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					                        Sixth International Symposium on Mixing in Industrial Process Industries- ISMIP VI
                                       Niagara on the Lake, Niagara Falls, Ontario, Canada
                                                        August17-21, 2008

                               N.V.RAMA RAO, M.H.I.BAIRD, A.N.HRYMAK, and P.E. WOOD

                 Dept. of Chemical Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada

                                                                (1995) found that the breakup of a drop moving at its
ABSTRACT                                                        terminal velocity tended to occur at the leading edge of
                                                                the packing and depended on the balance between kinetic
The dispersed phase drop size distribution has been
                                                                energy and surface energy. In the case of a single drop
measured for flow through SMX static mixer elements, in
                                                                carried across SMX packing in a viscous liquid at high
columns of diameter 41.18 mm and 15.75 mm, for a
                                                                velocity, Liu et al. (2005) found that breakup occurred
continuous phase of aqueous corn syrup and a dispersed
                                                                mainly within the packing at the crossover points, with
phase of silicone oil. Droplet size distributions were
                                                                relatively smaller effects at the leading edges and in the
measured by the computer-aided analysis of images from
                                                                gaps between packing elements
a digital camera. The mean drop sizes have been
compared with a new model based on drop formation at
equivalent point sources within the packing.                    EXPERIMENTAL
                                                                The continuous phase was an aqueous solution of high
NOMENCLATURE                                                    fructose corn syrup adjusted to three different viscosity
                                                                values (22, 68 and 205 mPa.s). The dispersed phase was a
D column diameter, m
                                                                Dow Corning standard silicone oil with a rated kinematic
D43 mean drop diameter, 3Di4 / 3Di3 , m
                                                                viscosity of 500 and 1000 cSt. The viscosities of the oils
D’ drop diameter at detachment (in model), m
                                                                were, respectively, 978 mPa.s and 489 mPa.s with density
h dispersed phase volume fraction
                                                                978 kg/m3 in both cases. The interfacial tension between
Kd constant in equation 5
                                                                the silicone oil and the 22 cP (22 mPa.s) aqueous fructose
N number of elements
                                                                was measured by pendant drop shape analysis using a
p viscosity ratio, μd /μc
                                                                DSA10 unit (Krüss, Germany) and found to be 38.5
Q volume flow rate, m3/s
                                                                mN/m. Figure 1 shows the system for measurement of
Q’ volume flow rate per point source (in model), m3/s
                                                                pressure drop and drop sizes under continuous flow
Rec Reynolds number based on continuous phase
                                                                conditions. Two column sizes have been studied, with
                                                                diameters 41.18 and 15.75 mm. The column packing
t   time, s
                                                                consists of a number (6, 8 or 10) of SMX elements fitting
u superficial velocity, m/s
                                                                snugly into the column. In this work the packing element
u’ velocity of drop before detachment (in model), m/s
                                                                sizes are scaled to the column diameter, in order to
                                                                maintain geometric similarity. As intuitively expected,
ρ   density, kg/m3
                                                                the mean drop diameter is reduced by increasing the total
μ   viscosity, Pa.s
                                                                superficial velocity (u) of the flow, but it is increased
                                                                when the dispersed phase flow fraction (h) is increased.
                                                                As the number of mixer elements (N) is increased the drop
c continuous phase
                                                                diameter is at first reduced, but the results suggest that
d dispersed phase
                                                                with more than about 8 elements there is little further
Industrially, static mixers are preferable to rotating
impellers for dispersion because they are simpler and
because the energy dissipation is more uniform, favouring
the formation of more uniformly sized drops.                                               A
Grace (1982) has presented a very thorough review of the                             SMX
fundamentals of single droplet dispersion in a number of                             elements

flows, including multiple drop breakup in several
commercial types of static mixer and found that the mean
drop sizes varied inversely with pressure drop and with                          Thermometer
energy dissipation rates. The experimental significance of          Continuous
                                                                                                B        feed
energy dissipation rates was also noted by Streiff et al            feed
(1997, 1999). Rauline et al (1998) used a numerical                                                                              Separating
simulation approach to make a comparison between six                                                                             tank

types of commercial static mixer, under laminar creeping
flow conditions.

There is little information on the behaviour of a single        Figure 1: General flow diagram
moving drop in contact with fixed packing. Mao et al.

