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Dynamics of emulsification
Chapter 14
Dynamics of emulsification D C Peters
Vinamul Group Ltd, Warrington, Cheshire
14.1
Introduction
In preparing emulsions, many factors concerning their end-use have to be taken into account. Often, their preparation is merely an enabling step towards, for example, formation of disperse polymer systems, or enhancing a liquid/liquid extraction process. However, in many situations the emulsion is itself the end product. Abundant examples of where this is the case are to be found in the pharmaceutical, food, paint, dairy, agrichemical, cosmetic, adhesives and detergents industries. In these situations, as opposed to those where emulsification is an intermediate step, we are invariably more closely concerned with two key properties - rheology and stability. To the consumer, consistency of product properties represents quality and assurance that the product is functioning in the regularly expected way. The recent introduction of British Standard 5750 has strengthened the need for companies to be certified as approved in order to demonstrate their commitment to quality assurance. Quite apart from this marketing appeal, there can be a very real danger from incorrect function in an unstable product. This is recognized in law in many countries. For example, cosmetics and pharmaceutical products may be legally required to have a specific shelf life during which they show no physical changes. Product liability can lead to expensive litigation and claims. Accepting that rheology and stability are critical properties, how can largescale processes be designed and operated to achieve a given product specification? The approaches open are either (a) to adopt an empirical approach using experimental designs for formulation and processing which allow the 'best' combination to be selected or (b) to use an approach based on understanding the basic physical and chemical phenomena operating within the systems, and hope to build these into a formulation/process. There are advantages and disadvantages to both. The former will generally yield a compromise solution but the gaining of that solution offers little insight into how new products and processes might be developed. On the other hand, the latter should yield a solution, plus an expertise for future developments, but only with a considerable investment in skill and equipment. Of course, neither approach is exclusive and many 'empiricists' make excellent use of basic technology; similarly, no process has yet been designed purely 294
Rheology and stability
295
from a set of equations. The purpose of this chapter is to try and bridge the gap by presenting, in a semi-quantitative way, some of the concepts which are essential to a fuller understanding of emulsification. The approach taken does not purport to provide answers, but perhaps exposes some basic questions. Firstly, the aspects of an emulsion which are important to the product rheoIogy and stability are briefly considered.
14.2 Rheology and stability Most emulsions of interest show shear-thinning rheological behaviour, and many show other non-Newtonian features such as elasticity, yield stress or time dependent effects. All of these can be explained qualitatively, and sometimes quantitatively, by a relatively simple set of parameters (Table 14.1) which takes into acco.unt; droplet size, dd, droplet phase volume 4', conservative (colloid interaction) forces, hydrodynamic forces and interfacial properties tr (Figures
14.2a). Table 14.1 The key properties of emulsions and the basic parameters that control them
Property Viscosity Yield stress Stability Appearance
Relevant parameters do do dd dd
/.to 4~ A0 cb
~b A0
HO
H,,
Individual drops do not interfere with each other appreciably when the internal phase volume ~bis less than 30%. However, as the phase volume increases from 30% to 74% there are increasingly more frequent points of contact and the viscosity increases. For phase volumes between 50% and 52%, the spherical drops are stacked in a cubical arrangement; and as 4) increases to 68%, they arrange themselves in a tetrakaidecahedral array. Above 74% phase volume, the liquid drops are no longer spheres but multifaceted in shape (Figure 14. I). Shear-thinning behaviour is the result of either droplet distortion and alignment with the flow, or due to increasing shear stress causing breakdown of weak flocs with a consequent decrease in effective phase volume (Figure 14.2b). If a system is to possess elasticity, it must possess a physical mechanism for storing energy. In flow, droplet distortion causes an increase in surface free energy (crAA) which is released on cessation of flow, manifesting as, for example, recoverable shear compliance. The magnitude of the (dimensionless) droplet distortion is proportional to the Weber number, We, which is the ratio of the deforming stress (~t~) to the restoring Laplace stress (4tr/dd). Thus, the drop Weber number based on drop radius is defined as:
296
Dynamics of emulsification
Figure 14.1 Multifaceted particles of a Tetradecane/water 77% phase volume emulsion 5 mm = 10 Ixm
We -
~cTdd 20-
(14.1)
where ~ is the viscosity, 7 is the local shear rate and d d is the droplet diameter. Accordingly, small drops deform proportionately less; with the implication therefore that droplet size is important to the elastic flow properties. The property of yield stress arises in the case where the droplets form a continuous network throughout the system; for instance, under the influence of van der Waals forces I. In such a case, a complete network of deformable droplets has a yield stress, Zy, given by:
Aj 4'
7.v ~ ___;m " H 0 dd
(14.2)
and the network modulus, G, is a function of the interracial tension:
4,
G ~ tr-do
(14.3)
Time and stress dependent flow properties are generally modelled 2 via the making and breaking of inter-droplet bonds, such that at any given stress level there is a distribution of singlets, doublets, triplets, etc. and the rate of change of
297
Rheology and stability (a)
(b)
no flow condition
flow condition shear
attractive/repulsive forces
,
0 Q
,,,
,
00 0
drop size and phasevolume
floc distortion and breakdown; droplet collision
to, ;phase )cculated tops
Q internal circulation .
network connectivity
.
.
.
.
distortion to the point of break-up .
.
.
.
.
.
.
.
.
_
_
shear
Figure 14.2 Diagrammatic representation of the parameters which are important to emulsion properties
distribution on changing the stress level reflects the thixotropic change noted experimentally. Clearly, both droplet-droplet collision, essential for 'making' bonds, and multiplet breakdown (controlled by attractive forces) are strong functions of droplet size. The stability of an emulsion depends largely on the balance of attractive and repulsive forces and Figure 14.3 shows a primary and secondary minima system being the stablizing mechanism. If only attractive forces exist then the droplets coalesce, thereby lowering the system free energy (except in the event of extremely small particles where the entropic form can dominate). On the other hand, if repulsive forces dominate, then even very small density differences are sufficient to give rise to creaming, i.e. phase separation, for example. What is required for stability is a flocculation of droplets, which does not give rise to coalescence, to form a network throughout the system 3. The role of colloidal forces in the rheology of suspensions has recently been reviewed by Russel4. In Figure 14.3 a secondary minimum stabilizing mechanism is shown where stability is achieved simply through an appropriate balance of attractive and
298
Dynamics of emulsification +re
% short-range Born effects
~~
electrical %% (repulsive) energy %
energy I
% % % % % % %
repulsive
%
%
distance
0
energy I f I
/
attractive
/
.~. ~ "
~ ~ van der Waals (attractive) energy
.="on'4ar" --~ -minimum
f
f
/ /
\, -ve
/
primary minimum (often leads to coalescence)
Figure 14.3 Attraction and repulsion forces as a function of surface-surface distance
repulsive electrical forces. Other stabilizing mechanisms exist such as the use of absorbed species which present an entropic hindrance to the close approach of droplets; or the use of bridging polymers or finely divided solids. Each mechanism leads to a different strength of interaction and, generally, a different response to stress. In practice the stabilization of emulsions is induced by the use of surface active agents or emulsifiers. The surfactant structure can be represented diagrammatically and the way it orientates itself at an oil/waterphase interface is shown in Figure 14. 4. The application of mechanical energy creates droplets and for an oil-in-water emulsion the tails of the surfactant molecule point into the oil droplets and vice versa for a water-in-oil emulsion. These structures are known as spherical micelles. With increased concentration of the surfactant in the system, the micelles can become highly ordered and produce complex rheologicai behaviours through the formation of various phases 5. A schematic illustration of the various phases produced from increasing surfactant concentration are given in Figure 14.5. The interfacial aspects of emulsification, including thermodynamics of emulsion formation and breakdown, have been reviewed and described recently by Tadros 6. The role of emulsifiers is discussed in detail and the mechanisms outlined, although complex, are related to the particle size if for no other reason than that the number density is proportional to 1/(dd).
Rheology and stability
299
Figure 14.5 Structure produced by ordering of micelles 1. Molecular solution. 2. Spherical micelles. 3. Aggregated spherical micelles. 4. Cylindrical micelles. 5. Hexagonal phase. 6. Disc micelles. 7. Vesicles. 8. Lamellar sheets. 9. Crystalline
There are a large number of emulsifiers available and making a choice for a new formulation or product is difficult. The use of the hydrophilic-lipophilic balance (HLB) concept originated by Griffin in 19497 can be a useful empirical method of preliminary selection. The HLB number quantifies the balance of hydrophilic-lipophilic characteristics of the surfactant molecule on an arbitrary numerical scale. The least hydrophilic surfactants are assigned the lowest HLB (see Table 14.2). The use of the HLB balance is also well described by Davies and Ridea142.
300
Dynamics of emulsification
Table 14.2 HLB values of surfactants
HLB
Application
3-6 7-9 8--15 13-15 15-18
W/O emulsions Wetting agents O/W emulsions Detergents Solubilizers
Dispersibility in water 1-4 3-6 6-8 8-10 10-13 13-
Nil Poor Unstable milky dispersions Stable milky dispersions Translucent dispersions/solution Clear solution
Reconsidering the factors which can be seen to be important to rheology and stability, and over which some control is possible during processing, then clearly droplet size is a key parameter (Table 14.1). In the following sections, some fundamental aspects of droplet formation are examined, in order to yield some insight into the design and operation of processes for controlled droplet formation. However, in passing it should be noted that the way in which the emulsifier is added is important and that for high internal phase emulsions the material of construction of the making vessel is critical, i.e. the solid-liquid interface is important too.
14.3 Droplet formation In very general terms, droplets are formed by stress being imparted to a large primary drop, causing elongation of all or part of it, followed by development of surface wave growth to the point of instability, whereby the primary drop breaks into droplets and, often smaller satellite droplets. The disruption of a plane surface separating two bulk phases, necessary to produce drops, threads or mis-shaped lumps of dispersed phase can be achieved 8 via turbulent eddies 9, surface ripples, Rayleigh-Taylor instabilities l~ and Kelvin-Helmholtz instabilities I~. The factors important to these processes are" (a) the viscous and elastic properties of the disperse and continuous phases; (b) the interfacial properties; (c) the flow conditions. There are a number of difficulties in examining the roles of these factors either experimentally or theoretically. The key difficulty is that, in a practical sense, emulsification does not take place under steady conditions, but under dynamic conditions where time scales may be of the order of seconds down to 10-6 seconds s. However, it has to be assumed that the direction of effects is independent of time scale. Then, a combination of steady-state effects with a knowledge of how time scales might modify their magnitude can be used.
