The formation of fine droplets in a photographic emulsion which is forced through an orifice disperser consisting of a tube with one or more abrupt constrictions is considered. Some design ideas for reducing the droplet size are presented

Carnie, S.

Davidson, M.

English

Mathematics in Industry Information Service Eprints Archive

70 Kodak
EFFICIENT HOMOGENISATION OF PHOTOGRAPHIC
DISPERSIONS
The formation of fine droplets in a photographic emulsion which
is forced through an orifice disperser consisting of a tube with one
or more abrupt constrictions is considered. Some design ideas for
reducing the droplet size are presented.
1. Introduction
During the manufacture of colour photographic paper, the paper surface is
overlaid with expensive dye-forming chemicals. These chemicals are dissolved in
oil and are applied to the paper surface as an emulsion of small droplets in an
aqueous phase. It is desirable that the droplets be made as small as possible.
Smaller droplets would result in higher quality photographs for a given quantity
of chemicals. Alternatively, smaller droplets would achieve a cost saving for a
given photographic quality, by requiring a smaller quantity of chemicals.
The final homogenisation of the emulsification is achieved by forcing a rela-
tively coarse oil-aqueous mixture containing 1-2 pm drops through an orifice dis-
perser (Figure 1) to create droplets with a mean diameter of about 0.1-0.2 pm.
The tube diameter D = 5 mm, and the orifice typically has diameter dQ = 1mm,
giving an area contraction ratio of 25:1. Strength requirements for the disperser
dictate that the orifice length must be at least 3 mm. The available pressure
drop is limited to 50 MPa.
The MISG problem, presented by Kodak (Australasia) Pty Ltd, was to iden-
tify changes to the disperser geometry, subject to a fixed pressure "budget",
which would result in a lower mean drop size. Because of the cost of the pho-
tographic chemicals and the quantity used world wide by Kodak, even a small
reduction in the final mean drop diameter from, say, 0.11 to 0.10 pm would be
significant.
orifice diameter do
coarse
droplets
fine
droplets
D
Figure 1: Schematic of the Kodak orifice disperser.
Eflicient homogenisation of photographic dispersions 71
2. Notation
D tube diameter
do orifice diameter
Q volumetric flow rate
!l.P pressure drop
CD discharge coefficient
Ao orifice area
p mixture density
Reo orifice Reynolds number for the mixture
Ca Capillary number
G shear rate
TI bulk viscosity
TIc viscosity of continuous (aqueous) phase
TlD viscosity of droplet phase
'Y interfacial tension
r droplet radius
Uo mean velocity in the orifice
t time
tdeJ time scale for drop deformation
o ratio of final to initial drop radius
R orifice radius Odo)
y boundary layer thickness
z axial distance
(1' Cavitation number
Pv vapour pressure
3. Material properties
The photographic chemicals are dissolved in the oil phase. The aqueous (gel)
phase consists of water with about 15 per cent (by volume) of dissolved gelatine
together with about 0.5 per cent surfactant. The oil-aqueous mixture (emulsion)
which is passed through the disperser contains 10-20 per cent by volume of oil.
The temperature of the flow is given as BODC. Table 1 gives the viscosity and
density range of each phase at this temperature. The deliberations of the MISG
were based on the specific typical values also shown in Table 1. Experiments
performed by Kodak showed that the mixture is only slightly shear thinning
and that the viscosity is almost constant for shear rates greater than 103 s-l.
Calculations by the finite element numerical package Fastflo (Luo et al., 1996)
indicated that the shear rates in the orifice are typically orders of magnitude
greater than this value. We therefore concluded, with some relief, that the
mixture is approximately Newtonian for the shear rates obtained in the disperser.
72 Kodak
viscosity (aqueous)
viscosity (oil)
viscosity (bulk)
specific gravity (aqueous)
specific gravity (oil)
Interfacial tension
5-20 cP
10-500 cP
(15 cP)
(50 cP)
(40 cP)
(1.0)
(1.0)
(10 dyne/cm)
1.0-1.2
0.9-1.2
3-20 dyne/cm
Table 1: Ranges of physical properties. Typical values used by the MISG are
bracketed.
4. Orifice flow
The flow rate (Q) through the orifice depends on the applied pressure drop
(AP) across the orifice and the flow resistance which is usually expressed by
an empirical discharge coefficient CD. For a narrow, sharp-edged orifice hav-
ing a small orifice area (Ao) compared with the area of the pipe (Coulson and
Richardson, 1990; p. 210)
(1)
where p denotes the fluid density. For orifice Reynolds numbers (Reo = pUodo/TJ)
exceeding 1000, CD lies in the range 0.6-0.65. Appropriate CD values are not
known for the Kodak disperser with its much longer orifice (at least 3 mm), but
we expect them to be smaller because of increased frictional resistance.
A consequence of equation (1) is that the fluid velocity in the orifice (Uo =