MODELING                                                                        Qd                                       (3)
                                                                              Qc + Qd
From the experimental studies droplets interact with the
packing in two ways: sometimes they adhere momentarily                   and the definition of superficial velocity is
and then form films which break away to form smaller
drops; and, a single drop breaks up when it impinges on
the cross-over points of the SMX elements and undergoes                  u=
                                                                              (Qc + Qd )                                 (4)
deformation and splitting (Liu et al., 2005). This differs                     ⎛ πD   2
                                                                               ⎜          ⎟
from the classical turbulent break-up model which has                          ⎜ 4        ⎟
been applied in low viscosity two-phase systems,                               ⎝          ⎠
(Kolmogoroff, 1941) in which turbulent eddies are
considered to act against surface forces. Such a model                   Hence, in dimensionless form,
cannot apply in very high viscosity systems.            A
mechanistic model based on drop detachment has been                          = K d [1.5h(1 + p )]
                                                                                                 1/ 2
developed by considering the balance of frictional forces                D
acting on a drop as it grows and detaches from a packing
element within the array of mixing elements.          The                Values of Kd can be obtained from individual
detachment model follows the approach of Davidson and                    experimental values of D43 on the basis of either equation
Schuler (1960) and Scheele and Meister (1968).                           (2) or equation (5).
According to the present model the drop is formed as a
hemisphere at a point source on a vertical plate as shown                RESULTS
in Figure 2.     As the drop grows it is subjected to two
                                                                         Figure 3 shows the effect of continuous phase viscosity
vertical forces: (1) drag from the moving continuous
                                                                         for a fixed number (6) and diameter (4.118 cm) of
phase, depending on drop size, superficial velocity u and
                                                                         elements. It is seen that for a given liquid viscosity the
continuous phase viscosity μc .and (2) opposing drag from
                                                                         mean diameters data are linearly proportional to h1/2 .
the plate, depending on drop size, drop velocity u΄ and
                                                                         Moreover the slope of the trend lines decrease as μc is
dispersed phase viscosity μd.
                                                                         increased. It is also notable that there is no effect of the
                                                                         superficial velocity u except to the extent that it is
                                                                         included in the calculation of h. According to equation
                                                                         (5), drop size should be proportional to the diameter D,
                                                                         assuming no change in the geometry. This effect is
                                                                         examined in Figure 4 for the two different diameters of
                                                                         elements. The ratios of slopes are about 3.5 while the
     Initial                    Growing          Detaching
                                                 from source             ratio of cell diameters is 2.61. This is reasonably
                                                                         consistent with the model. All data were analysed in
     3 stages in the formation of a drop                                 terms of the values Kd obtained from equation (5) and the
     at a fixed point source in a vertical plate                         measured drop diameters and conditions. According to
     in a moving liquid                                                  the present model, the values of Kd should be
                                                                         approximately constant. However it was noted that at
Figure 2: Sketch for hemispherical drop detachment                       high flow velocities and low continuous phase viscosities,
model derivation                                                         lower values of Kd were obtained. This effect is shown
                                                                         by comparing the dimensionless Kd values with the flow
A detailed derivation is available in Rama Rao et al                     Reynolds number, based on the superficial velocity, the
(2007).                                                                  continuous phase properties and the diameter of the cell.
                                                                         The comparison is shown in Figure 5. It is seen that at
The diameter at detachment is:                                           values of Rec < 40, Kd ~ 0.035. At higher Reynolds
                                                                         numbers, enhanced breakup may occur because of
     ⎡ 6Q ' (1 + p ) ⎤
                         1/ 2                                            eddying, so the conditions no longer correspond to
D' = ⎢                                                   (1)
                     ⎥                                                   creeping flow and Kd trends lower. It may also be noted
     ⎣      πu       ⎦                                                   that from Figure 5 that larger Kd values generally occurred
                                                                         when only 6 elements were present, due to lack of
For the purposes of correlating the experimentally                       dynamic equilibrium.
measured values of the mean diameter D43 with operating
conditions, we assume that the volume flow rate per unit
source (Q’ ) is proportional to the volumetric flowrate of
the dispersed phase (Qd ) and introduce a dimensionless
empirical constant Kd to take account of this.                           A droplet breakup model has been proposed for high
                                                                         viscosity liquid-liquid dispersions in an SMX static mixer
                                          1/ 2                           based on the analogy of a hemispherical droplet growing
                    ⎡ 6Q (1 + p ) ⎤                            (2)
D43 = K d D ' = K d ⎢ d           ⎥                                      and detaching from a surface. The model provides a mean
                    ⎣    πu       ⎦                                      drop diameter D43 that agrees well with experimental data
                                                                         as a function of holdup ratio, continuous phase viscosity
This model-based equation includes the effects of both                   and mixer diameter.
phase flows and viscosities        An alternative form of
equation (2) can be obtained in terms of the volume
fraction of the flowing dispersed phase, defined as

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Figure 3: Effect of holdup (h) and continuous phase            (1995) “Single liquid drop velocities and breakage
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Figure 4: Effect of holdup (h) and mixer diameter on
mean drop diameter, for 10 elements and continuous
phase viscosity 2.00 Pa.                                       Financial support from the Procter and Gamble Company
                                                               through a grant-in-aid of research for partial support of
                                                               this work is gratefully acknowledged. The authors thank
                                                               Dr. Shaffiq Jaffer for his insights and comments in the
                                                               development of this work. The authors also thank Dr.
                                                               Konstantinos Apostolou for assistance with figure

Figure 5: Values of Kd, plotted against Rec . Symbols: ○
6 elements, ● 8 and 10 elements.