Droplet formation 14.3.1
301
Deformation and breakup in steady flows
This problem is first studied by Taylor t2 and more recently by others ~3-~5. Basically the approaches made consider how the shearing forces acting on the droplet give rise to a pressure field around it, which can then be translated into a shape by using the Laplace equation,
AP =
o"
+
(14.4)
(see Figure !4.6). The deformation of droplets (Figure 14.6) depends on two parameters, the Weber number, defined earlier, and the viscosity ratio, R, where: R=
viscosity of drop phase
~d = -viscosity of continuous phase ~
(14.5)
At low viscosity ratios, R < 0.2, drops in shear flow first align themselves at 45~ to the flow in the form of ellipsoidai drops. Increasing flow and hence shear creates drops with pointed tails that break off to create satellite drops. Drops with viscosity ratios of ~ 1.0 form dumb-bells that with increases shear burst into two spherical drops with some satellite drops. Increasing R tends to produce long slender drops, R > 3.8 drops do not break but form ellipsoidal drops aligned with the shear field. In elongational flow (Figure 14. 7) and small R, ellipsoidal drops parallel to the flow and with trailing ends are formed, sometimes accompanied with smaller satellite drops. Drops with large R in elongational flow give ellipsoidal drops aligned with the flow and no breakup. Taylor 12obtained the following relationship for deformation of an ellipsodial drop: Deformation, D' -
=We
L-
B
L+B
-
~'ddl~c(19/16R + l ) 2o-
9
R+I
19/16R + 1) R+I
(14.7)
v
~
(14.6)
Laplaceforces fluidforces
Fisure 14.6 Relationshipbetweenfluid shear and Laplacesurfaceforces
302
Dynamics of emulsification
''
J
o O O J t~
0.,. 0
6.,,=.,, 0
"
6. Rlarge @
O
~~-..--..J
Figure 14.7 Possible de[ormation patterns (1--4 shear, 5--6 extension) 13.14
Drop rupture occurred when D' > D~ and this critical deformation was a singled valued function of R. Karam 16has shown experimentally (Figure 14.8) the form of this function. Similar results have been reported by Grace 17. In general there is a minimum as well as a maximum viscosity ratio beyond which the drop does not burst in
1210+
rr
rA
8-
drop deformation only
+ 6 420
,
0.001
.
.,I
0.01
.
---~----:'-~-,,i---,---,'r''T
0.1
....
1
viscosity ratio, pd/Ptc
10 =
R
Figure 14.8 Critical shear rate ('j,) for drop break-up versus R 17 ( --- extension)
100
....... s h e a r ,
Droplet formation
303
laminar shear fields. In addition for an ellipsoidal drop (Figure 14.7) for large We and small R, the deformation, D', is given by:
L-B L+B
D'-
oc
Wc
(14.S)
whereas for R >> 1 and small We, then: 1
O' oc-R
(14.9)
Grace ~7has also reported comparisons between simple shear (Couette flow) versus extensional irrotational shear (4 roll mill). These are shown in Figure 14.8 with the full line representing shear and dotted line extension. It can be seen that for equal shear rates extensional shear produces more effective break-up and dispersion than simple rotational shear, even at low viscosity ratios. These findings are substantiated in a recent review by Rallinson29of viscous drop deformation in shear flows. Figure 14. 9 summarizes the experimental and theoretical data for extensional flows. The corresponding comparison for simple shear is given in Figure 14.10. The comparison of Figures 14.9 and 14.10 substantiate that extensional shear is more effective than simple shear in promoting drop break-up. The break-up of droplets in steady flow is generally approached through consideration of the hydrodynamic stability of liquid cylinders, which are assumed to be a precursor of break-up. The critical factor for the rate of break-up of a given cylinder is the rate at which instabilities grow, whereas for the ultimate drop size, it is the wavelength of the instability which dominates (Figure 14.11). 0.5
-
0.4
".."%,,
0.3 0 *U
o.,
%
0.1
.
J
0 *
J
10 -4
10 -:~
,
J
10 -2
4. . . . . .
10 -1
! .....
1
,,,,0 ~ g " i
J
I
10
10 2
10 3
Figure 14.9 Comparison of critical We for extensional flow for experimental data and theory (..... experimental1?; asymptotictheory3~ o theory31)
304
Dynamics of emulsification 10 3
,
!
10 -
I-
~
I ! ! !
o
0
10 -1
I
I
I
10 - 3
10 - 4
10 - 3
,
|
10 - 2
,
I
I
10 -'t
1
,
I 10
R
Figure 14.10 Comparison of Wet for simple shear for experimental data and theory (..... experimental nT; ~ asymptotic theory3~ 0 theory 3~)
Figure 14.11
Waves on a cylinder
A typical theoretical result for the rate of growth of instability of a cylinder ~s is given in Figure 14.12. The ordinate, Sp, represents growth rate of instabilities while a, the wave number, represents the wavelength of the instability expressed as a ratio to the cylinder circumference. There is clearly a strong dependence on viscosity ratio; the larger the ratio the smaller the growth rate. The Weber number again plays an important role (Figure 14.13) with decreasing We leading to an increasing instability growth rate, i.e. increasing interfacial tension destabilizes the system. However, decreasing We also causes a reduction in deformation and therefore, with respect to drop break-up, there are two competing effects. Turning to the ultimate drop size, it is generally assumed that the wavelength, Am, corresponding to the maximum instability growth rate dominates break-up. The critical deformation, D,~t, can be expressed as: oc,, = Lc, - ,
L.it + B =
+
(14.10)
Droplet formation Sp
305
R " 0.001
0.01
0.1
1.0 1.0
0
Figure 14.12 Stability of Newtonian systems 18
Sp
W, - 1 0
1G 20 2G
0
1.0
Figure 14.13 Effect of Weber number on the stability of Newtonian systems 18
where am = Am/2~"to. From this analysis, the resulting drop size, rid, is given by" do = 2to
[
32~m] t/3
(14.11)
306
Dynamics of emulsification
which is a very weak function of am but a strong function of the cylinder radius, to. Control of drop size is therefore best effected by control of ro and, assuming a 1:1 relationship between ro and the deformability of the primary droplet, it can be concluded that Weber number and viscosity ratio are critical parameters.
14.3.2 Dynamic effects The most important practical dynamic effects are effects associated with rapid stretching of a droplet, i.e. surface dilation and bulk elastic effects. The former is important where surface active materials are present. In the equilibrium state, the droplet will have an adsorbed monolaycr of surfactant and, in consequence, the interfacial tension will be lowered. On stretching, the interfacial tension rises because the surfactant molecules cannot respond instantaneously, and therefore become depleted in terms of the number/unit area. In this way, the Weber number falls and deformation is less than might be expected from a knowledge of the equilibrium interracial tension alone. As time passes, further adsorption from the bulk may occur, or the droplet may contract back to a spherical shape under the influence of its new-found high interfacial tension. The exact circumstances will depend strongly on surfactant properties and concentration. A further manifestation of this effect is discussed in detail elsewhere 19. If the droplet material has viscoelastic properties, characterized by a relaxation time At, then the behaviour of the drop will depend on the timescale of the applied deforming stress. For very short times (<~ At), it might be expected that the drop will appear as an elastic solid, whereas at long times (>~ At) the viscous nature will dominate. A knowledge of the material rheology across a range of timescales is therefore important for a fuller understanding.
14.3.3 Turbulence So far steady flow conditions have been considered but in practice emulsification is very often carried out under turbulent conditions. Fortunately, the same physical concepts apply but this time in the context of a rapidly changing flow field. Under such conditions the Weber number cannot be defined using a steady-state shear rate and is reformulated as: We' = pcfl2(dd) dd O" -
-
(14.12)
where Pc is the continuous phase density and I)2(dd) is the root mean square of the different of velocities at a distance apart do of the turbulent fluctuation velocity.
Droplet formation
307
Kolmogoroff 2~ suggested that a drop will break if the pressure across it is similar to the pressure due to surface tension holding the drop together; that is, (dd)ma~ = C (--~--c)0"6 ef ~
(14.13)
where eT is the mean turbulent energy dissipation rate or the power dissipated per unit masg of fluid in the turbulent region. In Couette flow, Hinze 2~ found that the constant in the above equation was 0.725 and for turbulent flow in pipes, other workers have confirmed this constant 22'23. In stirred vessels the majority of workers have found that
dd oc E.170"4
(14.14)
with the exception of Shinnar 24 who reported dd oc ef 0.25. The impeller design has been shown to vary the constant C, and the Sauter mean drop diameter was found to be proportional to a circulation time tc25 thus:
d32 oc
(14.15)
The data for turbine impellers have all been collated together by McManamey 26 using the concept of the swept out mass of liquid by the rotating impeller when calculating eT.
d32 = C(~c ) 0.6N-l'2D-~
(14.16)
This equation can be expressed in the form of dimensionless groups:
d32 -- C!Po-~ D
0'6
(14.17)
P pcN3D5 ' pcN2D 3 and We~ = agitation Weber number = tr
where Po = power number =
In the theories used to obtain equation (14.13), the viscosity of the dispersed phase was neglected, but Hinze 21 has pointed out that if the viscosity of the dispersed phase within the drop is significant but the viscosity of the continuous phase is not, then the turbulent flow is still dominated by inertial forces. Then, a viscosity group is needed to be introduced, such as:
308
Dynamics of emulsification
/xa Nvi - (PdO.dd)I/2
(14.18)
This viscosity group is introduced to modify the We number by We'(1 + f(Nvi)) where f(N~i) ~ 0 as N~i ~ 0. A reasonably successful approach to finding the maximum stable drop size (dd)m~ in turbulence using this approach has been made by Arai et al. 27. For drop sizes much larger than the Kolmogoroff length scale, i.e.
']
1/4
10-4 m
(14.19)
in aqueous systems, they showed that theoretically ,,.2/3/',4 /5/3 PceT ~.t~dhnax
,,(1
+
f(Nv0)
= const.
(14.20)
eT is the energy dissipation rate per unit mass (aN3D 2 for fully developed turbulence flow in baffled vessels) and N~i = #der1/3(dd)ma~/tr 1/3 allows for the effect of a finite disperse phase viscosity, f (N,i) has to be determined experimentally. In attempting to confirm their expression experimentally, Arai et al. noted a strong interaction between drop viscosity and interfacial tension. Thus, for a low viscosity drop, one might expect a rapid response to the periodic turbulent stress, i.e. the droplet would rapidly adjust its shape. On the other hand, viscous drops having a response time longer than the periodicity will be subject to enhanced deformation by consecutive stress cycles. For drops smaller than the Kolmogoroff length scale, shear forces rather than periodic inertial forces dominate the break-up mechanism and theoretically28
erPc)
= const.
