INVESTIGATION OF LAMINAR MICROMIXING OF LIQUIDS DIFFERING IN VISCOSITY - investigation of laminar micromixing of liquids ...
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INŻYNIERIA CHEMICZNA I PROCESOWA
26, 649–670 (2005)
ANTONI ROŻEŃ
INVESTIGATION OF LAMINAR MICROMIXING
OF LIQUIDS DIFFERING IN VISCOSITY
Faculty of Chemical and Process Engineering, Warsaw University of Technology
Two competitive-parallel chemical reactions proceeding between HCl, NaOH and CH2ClCOOC2H5
in water–polyethylenepolypropylene glycol systems are applied to study micromixing of liquids of dif-
ferent viscosities in a reactor with the Couette flow. A new micromixing model is proposed to interpret
experimental results. Additionally a full set of physicochemical data characterizing the test reaction
system is presented.
Dwie równoległe i konkurencyjne reakcje chemiczne zachodzące między HCl, NaOH i CH2ClCOOC2H5
w układach woda–glikol polietylenopolipropylenowy użyto do badania mikromieszania cieczy różniących
się lepkością w reaktorze z przepływem Couette’a. Zaproponowano nowy model mikromieszania do
interpretacji wyników doświadczeń. Dodatkowo przedstawiono pełny zestaw danych fizykochemicznych
charakteryzujących układ reakcji testowych.
1. INTRODUCTION
Processes of mixing of liquids in a laminar flow regime are met in many industrial
applications in the chemical and biochemical technology when mixed liquids are very
viscous. Mixing in laminar flow is slower and less effective than mixing in turbulent
flow especially if good homogenization on the molecular level (micromixing) is re-
quired. Fast micromixing is essential when complex chemical reactions of nonlinear
kinetics are carried out between initially segregated reactants, because it can prevent
slower side reactions to occur and change composition and properties of a final prod-
uct. Hence, it is important to be able to identify and correctly model mixing processes
in a laminar flow.
Micromixing in the laminar flow proceeds by deformation of fluid elements and
molecular diffusion [1, 2]. Deformation of fluid elements generates a new contact
surface area, decreases spatial segregation scales in a reactor and keeps local concen-
tration gradients at high levels. Molecular diffusion is responsible for decreasing local
concentration variances and the final homogenization on the molecular level. Many
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factors affect deformation and molecular diffusion in a mixer. The most important
ones are: the mixer geometry and power input, properties of mixed fluids such as den-
sity, viscosity, interfacial tension and coefficients of molecular diffusion of system
components. The present work is concerned with the influence of viscosity differ-
ences on micromixing with instantaneous and fast chemical reactions. The study is
carried out for isothermal mixing of Newtonian and fully miscible liquids (zero static
interfacial tension) in the Couette flow, which is predominant in many mixing devices
working in the laminar flow regime. The effect of viscosity differences on the course
of micromixing in the Couette flow has been investigated previously [3–7]. It was
found that viscosity differences change the rate of deformation fluid elements and
lead to flow destabilization, which results in formation of segregated structures and
severe deterioration of mixing conditions. This work shows how to model complex
mixing phenomena during mixing of unequally viscous liquids. Micromixing is also
investigated experimentally by means of a system of two competitive-parallel chemi-
cal reactions:
NaOH + HCl ⎯⎯→
k1
NaCl (1)
(A) (B)
k2
NaOH + CH 2ClCOOC2 H 5 ⎯⎯→ CH 2ClCOONa + C 2 H 5OH (2)
(A) (C)
and a viscosity controlling agent – polyethylenepolypropylene glycol. This reactive
tracer method is sensitive to mixing conditions in the laminar flow [2, 6–8]. Addition-
ally, the results of new measurements of diffusion coefficients of the polymer and the
test reactants, and the reaction rate constants in water-polymer systems are presented
together with the earlier findings [2, 6] to fully characterize the reactive tracer
method.
2. MEASUREMENTS OF TRANSPORT PROPERTIES
AND REACTION KINETICS
The competitive-parallel reactions (1), (2), used to investigate micromixing of liq-
uids differing in viscosity in the laminar flow, were carried out in aqueous solutions.
Polyethylenepolypropylene glycol (Breox 75W18000) was used to adjust viscosity of
these solutions. This polymer is chemically stable in dilute solutions of the test reac-
tants and mixes in all proportions with water giving transparent Newtonian liquids
[2,6]. Measurements conducted by means of a capillary viscometer and a rotation
viscometer Rheotest RN 4.1 showed that the viscosity of water-polymer solutions
depends on the polymer mass fraction, cP and can be approximated for 0≤ cP ≤0.9 by
the following equation:
log µ = −3.0393 + 11.0149c P − 27.0860c P2 + 56.7926c P3 − 56.9419c P4 + 20.9071c P5 (3)
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Increasing the polymer content decreases the coefficients of molecular diffusion of
all test reactants [2,6]. Hence, if solutions of different polymer content are mixed the
reactant diffusivities will vary in the vicinity of the contact surface, which may affect
mixing and chemical reactions. Measurements of coefficients of molecular diffusivity
in the water-polymer systems were carried out at 298 K in small cylindrical chamber
(diameter, 2cm; length 6 cm) consisting of two equal compartments: upper and lower.
