for coalescence

8/19/2019 For coalescence

1/8

Chemfcol Engineering Scknce, Vol. 41, No. I, pp. 65-72, 1986.

ocQ9-2509/86 s3.00 + 0.00

Printed in Great Britain.

0 1986. Pergamon Press Ltd.

BREAKAGE OF VISCOUS AND NON NEWTONIAN DROPS

IN STIRRED DISPERSIONS

J. S. LAGISET TY, P. K. DAS and R. KUMAR

Department of Chem ical En gineering, Indian Institute of Science, Bangalore 560 012, India

and

K. S. GANDHI

Department of Chem ical Engineering, Indian Institute of Technology. Kanpur 20801 6, India

(Received 15 February 1985)

Abstract-A model of breakage of drops in a stirred vessel has been proposed to account for the effect of

rheology of the dispersed phase. The deformation of the drop is represented by a Voigt element. A realistic

description of the role of interracial tension is incorporated by treating it as a restoring force w hich passes

through a maxim um as the drop deforms and eventually reaching a zero value at the break point. It is

considered that the drop will break when the strain of the drop has reached a value equal

to

its diameter. An

expression for maxim um stable drop diameter, d,.

is derived from th e model and found to be applicable

over a wide range of variables, as well as to data already existing in literature. The model could be naturally

extended to predict observed values of d,, when the dispersed ph ase is a power law fluid or a Bingham

plastic.

INTRODUCTION

The analysis of rate processes in liquid-liquid disper-

sions requires knowledge of the sizes of the drops

existing in the vessel To estimate the drop sizes, it is

essential to know the break-up mecha nism of a drop in

a turbulent dispersion. Hinze (1955) suggested a model

for predicting maximum stable drop diameter by

comparing the restoring elastic stress in

the

drop due

to interfacial tension with the inertial stress across the

drop diame ter. He proposed that break-up of a drop

occurs w hen the ratio of the inertial stress to the elastic

stress, i.e.

PCu* (4 40,

exceeds a critical value.

Assum ing that turbulence is isotropic a nd that the

diameter

d

1 (Kolmogoroff length), the mean square

velocity fluctuation across distance, d, is given by

UZ(4) cc (& ‘3.

(1)

If the Reynolds number is sufficiently large, then for a

fully baffled vessel, the power dissipation per unit

mass, E, can be expre ssed a s

E a N3D2 .

(2)

Substituting eqs (1) and (2) in Hinze’s proposed

criterion, Shinnar (1961) derived the following equa-

tion for the maxim um stable drop diameter,

d ,,,:

d

-225 = constant (We)-o.6

D

(3)

where We is the Weber number.

Equation (3) has been used by a numbe r of investi-

gators. Coulaloglou and Tavlarides (1976) have dis-

cussed exhaustively all the correlations available to

predict

d,

in a turbulent dispersion.

Equation (3) is based on the assump tion that there is

hardly any difference in densities and viscosities of the

two phases. The viscosity of the dispersed phase has,

however, been found to have a significant influence on

the drop size (Arai et al. , 1977; Ko nno et al ., 1982). The

viscous stress resists the flow inside the drop, leading

to

its breakage, and som etimes this stress has the same

order of magnitude as the inertial stress due to pressure

fluctuation. There exists very limited information (Arai

et al . , 1977; Konno et al. 1982) on the prediction of

maxim um stable drop size taking into account the

effect of dispersed phase viscosity. Konno et a l . (1977)

were the first to propose a model for predicting the

maxim um stable drop diameter incorporating the

effect of dispersed phase viscosity. They considered

that the deformation of a drop under external stress

can be described by the Voigt model which simul-

taneously takes both interfacial tension (restoring

force) and viscous dissipation (due to resistance to flow

inside the drop) into account. In deriving the model

equations, they assumed that the pressure fluctuation

across a distance due to turbulent flow is periodic and

the break-up of a drop occurs when the deformation

strain (0) reaches a critical value, 13~~. They have

obtained a semi-empirical correlation for the maxi-

mum stable drop diameter in terms of two dimension-

less groups, Weber numbe r and viscous number.

Although their semi-empirical expression predicts

their experimental re sults reasonably well, the model is

open to criticism on several grounds. The Voigt mode l

has a maxim um equilibrium deformation and it is

reversible. As such, it is hard to v isualize the breaking

of a drop through a classical Voigt model. Arai ef al.