(14.21)
Other authors have attempted to allow for the influences of viscosity by introducing a factor (#dgc) '~where a is found empirically. Calderbank 32found that a = 0.25 for the mean drop size in his agitation system. In order to arrive at a drop size distribution, it is necessary to take coalescence into account and to ascribe an energy spectrum to the turbulent flow. Lee 33 reviews and presents results for drop size distribution produced by turbine agitators. Dibutyl phthalate agitated for 60 minutes gave the following correlation.
Implications for process design d32 D
- 0.184(1 + 5.4 ~b)We~~
309 (14.22)
where ck is the volume of the dispersed phase. Other workers 34-36 have reported that drop sizes close to the impeller were some 10% smaller than further away, which suggests that for the systems they studied the effect of coalescence is not large. Park 37 found that once equilibrium has been established, break-up occurred only close to the impeller, with coalescence predominating further away. This would be expected when energy dissipation is non-uniform as found in a stirred vessel. All authors find that the drop diameter increases with volume fraction ~ of the dispersed phase provided ~k > ---0.01. A review of disruption of drops by cavitation is given by Walstra s.
14.4
Implications for process design
A strong semi-quantitative link has been made between rheology/stability and drop size and between drop size and processing conditions. A fundamental approach therefore requires a study of both links. The first is outside the scope of these notes, but the literature contains a great deal of useful information 4~ Some comments on the second are made below.
14.4.1 Batch processing The variables available in a typical baffled batch mixer of standard geometry are impeller geometry and speed, temperature, time and the entry point of the second phase. Each of these is considered below. Impellers are available which provide for different distribution and stress patterns. For achieving small drops, an impeller having a high power number/ discharge ratio is preferred and vice versa for large drops. However, where some control of drop size distribution is required, consideration might be given to recirculation via a high energy device, or discharge via such a device. Temperature may play a strong role in changing the viscosity ratio of the two phases. For two Newtonian fluids obeying the Arrhenius-type equation, temperature independent activation energies Ed and Ec, we find:
ddo (InR) =
tER-go- '/
(14.23)
where 0 is the absolute temperature and Rg is the gas constant. In many systems, either or both E~ and Ed may be strongly temperature dependent. Such would particularly be the case where a phase change occurs. Again use may be made of such phenomena to control ultimate drop size. Temperature effects will be more complicated where emulsifiers are present because of
310
Dynamics of emulsification
temperature-dependent changes in adsorption, configuration at the interface, interfacial tension, phase structure, etc. and these effects must be studied separately. Time of mixing is important from the point of view of (a) ensuring adequate gross mixing, (b) ensuring equilibrium drop-size distribution 33'38 is achieved and (c) avoidance of overprolonged mixing with its penalties of higher energy costs, capacity or throughput problems, and possible damage to the product. The effect of prolonged agitation at different speeds on the mean droplet size dd is shown in Figure 14.144:. A model oil-in-water emulsion of 20% phase volume was used. In this agitation system, increasing the speed from 350 to 500 rpm did not produce further reduction in the mean droplet diameter. For these reasons, it is important to have an idea as to the limits to which a product can be 'pushed' in terms of intensity/time of mixing together with a good knowledge of the mixing time for systems of comparable rheology. For the latter, model studies using tracers, dyes, etc. are widely used. A numerical example of process design is given in Appendix 1. The entry point of the second phase can have a pronounced effect. For rate-of-mixing purposes and uniformity of drop-size distribution, it is claimed that entry close to the impeller tip is advantageous. This effect becomes particularly important where the temperatures of the two liquids are not equal.
40 N (rpm)
-'0"9- d k - ,
E
2OO 280 350
-.0-
5oo
.N -~ Q.
20
0 I 0
I
I 40
....
I
I .......... 80
I
! 120
mixing time t (min)
Figure 14.14 The variation of average droplet diameter, dd, with mixing time t at various angular speeds, N, when T = 21.5 cm, D = 0.67 T using an inclined paddle and three baffles.
Implications ]'or process design
311
The scaling up or down of mixing systems with Newtonian rheological characteristics is comprehensively covered in the literature a6 and in this book. However, the majority of emulsions with medium phase volume (30-55%) exhibit non-Newtonian behaviour. Literature is bereft of methods to scale up non-Newtonian systems. An approach is presented in Appendix 2 based on a generalized scale-up equation for agitated systems where: N2 = N~(D~/D2) x
(14.24)
with N2 and D2 being the new rotational speed and vessel diameter, N~ and D~ the old vessel speed and diameter and x the scaling factor.
14.4.2 ContinuoUs processing It is probably in continuous processing that a deeper knowledge of the fundamental physiochemical molecular effects pays dividends. Generally, a continuous emulsification process consists of two essential elements: a proportioning pump to ensure correct formulation, and a shearing device to ensure correct dispersion of the phases. This apparent simplicity belies the true complexity of such a process. For instance, where an emulsifier is present in one or both of the liquid phases, the state of adsorption at the point of shear will play an important role in determining the resulting drop size. Thus a dependence can be clearly seen on the contact time (and contact area) available between the pump and the shearing device. Davies has shown that many of the equations applied to stirred tanks are applicable here too 42'43. This will depend on pipe dimensions, on the use of in-line devices to produce a coarse initial dispersion ~7 and on the flowrate which, typically for a positive displacement pump, is pulsed! These are some of the factors upstream of the shearing device; downstream one has to consider the flow conditions pertaining immediately after the stretching zone, and the relationship between pipe dimensions and flow rate and the coalescence or further break-up of droplets. There are no simple answers in seeking a design for either batch or continuous processes. However, research and development can usefully be guided by fundamental studies of equilibrium and dynamic effects found from model studies.
Notation A Ad B C D
Hamaker constant area of droplets, m2 breadth of an ellipsoidal drop, m constant impeller diameter, m
312 D
F
D~
dd
(dd)max d32 E G
Ho L Lcrit
N n
ziP Po R
ro
RIR2 Rg Sp T tc
2(dd)
V We We' Wea ot fitm
ET
5,
K A Am At # #e P o"
0 ry X
Dynamics of emulsification deformation, dimensionless critical deformation leading to breakage, dimensionless droplet size, m maximum droplet size, m Sauter Mean Diameter, m activation emergency for viscous flow (J mol -~) network modulus, Nm -2 surface-surface separation between drops, m length of an ellipsoidai drop, m critical length of an ellipsoidal drop, m impeller speed, rev/s viscosity correlation number, dimensionless powder law index, dimensionless pressure difference, Pa power number, dimensionless #d/#r dimensionless diameter of a deforming cylinder, m principal radii of an ellipsoidal surface, m gas constant (J mol -~ K -~) growth rate of instability, s -~ tank diameter,, m circulation time, s mean square of the difference in velocity over a distance dd, (m/s) 2 vessel volume, m 3 laminar shear Weber number, dimensionless turbulent Weber number, dimensionless agitator Weber number, dimensionless A/2crro, dimensionless Am/2crro, dimensionless mean turbulent energy dissipation rate, W/kg phase volume extensional shear rate, s -~ laminar shear rate, s -~ consistency Pa.s '~ wavelength of an instability, m maximum wavelength before break-up, m relaxation time, s dynamic viscosity, Pa s elongational viscosity, Pa s density, Kg/m3 surface tension, N/m absolute temperature, K yield stress, Pa exponential scaling factor.
References
313
Subscripts d c
1 2
dispersed phase or droplet continuous phase old scale parameter new scale parameter
References 1 van den TEMPEL, M. (1961) J. Coll. Sci., 16, 284; PAPENHUIZEN, J. M. P. (1972) Rheol. Acta., 11, 73. 2 MICHAELS, A. S. and BOLGER, J. C. (1962) I.&.E.C. Fund., 1,153; FRIEND, J. P. and HUNTER, R. J. (1971) J. Coll. Int. Sci., 37, 548. 3 SHERMAN, P. (1963) in Rheology of Emulsions (ed. Sherman, P.), Pergamon Press; SHERMAN, P. (1983) in Encyclopedia of Emulsion Technology, Vol. 1, Chapter 7 (ed. P. Becher), Marcel Dekker Inc. 4 RUSSEL, W. B. (1980)J. Rheology, 24, 287--317. 5 BARNES, H. A. (1981) Dispersion Rheology. Sponsored by the Process Technology Group of the Royal Society of Chemistry's Industrial Division. 6 TADROS, Th. F. (1984) Symposium on The Formation of Liquid-Liquid Dispersions: Chemical and Engineering Aspects, Inst. Chem. Eng./Society Chemical Industry, London, (May). TADROS, Th. F. and VINCENT, B. (1983) Encyclopedia of Emulsion Technology, Vol. 1, Chapter 1 (ed. P. Becher), Marcel Dekker Inc. 7 GRIFFIN, W. C. (1949)J. Soc. Cosmetic Chemists, 1,311. 8 WALSTRA, P. (1983) Encyclopedia of Emulsion Technology, Vol. 1, Chapter 2 (ed. P. Becher), Marcel Dekker Inc. 9 DAVIES, J. T. (1972) Turbulence Phenomena, Chapters 8 to 10, Academic Press. 10 GOPAL, E. S. R. (1968) in Emulsion Science (ed. P. Sherman), Chapter 1, Academic Press; GOPAL, E. S. R. (1963) Rheology of Emulsions (ed. P. Sherman), pp. 15-25, Pergamon Press. 11 CHANDRASEKHAR, S. (1961) Hydrodynamic and Hydromagnetic Stability, Chapters 10 to 12, Clarendon Press. 12 TAYLOR, G. I. (1934) Proc. R. Soc., A146, 501. 13 RUMSCHEIDT, F. D. and MASON, S. G. (1972) J. Coll. Int. Sci., 38, 395. 14 TORZA, S., COX, R. G. and MASON, S. G. (1972) J. Coll. Int. Sci., 38, 395. 15 BARTHES-BIESEL, D. (1972) Ph.D. Dissertation, University of Stanford. 16 KARAM, H. J. and BELLINGER, J. C. (1968) I.&.E.C. Fund, I, 576. 17 GRACE, H. P. (1982) Chem. Eng. Commun., 14, 225. 18 CHIN, H. B. (1978) Ph.D. Dissertation, Polytechnic Institute of New York. 19 CARROLL, B. J. and LUCASSEN, J. (1970) Theory and Practice of Emulsion Technology, p. 29 (ed. A. L. Smith), Academic Press, London. 20 KOLMOGOROFF, A. N. (1949) Dokl. Akad. Nauk. S.S.S.R. (N.S.), 66, 825. 21 HINZE, J. O. (1955) A.I.Ch.E.J., 1,289. 22 KARABELAS, A. J. (1978) A.I.Ch.E.J., 24, 170. 23 KUBIE, J. and GARDENER, G. C. (1977) Chem. Eng. Sci., 32, 195. 24 SHINNAR, R. (1961) J. Fluid. Mech., 10, 259. 25 BROWN, D. E. and PITT, K. (1974) Chem. Eng. Sci., 29, 345. 26 McMANAMEY, W. J. (1979) Chem. Eng. Sci., 34, 432. 27 ARAI, K., KONNO, K., MATNUGA, Y. and SAITO, S. J. (1977) Chem. Eng. Jap., 10, 325. 28 SPROW, F. B. (1967) A.I.Ch.E.J.., 13, 995. 29 RALLINSON, J. M. (1984) Ann. Rev. Fluid Mech., 16, 45. 30 HINCH, E. J. and ACRIVOS, A. (1979) J. Fluid Mechanics, 98, 305. 31 BARTHES-BIESEL, D. and ACRIVOS, A. (1973) J. Fluid Mechanics, 61, 1. 32 CALDERBANK, P. H. (1958) Trans. Inst. Chem. Eng., 36, 443.