The chamber was made of brass and its inner walls were tiled with PVC. The chamber
compartments could be separated, independently filled, joined together to allow mo-
lecular diffusion to proceed for a specified period of time and finally separated again.
During each experiment the diffusion chamber was positioned vertically in a thermo-
stating bath with denser solution always in the lower compartment to avoid uncon-
trolled buoyancy driven flows.
Measurements of the polymer diffusivity were performed in solution systems:
(0, 0.05), (0.1, 0.15), (0.2, 0.25), (0.3, 0.35), (0.4, 0.45), (0.5, 0.55) and (0.6, 0.65),
where numbers in brackets denote the initial polymer mass fraction in the upper and
lower compartment respectively. The final average polymer concentration in each
compartment was determined by means the refractive index measurement. The initial
and final polymer concentrations in the compartments and the total diffusion time
were then used to calculate diffusivity coefficients [6]. The calculations were con-
ducted under assumption that the polymer diffusivity does not change significantly in
the range of the polymer concentrations applied in each experiment. Obtained in this
way diffusivity coefficients are plotted in Fig. 1 vs. the mean polymer mass fraction in
the diffusion chamber. As it can be seen, the polymer diffusivity has a maximum for
cP≈0.45 and it can be approximated for 0≤ cP ≤0.6 by the expression
DP ⋅1010 = −4.412 + 7.739(1.0957 − c P ) −1 − 2.402(1.0957 − c P ) −2 (4)
Fig1. Diffusion coefficient of the polymer in water–polymer systems at 298 K
Rys. 1. Współczynnik dyfuzji polimeru w układach woda–polimer w 298 K
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Diffusivity coefficients of the test reactants: NaOH, HCl and CH2ClCOOC2H5 are
known for the water–polymer systems of viscosity higher than 0.1 Pa·s [2,6]. To de-
termine the diffusivity coefficients in solutions of lower viscosity additional meas-
urements had to be carried out at 298 K. In these experiments the initial reactant con-
centration in the lower compartment was set to 10 mol/m3, while in the upper
compartment was equal to 0. The results presented in Fig. 4 confirm that increasing
the solution viscosity decreases the diffusivity coefficients of the reactants. Combin-
ing the present and earlier results allowed formulation of new correlations for the
diffusivity coefficient:s
a) sodium hydroxide for 0.001≤ µ ≤10 Pa·s (0≤ cP ≤0.8)
log D A = −9.5105 − 0.4408 log µ − 0.05424(log µ ) 2 (5)
b) hydrochloric acid for 0.001≤ µ ≤10 Pa·s (0≤ cP ≤0.8)
log DB = −9.0342 − 0.2715 log µ − 0.02869(log µ ) 2 (6)
c) ethyl chloroacetate for 0.001≤ µ ≤2 Pa·s (0≤ cP ≤0.6)
log DC = −10.2275 − 0.8147 log µ − 0.13896(log µ ) 2 (7)
Fig. 2. Diffusion coefficients of the reactants in water–polymer systems at 298 K
Rys. 2. Współczynniki dyfuzji reagentów w układach woda–polimer w 298 K
The selectivity of the test reactions (1) and (2) defined as
NC 0 − NC
X = (8)
N A0
is directly related to the mixing conditions [9]; X = 0 for perfect mixing and X = Xmax
for very slow mixing. The rate constants of the test reactions (1) and (2) must be
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known to predict the course of the competitive-parallel reactions. The reaction con-
stant of the acid-base neutralization (1) is so high in pure water (k1 ≈ 108 m3/(mol·s) at
298 K [10]) that even if its value is changed by one or two orders of magnitude in the
water–polymer system, this reactions can be still considered as an instantaneous and
completely controlled by mixing. In fact only the reaction rate constant of the ester
hydrolysis (2) should be determined exactly. Earlier measurements conducted for cP
equal to 0.3 and 0.4, showed that the reaction rate constant is almost the same as in
Fig. 3. Scheme of the experimental system used to measure kinetics of ester hydrolysis
Rys. 3. Schemat układu doświadczalnego do pomiarów kinetyki hydrolizy estru
pure water (k2 = 0.033 m3/(mol·s) at 298K [11]). In the present work, kinetic meas-
urements were carried out for 0 ≤ cP ≤ 0.5. During measurements 375 cm3 of the di-
luted NaOH solution (cA0=6.67 mol/m3) was poured into a small tank reactor (77 mm
diameter). The reactor was equipped with a double screw agitator (50 mm in diame-
ter) and put into a thermostating bath as schematically shown in Fig. 3. Then 25 cm3
of the concentrated CH2ClCOOC2H5 solution (cC0 = 100 mol/m3) was rapidly injected
(in less than 1 s) using compressed air into the intensively agitated NaOH solution (up
to 700 rpm). Depending on the viscosity of the mixed reactant solutions the approxi-
mate mixing time varied from 5 s for the least viscous solutions up to 15 s for the
most viscous solutions. The progress of mixing and the ester hydrolysis was moni-
tored by measuring mixture conductivity. The analog output signal from a conductiv-
ity meter (Metrohm 660) was digitized with 100 Hz sampling rate and stored in
a computer hard disc. Then determined independently relation between the conductiv-
ity and the mixture composition was used to find out how NaOH concentration
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changed in time. Finally, assuming a second order reaction kinetics the reaction rate
constant was calculated [6].
