodynamics of stirred gas-liquid dispersions PhD Thesis Delft University of Technology The

Netherlands 1992

Bakker A Van Den Akker HEA A computational model for the gas-liquid flow in stirred reactors Trans IchemE

A72 (1994) pp 594-606

Bapat PM Tavlarides LL Smith GW Monte Carlo simulation of mass transfer in liquid-liquid dispersions Chem

Eng Sci 38 (1983) pp 2003-2013

Barigou M Greaves M Bubble size in the impeller region of a Rushton turbine Trans IChemE 70A (1992a) pp 153-

160

Barigou M Greaves M Bubble-size distributions in a mechanically agitated gas-liquid contactor Chem Eng Sci 47

(1992b) pp 2009-2025

Brucato A Grisafi F Montante G Particle drag coefficients in turbulent fluids Chem Eng Sci 53(18) (1998) pp

3295-3314

Chaouki J Larachi F Dudukovic MP Noninvasive and velocimetric monitoring of multiphase flows Ind Eng

Chem Res 36 (1997) pp 4476-4503

Clift R Grace JR Weber ME Bubbles Drops and Particles Academic Press New York 1978 380 p

Coulaloglou CA Tavlarides LL Description of Interaction Processes in Agitated Liquid-Liquid Dispersions Chem

Eng Sci 32 (1977) pp 1289-1297

Deen NG Westerweel J Delnoij E Two-phase PIV in bubbly flows Status and trends Chem Eng Technol

25(2002) pp 97-101

Derksen JJ Venneker BCH Van Den Akker HEA Population balance modeling of aerated stirred vessels based on

CFD AIChE J 48(4) (2002) pp 673-685

Geary NW Rice RG Bubble size prediction for rigid and flexible spargers AIChE J 37(2) (1991) pp 161-168

Gosman AD Lekakou C Politis S Issa RI Looney MK Multidimensional modeling of turbulent two-phase

flows in stirred vessels AIChE J 38(2) (1992) pp 1946-1956

25



Hinze JO Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes AIChE J 1(3) (1955) pp

289-295

Honkanen M Saarenrinne P Turbulent bubbly flow measurements in a mixing vessel with PIV 11th Int Symposium on

Applications of Laser Techniques to Fluid Mechanics Lisboa 2002 Paper 32

Laakkonen M Honkanen M Saarenrinne P Aittamaa J Determination of local gas-liquid interfacial area and gas

hold-up profiles for air-water and CO2-n-butanol systems in a stirred tank from Particle Image Velocimetry results To be

published

Lane GL Schwarz MP Evans GM Modeling of the interaction between gas and liquid in stirred vessels 10th

European Conference on Mixing Delft The Netherlands 2000 pp 197-204

Lasheras JC Eastwood C Martiacutenez-Bazaacuten C Montantildeeacutes JL A review of statistical models for the break-up of an

immiscible fluid immersed into a fully developed turbulent flow Int J Multiphase Flow 28 (2002) pp 247-278

Litster JD Smit DJ Hounslow MJ Adjustable discretized population balance for growth and aggregation AIChE J

41(3) (1995) pp 591-603

Luo H Svendsen HF Theoretical model for drop and bubble breakup in turbulent dispersions AIChE J 42(5) (1996)

pp 1225-1233

Machon V Pacek AW Nienow AW Bubble sizes in electrolyte and alcohol solutions in a turbulent stirred vessel

Trans IChemE 75A (1997) pp 339-348

Magelli F Fajner D Noncentini M Pasquali G Solid distribution in vessels stirred with multiple impellers Chem

Eng Sci 45(3) (1990) pp 615-625

Martiacutenez-Bazaacuten C Montantildeeacutes JL Lasheras JC On the breakup of an air bubble injected into a fully developed

turbulent flow Part 1 Breakup frequency J Fluid Mech 401 (1999) pp 157-182

Mavros P Flow visualization in stirred vessels ndash A review of experimental techniques Trans IChemE 79A (2001) pp

113-127

Maxey MR Chang EJ Wang L-P Simulation of interactions between microbubbles and turbulent flows Appl