314
Dynamics of emulsification
33 LEE, J. C. and TASAKORN, P. (1984) Symposium on The Formation of Liquid-Liquid Dispersions: Chemical and Engineering Aspects. Inst. Chem. Eng./Society Chemical Industry, London, May. 34 WEINSTEIN, B. and TREYBAL, R. E. (1973) A.LCh.E.J., 19, 304. 35 COULALOGLOU, C. A. and TAVLARIDES, L. L. (1976) A.L Ch.E.J., 22, 289. 36 NAGATA, S. (1975) Mixing--Principles and Applications, John Wiley and Sons. 37 PARK, J. Y. and BLAIR, L. M. (1975) Chem. Eng. Sci., 30, 1057. 38 CHAVAN, V. V. (1983) J. Dispersion Science and Technology, 4, 47. 39 FORD, D. E., MASHELKER, R. A. and ULBRECHT, J. (1972) Process Technology, 17, 803. 40 MEWIS, J. and SPAULL, A. J. B. (1976) Coll. Int. Sci., 6, 173. 41 PETERS, D. C., AKAY, G. and STALLER, K. G. (1985) I. Chem. E. Symposium series No. 94., April. 42 DAVIES, J. T. and RIDEAL, E. (1963) Interracial Phenomena, Academic Press, New York. 43 DAVIS, J. T. (1987) Chem. Eng. Sci., 42, 1671.
Appendix 14.1
315
Appendix 14.1:
Numerical example of process design
Consider a process in which the vessel is 2 m in diameter and fully baffled with a 0.7 m diameter, six-bladed turbine agitator. The vessel is filled to the same depth with a mixture of 40% of acrylonitrile (surface tension 26.63 mN.m -~) in 98% sulphuric acid (surface tension 72.0 mN.m -x and density 1,830 kg/rn3). What should be the rotational speed of the agitator to give a d32 of 200/zm? Assume (i) that the temperature is 20"C and that the phase volume ~b = 0.40. (ii) that the surface tension between two immiscible liquids that are mutually saturated with each other can be estimated from the relationship of Good and Girifalco (J. Phys. Chem., 64,561, 1960). trin- nv = trill + crnl- 2K V'(crnn x trn) K is a unique property of components II and III and can be calculated. However, experimental values for different liquid-liquid systems are given in the following table (cf. Good and EIbing, Ind. Eng. Chem., 1970).
Table 14.A.I Liquid-liquid system Fluorocarbonmhydrocarbon near critical mixing temperature Liquid metal-liquid metal Hydrogen-bonded organic liquidmwater Non-polar saturated organicmwater Aromatic hydrocarbon/water
Solution crluu- nl = crni + crll- 2K V'(trtln x trut) = 26.6 + 72.0 - 2(0.55)X/[(26.6)(72)1 - 98.6 - 48.14 = 50.46 m N.m -n Applying the correlation of Lee and Tasakorn 33 (see equation (14.22))
d32
= 0.184(1 + 5.4 ~)Wea ~
D 200 x 10 -6 0.7
= 0.184(1 + 5.4(0.4))We~ ~
K exp. 1.0 1.0 1.0 0.55 0.7
316
Dynamics of emulsification 0.58 x 0.7 x 106) 1.667
t
Wea Wea
200
= (2030) 1"667 = 326 284
Substituting for
Wea,
N2D 3
Pc
or
= 326 284 3.26 x 105 x 50.4
x 10 -3 = 2.62 s -2
i0.7)3 X 1830 N
= 1.62s -~
N
= 97 rpm
.'. N
-- 100 rpm
J ,
(a)
,
,
,
If the phase volume is reduced to 10%, what is the effect on the drop size d32 .9 d32 -
D
0.184 (1 + 5.4 ~)We -~
d32 = 0.7 x 0.184(1 + 5.4 x 0.1) (326 284)-~ m d32 =
d32
-
1.288 x 1.54 x 10-1 (3.263 x 105)o.6 1.98 2.033
x
X 106/xm
102 t~m
d32 = 97.4/~m -- 100/zm The Sauter mean drop size is reduced to ~- 100 Itm or halved by reducing the phase volume from 40-10%. (b)
Do we have turbulent flow? Assume the viscosity of 98% sulphuric acid is 25 mPa.s and that Einstein's equation holds for the viscosity of the dispersion, i.e. /t --= /~o
1 + a~b
where for spheres, a = 5/2.
Appendix 14.1
317
Viscosity of the dispersion is p -
-
(1
+
2.5 ok)
= 25 x (1 + 2.5 x 0.4) = 50 mPa.s. ND 2 Reynolds No = P c - - - 1830 x 1.62 x (0.7) 2 Re = 50 X 10 -3
1.83 x 1.62 x 4.9 x 104 R e --
5.0 Re--- 30000 .'. dispersion is in turbulent flow.
(c)
What is the power per unit volume used to create these emulsions? Since the flow is turbulent, then for a six-bladed turbine the power number is a constant, Po = 5.5. P = P o x Pc x N 3 • D 5 P = 5.5 x 1830 x (1.62) 3 x (0.7) 5 watts P = 5.5 x 1.83 x 4.25 x 0.168 kW P = 7.2 kW 7rxT 3
Filled volume of vessel -
~rx8 V
m
- 6.25 m 3
7.2 Then, power/volume = - - - - kW/m 3 = 1.2 kW/m 3 6.25 Normally for emulsification the power per unit volume is of the order of 1 to 2 kW/m 3.
318
Dynamics o f emulsification
Appendix 14.2 A procedure for scaling up or down non-Newtonian processes In a typical process involving emulsification, there are usually a large number of operations involved; these could all generally be carried out in the same agitated vessel. These operations vary from simple blending to dissolution, reactions, emulsification, solid suspension and to heating and cooling. It is also common for the viscosity and rheology to vary through the process and during each operation. The first step in the scale-up procedure is to identify the individual steps in the process. For each operation, a scale-up equation can be obtained depending on which parameter it is important to keep constant for that operation, Table 1 4 . A . 2 is based on applying the generalized scale-up equation N2 = NI(DIID2) x
to the parameter to be kept constant and defining a value for x.
Table 14.A.2 Generalized scale-up equation Parameter to be kept constant
1. Blend time 2. Froude no.
Function
Scale-up equation
Scale-up exponent x
1/N
N2 = NI
x = 0
DN2/g
N2 - (Dl/O2) 112 Ni
x
--"
112
3. Power/volume 4. Solids suspension
N3DS/D 3
N2 = (DI/D2) 213 N!
x = 2/3
N D 314
N2 = (DIID2) 314 Ni
x = 314
5. Tip speed
ND
N2 = (Dr~D2) Ni
x = 1
N2
x
N2D3p
6. Weber no.
o-
=
(DI/D2) 3/2 Ni
=
3/2
pND 2
7. Reynolds no.
/.~
N2 = (DI/D2) 2 N~
x = 2
8. Pumping no.
Q/ND 3
N2 = (DI/D2) 3 NI
x = 3
The recommended scale-up exponents for various operations outlined in the literature are shown in Table 14.A.3.
Figure 14.A.1 shows how for a pseudoplastic emulsion product the apparent viscosity, #a and shear thinning behaviour expressed as a power law fluid, altered during ten operations during its manufacture"
g . = K(5,).-,
Appendix 14.2 Table 14.A.3
319 Recommended scale-up exponents for various processing operations
Operation
Scale-up exponent, x
Blending Surface behaviour Mass transfer/droplet size Solids suspension Constant velocities
0 l/2 2/3 3/4 1
1000 n = 0.3
-
0.4
800
C
C 0 U tO
soo n = 0.6
m m IX.
E
/
400
0 .m
n =
200
I '0 1
2
3
4
5
6
7
i
I
m
8
9
10
process step
Figure 14A. 1 Viscosity profile during processing where K is the consistency index, Pa.s' and n the power law index. It can be seen that the process starts with a water phase at low viscosity, and up to step 5 the vessel contents are Newtonian with A = 1.0. Beyond step 5, the emulsification process causes the liquid to become increasingly more viscous and it starts to become pseudoplastic with a power law index of 0.6. As the process proceeds to steps 9 and 10, the viscosity increases further and the emulsion becomes very shear thinning in behaviour. The effect on the Reynolds no. in the vessel can be seen in Figure 14.A.2 and it can be seen that in the first three operations the vessel contents are in highly
320
Dynamics of emulsification
10 5
L0 .,0
E
c: 104 'I0 0 C >. rr
10 3
10 2
' 1
i 2
I 3
I 4
,I 5
I 6
a 7
, 8
I, 9
i 10
process step
Figure 14,4.2 Reynolds number for each process step
turbulent motion. However, from steps five onwards, the motion becomes less and less turbulent and more transitional. The next step in the scale-up procedure is to carry out, at three different scales, experiments at each step to determine the agitator speed required to create a constant process result for that step. For example, if it is a dissolution step, introducing solids to the top of the vessel, then the rate or time of dissolution can be chosen. In this way, the scale-up exponent can be established for each stage. Figure 14.A.3 illustrates the scale-up exponents found for the various process steps under consideration. It can be seen from Figure 14.A.3 that up to step 6, a scale-up exponent of 0.67 can be chosen. This corresponds to a power/unit volume scale-up criteria in line with mass transfer and equal droplet sizes in turbulent flow. Above step 6, then, an exponent of 1.0 can be chosen corresponding to a tip speed criteria for scale-up (similar to that found for motion in yield stress fluids). It is important that having established the more demanding scale-up criteria, in this case power/volume is to be held constant. The effect on the other process steps that need equal tip speed as a criteria are not compromised.
Appendix 14.2
321
1.1 1.0 0.9 0.8 0.7 >~ 0.6 0.5 0.4 0.3 0.2 0.1 . I
1
I
I
I
I,
2
3
4
5 process
,.
I
I,,
6
7
.
I
8
..