Fig. 4. Reaction rate constant of ester hydrolysis in water–polymer systems at 298 K
Rys. 4. Stała szybkości reakcji hydrolizy estru w układach woda–polimer w 298 K
Results of the kinetic measurements, shown in Fig. 4, indicate that the reaction rate
constant, k2, increases linearly from 0.035 (pure water) to 0.052 m3/(mol·s) for cP≈0.4
k2 = 0.0412cP + 0.03487 for 0 ≤ cP ≤ 0.4 (9)
Then the reaction rate constant decreases to 0.031 m3/(mol·s) for cP≈0.5
k2 = −0.207cP + 0.1351 for 0.4 ≤ cP ≤ 0.5 (10)
Comparison of Figs. 1 and 4 allows concluding that the reaction rate constant, k2,
depends on the polymer mass fraction in a similar way as the polymer molecular dif-
fusivity. Difference between the recent results and earlier ones, implying that k2
= const [6], can be explained as the effect of application of better stirrer, especially
designed to mix highly viscous liquids, and performing experiments in the wider
range of the polymer concentrations.
3. SELECTIVITY MEASUREMENTS
OF COMPETITIVE-PARALLEL REACTIONS
The competitive-parallel reactions (1) and (2) and the viscosity regulating polymer
were used to investigate mixing in the shear flow. Experiments were conducted ina
reactor consisting of two coaxial cylinders (Fig. 1 in [7]). The outer cylinder was sta-
tionary, while the inner one could rotate and produce Couette flow. The annular gap
between the cylinders was initially divided into two compartments by thin vertical
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plates (Fig. 5). After filling the compartmenst with reactant solutions the separating
plates were carefully drawn out and the inner cylinder started to rotate with a constant
speed. All tests were conducted at 298 K.
Fig. 5. Initial distribution of the reactant solutions in the mixer
Rys. 5. Początkowe rozmieszczenie roztworów reagentów w mieszalniku
In the first group of experiments, a concentrated NaOH solution (100 mol/m3) was
mixed with a diluted premixture of HCl and CH2ClCOOC2H5 (14.3 mol/m3). In the
second group of tests, the concentrated premixture of HCl and CH2ClCOOC2H5 (100
mol/m3) was mixed with the diluted NaOH solution (14.3 mol/m3). The volume ratio
of the mixed solutions was 1/7, which means that chemically equivalent amounts of
substrates were mixed. Viscosity of the reactant solutions was adjusted by polyethyl-
enepolypropylene glycol, their density was equalized by potassium chloride [7].
Table 1. Composition of the reactant solutions used in 1st group of experiments
Tabela 1. Skład roztworów reagentów użytych w pierwszej grupie eksperymentów
Exp. Solution NaOH HCl Ester KCl Polymer µ ρ
No. No. [mol/m3] [mol/m3] [mol/m3] [kg/kg] [kg/kg] [Pa⋅s] [kg/m3]
1 0 14.60 14.04 0.034 0.420 0.314
1, 2 1086
2 100.2 0 0 0 0.560 1.454
1 0 14.51 13.58 0.021 0.420 0.319
3, 4 1077
2 99.04 0 0 0 0.488 0.690
1 0 13.98 14.03 0.015 0.415 0.308
5, 6 1072
2 101.8 0 0 0 0.454 0.485
1 0 14.27 13.79 0.006 0.420 0.331
7, 8 1067
2 98.40 0 0 0 0.420 0.342
1 0 14.41 14.50 0 0.420 0.339
9, 10 1063
2 100.1 0 0 0.002 0.386 0.228
1 0 14.30 13.72 0 0.420 0.340
11, 12 1063
2 97.27 0 0 0.011 0.352 0.172
1 0 14.24 14.20 0 0.420 0.345
13, 14 1063
2 98.70 0 0 0.028 0.284 0.076
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When the blue color of the pH indicator (bromothymol blue) added to the reactant
solutions faded out a sample of the mixture was taken and analyzed by HPLC to de-
termine the final ester concentration [7]. Compositions, volumes, viscosities and den-
sities of the reactant solutions are given in tables 1–4. The final selectivities are pre-
sented in Figs. 6 and 7.
