Kinetics of drop breakup during

emulsification in turbulent flow

N. Vankova, S. Tcholakova,

N. Denkov, I. B. Ivanov, and T. Danner*

Faculty of Chemistry, Sofia University,Sofia, Bulgaria,

and *2BASF Aktiengesellschaft, Ludwigshafen, Germany

Aims

2. Formulation of kinetic scheme, which accounts for:

• Generation of drops with given size from larger ones

• Their breakage into smaller drops

1. To elucidate the effects of drop size, oil viscosity, and hydrodynamic conditions on drop breakage process:

• Breakage rate constant

• “Daughter” drop size distribution

3. Analysis of the process of drop breakage (comparison of the experimental results with theoretical models)

• Drop-eddy collision frequency

• Breakage efficiency

Materials

Aqueous phase - 1 wt % Brij 58 + 150 mM NaCl

Oil phases:

Soybean oil (SBO): OW = 7. 4 mN/m; D = 50 mPa.s

Silicone oil: OW = 10.5 mN/m; D = 50 mPa.s

Silicone oil: OW = 10.4 mN/m; D = 100 mPa.s

Experimental method

Narrow-gap homogenizer

Hydrodynamic conditions

Oil phase p105,

Pa

Q10-3,

m3/s

108,

J/(m3.s)

10-3,

s

SBO 0.98 0.086 0.49 2.0

SBO 2.20 0.15 1.92 1.1

Silicone oil 1.00 0.092 0.53 1.9

Silicone oil 1.01 0.096 0.56 1.8

Initial premix

Final emulsion

Prepared by membrane emulsification

Mean drop size vs number of passesEffect of oil viscosity

Number of passes, u0 10 20 30 40 50 60 70 80 90 100

d3

2,

m

0

5

10

15

20

25

30

35

401 wt % Brij 58 + 150 mM NaCl + silicone oil

D = 50 mPa.s

D = 96 mPa.s

dK = 12.8 m

dK = 13.6 m

The mean drop size decreases more rapidlyfor oils with lower viscosity

Mean drop size vs number of passesEffect of hydrodynamic conditions

Number of passes, u

0 10 20 30 40 50 60 70 80 90 100

d 32,

m

0

5

10

15

20

25

30

35

401 wt % Brij 58 (150 mM NaCl) + SBO

p = 0.98*105 Pa= 0.49*108 kg/(m.s3)

p = 2.2*105 Pa= 1.92*108 kg/(m.s3)

dK = 11.2 m

dK = 6.4 m

The mean drop size decreases more rapidlywhen the emulsification is performed at higher applied pressure.

Formulation of a kinetic scheme for data analysis

System under consideration

Discrete set of sizes in the system

v0 - volume of the smallest drops; vN - volume of the largest drops

vS = 2Sv0

•Drops with vN - only break into smaller drops

•Drops with vK < v < vN - break and form from larger drops

•Drops with v vK - only form from larger drops

Product of drop breakage

ps,m - fraction of volume of the “mother” drop, ds, which is transformed into

daughter drops with diameter dm

2s-mps,m - average number of drops with diameter dm, formed after breakage of

drop with diameter dS1

,0

1q

S qq

p

Mass balance

Kinetic Scheme

For largest drops, steady-state

For drops having volume vN-1 = vN/2

1N

N N

dn xV k n x

dx 0

1expN N Nx L

n n k 0 expN N Nn u n uk

11 1 1 , 12N

N N N N N N

dn xV k n x p k n x

dx

0 0

, 1 , 101 1 1

1 1

2 2exp expN N N N N N N N

N N N NN N N N

p k n p k nn u n uk uk

k k k k

V1 - linear velocity of the fluid

Processing element

inlet outlet

0 xL1

After 1 pass After u passes

1 ,1

2N

S q SS S q S q q

q S

dn xV k n x p k n x

dx

n x

10-1

0 , m-3

0 10 20 30 40 500

200

400

600

800

1000

1200

Drop diameter, m

Pass 1001 wt % Brij 58 + Silicone oil

u = 100

Experimental Results

Drop diameter, m0 10 20 30 40 50

0

20

40

60

80

100Pass 2

1 wt % Brij 58 + Silicone oil

n x

10-1

0 , m-3

u = 2

Premix1 wt % Brij 58 + Silicone oil

Drop diameter, m0 10 20 30 40 50

n x

10-1

0 , m-3

0

5

10

15

20

25

u = 0nN-1(0)

nN-1(u = 2)

nN-1(u = 100)