I
9
.,
I
10
step
Figure 14,4.3 Scale-up exponent for the various process steps
Dynamics of emulsification D C Peters
Vinamul Group Ltd, Warrington, Cheshire
14.1
Introduction
In preparing emulsions, many factors concerning their end-use have to be taken into account. Often, their preparation is merely an enabling step towards, for example, formation of disperse polymer systems, or enhancing a liquid/liquid extraction process. However, in many situations the emulsion is itself the end product. Abundant examples of where this is the case are to be found in the pharmaceutical, food, paint, dairy, agrichemical, cosmetic, adhesives and detergents industries. In these situations, as opposed to those where emulsification is an intermediate step, we are invariably more closely concerned with two key properties - rheology and stability. To the consumer, consistency of product properties represents quality and assurance that the product is functioning in the regularly expected way. The recent introduction of British Standard 5750 has strengthened the need for companies to be certified as approved in order to demonstrate their commitment to quality assurance. Quite apart from this marketing appeal, there can be a very real danger from incorrect function in an unstable product. This is recognized in law in many countries. For example, cosmetics and pharmaceutical products may be legally required to have a specific shelf life during which they show no physical changes. Product liability can lead to expensive litigation and claims. Accepting that rheology and stability are critical properties, how can largescale processes be designed and operated to achieve a given product specification? The approaches open are either (a) to adopt an empirical approach using experimental designs for formulation and processing which allow the 'best' combination to be selected or (b) to use an approach based on understanding the basic physical and chemical phenomena operating within the systems, and hope to build these into a formulation/process. There are advantages and disadvantages to both. The former will generally yield a compromise solution but the gaining of that solution offers little insight into how new products and processes might be developed. On the other hand, the latter should yield a solution, plus an expertise for future developments, but only with a considerable investment in skill and equipment. Of course, neither approach is exclusive and many 'empiricists' make excellent use of basic technology; similarly, no process has yet been designed purely 294
Rheology and stability
295
from a set of equations. The purpose of this chapter is to try and bridge the gap by presenting, in a semi-quantitative way, some of the concepts which are essential to a fuller understanding of emulsification. The approach taken does not purport to provide answers, but perhaps exposes some basic questions. Firstly, the aspects of an emulsion which are important to the product rheoIogy and stability are briefly considered.
14.2 Rheology and stability Most emulsions of interest show shear-thinning rheological behaviour, and many show other non-Newtonian features such as elasticity, yield stress or time dependent effects. All of these can be explained qualitatively, and sometimes quantitatively, by a relatively simple set of parameters (Table 14.1) which takes into acco.unt; droplet size, dd, droplet phase volume 4', conservative (colloid interaction) forces, hydrodynamic forces and interfacial properties tr (Figures
14.2a). Table 14.1 The key properties of emulsions and the basic parameters that control them
Property Viscosity Yield stress Stability Appearance
Relevant parameters do do dd dd
/.to 4~ A0 cb
~b A0
HO
H,,
Individual drops do not interfere with each other appreciably when the internal phase volume ~bis less than 30%. However, as the phase volume increases from 30% to 74% there are increasingly more frequent points of contact and the viscosity increases. For phase volumes between 50% and 52%, the spherical drops are stacked in a cubical arrangement; and as 4) increases to 68%, they arrange themselves in a tetrakaidecahedral array. Above 74% phase volume, the liquid drops are no longer spheres but multifaceted in shape (Figure 14. I). Shear-thinning behaviour is the result of either droplet distortion and alignment with the flow, or due to increasing shear stress causing breakdown of weak flocs with a consequent decrease in effective phase volume (Figure 14.2b). If a system is to possess elasticity, it must possess a physical mechanism for storing energy. In flow, droplet distortion causes an increase in surface free energy (crAA) which is released on cessation of flow, manifesting as, for example, recoverable shear compliance. The magnitude of the (dimensionless) droplet distortion is proportional to the Weber number, We, which is the ratio of the deforming stress (~t~) to the restoring Laplace stress (4tr/dd). Thus, the drop Weber number based on drop radius is defined as:
296
Dynamics of emulsification
Figure 14.1 Multifaceted particles of a Tetradecane/water 77% phase volume emulsion 5 mm = 10 Ixm
We -
~cTdd 20-
(14.1)
where ~ is the viscosity, 7 is the local shear rate and d d is the droplet diameter. Accordingly, small drops deform proportionately less; with the implication therefore that droplet size is important to the elastic flow properties. The property of yield stress arises in the case where the droplets form a continuous network throughout the system; for instance, under the influence of van der Waals forces I. In such a case, a complete network of deformable droplets has a yield stress, Zy, given by:
Aj 4'
7.v ~ ___;m " H 0 dd
(14.2)
and the network modulus, G, is a function of the interracial tension:
4,
G ~ tr-do
(14.3)
Time and stress dependent flow properties are generally modelled 2 via the making and breaking of inter-droplet bonds, such that at any given stress level there is a distribution of singlets, doublets, triplets, etc. and the rate of change of
297
Rheology and stability (a)
(b)
no flow condition
flow condition shear
attractive/repulsive forces
,
0 Q
,,,
,
00 0
drop size and phasevolume
floc distortion and breakdown; droplet collision
to, ;phase )cculated tops
Q internal circulation .
network connectivity
.
.
.
.
distortion to the point of break-up .
.
.
.
.
.
.
.
.
_
_
shear
Figure 14.2 Diagrammatic representation of the parameters which are important to emulsion properties
distribution on changing the stress level reflects the thixotropic change noted experimentally. Clearly, both droplet-droplet collision, essential for 'making' bonds, and multiplet breakdown (controlled by attractive forces) are strong functions of droplet size. The stability of an emulsion depends largely on the balance of attractive and repulsive forces and Figure 14.3 shows a primary and secondary minima system being the stablizing mechanism. If only attractive forces exist then the droplets coalesce, thereby lowering the system free energy (except in the event of extremely small particles where the entropic form can dominate). On the other hand, if repulsive forces dominate, then even very small density differences are sufficient to give rise to creaming, i.e. phase separation, for example. What is required for stability is a flocculation of droplets, which does not give rise to coalescence, to form a network throughout the system 3. The role of colloidal forces in the rheology of suspensions has recently been reviewed by Russel4. In Figure 14.3 a secondary minimum stabilizing mechanism is shown where stability is achieved simply through an appropriate balance of attractive and
298
Dynamics of emulsification +re
% short-range Born effects
~~
electrical %% (repulsive) energy %
energy I
% % % % % % %
repulsive
%
%
distance
0
energy I f I
/
attractive
/
.~. ~ "
~ ~ van der Waals (attractive) energy
.="on'4ar" --~ -minimum
f
f
/ /
\, -ve
/
primary minimum (often leads to coalescence)
Figure 14.3 Attraction and repulsion forces as a function of surface-surface distance
repulsive electrical forces. Other stabilizing mechanisms exist such as the use of absorbed species which present an entropic hindrance to the close approach of droplets; or the use of bridging polymers or finely divided solids. Each mechanism leads to a different strength of interaction and, generally, a different response to stress. In practice the stabilization of emulsions is induced by the use of surface active agents or emulsifiers. The surfactant structure can be represented diagrammatically and the way it orientates itself at an oil/waterphase interface is shown in Figure 14. 4. The application of mechanical energy creates droplets and for an oil-in-water emulsion the tails of the surfactant molecule point into the oil droplets and vice versa for a water-in-oil emulsion. These structures are known as spherical micelles. With increased concentration of the surfactant in the system, the micelles can become highly ordered and produce complex rheologicai behaviours through the formation of various phases 5. A schematic illustration of the various phases produced from increasing surfactant concentration are given in Figure 14.5. The interfacial aspects of emulsification, including thermodynamics of emulsion formation and breakdown, have been reviewed and described recently by Tadros 6. The role of emulsifiers is discussed in detail and the mechanisms outlined, although complex, are related to the particle size if for no other reason than that the number density is proportional to 1/(dd).
Rheology and stability
299
Figure 14.5 Structure produced by ordering of micelles 1. Molecular solution. 2. Spherical micelles. 3. Aggregated spherical micelles. 4. Cylindrical micelles. 5. Hexagonal phase. 6. Disc micelles. 7. Vesicles. 8. Lamellar sheets. 9. Crystalline
There are a large number of emulsifiers available and making a choice for a new formulation or product is difficult. The use of the hydrophilic-lipophilic balance (HLB) concept originated by Griffin in 19497 can be a useful empirical method of preliminary selection. The HLB number quantifies the balance of hydrophilic-lipophilic characteristics of the surfactant molecule on an arbitrary numerical scale. The least hydrophilic surfactants are assigned the lowest HLB (see Table 14.2). The use of the HLB balance is also well described by Davies and Ridea142.
300
Dynamics of emulsification
Table 14.2 HLB values of surfactants
HLB
Application
3-6 7-9 8--15 13-15 15-18
W/O emulsions Wetting agents O/W emulsions Detergents Solubilizers
Dispersibility in water 1-4 3-6 6-8 8-10 10-13 13-
Nil Poor Unstable milky dispersions Stable milky dispersions Translucent dispersions/solution Clear solution
Reconsidering the factors which can be seen to be important to rheology and stability, and over which some control is possible during processing, then clearly droplet size is a key parameter (Table 14.1). In the following sections, some fundamental aspects of droplet formation are examined, in order to yield some insight into the design and operation of processes for controlled droplet formation. However, in passing it should be noted that the way in which the emulsifier is added is important and that for high internal phase emulsions the material of construction of the making vessel is critical, i.e. the solid-liquid interface is important too.
14.3 Droplet formation In very general terms, droplets are formed by stress being imparted to a large primary drop, causing elongation of all or part of it, followed by development of surface wave growth to the point of instability, whereby the primary drop breaks into droplets and, often smaller satellite droplets. The disruption of a plane surface separating two bulk phases, necessary to produce drops, threads or mis-shaped lumps of dispersed phase can be achieved 8 via turbulent eddies 9, surface ripples, Rayleigh-Taylor instabilities l~ and Kelvin-Helmholtz instabilities I~. The factors important to these processes are" (a) the viscous and elastic properties of the disperse and continuous phases; (b) the interfacial properties; (c) the flow conditions. There are a number of difficulties in examining the roles of these factors either experimentally or theoretically. The key difficulty is that, in a practical sense, emulsification does not take place under steady conditions, but under dynamic conditions where time scales may be of the order of seconds down to 10-6 seconds s. However, it has to be assumed that the direction of effects is independent of time scale. Then, a combination of steady-state effects with a knowledge of how time scales might modify their magnitude can be used.