Table 2. Solution volumes and final ester concentrations; 1st group of experiments
Tabela 2. Objętości roztworów i końcowe stężenia estru; pierwsza grupa eksperymentów
Exp. Ester final
No. n [rpm] V1 [dm3] V2 [dm3] [mol/m3]
1 6 1.618 0.231 8.409
2 60 1.599 0.233 8.141
3 6 1.599 0.236 7.958
4 60 1.598 0.227 8.236
5 6 1.612 0.235 8.390
6 60 1.640 0.238 8.219
7 6 1.605 0.226 8.910
8 60 1.604 0.233 8.590
9 6 1.591 0.232 7.955
10 60 1.603 0.220 8.642
11 6 1.606 0.245 7.054
12 60 1.619 0.233 8.186
13 6 1.607 0.224 7.570
14 60 1.607 0.216 8.544
Table 3. Composition of the reactant solutions used in 2nd group of experiments
Tabela 3. Skład roztworów reagentów użytych w drugiej grupie eksperymentów
Exp. Solution NaOH HCl Ester KCl Polymer µ ρ
No. No. [mol/m3] [mol/m3] [mol/m3] [kg/kg] [kg/kg] [Pa⋅s] [kg/m3]
1 14.07 0 0 0.034 0.415 0.298
15, 16 1084
2 0 97.83 89.66 0 0.560 1.433
1 13.40 0 0 0.020 0.420 0.318
17, 18 1076
2 0 98.62 81.98 0 0.488 0.691
1 13.79 0 0 0.012 0.420 0.325
19, 20 1072
2 0 96.94 91.94 0 0.454 0.494
1 13.23 0 0 0.004 0.420 0.330
21, 22 1067
2 0 97.51 86.77 0 0.420 0.350
1 14.32 0 0 0.004 0.420 0.341
23, 24 1064
2 0 97.36 90.53 0 0.386 0.245
1 13.32 0 0 0 0.420 0.340
25, 26 1064
2 0 96.89 72.42 0.013 0.352 0.167
1 13.67 0 0 0 0.420 0.339
27, 28 1064
2 0 97.44 79.06 0.030 0.285 0.077
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Table 4. Solution volumes and final ester concentrations; 2nd group of experiments
Tabela 4. Objętości roztworów i końcowe stężenia estru w 2 grupie eksperymentów
Ester final
Exp. No. n [rpm] V1 [dm3] V2 [dm3] [mol/m3]
15 6 1.590 0.231 7.072
16 20 1.592 0.230 7.121
17 6 1.574 0.234 8.676
18 20 1.569 0.230 8.364
19 6 1.613 0.227 8.938
20 20 1.620 0.232 8.945
21 6 1.622 0.226 10.25
22 20 1.598 0.228 10.38
23 6 1.610 0.228 9.560
24 20 1.613 0.228 8.727
25 6 1.591 0.230 7.669
26 20 1.594 0.231 7.433
27 6 1.598 0.230 6.672
28 20 1.589 0.229 6.191
Fig. 6. Selectivities of the parallel reactions measured in 1st group of experiments
Rys. 6. Selektywności reakcji równoległych zmierzone w 1 grupie eksprymentów
The lowest selectivities were obtained when equally viscous solutions were mixed,
whereas the highest selectivities were obtained when the solution viscosities differed
most. These results agree with the earlier ones [6,7] and show a drastic change in the
mixing mechanism. The effect of the viscosity ratio on the final selectivity is the
strongest in 2nd group of tests. In 1st group of tests this effect is weaker but still visi-
ble. The highest selectivity growth is observed when the viscosity difference is small.
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In most cases further increase of the viscosity difference slightly decreases the final
selectivity but when the viscosity ratio becomes larger than 2 or smaller than 0.5 the
selectivity increases again. The effect of the rotational speed on the final selectivity is
inconsistent. In 1st group of the tests increasing the rotational speed either decreases
the selectivity for µ2/µ1 1. In 2nd group of
the tests increasing the rotational speed either increases the selectivity for µ2/µ1 < 1 or
has a practical no effect on it for µ2/µ1 > 1.
Fig. 7. Selectivities of the parallel reactions measured in 2nd group of experiments
Rys. 7. Selektywności reakcji równoległych zmierzone w 2 grupie eksperymentów
Visual observations conducted during the experiments showed that when equally
viscous solutions were mixed a spiral structure of the solution layers was formed.
Depending on the rotation speed decolorization of the pH-indicator was achieved after
5 to 10 minutes of mixing. The reaction time could not be determined more precisely
due to very weak color intensity in the final stages of mixing. When unequally viscous
solutions were mixed the shear flow became unstable and the lamellar structure was
disintegrating into small irregular streaks and drops of the “minor” component (Figs.
3 and 4 in [7]). It was also found that the less viscous solution migrated to the center
of the annular gap and formed several bands around the inner cylinder at various
heights. On the contrary, the more viscous solution migrated towards the reactor
walls. Most of the colored bands were disappearing after 30 to 60 minutes of continu-
ous mixing. At revolution speeds equal to 20 and 60 rpm small vortices were formed
near the reactor bottom and close to the liquid surface. Decolorization of pH-indicator
in these vortices was very slow.
It should be noted that migration of the low viscosity liquid drops immersed in the
more viscous liquid phase is known to occur in the shear flow. The process is caused
by the anisotropy of the flow field around a drop moving nearby a rigid system wall
[12, 13]. The present work confirms that a similar phenomenon may occur in a single
phase system.
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4. MODELLING OF MICROMIXING IN THE COUETTE FLOW
It was assumed in the previous work [7] that flow destabilization can be accounted
by modifying the average shear rate. Now, it is proposed to divide the whole mixing
process into two stages. Initially mixing proceeds in the stable flow. Then at a certain
moment the flow destabilization occurs and the regular mixture structure is destroyed.