(1977) overcame this problem through the artifice of

associating drop breakage with a maxim um deforma-

tion, Qr,

which is left as an arbitrary parameter.

Further, the regular and periodic pattern assigned by

them to the turbulent flow around the drop does no t

CES 41:1-E

65


2/8

66

J. S. LAGISEITY et al.

appear realistic, especially on the length scale of drop

size being dealt with. Apart from this, their mode l does

not give rise to the low viscosity limit of the maximum

stable drop size naturally and had to be introduced in

an ad hoc mann er. Finally, their model does not yield a

final expression which can be use d directly for evaluat-

ing the two constants contained in the model. Instead,

the final expression is assumed and the two constants

involved in it are evaluated from their own experimen-

tal data. Such a procedure does not allow extension of

their model to other rheologically complex fluids.

The present work aims at proposing a new phenom-

enological model which accounts for the effect of

viscosity of the dispersed phase in a more ra tional way.

It also tests the validity of the model against exper-

imental results.

EXPERIMENTAL

The experimental apparatus consisted of a glass

vessel of 14.5 cm i-d. and 20 cm height. The impeller

used was a six-bladed disk turbine, placed centrally in

the mixing vessel through a stainless-steel shaft, which

in turn was connected to a motor, the speed of which

could be regulated. A set of four, equally spaced

stainless-steel balIles, arranged vertically at the wa ll of

the vessel, was used. The width o f each baffle was equal

to one-tenth of the diameter of the vessel. A schem atic

diagram of the equipment is shown in Fig. 1.

The continuous phase was taken in the stirred vessel

and the stirrer speed ra ised to the desired value. The

dispersed phase w as then added to the vessel. The

dispersed phase volume fraction was kept at less than

0.02 to minimize coalescence. The equipment was run

for 1 h to achieve steady state. Samples were then

D.14.5 cm

Fig. 1. Diagram of the stirredvessel.

drawn from the vessel and the particle sizes measu red

with an optical microscope. At least 150 drop di-

ameters were measu red to obtain the value of d,. To

avoid coalescence of drops during sampling and

subsequent size measu rements, polyvinyl alcohol

(PVA) was added immediately before drawing the

sample. This was done when the aqueous phase was

continuous_ When the aqueous phase was dispersed,

PVA was added to it at the beginning itself.

The liquids used together with their properties and

experimental conditions are listed in Tables 1 and 2,

respectively. Interfacial tensions were determined by

the pendant drop m ethod. Viscosities w ere measu red

with a coaxial cylinder viscometer.

DEVELOPMENT OF THE MODEL

When a viscous droplet becomes deformed in a

turbulent flow field, the viscous stress due to the

internal flow will act simultaneously with the interfa-

cial tension force to resist the deformation of the drop

against the external inertial stress arising from the

turbulent p ressure fluctuations. If the inertial stress is

sufficiently large and sustained over a long enough

time interval, the drop w ill deform and a thin column

of liquid will appear so mew here in its bulk. The

column will break due to instability producing two

daughter droplets. This has been assumed to occur

when the magnitude of deformation is of the order of a

drop diameter. As the interfacial tension restoring

force is absent both at zero deformation and at the

breakage point, this restoring force should pass

through a maxim um as deformation proceeds. In this

connection, it is interesting to note that based on

calculations for low Reynolds numbers, Rallison

(1984) states that for globular drops, surface tension

acts as a restoring force but that surface tension may

even promote breakage once the drop becom es

elongated. As the interfacial tension and viscous stress

simultaneously oppose the inertial stress, the Voigt

model offers a suitable description of drop deforma-

tion. How ever, the elastic stress due to interfacial

tension needs to be represented in a more realistic

fashion. The Voigt m odel corresponding to the process

of deformation is presented in Fig. 2.

In accordance with the Voigt. model, the applied

stress (T) must be equal to the opposing elastic stress

due to interfacial tension (rJ plus the viscous stress 7,).

The governing equation therefore has the following

form

7 = 7, + 7”.