Droplet formation 14.3.1
301
Deformation and breakup in steady flows
This problem is first studied by Taylor t2 and more recently by others ~3-~5. Basically the approaches made consider how the shearing forces acting on the droplet give rise to a pressure field around it, which can then be translated into a shape by using the Laplace equation,
AP =
o"
+
(14.4)
(see Figure !4.6). The deformation of droplets (Figure 14.6) depends on two parameters, the Weber number, defined earlier, and the viscosity ratio, R, where: R=
viscosity of drop phase
~d = -viscosity of continuous phase ~
(14.5)
At low viscosity ratios, R < 0.2, drops in shear flow first align themselves at 45~ to the flow in the form of ellipsoidai drops. Increasing flow and hence shear creates drops with pointed tails that break off to create satellite drops. Drops with viscosity ratios of ~ 1.0 form dumb-bells that with increases shear burst into two spherical drops with some satellite drops. Increasing R tends to produce long slender drops, R > 3.8 drops do not break but form ellipsoidal drops aligned with the shear field. In elongational flow (Figure 14. 7) and small R, ellipsoidal drops parallel to the flow and with trailing ends are formed, sometimes accompanied with smaller satellite drops. Drops with large R in elongational flow give ellipsoidal drops aligned with the flow and no breakup. Taylor 12obtained the following relationship for deformation of an ellipsodial drop: Deformation, D' -
=We
L-
B
L+B
-
~'ddl~c(19/16R + l ) 2o-
9
R+I
19/16R + 1) R+I
(14.7)
v
~
(14.6)
Laplaceforces fluidforces
Fisure 14.6 Relationshipbetweenfluid shear and Laplacesurfaceforces
302
Dynamics of emulsification
''
J
o O O J t~
0.,. 0
6.,,=.,, 0
"
6. Rlarge @
O
~~-..--..J
Figure 14.7 Possible de[ormation patterns (1--4 shear, 5--6 extension) 13.14
Drop rupture occurred when D' > D~ and this critical deformation was a singled valued function of R. Karam 16has shown experimentally (Figure 14.8) the form of this function. Similar results have been reported by Grace 17. In general there is a minimum as well as a maximum viscosity ratio beyond which the drop does not burst in
1210+
rr
rA
8-
drop deformation only
+ 6 420
,
0.001
.
.,I
0.01
.
---~----:'-~-,,i---,---,'r''T
0.1
....
1
viscosity ratio, pd/Ptc
10 =
R
Figure 14.8 Critical shear rate ('j,) for drop break-up versus R 17 ( --- extension)
100
....... s h e a r ,
Droplet formation
303
laminar shear fields. In addition for an ellipsoidal drop (Figure 14.7) for large We and small R, the deformation, D', is given by:
L-B L+B
D'-
oc
Wc
(14.S)
whereas for R >> 1 and small We, then: 1
O' oc-R
(14.9)
Grace ~7has also reported comparisons between simple shear (Couette flow) versus extensional irrotational shear (4 roll mill). These are shown in Figure 14.8 with the full line representing shear and dotted line extension. It can be seen that for equal shear rates extensional shear produces more effective break-up and dispersion than simple rotational shear, even at low viscosity ratios. These findings are substantiated in a recent review by Rallinson29of viscous drop deformation in shear flows. Figure 14. 9 summarizes the experimental and theoretical data for extensional flows. The corresponding comparison for simple shear is given in Figure 14.10. The comparison of Figures 14.9 and 14.10 substantiate that extensional shear is more effective than simple shear in promoting drop break-up. The break-up of droplets in steady flow is generally approached through consideration of the hydrodynamic stability of liquid cylinders, which are assumed to be a precursor of break-up. The critical factor for the rate of break-up of a given cylinder is the rate at which instabilities grow, whereas for the ultimate drop size, it is the wavelength of the instability which dominates (Figure 14.11). 0.5
-
0.4
".."%,,
0.3 0 *U
o.,
%
0.1
.
J
0 *
J
10 -4
10 -:~
,
J
10 -2
4. . . . . .
10 -1
! .....
1
,,,,0 ~ g " i
J
I
10
10 2
10 3
Figure 14.9 Comparison of critical We for extensional flow for experimental data and theory (..... experimental1?; asymptotictheory3~ o theory31)
304
Dynamics of emulsification 10 3
,
!
10 -
I-
~
I ! ! !
o
0
10 -1
I
I
I
10 - 3
10 - 4
10 - 3
,
|
10 - 2
,
I
I
10 -'t
1
,
I 10
R
Figure 14.10 Comparison of Wet for simple shear for experimental data and theory (..... experimental nT; ~ asymptotic theory3~ 0 theory 3~)
Figure 14.11
Waves on a cylinder
A typical theoretical result for the rate of growth of instability of a cylinder ~s is given in Figure 14.12. The ordinate, Sp, represents growth rate of instabilities while a, the wave number, represents the wavelength of the instability expressed as a ratio to the cylinder circumference. There is clearly a strong dependence on viscosity ratio; the larger the ratio the smaller the growth rate. The Weber number again plays an important role (Figure 14.13) with decreasing We leading to an increasing instability growth rate, i.e. increasing interfacial tension destabilizes the system. However, decreasing We also causes a reduction in deformation and therefore, with respect to drop break-up, there are two competing effects. Turning to the ultimate drop size, it is generally assumed that the wavelength, Am, corresponding to the maximum instability growth rate dominates break-up. The critical deformation, D,~t, can be expressed as: oc,, = Lc, - ,
L.it + B =
+
(14.10)
Droplet formation Sp
305
R " 0.001
0.01
0.1
1.0 1.0
0
Figure 14.12 Stability of Newtonian systems 18
Sp
W, - 1 0
1G 20 2G
0
1.0
Figure 14.13 Effect of Weber number on the stability of Newtonian systems 18
where am = Am/2~"to. From this analysis, the resulting drop size, rid, is given by" do = 2to
[
32~m] t/3
(14.11)
306
Dynamics of emulsification
which is a very weak function of am but a strong function of the cylinder radius, to. Control of drop size is therefore best effected by control of ro and, assuming a 1:1 relationship between ro and the deformability of the primary droplet, it can be concluded that Weber number and viscosity ratio are critical parameters.
14.3.2 Dynamic effects The most important practical dynamic effects are effects associated with rapid stretching of a droplet, i.e. surface dilation and bulk elastic effects. The former is important where surface active materials are present. In the equilibrium state, the droplet will have an adsorbed monolaycr of surfactant and, in consequence, the interfacial tension will be lowered. On stretching, the interfacial tension rises because the surfactant molecules cannot respond instantaneously, and therefore become depleted in terms of the number/unit area. In this way, the Weber number falls and deformation is less than might be expected from a knowledge of the equilibrium interracial tension alone. As time passes, further adsorption from the bulk may occur, or the droplet may contract back to a spherical shape under the influence of its new-found high interfacial tension. The exact circumstances will depend strongly on surfactant properties and concentration. A further manifestation of this effect is discussed in detail elsewhere 19. If the droplet material has viscoelastic properties, characterized by a relaxation time At, then the behaviour of the drop will depend on the timescale of the applied deforming stress. For very short times (<~ At), it might be expected that the drop will appear as an elastic solid, whereas at long times (>~ At) the viscous nature will dominate. A knowledge of the material rheology across a range of timescales is therefore important for a fuller understanding.
14.3.3 Turbulence So far steady flow conditions have been considered but in practice emulsification is very often carried out under turbulent conditions. Fortunately, the same physical concepts apply but this time in the context of a rapidly changing flow field. Under such conditions the Weber number cannot be defined using a steady-state shear rate and is reformulated as: We' = pcfl2(dd) dd O" -
-
(14.12)
where Pc is the continuous phase density and I)2(dd) is the root mean square of the different of velocities at a distance apart do of the turbulent fluctuation velocity.
Droplet formation
307
Kolmogoroff 2~ suggested that a drop will break if the pressure across it is similar to the pressure due to surface tension holding the drop together; that is, (dd)ma~ = C (--~--c)0"6 ef ~
(14.13)
where eT is the mean turbulent energy dissipation rate or the power dissipated per unit masg of fluid in the turbulent region. In Couette flow, Hinze 2~ found that the constant in the above equation was 0.725 and for turbulent flow in pipes, other workers have confirmed this constant 22'23. In stirred vessels the majority of workers have found that
dd oc E.170"4
(14.14)
with the exception of Shinnar 24 who reported dd oc ef 0.25. The impeller design has been shown to vary the constant C, and the Sauter mean drop diameter was found to be proportional to a circulation time tc25 thus:
d32 oc
(14.15)
The data for turbine impellers have all been collated together by McManamey 26 using the concept of the swept out mass of liquid by the rotating impeller when calculating eT.
d32 = C(~c ) 0.6N-l'2D-~
(14.16)
This equation can be expressed in the form of dimensionless groups:
d32 -- C!Po-~ D
0'6
(14.17)
P pcN3D5 ' pcN2D 3 and We~ = agitation Weber number = tr
where Po = power number =
In the theories used to obtain equation (14.13), the viscosity of the dispersed phase was neglected, but Hinze 21 has pointed out that if the viscosity of the dispersed phase within the drop is significant but the viscosity of the continuous phase is not, then the turbulent flow is still dominated by inertial forces. Then, a viscosity group is needed to be introduced, such as:
308
Dynamics of emulsification
/xa Nvi - (PdO.dd)I/2
(14.18)
This viscosity group is introduced to modify the We number by We'(1 + f(Nvi)) where f(N~i) ~ 0 as N~i ~ 0. A reasonably successful approach to finding the maximum stable drop size (dd)m~ in turbulence using this approach has been made by Arai et al. 27. For drop sizes much larger than the Kolmogoroff length scale, i.e.
']
1/4
10-4 m
(14.19)
in aqueous systems, they showed that theoretically ,,.2/3/',4 /5/3 PceT ~.t~dhnax
,,(1
+
f(Nv0)
= const.
(14.20)
eT is the energy dissipation rate per unit mass (aN3D 2 for fully developed turbulence flow in baffled vessels) and N~i = #der1/3(dd)ma~/tr 1/3 allows for the effect of a finite disperse phase viscosity, f (N,i) has to be determined experimentally. In attempting to confirm their expression experimentally, Arai et al. noted a strong interaction between drop viscosity and interfacial tension. Thus, for a low viscosity drop, one might expect a rapid response to the periodic turbulent stress, i.e. the droplet would rapidly adjust its shape. On the other hand, viscous drops having a response time longer than the periodicity will be subject to enhanced deformation by consecutive stress cycles. For drops smaller than the Kolmogoroff length scale, shear forces rather than periodic inertial forces dominate the break-up mechanism and theoretically28
erPc)
= const.