Mech Rev 47 (1994) pp S70-S74

26



Miettinen T Laakkonen M Aittamaa J The applicability of various flow visualization techniques for the

characterization of gas-liquid flow in a mixed tank To be presented in AIChE Annual Meeting 2002 Indianapolis Nov

3-11

Morud KE Hjertager BH LDA measurements and CFD modeling of gas-liquid flow in a stirred vessel Chem Eng

Sci 51(2) (1996) pp 233-249

Narsimhan G Gupta JP Ramkrishna D A model for transitional breakage probability of droplets in agitated lean

liquid-liquid dispersions Chem Eng Sci 34 (1979) pp 257-265

Prince MJ Blanch HW Bubble Coalescence and Break-up in Air-Sparged Bubble Columns AIChE J 36(10) 1990

pp 1485-1499

Schaumlfer M Waumlchter P Durst F Experimental investigation of local bubble size distributions in stirred vessels using

Phase Dobbler Anemometry 10th European Conference on Mixing 2000 pp 205-212

Spelt PDM Biesheuvel A On the motion of gas bubbles in homogenous isotropic turbulence J Fluid Mech 336

(1997) pp 221-244

Takahashi K McManamey WJ Nienow AW Bubble size distributions in impeller region in a gas-sparged vessel

agitated by a Rushton turbine J Chem Eng Jpn 25(4) (1992) pp 427-432

Tatterson GB Fluid mixing and gas dispersion in agitated tanks McGraw-Hill New York 1991 548 p

Tsouris C Tavlarides LL Breakage and Coalescence Models for Drops in Turbulent Dispersions AIChE J 40(3)

(1994) pp 395-406

Tsuke H Hydrodynamics of bubble formation from submerged orifices in N P Cheremisinoff (Ed) Encyclopedia of

Fluid Mechanics vol 3 (p 191) Houston 1986 Gulf Publishing Company

Wang CY Calabrese RV Drop breakup in turbulent stirred-tank contactors Part II Relative influence of viscosity

and interfacial tension AIChE J 32 (1986) pp 667-676



Abstract

Local time-averaged bubble size distributions (BSD) have been measured for air-water and CO2-n-butanol dispersions in

a baffled 138 dm3

mixed tank by Particle Image Velocimetry (PIV) technique BSD measurements were performed at six

locations in the tank at various gas feeds and mixing speeds The fundamental bubble breakage and coalescence

phenomena have been studied using the measured BSDs and a multi block stirred tank model which consists of a limited

amount of ideally mixed connected units in the tank

The multi block stirred tank model including the population balance equations for bubbles has been created It has been

used to fit the parameters of bubble breakage and coalescence functions because it is simple enough for the fitting but

capable in predicting inhomogeneities occurring in the stirred tank Computational fluid dynamics (CFD) has been used to

determine the flow patterns and local dissipations of turbulent energy in the mixed tank The flow pattern is needed to

estimate the flow rates between the blocks PIV technique has been applied for the verification of the flow pattern Local

dissipations of turbulent energy are required for the bubble breakage and the coalescence models The applicability of

various breakage and coalescence models found from the literature has been tested

The BSDs at the gas inlet are important for the fitting process They have been estimated from the PIV bubble size

distributions and from the pictures of high-speed camera The arrangement of the blocks in the tank and the selection of

the bubble velocity model which describes the relative velocities between the bubbles and the liquid are other important

issues in the fitting

The fitted bubble population balance model predicts the dependence of the bubble populations on the local turbulence

energy dissipation If local dissipation of turbulent energy is known all over the vessel or predicted by CFD then local

bubble sizes and mass transfer areas can be estimated and used to scale up the mixed tanks

1



Introduction




















Derksen et al 2002)








2















tank model

Population balances







1999)

( ) ( ) ( )( )( )

( ) ( )( )

1

2

1

3133

1

j

V V

j

ji jiiout i

ji

V

j

j ji j

NC

i j

j jiinii

Y aa F Y Y a g Y


dY

i NC

i

sum

sumsumminus

=

=+=

minusminusminus

minus+∆+= β

(1)




3






Stirred tank model

























4







aveε ε ϕ = (3)



so that

1=

sum

NB

iiV (4)

and

(5)1=sum i

NB

i

iV ϕ



42

i

ij

ij D A A

π sdot= (6)