(4)

Table 1. Experimental conditions

Impeller speed

Reynolds number of the

continuous phase

Dispersed phase volume

fraction

Temperature

3.33-iOrev/s

1.5 x l@-5 x 104

0.02

26°C


3/8

Breakage of drops in a stirred suspension

Table 2. Properties of the $ontinuous and dispersed phases for the systems studied

67

Continuous phase

Dispersed phase

Description

c1

P

Description

K

”

T

(Poise)

(Mn3)

13/(cn=2-“)

(dynes/cm) (g/&3)

Water

0.01

1.0

Polystyrene

0.43-

1 20

0X8-0.92

in styrene

37.50

lO-30% by

wt

Kerosene 0.021 0.78 lOOm1 of 14.5 2/3 50 1

CMC in water

(2.5 %I +

60ml of 2%

PVA

CaCO, aqueous

0.137 1 45.2 1.47

suspension

(59.5 % Caco3 +

2.00 y0 polyvinyl

alcohol)

-

i

Fig. 2. Voigt model for the deformation of a drop.

Constitutive equation for T,

When the drop suffers a small but finite strain, the

interfacial tension force generates a restoring stress

proportional to the strain. However, as the drop

undergoes further deformation and approaches the

breaking point, its structure will be more like that of a

dumb bell: two “yet to be born” daughter droplets

connected by a thin filament of liquid. The filament will

break because of its inherent instability. Thus near the

break point,- surface tension has no restoring effect.

Hence a more realistic description of the role of surface

tension must be to assign to it a retractive force that

increases for small deformations of the drop but

decreases afterwards and e ventually reaches zero at the

break point. Assuming that the total deformation

undergone by the drop up to the point of breakage is of

the order of ma gnitude of

d,

the following simple

functional relationship between 5, and 0, has been

assumed.

TS= $ es (1 -e,>

e, c 1

=

0

e 3 I.

(5)

The condition that 7, is zero for 0, > 1 explicates the

essential feature of the model, i.e. the drop has reached

its break point when the dimensionless strain has

reached the value of unity and surface tension can no

longer bring it back to its original state. Thus the Voigt

model proposed in the present work cannot retract to

its original state, once 0, & 1, even when the applied

stress is withdrawn.

Constitu tive equation for TV

For the fluids considered in this work, the constitut-

ive equation for viscous stress can be written in a

generalized form as

7v = TV + K (de,/dtY.

(6)

Equation (6) simplifies to the case of Newtonian,

Bingham plastic and power law models when ap-

propriate simplifications are made.

Model equation

For a Voigt element, the total strain 8 = 8, = 0,.

Noting this and substituting eqs (5) and (6), respect-

ively, in eq. (4), we obtain

r-=,--ae l-e8)+K

d

c

1.

(7)

The flow inside the drop during breakage will be quite

complex. In the present model, however, it has been

assumed to be a simple shear flow.

Description of z

In eq. (4), T represents the dynamic pressure dif-

ference across the drop diam eter. In a turbulent flow

field the pressure difference normally associated with

the arrival of eddies is a quantity which fluctuates with

time and distance over which it operates. However,

exact knowledge of the fluctuations is not yet available.

Therefore, it is assum ed that the deformation takes

place under the influence of a mean stress which

remains constant over the average life time of an eddy.


4/8

68

J. S. Llsorszrrv et al.

Expr ess ions for aver age va lu es of 7 and T

Let the average values of 7 be + and the average life

time of an eddy beF The expressions for these average

values given by Hinze (1955) a nd Coulaloglou and

Tavlarides (1977) have been used in the present

investigation. They are

and

7 ~PcU2W (8)

=&*

(9)

It is necessary to express u2 (d) in terms of the

parameters associated with the stirred vessel. The

expressions for the rate of energy dissipation per unit

mass, E, and u2 (d) given by Coulaloglou and

Tavlarides (1977) are

E = 0.407 IV= D 2

(10)

and

u2 (d ) = 1.88 s2’3 d2’3.

(11)

Therefore + can be expressed as:

+ -_ C pc N2 D4/= d2j3 (12)

where C is a constant dependent only on the geometry

of the tank and agitator. The average life time is given

by

(13)

The drop experiences no stress prior to its coming

under the influence of an eddy a nd from then onwards,

a constant stress equal to + for a tim e interval of T

Thus

r = O t t o

=p

o< t gT

Substituting these into eq. (7), we obtain

C PcN2 D413 ‘ 213 _

de n

Tg +1 -e)e+K -&

.