A more viscous solution migrates towards the cylinder walls and a less viscous solu-
tion migrates to the gap centre. This secondary segregation marks the beginning of the
second stage of mixing, when small portions of the “minor” component (solution of
a small initial volume, V2) enter the “major” component (solution of a large initial
volume, V1) and mix with it. The elements of the “minor” component may have dif-
ferent shapes i.e.: streaks, drops or thin rims surrounding larger unmixed lumps of
liquid [7]. These elements are gradually elongated, form striation-like shapes and
“dissolve” in the surrounding liquid. Therefore, similarly as in the first stage, mi-
cromixing in the second stage can be modeled as one-dimensional process. The mate-
rial balance written in a local coordinate system (x′,y′) attached to the center of the
deformed striation (Fig. 6 in [7]) reads
∂ci 1 ds ∂c ∂ ⎛ ∂ci ⎞ ~
+ = ⎜ Di ⎟ + ri , i = A, B, C , P (11)
∂t s dt ∂x ′ ∂x ′ ⎝ ∂x ′ ⎠
where the reaction term, ~
ri , is given by the following expressions:
~
rA = −k1c A c B − k 2 c A cC , ~
rB = −k1c A c B , ~
rC = −k 2 c A cC , ~
rP = 0 (12)
In the first stage of mixing boundary conditions are given by:
∂ci ∂ci
= 0, = 0, i = A, B, C , P (13)
∂x ′ x′= 0
∂x ′ x ′ =δ
where δ is the distance between the centers of the two neighboring liquid striations:
one containing A-rich solution and the other one with B and C premixture. The aver-
age value of the segregation scale δ in the first stage of mixing can be estimated as [7]
δ0 2π r12 + r1 r2 + r22
δ = , δ0 = (14)
1 + G 2t 2 3 r1 + r2
where the mean shear rate, G , is given by Eq. (A4) in Appendix A. The mean stria-
tion thickness s is related to the segregation scale δ, and the initial volume ratio
⎛ V ⎞δ
s = ⎜⎜1 + 1 ⎟⎟ (15)
⎝ V2 ⎠2
The initial conditions depend on the initial distribution of the reactants in the
mixer (Fig. 5).
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In the second stage of mixing the boundary conditions read:
∂ci
= 0 , c i∞ = c i ( x ′ → ∞ ) , i = A, B, C , P (16)
∂x ′ x′= 0
The far field concentrations ci∞ gradually change as subsequent elements of the
“minor” component are mixed with the liquid matrix. On the other hand, the reactant
and the polymer concentrations in the “minor” component volume remain constant,
provided that the elements of the liquid matrix can not mix with the “minor” compo-
nent.
If equally viscous reactant solutions are mixed, the final selectivity can be found
by solving the system of Eqs. (11) only once (details of the numerical solution are
given in [7]) with the deformation rate of the liquid striations approximated by
1 ds G 2t
=− (17)
s dt 1 + G 2t 2
The computation results shown in table 5 agree with the selectivities obtained in
1st group of tests (mixing of the concentrated base solution with the diluted acid and
ester premixture). The agreement is worse for 2nd group of tests (mixing of the con-
centrated acid and ester premixture with the diluted base solution) but both measured
and calculated selectivities are very low. The model predicts that increasing the revo-
lution speed decreases the selectivity. Visual observations revealed, however, forma-
tion of small vortices close to reactor bottom and the liquid surface at higher revolu-
tion speeds. Mixing in these vortices was very slow and this probably caused the final
selectivity to rise in the experiments.
Table 5. Selectivities computed for mixing in the stable shear flow.
Tabela 5. Selektywności obliczone dla mieszania w stabilnym przepływie ścinającym
Exp. no. 7 8 21 22
µ2/µ1 1.033 1.033 1.061 1.061
Xexp 0.261 0.277 0.032 0.042
Xmodel 0.264 0.239 0.014 0.010
When modeling mixing of unequally viscous liquids, one should first determine
reactant concentration profiles at the end of the first stage of mixing and then use
these profiles as the initial condition in the second stage of mixing. In the first stage
calculations are performed as for mixing of equally viscous liquids (the shear flow is
stable, mixing proceeds in a batch mode). Modeling of the second stage depends on
a reactant distribution between the “major” and “minor” components. If small por-
tions of the acid and ester premixture (“minor” component) subsequently mix with the
large volume of the base solution (“major” component), both acid and ester contained
in each pocket will be completely consumed. Hence, earlier the second stage of semi-
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batch mixing begins, the higher chances for ester to react with base are and the higher
final selectivity becomes. Figure 8 shows how the final selectivity depends on the
amount of shear applied in the stable flow. Comparing the calculated selectivities with
these measured in 2nd group of experiments one can estimate the critical amount of
shear required to destabilize the flow. Results presented in table 6 show that the low-
est critical shear amounts were obtained for the lowest and the highest viscosity ra-
tios. Decreasing the difference between the solution viscosities delayed the onset of
the secondary segregation in the system and lowered the final selectivity. A detailed
algorithm of calculation of the final selectivity for 2nd group of experiments is given
in Appendix B.