5







subregioniV iϕ

1 0113581 0025158

2 0201753 0015869

3 0166078 009479

4 0093496 0175606

5 0063023 0515762

6 0068275 0264278

7 0007222 2121584

8 0100999 15867569 0086571 0821006

10 0028857 0683946

11 0009096 4904083

12 0014878 0931224

13 0012505 1491787

14 0033666 7780494



A 03600 06400 02500 03900 03600 03600 03900


A 01109 07500 01391 01109 01391 02500 02500


3




sumsum==

++= NB

k j

k ik ikj NB

k j

k ikj

feed jiin jiV

Y U A

V

Y QY Y

1

1

(8)

and

1

1

sumsum == ++=

NB

k j

ji ji jk NB

k j

ji jk

prod jiout ji V

Y U A

V

Y Q

Y Y (9)

6






ji

ji


from to 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 0396

2 0396

3 0124 0815

4 0396 0543

5 0939 0149

6 0059 0102 0158

7 0810

8 1088 0004

9 0004

10 0004

11 0123 0265

12 0687 0286

13 0046 1029

14 1033

Experimental














7








Laakkonen et al







3 =)( 2CO D ρ 183 kgm

3and

















q

ii aa 133

1 2 =+

8






NC mina

maxa

ia

ia∆

1a

a

+ min

a

a

min1

minus

=

2

1

max

min

min

max aa

NC NC

i

i

= (10)

1

max

min

min

max aa

a

a

aa

NC NC

i

i

∆ (11)











9





















10

























11



( ) 3431335322

31

1

+=

j DC

C

jC

ja

C a


micro

ε ρ

σ ε (12)


( ) ( ) ( ) 312132322







( )2

3

32

3

3

3

2

190

minus

=

j

i

j

i

j

i ji

a

a

a

a

a

aaa β (14)















t U

t t U

12
















1

21

minus

+=

λ

i

it

i a K

U

U (15)


the equation (16)

413

=

ε

ν λ (16)


(1978 s 114 Table 53)






(17)


3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13








Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406







Introduction




















Derksen et al 2002)








2















tank model

Population balances







1999)

( ) ( ) ( )( )( )

( ) ( )( )

1

2

1

3133

1

j

V V

j

ji jiiout i

ji

V

j

j ji j

NC

i j

j jiinii

Y aa F Y Y a g Y


dY

i NC

i

sum

sumsumminus

=

=+=

minusminusminus

minus+∆+= β

(1)




3






Stirred tank model

























4







aveε ε ϕ = (3)



so that

1=

sum

NB

iiV (4)

and

(5)1=sum i

NB

i

iV ϕ



42

i

ij

ij D A A

π sdot= (6)

5







subregioniV iϕ

1 0113581 0025158

2 0201753 0015869

3 0166078 009479

4 0093496 0175606

5 0063023 0515762

6 0068275 0264278

7 0007222 2121584

8 0100999 15867569 0086571 0821006

10 0028857 0683946

11 0009096 4904083

12 0014878 0931224

13 0012505 1491787

14 0033666 7780494



A 03600 06400 02500 03900 03600 03600 03900


A 01109 07500 01391 01109 01391 02500 02500


3




sumsum==

++= NB

k j

k ik ikj NB

k j

k ikj

feed jiin jiV

Y U A

V

Y QY Y

1

1

(8)

and

1

1

sumsum == ++=

NB

k j

ji ji jk NB

k j

ji jk

prod jiout ji V

Y U A

V

Y Q

Y Y (9)

6






ji

ji


from to 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 0396

2 0396

3 0124 0815

4 0396 0543

5 0939 0149

6 0059 0102 0158

7 0810

8 1088 0004

9 0004

10 0004

11 0123 0265

12 0687 0286

13 0046 1029

14 1033

Experimental














7








Laakkonen et al







3 =)( 2CO D ρ 183 kgm

3and

















q

ii aa 133

1 2 =+

8






NC mina

maxa

ia

ia∆

1a

a

+ min

a

a

min1

minus

=

2

1

max

min

min

max aa

NC NC

i

i

= (10)