(14.21)
Other authors have attempted to allow for the influences of viscosity by introducing a factor (#dgc) '~where a is found empirically. Calderbank 32found that a = 0.25 for the mean drop size in his agitation system. In order to arrive at a drop size distribution, it is necessary to take coalescence into account and to ascribe an energy spectrum to the turbulent flow. Lee 33 reviews and presents results for drop size distribution produced by turbine agitators. Dibutyl phthalate agitated for 60 minutes gave the following correlation.
Implications for process design d32 D
- 0.184(1 + 5.4 ~b)We~~
309 (14.22)
where ck is the volume of the dispersed phase. Other workers 34-36 have reported that drop sizes close to the impeller were some 10% smaller than further away, which suggests that for the systems they studied the effect of coalescence is not large. Park 37 found that once equilibrium has been established, break-up occurred only close to the impeller, with coalescence predominating further away. This would be expected when energy dissipation is non-uniform as found in a stirred vessel. All authors find that the drop diameter increases with volume fraction ~ of the dispersed phase provided ~k > ---0.01. A review of disruption of drops by cavitation is given by Walstra s.
14.4
Implications for process design
A strong semi-quantitative link has been made between rheology/stability and drop size and between drop size and processing conditions. A fundamental approach therefore requires a study of both links. The first is outside the scope of these notes, but the literature contains a great deal of useful information 4~ Some comments on the second are made below.
14.4.1 Batch processing The variables available in a typical baffled batch mixer of standard geometry are impeller geometry and speed, temperature, time and the entry point of the second phase. Each of these is considered below. Impellers are available which provide for different distribution and stress patterns. For achieving small drops, an impeller having a high power number/ discharge ratio is preferred and vice versa for large drops. However, where some control of drop size distribution is required, consideration might be given to recirculation via a high energy device, or discharge via such a device. Temperature may play a strong role in changing the viscosity ratio of the two phases. For two Newtonian fluids obeying the Arrhenius-type equation, temperature independent activation energies Ed and Ec, we find:
ddo (InR) =
tER-go- '/
(14.23)
where 0 is the absolute temperature and Rg is the gas constant. In many systems, either or both E~ and Ed may be strongly temperature dependent. Such would particularly be the case where a phase change occurs. Again use may be made of such phenomena to control ultimate drop size. Temperature effects will be more complicated where emulsifiers are present because of
310
Dynamics of emulsification
temperature-dependent changes in adsorption, configuration at the interface, interfacial tension, phase structure, etc. and these effects must be studied separately. Time of mixing is important from the point of view of (a) ensuring adequate gross mixing, (b) ensuring equilibrium drop-size distribution 33'38 is achieved and (c) avoidance of overprolonged mixing with its penalties of higher energy costs, capacity or throughput problems, and possible damage to the product. The effect of prolonged agitation at different speeds on the mean droplet size dd is shown in Figure 14.144:. A model oil-in-water emulsion of 20% phase volume was used. In this agitation system, increasing the speed from 350 to 500 rpm did not produce further reduction in the mean droplet diameter. For these reasons, it is important to have an idea as to the limits to which a product can be 'pushed' in terms of intensity/time of mixing together with a good knowledge of the mixing time for systems of comparable rheology. For the latter, model studies using tracers, dyes, etc. are widely used. A numerical example of process design is given in Appendix 1. The entry point of the second phase can have a pronounced effect. For rate-of-mixing purposes and uniformity of drop-size distribution, it is claimed that entry close to the impeller tip is advantageous. This effect becomes particularly important where the temperatures of the two liquids are not equal.
40 N (rpm)
-'0"9- d k - ,
E
2OO 280 350
-.0-
5oo
.N -~ Q.
20
0 I 0
I
I 40
....
I
I .......... 80
I
! 120
mixing time t (min)
Figure 14.14 The variation of average droplet diameter, dd, with mixing time t at various angular speeds, N, when T = 21.5 cm, D = 0.67 T using an inclined paddle and three baffles.
Implications ]'or process design
311
The scaling up or down of mixing systems with Newtonian rheological characteristics is comprehensively covered in the literature a6 and in this book. However, the majority of emulsions with medium phase volume (30-55%) exhibit non-Newtonian behaviour. Literature is bereft of methods to scale up non-Newtonian systems. An approach is presented in Appendix 2 based on a generalized scale-up equation for agitated systems where: N2 = N~(D~/D2) x
(14.24)
with N2 and D2 being the new rotational speed and vessel diameter, N~ and D~ the old vessel speed and diameter and x the scaling factor.
14.4.2 ContinuoUs processing It is probably in continuous processing that a deeper knowledge of the fundamental physiochemical molecular effects pays dividends. Generally, a continuous emulsification process consists of two essential elements: a proportioning pump to ensure correct formulation, and a shearing device to ensure correct dispersion of the phases. This apparent simplicity belies the true complexity of such a process. For instance, where an emulsifier is present in one or both of the liquid phases, the state of adsorption at the point of shear will play an important role in determining the resulting drop size. Thus a dependence can be clearly seen on the contact time (and contact area) available between the pump and the shearing device. Davies has shown that many of the equations applied to stirred tanks are applicable here too 42'43. This will depend on pipe dimensions, on the use of in-line devices to produce a coarse initial dispersion ~7 and on the flowrate which, typically for a positive displacement pump, is pulsed! These are some of the factors upstream of the shearing device; downstream one has to consider the flow conditions pertaining immediately after the stretching zone, and the relationship between pipe dimensions and flow rate and the coalescence or further break-up of droplets. There are no simple answers in seeking a design for either batch or continuous processes. However, research and development can usefully be guided by fundamental studies of equilibrium and dynamic effects found from model studies.
Notation A Ad B C D
Hamaker constant area of droplets, m2 breadth of an ellipsoidal drop, m constant impeller diameter, m
312 D
F
D~
dd
(dd)max d32 E G
Ho L Lcrit
N n
ziP Po R
ro
RIR2 Rg Sp T tc
2(dd)
V We We' Wea ot fitm
ET
5,
K A Am At # #e P o"
0 ry X
Dynamics of emulsification deformation, dimensionless critical deformation leading to breakage, dimensionless droplet size, m maximum droplet size, m Sauter Mean Diameter, m activation emergency for viscous flow (J mol -~) network modulus, Nm -2 surface-surface separation between drops, m length of an ellipsoidai drop, m critical length of an ellipsoidal drop, m impeller speed, rev/s viscosity correlation number, dimensionless powder law index, dimensionless pressure difference, Pa power number, dimensionless #d/#r dimensionless diameter of a deforming cylinder, m principal radii of an ellipsoidal surface, m gas constant (J mol -~ K -~) growth rate of instability, s -~ tank diameter,, m circulation time, s mean square of the difference in velocity over a distance dd, (m/s) 2 vessel volume, m 3 laminar shear Weber number, dimensionless turbulent Weber number, dimensionless agitator Weber number, dimensionless A/2crro, dimensionless Am/2crro, dimensionless mean turbulent energy dissipation rate, W/kg phase volume extensional shear rate, s -~ laminar shear rate, s -~ consistency Pa.s '~ wavelength of an instability, m maximum wavelength before break-up, m relaxation time, s dynamic viscosity, Pa s elongational viscosity, Pa s density, Kg/m3 surface tension, N/m absolute temperature, K yield stress, Pa exponential scaling factor.
References
313
Subscripts d c
1 2
dispersed phase or droplet continuous phase old scale parameter new scale parameter
References 1 van den TEMPEL, M. (1961) J. Coll. Sci., 16, 284; PAPENHUIZEN, J. M. P. (1972) Rheol. Acta., 11, 73. 2 MICHAELS, A. S. and BOLGER, J. C. (1962) I.&.E.C. Fund., 1,153; FRIEND, J. P. and HUNTER, R. J. (1971) J. Coll. Int. Sci., 37, 548. 3 SHERMAN, P. (1963) in Rheology of Emulsions (ed. Sherman, P.), Pergamon Press; SHERMAN, P. (1983) in Encyclopedia of Emulsion Technology, Vol. 1, Chapter 7 (ed. P. Becher), Marcel Dekker Inc. 4 RUSSEL, W. B. (1980)J. Rheology, 24, 287--317. 5 BARNES, H. A. (1981) Dispersion Rheology. Sponsored by the Process Technology Group of the Royal Society of Chemistry's Industrial Division. 6 TADROS, Th. F. (1984) Symposium on The Formation of Liquid-Liquid Dispersions: Chemical and Engineering Aspects, Inst. Chem. Eng./Society Chemical Industry, London, (May). TADROS, Th. F. and VINCENT, B. (1983) Encyclopedia of Emulsion Technology, Vol. 1, Chapter 1 (ed. P. Becher), Marcel Dekker Inc. 7 GRIFFIN, W. C. (1949)J. Soc. Cosmetic Chemists, 1,311. 8 WALSTRA, P. (1983) Encyclopedia of Emulsion Technology, Vol. 1, Chapter 2 (ed. P. Becher), Marcel Dekker Inc. 9 DAVIES, J. T. (1972) Turbulence Phenomena, Chapters 8 to 10, Academic Press. 10 GOPAL, E. S. R. (1968) in Emulsion Science (ed. P. Sherman), Chapter 1, Academic Press; GOPAL, E. S. R. (1963) Rheology of Emulsions (ed. P. Sherman), pp. 15-25, Pergamon Press. 11 CHANDRASEKHAR, S. (1961) Hydrodynamic and Hydromagnetic Stability, Chapters 10 to 12, Clarendon Press. 12 TAYLOR, G. I. (1934) Proc. R. Soc., A146, 501. 13 RUMSCHEIDT, F. D. and MASON, S. G. (1972) J. Coll. Int. Sci., 38, 395. 14 TORZA, S., COX, R. G. and MASON, S. G. (1972) J. Coll. Int. Sci., 38, 395. 15 BARTHES-BIESEL, D. (1972) Ph.D. Dissertation, University of Stanford. 16 KARAM, H. J. and BELLINGER, J. C. (1968) I.&.E.C. Fund, I, 576. 17 GRACE, H. P. (1982) Chem. Eng. Commun., 14, 225. 18 CHIN, H. B. (1978) Ph.D. Dissertation, Polytechnic Institute of New York. 19 CARROLL, B. J. and LUCASSEN, J. (1970) Theory and Practice of Emulsion Technology, p. 29 (ed. A. L. Smith), Academic Press, London. 20 KOLMOGOROFF, A. N. (1949) Dokl. Akad. Nauk. S.S.S.R. (N.S.), 66, 825. 21 HINZE, J. O. (1955) A.I.Ch.E.J., 1,289. 22 KARABELAS, A. J. (1978) A.I.Ch.E.J., 24, 170. 23 KUBIE, J. and GARDENER, G. C. (1977) Chem. Eng. Sci., 32, 195. 24 SHINNAR, R. (1961) J. Fluid. Mech., 10, 259. 25 BROWN, D. E. and PITT, K. (1974) Chem. Eng. Sci., 29, 345. 26 McMANAMEY, W. J. (1979) Chem. Eng. Sci., 34, 432. 27 ARAI, K., KONNO, K., MATNUGA, Y. and SAITO, S. J. (1977) Chem. Eng. Jap., 10, 325. 28 SPROW, F. B. (1967) A.I.Ch.E.J.., 13, 995. 29 RALLINSON, J. M. (1984) Ann. Rev. Fluid Mech., 16, 45. 30 HINCH, E. J. and ACRIVOS, A. (1979) J. Fluid Mechanics, 98, 305. 31 BARTHES-BIESEL, D. and ACRIVOS, A. (1973) J. Fluid Mechanics, 61, 1. 32 CALDERBANK, P. H. (1958) Trans. Inst. Chem. Eng., 36, 443.