Fig. 8. Effect of the shear amount in the first stage of mixing
on selectivity as calculated for 2nd group of experiments (n = 6 rpm)
Rys. 8. Wpływ ilości ścinania w pierwszym etapie mieszania
na selektywność dla 2 grupy eksperymentów (n = 6 obr/min)
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Table 6. Critical shear amount causing flow destabilization during mixing
of the unequally viscous liquids (2nd group of experiments, n = 6 rpm)
Tabela 6. Krytyczna ilość ścinania wywołująca destabilizację przepływu
podczas mieszania cieczy o różnych lepkościach (2 grupa eksperymentów, n = 6 obr/min)
Exp. no. 15 17 19 23 25 27
µ2/µ1 4.81 2.17 1.52 0.72 0.49 0.23
Xexp 0.350 0.167 0.201 0.132 0.126 0.273
γcrit 185 300 295 378 331 224
If in the second stage of mixing small portions of the base solution (“minor”
component) subsequently mix with the large volume of the acid and ester premix-
ture (“major” component), both acid and ester will have to compete to react with
base. To find out the molar amount of acid and ester consumed in the reaction with
base one has to solve the system of Eqs. (11) with the boundary conditions defined
by Eq. (16). The initial conditions depend on how much of the “minor” component
was already mixed with the “major” component and are given by Eqs.(C5÷C7) in
Appendix C. Additionally, one should know the mean shear rate in the liquid
matrix to determine the deformation rate of the liquid striation in Eq. (11). As it
was already pointed out that after the onset of the flow instability the less viscous
solution migrates to the centre of the annular gap, whereas the more viscous solu-
tion migrates towards the reactor walls. The mean shear rate in the less viscous
liquid, G L , and in the more viscous liquid, G H , are given by Eqs. (A9) and (A10) in
Appendix A. These average shear rates gradually change as subsequent portions of
the “minor” component are mixed with the “major” component. The initial striation
thickness of the base solution is not known a priori. Therefore, modeling of the
second stage should be performed for different values of the initial striation thick-
ness, s0, characterizing small pockets of the “minor” component entering the liquid
matrix. Finally, one has to decide how to partition the “minor” component volume,
V2, to properly simulate semi-batch mixing. The usual practice is to divide volume
V2 into M equal parts, which are subsequently mixed with “major” component [9].
Results given in table 7 show that for M > 20 the effect of partition on the final
selectivity becomes negligible. Hence M = 20 was applied in all further simulations.
Table 7. Effect of partition of volume V2 on selectivity; n = 6 rpm, µ2/µ1=4.8, γcrit=52.1.
Tabela 7. Wpływ podziału objętości V2 na selektywność; n = 6 obr/min, µ2/µ1=4.8, γcrit=52.1.
M 2 5 10 20 50 100
X (s0=0.001 m) 0.229 0.256 0.265 0.269 0.271 0.272
X (s0=0.01 m) 0.274 0.298 0.305 0.309 0.311 0.311
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Fig. 9. Effect of the initial striation thickness in the second stage of mixing
on selectivity as calculated for 1st group of experiments (n = 6 rpm)
Rys. 9. Wpływ początkowej grubości warstewki w drugim etapie mieszania
na selektywność obliczony dla 1 grupy eksperymentów (n = 6 obr/min)
Figure 9 shows the effect of the initial striation thickness on the final selectivity of
the parallel reactions predicted by the micromixing model. Simulations were per-
formed only for these points from 1st group of experiments, when the unequally vis-
cous liquids were mixed and when the revolution speed was equal to 6 rpm. Values of
the critical shear amounts, listed in table 6, were interpolated to the new conditions.
The initial striation thickness in the second stage of mixing was changed from
0.001 m to 0.02 m equal to the width of the annular gap in the reactor. Further details
of calculations of the final selectivity for 1st group of experiments can be found in
Appendix C. As expected, increasing the initial striation thickness resulted in higher
selectivity in all considered cases. The highest growth of the selectivity was observed
for the lowest and the highest viscosity ratios, when the critical shear amount required
to destabilize the flow in the reactor was the lowest. Nevertheless, the final selectivi-
ties predicted by the micromixing model, even for s0=0.02 m, are still lower than
those achieved in the experiments, especially for µ2/µ1664 A. ROŻEŃ
then difference in viscosity of the mixed liquids can not affect its deformation in the
simple shear flow [14]. In the other words, the one-dimensional model overestimates
the deformation rate when the “minor” component is more viscous than the liquid
matrix. Hence, selectivities predicted for µ2/µ1 > 1 are too low. In the opposite case
i.e. µ2/µ1 < 1 the deformation rate is underestimated in the model but it should be not
a major problem, because then the “minor” component deforms only a little faster
then the liquid matrix [14]. Second, one can not entirely exclude that small pockets of
the diluted acid and ester premixture enter the concentrated base solution. If this hap-
pens even on a relatively small scale, when compared to the overall mass transfer, the
final selectivity will increase, because all ester contained in these small elements will
react with base. The best conditions for this reverse mass flow to occur exist when the
liquid matrix is more viscous the “minor” component and can be easily “dissolved” by
the less viscous solution.