1

max

min

min

max aa

a

a

aa

NC NC

i

i

∆ (11)











9





















10

























11



( ) 3431335322

31

1

+=

j DC

C

jC

ja

C a


micro

ε ρ

σ ε (12)


( ) ( ) ( ) 312132322







( )2

3

32

3

3

3

2

190

minus

=

j

i

j

i

j

i ji

a

a

a

a

a

aaa β (14)















t U

t t U

12
















1

21

minus

+=

λ

i

it

i a K

U

U (15)


the equation (16)

413

=

ε

ν λ (16)


(1978 s 114 Table 53)






(17)


3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13








Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406



















tank model

Population balances







1999)

( ) ( ) ( )( )( )

( ) ( )( )

1

2

1

3133

1

j

V V

j

ji jiiout i

ji

V

j

j ji j

NC

i j

j jiinii

Y aa F Y Y a g Y


dY

i NC

i

sum

sumsumminus

=

=+=

minusminusminus

minus+∆+= β

(1)




3






Stirred tank model

























4







aveε ε ϕ = (3)



so that

1=

sum

NB

iiV (4)

and

(5)1=sum i

NB

i

iV ϕ



42

i

ij

ij D A A

π sdot= (6)

5







subregioniV iϕ

1 0113581 0025158

2 0201753 0015869

3 0166078 009479

4 0093496 0175606

5 0063023 0515762

6 0068275 0264278

7 0007222 2121584

8 0100999 15867569 0086571 0821006

10 0028857 0683946

11 0009096 4904083

12 0014878 0931224

13 0012505 1491787

14 0033666 7780494



A 03600 06400 02500 03900 03600 03600 03900


A 01109 07500 01391 01109 01391 02500 02500


3




sumsum==

++= NB

k j

k ik ikj NB

k j

k ikj

feed jiin jiV

Y U A

V

Y QY Y

1

1

(8)

and

1

1

sumsum == ++=

NB

k j

ji ji jk NB

k j

ji jk

prod jiout ji V

Y U A

V

Y Q

Y Y (9)

6






ji

ji


from to 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 0396

2 0396

3 0124 0815

4 0396 0543

5 0939 0149

6 0059 0102 0158

7 0810

8 1088 0004

9 0004

10 0004

11 0123 0265

12 0687 0286

13 0046 1029

14 1033

Experimental














7








Laakkonen et al







3 =)( 2CO D ρ 183 kgm

3and

















q

ii aa 133

1 2 =+

8






NC mina

maxa

ia

ia∆

1a

a

+ min

a

a

min1

minus

=

2

1

max

min

min

max aa

NC NC

i

i

= (10)

1

max

min

min

max aa

a

a

aa

NC NC

i

i

∆ (11)











9





















10

























11



( ) 3431335322

31

1

+=

j DC

C

jC

ja

C a


micro

ε ρ

σ ε (12)


( ) ( ) ( ) 312132322







( )2

3

32

3

3

3

2

190

minus

=

j

i

j

i

j

i ji

a

a

a

a

a

aaa β (14)















t U

t t U

12
















1

21

minus

+=

λ

i

it

i a K

U

U (15)


the equation (16)

413

=

ε

ν λ (16)


(1978 s 114 Table 53)






(17)


3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13








Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406










Stirred tank model

























4







aveε ε ϕ = (3)



so that

1=

sum

NB

iiV (4)

and

(5)1=sum i

NB

i

iV ϕ



42

i

ij

ij D A A

π sdot= (6)

5







subregioniV iϕ

1 0113581 0025158

2 0201753 0015869

3 0166078 009479

4 0093496 0175606

5 0063023 0515762

6 0068275 0264278

7 0007222 2121584

8 0100999 15867569 0086571 0821006

10 0028857 0683946

11 0009096 4904083

12 0014878 0931224

13 0012505 1491787

14 0033666 7780494



A 03600 06400 02500 03900 03600 03600 03900


A 01109 07500 01391 01109 01391 02500 02500


3




sumsum==

++= NB

k j

k ik ikj NB

k j

k ikj

feed jiin jiV

Y U A

V

Y QY Y

1

1

(8)

and

1

1

sumsum == ++=

NB

k j

ji ji jk NB

k j

ji jk

prod jiout ji V

Y U A

V

Y Q

Y Y (9)