Q / Ao) is independent of the orifice diameter do (assuming a constant value of
CD), and hence Reo is linear in do.
For pressure drops ranging between 10 and 50 MPa, equation (1) predicts
flow rates in the range 4-10 l/min, and Reo in the range 2000-5000, for a single
1 mm diameter orifice and a bulk viscosity of 40 cP. The corresponding inlet
Reynolds number range is 400-1000. Obviously, smaller pressure drops will
result in lower values of flow rate and Reynolds number. If the orifice in the
Kodak disperser offers more flow resistance than this example case, as discussed
above, then Q and Re will again be lower. This is consistent with the example
flow rate of 3 l/min quoted by Kodak.
The mixture emerges from the orifice as a high speed jet ('" 100 m/s) sur-
rounded by one or more toroidal recirculation cells. Nevertheless the Reynolds
numbers in the orifice and upstream are such that the flow is in transition be-
tween laminar and turbulent flow, with the flow becoming fully turbulent down-
stream of the orifice. It is possible that pressures in the recirculation zone become
Efficient homogenisation of photographic dispersions 73
sufficiently low that cavitation occurs. The axial extent of the recirculation zone
is defined by the jet reattachment length which is expected to be about 8 tube
diameters for this contraction/expansion ratio (Agarwal, 1994). Downstream of
the jet reattachment, pressure recovery would result in the collapse of cavitation
bubbles. A schematic of the expected flow patterns is shown in Figure 2.
~I R~-3000 IC~/ ~
~nl~4
Elongational Turbulent Recirculation zone High-
Flow Boundary Layer and Cavitation Speed Jet
Figure 2: Schematic of likely flow patterns in the orifice disperser.
Thus, the mechanisms which can deform and subsequently break droplets in
the Kodak disperser include (i) laminar elongation and shear in the flow entering
the orifice, (ii) turbulent stresses in the developing jet, and (iii) turbulence and
(possibly) cavitation downstream of the recirculation zone.
5. Droplet break up
Walstra (1983) gave an excellent review of the physics of droplet formation in
emulsions, in viscous (laminar), turbulent, and cavitating flows. More recently,
Stone (1994) reviewed the dynamics of droplet breakup in viscous flow. We
consider, in turn, the implications of the following for droplet breakup: (i) the
laminar flow at the orifice entry, (ii) the growth of the turbulent boundary layer
within the orifice, and (iii) possible cavitation in the flow exiting the orifice.
5.1 Orifice entry flow
Drops in the Kodak disperser experience a time-dependent shear history,
whereas published experiments and theory which best describe drop deformation
relate to drops in steady shear. Thus only rough estimates of conditions for drop
breakup can be derived.
74 Kodak
A drop of radius T will break when the capillary number
Ca = GT/cr
I
(2)
exceeds a critical value Cacrit. The capillary number represents the ratio of
viscous stresses (T/cG) to the Laplace pressure difference (,/ r) which resists
drop deformation.
The critical capillary number depends on the flow type (e.g. simple shear
versus elongational flow) and on the viscosity ratio T/D/ ttc of the dispersed (oil)
to continuous (aqueous) phase. The converging flow entering the orifice is neces-
sarily elongational (i.e. there is a velocity gradient in the direction of flow) and,
typically, T/D/T/C '" 3. In that case, Cacrit '" 0.1 - 0.2 (Stone, 1994). Thus we
expect that the rate of strain G must satisfy
(3)
for drops larger than 0.1 J.Lm to break.
In elongational flow with viscosity ratios present in the Kodak disperser,
the drop is pulled into a thinning thread before breaking into droplets. Frag-
mentation can be triggered by capillary instabilities on very long threads, and
"end-pinching" during transient stretching and relaxation (Stone, 1994). Final
droplet size depends on the thread diameter at breakup. Higher viscosity drops
will result in thinner threads before breakup and hence smaller droplets.
Using G '" Uo/do gives a G value of the order of Gcrit for pressure drops in
the range 10-50 MPa when do = 1 mm, the orifice diameter currently used by
Kodak. Similarly, Fastflo laminar flow calculations for Reo = 2000 also predict
G '" Gcrit. (That calculation is not strictly valid since flow is in transition, and
the results are not considered completely reliable due to convergence difficulties).
Since the breakup of drops down to about 0.1 J.Lm diameter is known to occur,
the above estimates of G near the critical value in equation (3) suggests that
the orifice entry is responsible for the breakup of drops to some intermediate
size, and that factors such as turbulence and cavitation complete the final size
reduction.
A further requirement for the breakup of a drop is that it must experience
this rate of strain for a time t > tdej = T/Dr/ I where tdej is the time scale for drop