Experimental results - interpretation with binary breakage

Number of passes, u0 10 20 30 40 50

nN

-1x1

0-10 , m

-3

0

5

10

15

20

2531.45 m < d < 39.63m

dN-1 = 35.54 m

Theoretical fitkN-1 = 144 s-1

Number of passes, u

0 10 20 30 40 50

n N-2

x 10-1

0 , m

-3

0

10

20

30

40

25 m < d < 31.4 m

dN-2 = 28.2 m

kN-2 = 24 s-1

kN-2 = 50 s-1 kN-2 = 75 s-1

(C)

+dS dS-1 dS-1kS

Drop breakage is not a binary breakage

Experimental Results - interpretation with equal number probability for drop formation

Number of passes, u0 20 40 60 80 100

nN

-3x1

0-10 , m

-3

0

10

20

30

40

5019.8 m < d < 25 m

dN-3 = 22.4 m

kN-3 = 8 s-1

Theoretical fit

,

11 1

2S q S qp for q s

,0 1

1

2S Sp

Number of passes, u

0 20 40 60 80 100

n N-5

x 10-1

0 , m-3

0

20

40

60

80

100

120

140

12.5 m < d < 15.7 m

dN-5 = dK = 14.1 m

(D)

Kolmogorov-sized drops

+dS dS-1 dS-2

d0

+ +kS

Experimental results - interpretation by assuming self-similarity

,S S qp •The value of depends only from s-q

dM3/dS

3

2 4 6 8 10 12 14 16

2m-s

p M,S

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Very good agreement of the fits with all experimental data

dM3/dS

3

2 4 6 8 10 12 14 16

p M,S

0.0

0.1

0.2

0.3

0.4

0.5

Breakage rate constant vs drop diameter

Drop diameter, m

10 15 20 25 30 35 40 45 50

k BR, s

ec-1

0

50

100

150

200

250

300

350Silicone oilDmPa.s

dK = 13.6 m

The breakage rate constant rapidly decreases with the decrease of drop diameter and becomes virtually zero at d dK

Breakage rate constant vs drop diameter

Effect of oil viscosity

The breakage rate constant decreases more than 3 times when the oil viscosity increases 2 times.

Drop diameter, m

10 15 20 25 30 35 40 45 50

Bre

aka

ge

ra

te c

on

sta

nt,

se

c-1

0

50

100

150

200

250

300

350

Silicone oilmPa.s

SBOmPa.s

Silicone oilmPa.s

Breakage rate constant vs drop diameter

Effect of hydrodynamic conditions

Drop diameter, m5 10 15 20 25 30 35 40

k BR, s

ec-1

1

10

100

1000

10000

= 2x108 J/m3.s

= 0.5x108 J/m3.s

SBO + 1 wt % Brij 58

The breakage rate constant increases more than 40 times with a 4-fold increase of the rate of energy dissipation.

Interpretation of kBR - model byCoulaloglou and Tavlarides, 1977

1 1expBR

BREAKAGE

Efraction ofk drops breakingbreakage time E

Breakage efficiency

2 2 2 3 2 3~ ( ) ~TE d P d d d 2

2~ ~ OWCAP

dE d P

d

2 3 5 3exp ~ exp OWE

dE

Breakage rate constant

Unviscid drops, ReDR > 1

Viscid drops, ReDR < 1

1 3

2 3 2 3 5 3exp OW

BRk d

d d

2 3 2 3

2 3 5 3expC OW

BR

D

dk d

d

Re DR DDR

D

U d

and

Comparison of the experimental data with the expression for visccous drops

2 3 2 3

0 1 2 3 5 3expC OW

BR

D

dk d A A

d

(dk/d)5/3

0.0 0.2 0.4 0.6 0.8 1.0

ln(k

BR D

/( C

d2/

3 2/

3 ))