6






ji

ji


from to 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 0396

2 0396

3 0124 0815

4 0396 0543

5 0939 0149

6 0059 0102 0158

7 0810

8 1088 0004

9 0004

10 0004

11 0123 0265

12 0687 0286

13 0046 1029

14 1033

Experimental














7








Laakkonen et al







3 =)( 2CO D ρ 183 kgm

3and

















q

ii aa 133

1 2 =+

8






NC mina

maxa

ia

ia∆

1a

a

+ min

a

a

min1

minus

=

2

1

max

min

min

max aa

NC NC

i

i

= (10)

1

max

min

min

max aa

a

a

aa

NC NC

i

i

∆ (11)











9





















10

























11



( ) 3431335322

31

1

+=

j DC

C

jC

ja

C a


micro

ε ρ

σ ε (12)


( ) ( ) ( ) 312132322







( )2

3

32

3

3

3

2

190

minus

=

j

i

j

i

j

i ji

a

a

a

a

a

aaa β (14)















t U

t t U

12
















1

21

minus

+=

λ

i

it

i a K

U

U (15)


the equation (16)

413

=

ε

ν λ (16)


(1978 s 114 Table 53)






(17)


3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13








Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406











aveε ε ϕ = (3)



so that

1=

sum

NB

iiV (4)

and

(5)1=sum i

NB

i

iV ϕ



42

i

ij

ij D A A

π sdot= (6)

5







subregioniV iϕ

1 0113581 0025158

2 0201753 0015869

3 0166078 009479

4 0093496 0175606

5 0063023 0515762

6 0068275 0264278

7 0007222 2121584

8 0100999 15867569 0086571 0821006

10 0028857 0683946

11 0009096 4904083

12 0014878 0931224

13 0012505 1491787

14 0033666 7780494



A 03600 06400 02500 03900 03600 03600 03900


A 01109 07500 01391 01109 01391 02500 02500


3




sumsum==

++= NB

k j

k ik ikj NB

k j

k ikj

feed jiin jiV

Y U A

V

Y QY Y

1

1

(8)

and

1

1

sumsum == ++=

NB

k j

ji ji jk NB

k j

ji jk

prod jiout ji V

Y U A

V

Y Q

Y Y (9)

6






ji

ji


from to 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 0396

2 0396

3 0124 0815

4 0396 0543

5 0939 0149

6 0059 0102 0158

7 0810

8 1088 0004

9 0004

10 0004

11 0123 0265

12 0687 0286

13 0046 1029

14 1033

Experimental














7








Laakkonen et al







3 =)( 2CO D ρ 183 kgm

3and

















q

ii aa 133

1 2 =+

8






NC mina

maxa

ia

ia∆

1a

a

+ min

a

a

min1

minus

=

2

1

max

min

min

max aa

NC NC

i

i

= (10)

1

max

min

min

max aa

a

a

aa

NC NC

i

i

∆ (11)











9





















10

























11



( ) 3431335322

31

1

+=

j DC

C

jC

ja

C a


micro

ε ρ

σ ε (12)


( ) ( ) ( ) 312132322







( )2

3

32

3

3

3

2

190

minus

=

j

i

j

i

j

i ji

a

a

a

a

a

aaa β (14)















t U

t t U

12
















1

21

minus

+=

λ

i

it

i a K

U

U (15)


the equation (16)

413

=

ε

ν λ (16)


(1978 s 114 Table 53)






(17)


3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13








Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406











subregioniV iϕ

1 0113581 0025158

2 0201753 0015869

3 0166078 009479

4 0093496 0175606

5 0063023 0515762

6 0068275 0264278

7 0007222 2121584

8 0100999 15867569 0086571 0821006

10 0028857 0683946

11 0009096 4904083

12 0014878 0931224

13 0012505 1491787

14 0033666 7780494



A 03600 06400 02500 03900 03600 03600 03900


A 01109 07500 01391 01109 01391 02500 02500


3




sumsum==

++= NB

k j

k ik ikj NB

k j

k ikj

feed jiin jiV

Y U A

V

Y QY Y

1

1

(8)

and

1

1

sumsum == ++=

NB

k j

ji ji jk NB

k j

ji jk

prod jiout ji V

Y U A

V

Y Q

Y Y (9)