deformation. It is expected that the time needed for breakup is several times
that needed for deformation. For breakup of drops with diameters exceeding
0.1 J.Lm, we therefore require
t > tdej '" 10-6 s. (4)
Efficient homogenisation of photographic dispersions 75
We expect this condition to be well satisfied since a time scale for the flow based
on axial velocity and orifice diameter lies in the range 10-5-10-4 s.
A n optimal do?
From Walstra (1983; p 83), and since Ca ex: G,
() ex: G-~ (5)
where () denotes the ratio of final to initial drop radius based on the breakup
of extending liquid threads in elongational flow. IT the flow entering the orifice
(radius R = ~do) is modelled as a point sink then
(6)
Two limiting flow profiles within the orifice are considered: (i) plug flow to
represent the relatively flat velocity profile in fully developed turbulent pipe
flow, and (ii) Poiseuille flow corresponding to fully developed laminar flow in
a pipe. For plug flow, Q ex: R2, and for Poiseuille flow with a given pressure
gradient, Q ex: R4. Equations (5,6) then give
2
() ex: R"3 for plug (turbulent) flow
() ex: R- ~ for Poiseuille (laminar) flow.
(7)
(8)
Thus, reducing R is predicted to yield smaller droplets in plug flow, whereas
increasing R is predicted to do so in Poiseuille flow. For both cases Reo increases
with R, hence the flow is laminar when R is small and turbulent when R is large.
Thus, () first decreases (equation (8)) and subsequently increases (equation (7))
with increasing orifice radius. This implies that there exists an orifice diameter
which is optimal for droplet breakup.
5.2 Turbulent boundary layer
Within the orifice (Reo'" 2000-5000), it is expected that a turbulent bound-
ary layer grows from the entry corner. Its growth on a wall is given by (Davies,
1972; p. 45)
¥.. = 0.376Re;O.2 (9)
z
where z denotes the axial distance from the orifice entry and y denotes the
thickness of the boundary layer. For the orifice, the Reynolds number Rex is
approximated as pUoz/.,.,. Equation (9) then yields y/z '" 0.05 in which case the
entire orifice cross-section would become turbulent if the orifice was extended to
10-20 orifice diameters in length. Since turbulence is one of the mechanisms for
drop breakup, it may be desirable to lengthen the orifice from the minimum 3
mm permitted to, say, 10 mm for a Irnm diameter bore.
76 Kodak
5.3 Downstream cavitation
Along with turbulence in the downstream flow, the possibility of cavitation
exists in the low pressure recirculation adjacent to the jet exiting the orifice. It
occurs when the pressure falls below the vapour pressure of the liquid. Cavitation
bubbles will subsequently collapse, hence contributing to drop breakup, as they
move to regions of higher pressure. Evidence for the existence of cavitation in
the Kodak disperser is the occurrence of pitting on the upstream face of a second
orifice plate when installed. Other evidence derives from consideration of the
cavitation number
Pexit - P;u= (10)
1 U 2iP 0
Experiments on water flow through an orifice (Yan and Thorpe, 1990) show
cavitation occuring when a < 1.5 for Pexit measured 15 pipe diameters down-
stream of the orifice and an orifice/pipe diameter ratio equal to the present case
(do/D = 0.2). Values of a calculated for the Kodak disperser, assuming that
Pexit is atmospheric (free discharge), are about two orders of magnitude smaller
than 1.5; this gives strong additional evidence for the presence of cavitation.
6. Design ideas
6.1 Idea #1 (Remove upstream recirculation zone)
The converging flow into the orifice generates a recirculation zone in the
corner just before the orifice plate. The idea was to exclude this "wasted" re-
circulation from the flow field by altering the boundary shape to follow approxi-
mately (as a straight line) the outer converging streamline. Because energy was
not being expended driving the recirculation in the proposed new configuration,
it was thought that a given applied pressure drop would produce a higher flow
rate (hence larger shear rates, more downstream turbulence, and a smaller mean
droplet size). However, Fastfto numerical calculations showed the opposite effect
due to increased wall friction. Idea #1 was therefore abandoned in favour of #2
below.
6.2 Idea #2 (Rounded orifice)
One method of increasing the flow rate for a given pressure drop without
the above wall friction penality is to "round" the upstream corner of the orifice.
Perry's Chemical Engineers' Handbook shows that a rounded orifice will result
in about a 20 per cent increase in flow rate for the same pressure drop.
Efficient homogenisation of photographic dispersions 77
6.3 Idea #3 (Longer orifice)
In Section 5.2, we estimated that a turbulent boundary layer growing from
the orifice entry would fill an orifice of 10-20 diameters in length. Since increas-
ing the amount of turbulence should promote droplet breakup, it seems desirable