5. CONCLUSIONS
The selectivity of the competitive-parallel reactions proceeding between the ini-
tially unmixed reactants is strongly dependent on the viscosity ratio of the reactant
solutions mixed in the shear flow. The lowest selectivities, indicating good mixing,
were measured for mixing of equally viscous liquids, while the highest selectivities,
indicating poor mixing, were obtained for mixing of liquids, which differ most in
viscosity. Deterioration of the mixing conditions was caused by the flow destabiliza-
tion observed during mixing of unequally viscous liquids.
A new model of micromixing in the shear flow has been proposed. The model was
based on the assumption that the whole mixing process could be divided in two
stages: quick batch mixing in the stable flow and slow semi-batch mixing after the
flow destabilization. Comparison of selectivities predicted by the model with selectiv-
ities measured when the limiting reactant was contained in the “major” liquid compo-
nent allowed determining the critical amount of shear leading to the flow destabiliza-
tion. When the model was applied to predict the effect of the viscosity difference on
selectivity in experiments, when the limiting reactant was contained in the “minor”
mixture component, the calculated selectivities were underestimated. Causes of this
discrepancy were identified and will be taken into account when improving the model
in future.
The existing date base of the physicochemical properties of the test reactants
(NaOH, HCl, CH2ClCOOC2H5) and the viscosity increasing polymer (polyethylene-
polypropylene glycol) used to study micromixing in laminar flow conditions has been
updated. The new correlations for the diffusivity coefficients of the polymer and test
reactants and the reaction rate constant of the ester hydrolysis in the water–polymer
systems were proposed.
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APPENDIX A
A region bounded by two co-axial and infinitely long cylindrical surfaces: the inner one (r = r1) and
the outer one (r =r2) is filled with Newtonian fluid. If the cylindrical surfaces rotate with constant angular
velocities ω1 and ω2, and the non slip conditions are applied at these surfaces, the tangential velocity
profile will have the following form
r 2 − r12 r22 r2 − r2 r2
vθ (r ) = ω 2 + 22 2 1 ω1 (A1)
r2 − r1 r
2 2
r2 − r1 r
The local and average shear rates characterizing this flow read:
d ⎛ vθ ⎞ 2r 2 r 2 ω − ω
G (r ) = −r ⎜ ⎟ = − 2 1 22 2 2 1 (A2)
dr ⎝ r ⎠ r2 − r1 r
1
r2
− 4r 2 r 2 ⎛r ⎞
2 ∫
G= G (r )2πrdr = 2 1 22 2 (ω2 − ω1 ) ln⎜⎜ 2 ⎟⎟ (A3)
π (r − r1 ) r1
2
2
(r2 − r1 ) ⎝ r1 ⎠
If the inner surface rotates with a constant rotational speed n (ω1 = 2πn) and the outer surface is sta-
tionary (ω2=0), the average shear rate in the annular gap will be equal to
2
⎛ rr ⎞ ⎛ r2 ⎞
G = 8πn⎜⎜ 2 1 2 2 ⎟ ln⎜ ⎟
⎟ ⎜r ⎟ (A4)
⎝ r2 − r1 ⎠ ⎝ 1⎠
Let us know consider the same annular gap between two cylinders occupied by two Newtonian fluids
differing in viscosity. The fluid of low viscosity, µL, occupies the central region of the gap, r3 < r < r4,
whereas the fluid of high viscosity, µH, occupies two outer regions adjacent to the walls: r1 < r < r3 and
r4 < r < r2. The tangential velocities and stresses generated in the shear flow at two contact surfaces r = r3
and r = r4 must equalize:
2r12 r32 ω3 − ω1 2r 2 r 2 ω − ω
vθH (r3 ) = vθL (r3 ) = ω3 r3 , µ H = µL 2 3 4 2 4 2 3 (A5)
r3 − r1
2 2
r32
r4 − r3 r3
2r42 r22 ω2 − ω4 2r 2 r 2 ω − ω
vθH (r4 ) = vθL (r4 ) = ω4 r4 , µ H = µL 2 3 4 2 4 2 3 (A6)
r2 − r4
2 2
r42
r4 − r3 r4
If the inner cylinder rotates with a constant rotational speed n (ω1 = 2πn) and the outer one is station-
ary (ω2=0), then unknown a priori angular velocities will read:
⎡ ⎤
⎢ ⎥
1
ω3 = 2π n ⎢1 − ⎥ (A7)
⎢ r32 r12 r22 − r42 µ H r12 r42 − r32 ⎥
⎢ 1+ 2 2 2 + ⎥
⎣ r4 r2 r3 − r12 µ L r42 r32 − r12 ⎦
⎡ ⎤
⎢ ⎥
1
ω 4 = 2π n ⎢ ⎥ (A8)
⎢ r2 r4 r3 − r1
2 2 2 2
µ H r2 r4 − r3 ⎥
2 2 2
⎢ 1 + + ⎥
⎣ r12 r32 r22 − r42 µ L r32 r22 − r42 ⎦
The average shear rate in the fluid of low viscosity equals
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− 4r32 r42 ⎛r ⎞
GL = (ω4 − ω3 ) ln⎜⎜ 4 ⎟⎟ (A9)
(r42 − r32 ) 2 ⎝ r3 ⎠
The average shear rate in the fluid of high viscosity equals
− 4r12 r32 ⎛ r ⎞ − 4r 2 r 2 ⎛r ⎞
(ω3 − ω1 ) ln⎜⎜ 3 ⎟⎟ + 2 4 22 (−ω4 ) ln⎜⎜ 2 ⎟⎟
(r − r )
2 2
⎝ r1 ⎠ ( r2 − r4 ) ⎝ r4 ⎠
GH = 3 1 (A10)
(r32 − r12 ) + (r22 − r42 )
APPENDIX B
Mixing of a small volume of a concentrated acid and ester premixture, V2, with a large volume of
a diluted base solution, V1, is considered. The reactant solutions have unequal viscosities and mixing is
assumed to proceed in two stages as described in chapter 4.