6






ji

ji


from to 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 0396

2 0396

3 0124 0815

4 0396 0543

5 0939 0149

6 0059 0102 0158

7 0810

8 1088 0004

9 0004

10 0004

11 0123 0265

12 0687 0286

13 0046 1029

14 1033

Experimental














7








Laakkonen et al







3 =)( 2CO D ρ 183 kgm

3and

















q

ii aa 133

1 2 =+

8






NC mina

maxa

ia

ia∆

1a

a

+ min

a

a

min1

minus

=

2

1

max

min

min

max aa

NC NC

i

i

= (10)

1

max

min

min

max aa

a

a

aa

NC NC

i

i

∆ (11)











9





















10

























11



( ) 3431335322

31

1

+=

j DC

C

jC

ja

C a


micro

ε ρ

σ ε (12)


( ) ( ) ( ) 312132322







( )2

3

32

3

3

3

2

190

minus

=

j

i

j

i

j

i ji

a

a

a

a

a

aaa β (14)















t U

t t U

12
















1

21

minus

+=

λ

i

it

i a K

U

U (15)


the equation (16)

413

=

ε

ν λ (16)


(1978 s 114 Table 53)






(17)


3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13








Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406










ji

ji


from to 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 0396

2 0396

3 0124 0815

4 0396 0543

5 0939 0149

6 0059 0102 0158

7 0810

8 1088 0004

9 0004

10 0004

11 0123 0265

12 0687 0286

13 0046 1029

14 1033

Experimental














7








Laakkonen et al







3 =)( 2CO D ρ 183 kgm

3and

















q

ii aa 133

1 2 =+

8






NC mina

maxa

ia

ia∆

1a

a

+ min

a

a

min1

minus

=

2

1

max

min

min

max aa

NC NC

i

i

= (10)

1

max

min

min

max aa

a

a

aa

NC NC

i

i

∆ (11)











9





















10

























11



( ) 3431335322

31

1

+=

j DC

C

jC

ja

C a


micro

ε ρ

σ ε (12)


( ) ( ) ( ) 312132322







( )2

3

32

3

3

3

2

190

minus

=

j

i

j

i

j

i ji

a

a

a

a

a

aaa β (14)















t U

t t U

12
















1

21

minus

+=

λ

i

it

i a K

U

U (15)


the equation (16)

413

=

ε

ν λ (16)


(1978 s 114 Table 53)






(17)


3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13








Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406












Laakkonen et al







3 =)( 2CO D ρ 183 kgm

3and

















q

ii aa 133

1 2 =+

8






NC mina

maxa

ia

ia∆

1a

a

+ min

a

a

min1

minus

=

2

1

max

min

min

max aa

NC NC

i

i

= (10)

1

max

min

min

max aa

a

a

aa

NC NC

i

i

∆ (11)











9





















10

























11



( ) 3431335322

31

1

+=

j DC

C

jC

ja

C a


micro

ε ρ

σ ε (12)


( ) ( ) ( ) 312132322







( )2

3

32

3

3

3

2

190

minus

=

j

i

j

i

j

i ji

a

a

a

a

a

aaa β (14)















t U

t t U

12
















1

21

minus

+=

λ

i

it

i a K

U

U (15)


the equation (16)

413

=

ε

ν λ (16)


(1978 s 114 Table 53)






(17)


3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13








Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406










NC mina

maxa

ia

ia∆

1a

a

+ min

a

a

min1

minus

=

2

1

max

min

min

max aa

NC NC

i

i

= (10)

1

max

min

min

max aa

a

a

aa

NC NC

i

i

∆ (11)











9





















10

























11



( ) 3431335322

31

1

+=

j DC

C

jC

ja

C a


micro

ε ρ

σ ε (12)