(14)

(>

Equation (14) is more conveniently represented in

dimensionless form as

(dlJ/dt = C We (d/D)‘ / ’

- (q,d/o) - (0 - 02) (15)

where rl is the nondimen sional time given by

q = (o /dK)“” t .

(16)

Max i mu m s tab l e d r op s i ze

The drop is exposed to a stress of i only for a time

period T After this time, the eddy wou ld have dis-

sipated its energy and the external stress on the drop

becomes zero once again. If at the end of the time

interval T the value of B -Z 1, then the surface tension

spring w ould still have a finite retractive stress and the

drop would return to its original state. Thus for the

drop to break it is not on ly ne cessary that 8 = 1 is

reached but also that it should be attained at some

point during the time intervalT. The time required to

reach 0 = 1 can be obtained by integrating eq. (15); an

initial condition is needed to do so. The dashpo t

cannot show any instantaneous deformation on the

application of the stress. Hence

8=0 at q=O_

(17)

By separating the variables, eq. (15) can be written as

18)

where

a = C We (d/D )s’3 - (T,, d/u) - l/4.

(19)

The solution of eq. (18) w ith the initial condition

depends on the constant a in eq. (19). There are two

possible solutions corresponding to a < 0 or a > 0. If

a -Z 0, it can be shown that w hen t -B co, 8 --r [l/2

- fi]_ Thus, Bean never reach a value of unity, and

breakage is not possible in the finite time interval ofT_

But tI can reach unity in a finite time period when a

> 0, and therefore to predict the maxim um drop

diameter, only values of a greater than zero need to be

considered.

Our main concern is to know the max imum size of a

drop that can ex ist in the stirred vessel. All drops above

such a size will break. As stated earliei, the drop has

been assum ed to de form under the influence of T for a

period of?? T he max imum stable drop size therefore is

the max imum threshold drop size which reaches a

deformation of unity when exposed to a constant stress

F during the time interval F. Such a drop size can be

calculated as follows.

Equation (18) is integrated to find rl at 8 = 1. If rl at

8 = 1, the dimensionless time required to reach defor-

mation correspon ding to breakage, is more than the

nondimen sional life time of the eddy, breakage wo uld

not occur. Thus for breakage to occur, the following

condition must be satisfied:

q (0 = 1) <

(u /d IQ1 ’“Z

(20)

A s

the applied stress increases with diameter, q (0 = 1)

decreases with increasing diameter. The maximum

stable drop size is that for which

q (0 = 1) = (a/d_, K)““T.

(21)

Thus eq. (18) can be integrated up to rl(0 = 1)

= (a /d_K)““T to find the maxim um stable drop

size. Using the expression for T in eq. (9), ~(0 = 1) can

be written as

r.t(e = 1) =

(Re/We)“” (d/D)2’3-“n

where

Re =

D” NoI2 - n

,

2”-‘ (3+l /n r K

(23)

Forr,=Oandn= 1,1/3,1/2and2/3,eq.(18)canbe

integrated analytically; the solutions are listed in

Table 3. It should be noted that C = 8.0 has been used

in these results for both Newtonian and non-

Newtonian fluids. This will be discussed further in the

Results and Discussion.

It should be noted that the mode l developed in the


5/8


Table 3. Solution of eq. (18) for non-Newtonian fluids

n

Solution of the model equation

[32 We (d_/D)s’3 - I]“* tanm

=

(RejWe) (d-/D)- I i3

>I

Re/We)3/2 (d-/D)- 5’ 6

u2

1.6

51.2

We d_/D)“ ’ (32 We d_/D)513 -

1) +

[32 We d_/D)5’3 - l]3/2

x

tan-’

c32 We(d_;D)“” _ 1,~,2 = cRe/~~’ axP-4’3

1’3

2.66(44We(d_/D)5’ 3 -1

[(32We(d_/D)“ ” -l)* Wez(d_/D)‘ o’ “] + (32 We(d,,,f/ ” - 1)5f*

x tan-’

[32 We d_;D)= - l]“*

= [Re/

We]

’

(d- /D) - ‘ j3

present work involves a single parameter whereas

the model proposed by Arai et al. (1977) requires the

knowledge of two parameters to be evaluated from

experimental data. A positive

test that can be

applied to

the present mode l is that the constant C involved in eq.