-19

-18

-17

-16

-15

-14

-13

-12 Silicone oil, D = 50 mPa.s, p = 1 atm

Silicone oil, D = 96 mPa.s, p = 1 atm

SBO, D = 50 mPa.s, p = 2 atm

SBO, D = 50 mPa.s, p = 1 atm

5 3

0 12 3 2 3ln BR D K

C

k d dA A

d d

This model does not describe the dependence of kBR on oil viscosityand on the rate of energy dissipation

Model for kBR byPrince and Blanch, 1990

BRk eddy drop collision frequency breakage efficiency

Eddy-drop collision frequency – similar to kinetic theory of gases

1 22 2

e

eq e d d e

n

h d S u u n dn Breakage efficiency

2 3 5 3exp ~ exp OWE

dE

Breakage efficiency including the energy dissipation inside the drop (following the idea of Calabrese, 1986):

1 3 1 3

2 3 5 3

9.3 0.166

exp ~ exp

COW D

DDISS

C

dE E

dE

Comparison of the experimental data with the theoretical expression

5 3 4 32 3

1 28 3 2 30 11 3

ln ln 0.166BR K KD K C D

k d d d dA A d

d d

(dk/d)5/3-f(D, )x(dk/d)4/3

0.0 0.2 0.4 0.6 0.8 1.0 1.2

ln(k

BRd2/

3 /1/

3 )

-1

0

1

2

3

4

5

6 SBO, D = 50 mPa.s, p = 1 atm

SBO, D = 50 mPa.s, p = 2 atm

Silicone oil, D = 96 mPa.s, p = 1 atm

Silicone oil, D = 50 mPa.s, p = 1 atm

Comparison of the experimental data with the theoretical expression

Drop diameter, m0 10 20 30 40 50

Bre

akag

e ra

te c

onst

ant,

sec

-1

0

50

100

150

200

250

300 D, mPa.s

OW

mN/m

105

J/(kg.s)A0 A1

50SBO

7.4 0.49 0.08 0.03 1.2 0.2

50SBO

7.4 1.92 0.1 0.02 1 0.09

50silicone oil

10.9 0.53 0.1 0.01 1.0 0.05

96silicone oil

10.3 0.56 0.1 0.05 1.1 0.02

Average 0.1 0.05 1.1 0.1

All experimental data are described reasonably well under the assumption that the breakage rate is determined by:

•Drop-eddy collision frequency

•Breakage efficiency, accounting for (i) dissipated energy inside the drops and (ii) surface expansion energy.

Main Results

Experimental results for the mean drop size

• The mean drop size decreases much faster for emulsions prepared at

larger .

• The increased viscosity of the dispersed phase leads to much slower decrease of mean drop size.

Formulation of kinetic scheme for drop breakage

• The breakage process is considered as an irreversible reaction of first order

• A discrete set of drop sizes is considered

• The drop generation and drop breakage are taken into account

• The processing element is considered as a reactor with ideal displacement

• The formulated kinetic scheme allow us to determine KBR(d) and the probability for formation of smaller drops

Main results from the data interpretation with the kinetic scheme

1. The breakage process is not a binary breakage .

2. The probability for generation of smaller drops is determined.

3. KBR decreases with the decrease of drop diameter and approaches 0, when d approaches the Kolmogorov size.

4. KBR depends significantly on the hydrodynamic conditions and viscosity of the oil phase.

Main conclusions from the comparison of the experimental data and the theoretical models for kBR

All experimental data are described reasonably well under the assumption that the breakage probability is given by:

•Drop-eddy collision frequency

•Breakage efficiency, accounting for (i) the dissipated energy inside the drops and (ii) surface expansion energy.