( ) ( ) ( ) 312132322







( )2

3

32

3

3

3

2

190

minus

=

j

i

j

i

j

i ji

a

a

a

a

a

aaa β (14)















t U

t t U

12
















1

21

minus

+=

λ

i

it

i a K

U

U (15)


the equation (16)

413

=

ε

ν λ (16)


(1978 s 114 Table 53)






(17)


3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13








Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406

























10

























11



( ) 3431335322

31

1

+=

j DC

C

jC

ja

C a


micro

ε ρ

σ ε (12)


( ) ( ) ( ) 312132322







( )2

3

32

3

3

3

2

190

minus

=

j

i

j

i

j

i ji

a

a

a

a

a

aaa β (14)















t U

t t U

12
















1

21

minus

+=

λ

i

it

i a K

U

U (15)


the equation (16)

413

=

ε

ν λ (16)


(1978 s 114 Table 53)






(17)


3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13








Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406





























11



( ) 3431335322

31

1

+=

j DC

C

jC

ja

C a


micro

ε ρ

σ ε (12)


( ) ( ) ( ) 312132322







( )2

3

32

3

3

3

2

190

minus

=

j

i

j

i

j

i ji

a

a

a

a

a

aaa β (14)















t U

t t U

12
















1

21

minus

+=

λ

i

it

i a K

U

U (15)


the equation (16)

413

=

ε

ν λ (16)


(1978 s 114 Table 53)






(17)


3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13








Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406







( ) 3431335322

31

1

+=

j DC

C

jC

ja

C a


micro

ε ρ

σ ε (12)


( ) ( ) ( ) 312132322







( )2

3

32

3

3

3

2

190

minus

=

j

i

j

i

j

i ji

a

a

a

a

a

aaa β (14)















t U

t t U

12
















1

21

minus

+=

λ

i

it

i a K

U

U (15)


the equation (16)

413

=

ε

ν λ (16)


(1978 s 114 Table 53)






(17)


3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13








Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406




















1

21

minus

+=

λ

i

it

i a K

U

U (15)


the equation (16)

413

=

ε

ν λ (16)


(1978 s 114 Table 53)






(17)


3

3

4

C

C D

a N

micro

ρ ρ sdot∆=

13








Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406












Parameter fitting













)(

)()(

1 32

3232

1

1

32sum

sum

sum

=

=

=

minussdot

+minus

sdot

+minus

sdot=

NC

i MEAS

MODEL MEAS

ai

NC

i MEAS

i

MODEL

i

MEAS

ii

NC

i i

MEAS

rel

i

MODEL

rel i

MEAS

rel i

a

aaw

w

av

avavw RES

φ

φ φ φ

ν

(18)


)(

)()(

1

sum=

∆

= NC

j

j j

iirel

aav

avav (19)

14



























4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406































4






15








16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406












16













better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406

















better fittings

4

4 10628 minussdot=C

0890= K




impeller

17







butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406











butanol system

18







and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406











and03902 =C 2113 =C 4







properties



19










09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406














09770= K




t U



t U





20









Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406













Parameter number

Air-water fitting



C1 8658 4362 5867

C2 00267 00482 0039

C3 380 098 121C4 343 middot 10

-4862 middot 10

-4569 middot 10

-4

K 0124 0089 00977

Conclusions

















21

















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406





















Acknowledgement









Notation




22




a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406








a bubble diameter m




32 ii aaa ΣΣ=







-1 ia j










t time s



)(i


23










Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406














Greek symbols






-1







References





24




Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406








Netherlands 1992


A72 (1994) pp 594-606


Eng Sci 38 (1983) pp 2003-2013


160


(1992b) pp 2009-2025


3295-3314


Chem Res 36 (1997) pp 4476-4503



Eng Sci 32 (1977) pp 1289-1297


25(2002) pp 97-101


CFD AIChE J 48(4) (2002) pp 673-685




25




289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406








289-295





published






41(3) (1995) pp 591-603


pp 1225-1233




Eng Sci 45(3) (1990) pp 615-625




113-127


Mech Rev 47 (1994) pp S70-S74

26





3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406









3-11


Sci 51(2) (1996) pp 233-249




pp 1485-1499




(1997) pp 221-244





(1994) pp 395-406





parameter for bubble breakage and coalescence

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