(12), which relates the turbulent pressure fluctuations

to d, N and

D,

should not depend on the rheological

properties of the dispersed phase. If this is satisfied,

then clearly the present mod el offers a distinct ad-

vantage over the model of Arai et al. (1977) which

requires the evaluation of two parameters.

RESULTS

AND DISCUSSION

Newtonian fluids and inviscid limit

One of the interesting aspects of the model is that as

the dispersed phase viscosity becomes very small, i.e.

,ud+ 0, the solution of the model equation redu ces to

the well-known and widely tested equation

given

by

Shinnar (1961). For example, looking at the equation

for a Newtonian fluid (n = 1) presented in Table 3, the

only solution that is possible as pd - 0 or

Re -+ co

is

that We (d_/D)5’ 3

-+ constant. This result is identical

to eq. (3) reported by Shinnar (1961).

Using the data of drop diam eters against dispersed

phase viscosity for different speeds, the value of

constant C in eq. (12) has been determined. The value

of the constant worked out to be 8.0. Figures 3 and 4

show the experimental drop diameter, d,, against

dispersed phase viscosity, pd. and impeller speed. The

results predicted by the model are shown as lines

whereas the points are experimental results. The

theoretical predictions are in good agreement with the

observations. From this, it can be seen that C is

dependent on ly on the geometry of

the

vessel and

agitator, and not on cl,, . Further, it is worth noting that

the effect of dispersed phase viscosity on d, becomes

significant only after about 20 cP.

The predictions of the present model and the

numerical value of constant C can be further tested

using the experimental observations of other workers.

69

Fig. 3. Com parisonof experimental esultswith the present

model (effectof Jo)_

When the dispersed phase viscosity becomes small, i.e.

p, -V 0, the model predicts that

(d-/D) = 0.125 (We)-“* 6.

Sprow (1967) ha s given the following empirical re-

lationship when the dispersed ph ase viscosity is small:

(d-/D) = a1 (We)-“.6.

The value of a 1 ies between 0.126 and 0.15, w hich is in

agreement with the value found in the present work.

The present value of 0.125 is somew hat better because

it fits a large amo unt of other experimental data. For

instance, the experimental results of Arai et al. (1977)

also agree with the present model. This is shown in

Fig. 5.

The present model can be easily modified to in-

corporate the effect of the dispersed ph ase hold-up.

The value of the mean square velocity, u* (d), changes

with the change in the dispersed ph ase hold-up. The


6/8

J. S. LAGISETN et al.0

0

0.

E 0.

1

0.

0

I

I

L

I 300

Loo m 6

Impcllsr speed N (r.p.m 1

pd c ‘3 CP

o- z 20 dyneslcm

D = 4.5 cm

Fig. 4. Effect of impeller speed on the maximu m stable drop

diameter.

I

1.5

d

I

B

/

i L-0

d

I J

/ /

/’

d’

0.5

/‘)/’

_ A;./’

___--b-

_--,_A=

Experimental

Theoretical

l3.P.M

D

xl0

0

400

0 -.- 600

a- = 20 dyneslcm j D= 6. 5 c m

Fig. 5. Comparison of experimental results (Arai et al. with

the present model.

value of m

changes with 4 as (Coulaloglou and

Tavlarides, 1977)

m+=+ = (1

+a2+)-2.0[uz(d) ]+o.

(24)

Using the modified expression for u2 (d) , the equation

for maximum drop size for Newtonian fluids in the

limit of small viscosity becomes

d

z = 0.125 (1 + r~~+)“ ~ (We)-“.6.

D

c-w

The data available in the literature is in terms of dx2.

Sprow (1967) and Coulaloglou and Tavlarides (1976)

indicate that da2 is linearly related to d,. For

continuous system s, Coulaloglou and Tavlarides

(1976) indicate that d, = 1.5 dS2. Therefore

d

2 = 0.083 (1 + a2 )‘.” (We)-o.6.

D

(26)

It is found that for a2 = 4.0, the model predictions

given by eq. (26) fit the experimental data of

Coulaloglou and Tavlarides (1976) on continuous

systems reasonably well. This is shown in Fig. 6.