314
Dynamics of emulsification
33 LEE, J. C. and TASAKORN, P. (1984) Symposium on The Formation of Liquid-Liquid Dispersions: Chemical and Engineering Aspects. Inst. Chem. Eng./Society Chemical Industry, London, May. 34 WEINSTEIN, B. and TREYBAL, R. E. (1973) A.LCh.E.J., 19, 304. 35 COULALOGLOU, C. A. and TAVLARIDES, L. L. (1976) A.L Ch.E.J., 22, 289. 36 NAGATA, S. (1975) Mixing--Principles and Applications, John Wiley and Sons. 37 PARK, J. Y. and BLAIR, L. M. (1975) Chem. Eng. Sci., 30, 1057. 38 CHAVAN, V. V. (1983) J. Dispersion Science and Technology, 4, 47. 39 FORD, D. E., MASHELKER, R. A. and ULBRECHT, J. (1972) Process Technology, 17, 803. 40 MEWIS, J. and SPAULL, A. J. B. (1976) Coll. Int. Sci., 6, 173. 41 PETERS, D. C., AKAY, G. and STALLER, K. G. (1985) I. Chem. E. Symposium series No. 94., April. 42 DAVIES, J. T. and RIDEAL, E. (1963) Interracial Phenomena, Academic Press, New York. 43 DAVIS, J. T. (1987) Chem. Eng. Sci., 42, 1671.
Appendix 14.1
315
Appendix 14.1:
Numerical example of process design
Consider a process in which the vessel is 2 m in diameter and fully baffled with a 0.7 m diameter, six-bladed turbine agitator. The vessel is filled to the same depth with a mixture of 40% of acrylonitrile (surface tension 26.63 mN.m -~) in 98% sulphuric acid (surface tension 72.0 mN.m -x and density 1,830 kg/rn3). What should be the rotational speed of the agitator to give a d32 of 200/zm? Assume (i) that the temperature is 20"C and that the phase volume ~b = 0.40. (ii) that the surface tension between two immiscible liquids that are mutually saturated with each other can be estimated from the relationship of Good and Girifalco (J. Phys. Chem., 64,561, 1960). trin- nv = trill + crnl- 2K V'(crnn x trn) K is a unique property of components II and III and can be calculated. However, experimental values for different liquid-liquid systems are given in the following table (cf. Good and EIbing, Ind. Eng. Chem., 1970).
Table 14.A.I Liquid-liquid system Fluorocarbonmhydrocarbon near critical mixing temperature Liquid metal-liquid metal Hydrogen-bonded organic liquidmwater Non-polar saturated organicmwater Aromatic hydrocarbon/water
Solution crluu- nl = crni + crll- 2K V'(trtln x trut) = 26.6 + 72.0 - 2(0.55)X/[(26.6)(72)1 - 98.6 - 48.14 = 50.46 m N.m -n Applying the correlation of Lee and Tasakorn 33 (see equation (14.22))
d32
= 0.184(1 + 5.4 ~)Wea ~
D 200 x 10 -6 0.7
= 0.184(1 + 5.4(0.4))We~ ~
K exp. 1.0 1.0 1.0 0.55 0.7
316
Dynamics of emulsification 0.58 x 0.7 x 106) 1.667
t
Wea Wea
200
= (2030) 1"667 = 326 284
Substituting for
Wea,
N2D 3
Pc
or
= 326 284 3.26 x 105 x 50.4
x 10 -3 = 2.62 s -2
i0.7)3 X 1830 N
= 1.62s -~
N
= 97 rpm
.'. N
-- 100 rpm
J ,
(a)
,
,
,
If the phase volume is reduced to 10%, what is the effect on the drop size d32 .9 d32 -
D
0.184 (1 + 5.4 ~)We -~
d32 = 0.7 x 0.184(1 + 5.4 x 0.1) (326 284)-~ m d32 =
d32
-
1.288 x 1.54 x 10-1 (3.263 x 105)o.6 1.98 2.033
x
X 106/xm
102 t~m
d32 = 97.4/~m -- 100/zm The Sauter mean drop size is reduced to ~- 100 Itm or halved by reducing the phase volume from 40-10%. (b)
Do we have turbulent flow? Assume the viscosity of 98% sulphuric acid is 25 mPa.s and that Einstein's equation holds for the viscosity of the dispersion, i.e. /t --= /~o
1 + a~b
where for spheres, a = 5/2.
Appendix 14.1
317
Viscosity of the dispersion is p -
-
(1
+
2.5 ok)
= 25 x (1 + 2.5 x 0.4) = 50 mPa.s. ND 2 Reynolds No = P c - - - 1830 x 1.62 x (0.7) 2 Re = 50 X 10 -3
1.83 x 1.62 x 4.9 x 104 R e --
5.0 Re--- 30000 .'. dispersion is in turbulent flow.
(c)
What is the power per unit volume used to create these emulsions? Since the flow is turbulent, then for a six-bladed turbine the power number is a constant, Po = 5.5. P = P o x Pc x N 3 • D 5 P = 5.5 x 1830 x (1.62) 3 x (0.7) 5 watts P = 5.5 x 1.83 x 4.25 x 0.168 kW P = 7.2 kW 7rxT 3
Filled volume of vessel -
~rx8 V
m
- 6.25 m 3
7.2 Then, power/volume = - - - - kW/m 3 = 1.2 kW/m 3 6.25 Normally for emulsification the power per unit volume is of the order of 1 to 2 kW/m 3.
318
Dynamics o f emulsification
Appendix 14.2 A procedure for scaling up or down non-Newtonian processes In a typical process involving emulsification, there are usually a large number of operations involved; these could all generally be carried out in the same agitated vessel. These operations vary from simple blending to dissolution, reactions, emulsification, solid suspension and to heating and cooling. It is also common for the viscosity and rheology to vary through the process and during each operation. The first step in the scale-up procedure is to identify the individual steps in the process. For each operation, a scale-up equation can be obtained depending on which parameter it is important to keep constant for that operation, Table 1 4 . A . 2 is based on applying the generalized scale-up equation N2 = NI(DIID2) x
to the parameter to be kept constant and defining a value for x.
Table 14.A.2 Generalized scale-up equation Parameter to be kept constant
1. Blend time 2. Froude no.
Function
Scale-up equation
Scale-up exponent x
1/N
N2 = NI
x = 0
DN2/g
N2 - (Dl/O2) 112 Ni
x
--"
112
3. Power/volume 4. Solids suspension
N3DS/D 3
N2 = (DI/D2) 213 N!
x = 2/3
N D 314
N2 = (DIID2) 314 Ni
x = 314
5. Tip speed
ND
N2 = (Dr~D2) Ni
x = 1
N2
x
N2D3p
6. Weber no.
o-
=
(DI/D2) 3/2 Ni
=
3/2
pND 2
7. Reynolds no.
/.~
N2 = (DI/D2) 2 N~
x = 2
8. Pumping no.
Q/ND 3
N2 = (DI/D2) 3 NI
x = 3
The recommended scale-up exponents for various operations outlined in the literature are shown in Table 14.A.3.
Figure 14.A.1 shows how for a pseudoplastic emulsion product the apparent viscosity, #a and shear thinning behaviour expressed as a power law fluid, altered during ten operations during its manufacture"
g . = K(5,).-,
Appendix 14.2 Table 14.A.3
319 Recommended scale-up exponents for various processing operations
Operation
Scale-up exponent, x
Blending Surface behaviour Mass transfer/droplet size Solids suspension Constant velocities
0 l/2 2/3 3/4 1
1000 n = 0.3
-
0.4
800
C
C 0 U tO
soo n = 0.6
m m IX.
E
/
400
0 .m
n =
200
I '0 1
2
3
4
5
6
7
i
I
m
8
9
10
process step
Figure 14A. 1 Viscosity profile during processing where K is the consistency index, Pa.s' and n the power law index. It can be seen that the process starts with a water phase at low viscosity, and up to step 5 the vessel contents are Newtonian with A = 1.0. Beyond step 5, the emulsification process causes the liquid to become increasingly more viscous and it starts to become pseudoplastic with a power law index of 0.6. As the process proceeds to steps 9 and 10, the viscosity increases further and the emulsion becomes very shear thinning in behaviour. The effect on the Reynolds no. in the vessel can be seen in Figure 14.A.2 and it can be seen that in the first three operations the vessel contents are in highly
320
Dynamics of emulsification
10 5
L0 .,0
E
c: 104 'I0 0 C >. rr
10 3
10 2
' 1
i 2
I 3
I 4
,I 5
I 6
a 7
, 8
I, 9
i 10
process step
Figure 14,4.2 Reynolds number for each process step
turbulent motion. However, from steps five onwards, the motion becomes less and less turbulent and more transitional. The next step in the scale-up procedure is to carry out, at three different scales, experiments at each step to determine the agitator speed required to create a constant process result for that step. For example, if it is a dissolution step, introducing solids to the top of the vessel, then the rate or time of dissolution can be chosen. In this way, the scale-up exponent can be established for each stage. Figure 14.A.3 illustrates the scale-up exponents found for the various process steps under consideration. It can be seen from Figure 14.A.3 that up to step 6, a scale-up exponent of 0.67 can be chosen. This corresponds to a power/unit volume scale-up criteria in line with mass transfer and equal droplet sizes in turbulent flow. Above step 6, then, an exponent of 1.0 can be chosen corresponding to a tip speed criteria for scale-up (similar to that found for motion in yield stress fluids). It is important that having established the more demanding scale-up criteria, in this case power/volume is to be held constant. The effect on the other process steps that need equal tip speed as a criteria are not compromised.
Appendix 14.2
321
1.1 1.0 0.9 0.8 0.7 >~ 0.6 0.5 0.4 0.3 0.2 0.1 . I
1
I
I
I
I,
2
3
4
5 process
,.
I
I,,
6
7
.
I
8
..
I
9
.,
I
10
step
Figure 14,4.3 Scale-up exponent for the various process steps