to use a longer orifice than the minimum 3 mm required to maintain the struc-
tural strength of the device. A longer orifice will result in increased wall friction;
however the work done against friction is only about 10 per cent (based on an
orifice 10 mm long and 1mm diameter) of the work done in accelerating the flow
entering the orifice, and should not be a limiting factor.
The use of two closely spaced short orifices, rather than one long orifice, was
discussed. It was not clear whether one configuration has an advantage over the
other. However, two orifices spaced 1 mm apart, say, may conceivably provide
less resistance to flow. In view of the discussion above, the effect is likely to
be small, but even small increases in droplet breakup by increased flow rate (at
given LlP) are advantageous.
6.4 Discussion - second orifice plate
Use of a second orifice downstream of the first offers the prospect of ad-
ditional droplet breakup since it represents an additional pass through a high
shear / elongation zone. Fastflo calculations predicted that additional high shear
rates are indeed present at a second orifice placed 7 mm downstream, and that
about 80 per cent of the pressure drop occurs across the first orifice plate. The
extra plate will increase the pressure in the region near the front face which
will promote the collapse of cavitation bubbles (see Section 5.3). Any additional
droplet breakup achieved through exposure to extra strain rate, turbulence, and
cavitation must be balanced against the cost of generating the extra pressure
drop required.
6.5 Idea #4 (Multi-hole orifice)
Replacing a single hole orifice (diameter do) with a number of smaller holes
will increase the rate of strain which, in turn, will tend to produce smaller
droplets. ITthe total cross-sectional area is unchanged then the orifice velocity
is also unchanged for a fixed pressure drop (ignoring any differences in discharge
coefficient resulting from the changed orifice geometry). For four holes, the
diameter of each is then ~do. In that case G is increased by a factor of two.
However turbulence in the orifice, which can also contribute to droplet breakup,
will be reduced as Reo approaches sub-transitional values (constant velocity,
reducing do).
78 Kodak
7. Conclusions
The key issue is how to best use the fixed "budget" of available pressure
drop to maximise the influence offactors which promote droplet breakup. These
factors include laminar shear/elongation, turbulence, and cavitation. However,
the relative importance of these factors is not known in the present application
so it is not possible to be more quantitative. Nevertheless, some design ideas
have been presented based on simple calculations and qualitative argument. A
more detailed study could involve particle tracking methods to trace the history
of a representative number of drops within the flow with rules for drop breakup
(or otherwise) to be applied using the local conditions encountered by each drop
at every time step.
Acknowledgements
The moderators, Malcolm Davidson and Steven Carnie, were helped at all
stages by Samad Mahkri, the Kodak representative. The problem was at-
tended by Leonid Ant anovskii, Gleb Beliakov, Steven Brumby, Paul Cleary,
Ellis Cumberbatch, Neville Fowkes, Barry Hughes, Robert Lionberger, Bernhard
Neumann, Colin Please, Frank Polanco, Alexander Shugai, Nick Stokes, Elliot
Tonkes, Tim Tran, Ernie Tuck, Darrell Velegol, Andrew White, Bill Whiten,
Jinhong Xiao, Debbie Zhang, Zili Zhu.
References
N.K. Agarwal, "Mean separation and reattachment in turbulent pipe flow due
to an orifice plate", J. Fluids Eng. 116 (1994),373-376.
J.M. Coulson and J.F. Richardson, Chemical Engineering (Vol. 1, 4th ed., Perg-
amon, 1990).
J.T. Davies, Turbulence Phenomena (Academic Press, 1972).
X. Luo, Z. Zhu and N. Stokes, "Turbulence modelling in Fastflo", Proc. A us-
tralian Engineering Mathematics Conf. - AEMC96, Sydney 1996.
R.H. Perry, Perry's Chemical Engineers' Handbook (6th ed., McGraw-Hill,
1984).
H.A. Stone, "Dynamics of drop deformation and breakup in viscous fluids", Ann.
Rev. Fluid Mech. 26 (1994),65-102.
P. Walstra, "Formation of emulsions", Encyclopedia of Emulsion Technology,
Vol. 1. Basic Theory, P.Becher (Ed.) 1983,57-127.
Y. Yan and R.B. Thorpe, "Flow regime transitions due to cavitation in the flow
through an orifice", Int. J. Multiphase Flow 16 (1990),1023-1045.


Flow regime transitions due to cavitation in the flow through an orifice&quot;,

Formation of emulsions&quot;,

Mean separation and reattachment in turbulent pipe flow due to an orifice plate&quot;,

Perry's Chemical Engineers'

Turbulence modelling in Fastflo&quot;,

Turbulence Phenomena

Efficient homogenisation of photographic dispersions

Efficient homogenisation of photographic dispersions

Abstract

Similar works

Full text

Available Versions

Mathematics in Industry Information Service Eprints Archive