It can be concluded that the value of 8.0 assigned to

constant C is related only to the geometry of the vessel

and agitator, and arises from the nature of the

turbulent fluctuations. It is not affected by the prop-

erties of the dispersed phase, which have been varied

over a wide range. The m odel itself can then be seen to

account for the variation of the viscosity of the

dispersed phase. This can be tested further by examin-

ing non-Newtonian fluids.

Non -New ton i a n i d s

Figure 7 presents the experimental results of maxi-

mum stable drop diameter with change in impeller

speed for a power law fluid (2.5 0A CMC in water

+ PVA, n = 2/3). The grade of CM C used was the

same as that employed by Kumar and Saradhy (1972).

These authors u sed a test attributed to Philipoff to

ensure the absence of viscoelasticity in CMC solutions

up to a concentration of 4% by weight. The other

compon ent of the solution use d in the present work

was PVA, with a molecular weight of about 70,000. At

this molecular w eight and low concentrations, PVA

will also not impart an elastic character to the sol-

utions. Therefore the solution em ployed in this work

can be safely considered to be inelastic. The drop

diameter decreases with impeller speed in a nonlinear

fashion. As m entioned earlier, according to the model,

+

0 0.15

*

0.Y)

. 0 05

0 0.025

a2 z 4.0

Fig. 6. Comparison of

Coulaloglou and Tavlarides’ exper-

imental values (effect of 4) with those predicted by the present

model.


7/8


71

Impeller speed ti (r.p.m 1

Fig. 7. Comparison of experimental results with those pre-

dicted by the present model for a power law fluid.

constant C must be independent of the rheology of the

dispersed phase. The solid line represents the results

calculated from the model using constant C evaluated

with the experimental results of Newtonian fluids, i.e.

calculated from the second equation of Table 3. It was

found that the model predicts the results reasonably

well. This lends further credence to the model.

Gene ra l i zed rep resen t a t i on

The model predictions can be represented in gener-

alized graphs b y plotting

(d- /D)

against the Reynolds

number,

Re

defined on the basis of the viscosity of the

dispersed phase) with the Weber numb er, We, as a

parameter. Figure 8 shows plots of

d- /D

against

Re,

with We as a parameter for the values of n equal to I,

2j3 and 1 3. The plots show that at low Reynolds

numb ers, i.e. at high va lues of p,, all the curves merge,

indicating that interfacial tension does not play a

significant role in the breakage of drops. This result is

expected, since when the drop is sufficiently viscous,

the applied force essentially overcomes the mean

viscous resistance, compared to which the restoring

surface tension force is insignificant. On the other

hand, at high Reynolds numb ers, the restoring force

due to interfacial tension is predominan t compared to

the viscous forces and the applied force essentially has

to overcome the restoring interfacial tension forces.

That is why at high Reynolds numb ers different lines

are obtained for different W eber numb ers.

B i n am p l a s t i c s

For Bingh am plastic fluids the theoretical equations

can be easily developed by replacing

7

in the differential

equations for purely viscous fluids with 7 - 70)_ I t is

easily shown that this corresponds to replacing

[32 We (d _j D )s’3 - l]

by

[32 We (d _jD )5/3 - 4 7,, d-/a) -

l]

in equations of Table 3. This implies the usage of the

same value of C determined from New tonian fluids.

Figure 9 presents the experimental results of the

maximum stable drop diam eter with change in the

impeller speeds for a Bingham plastic fluid aqueous

CaCOS suspension with PVA, c = 45.2 dynes/cm T,,

= 75 dynes/cm’, n = 1, p,, = 13.7 cP). It is found that

the drop diameter decreases with increase in the

impeller speed in a nonlinear fashion. The solid line

represents the theoretical results calculated from the

first equation of Table 3 with the modification de-

scribed above. The dashed line represents the drop

diameters that would have been observed if the fluid

had been Newtonian, i.e.

7. =

0. The experimental

results are higher than those for a Newtonian fluid

since the effective applied stress is now lower. The

model reflects this and predicts the d , values quite

well, thus accruing more evidence in its favour.

Fig. 8. Effect of the Reynolds numb er on d /D with the Weber number as a parameter.


8/8

for coalescence

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