The initial molar amounts of the reactants are equal to:
N A0 = V1c A0 , N B 0 = V2 cB 0 , N C 0 = V2 cC 0 (B1)
The molar amounts of the reactants consumed in the first stage of mixing read:
∆N A1 = V1 (c A0 − < c A1 >V 1 ) (B2)
∆N B1 = V2 (cB 0 − < cB1 >V 2 ) − V1 < cB1 >V 1 (B3)
∆N C1 = V2 (cC 0 − < cC1 >V 2 ) − V1 < cC1 >V 1 (B4)
Concentration profiles, required to compute the mean reactant concentrations at the end of the first
stage, V1 and V2, are determined by solving Eqs. (11) with the boundary conditions defined by
Eq. (13) and the initial conditions reflecting the initial segregation in the reactor (Fig. 7b). If at the begin-
ning of the second stage of mixing
< c A1 >V 1 ≥ < cB1 >V 1 + < cC1 >V 1 (B5)
then the average base concentration will be reduced to
< c A 2 >V 1 = < c A1 >V 1 − < cB1 >V 1 − < cC1 >V 1 (B6)
while both acid and ester will be fully consumed in volume V1
< cB 2 >V 1 = < cC 2 >V 1 = 0 (B7)
Else if at the beginning of the second stage of mixing
< c A1 >V 1 < < cB1 >V 1 + < cC1 >V 1 (B8)
then the base will be fully consumed in volume V1.
< c A 2 >V 1 = 0 (B9)
In the latter case the mean acid and ester concentrations V1 and V1 should be determined by
solving the system of Eqs. (11) for an isolated striation. The initial concentration profiles and the initial
striation thickness are taken from the end of the first stage. The average shear rate in the liquid matrix is
given by Eq. (A9) or by Eq. (A10).
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Finally the molar amounts of the reactants consumed in the second stage of mixing read:
∆N A 2 = V1 < c A1 >V 1 (B10)
< c B1 > V 2
∆N B 2 = V1 (< cB1 >V 1 − < cB 2 >V 1 ) + V1 < c A 2 >V 1 (B11)
< cB1 >V 2 + < cC1 >V 2
< cC1 >V 2
∆N C 2 = V1 (< cC1 >V 1 − < cC 2 >V 1 ) + V1 < c A 2 >V 1 (B12)
< cB1 >V 2 + < cC1 >V 2
APPENDIX C
Mixing of a small volume of a concentrated base solution, V2, with a large volume of a diluted acid
and ester premixture, V1, is considered. The reactant solutions have unequal viscosities and mixing is
assumed to proceed in two stages as described in chapter 4.
The initial molar amounts of the reactants equal:
N A0 = V2 c A0 , N B 0 = V1cB 0 , N C 0 = V1cC 0 (C1)
The molar amounts of the reactants consumed in the first stage equal:
∆N A1 = V2 (c A0 − < c A1 >V 2 ) − V1 < c A1 >V 1 (C2)
∆N B1 = V1 (cB 0 − < cB1 >V 1 ) (C3)
∆N C1 = V1 (cC 0 − < cC1 >V 1 ) (C4)
Concentration profiles, required to compute the mean reactant concentrations at the end of the first
stage, V1 and V2, are determined by solving Eqs. (11) with the boundary conditions defined by
Eq. (13) and the initial conditions reflecting the initial segregation in the reactor (Fig. 7a). When at the
beginning of the second stage of mixing
< c A1 >V 1 > 0 (C5)
acid and ester will first react with base, which diffused into the liquid matrix. In this case the mean acid
and ester concentrations V1 and V1 should be determined by solving the system of
Eqs. (11) for an isolated striation of the liquid matrix. The initial concentration profiles and the initial
striation thickness are taken from the end of the first stage. The average shear rate in the liquid matrix is
given by Eq. (A9) or by Eq. (A10).
The semi-batch mixing is simulated by dividing volume V2 into M equal portions and mixing them sub-
sequently with volume V1. Each portion of V2 takes a form of a striation deformed in the liquid matrix,
where the average shear rate is given by Eq. (A9) or Eq. (A10). Selectivity, Xm, referring to mixing of mth
portion is determined by solving the system of Eqs. (11) with the boundary conditions given by Eq. (16).
The initial base concentration equals to V2, whereas the initial acid and ester concentrations n and
n change accordingly:
1 ⎡ V2 ⎤
< cB > m+1 = ⎢V1,m < c B > m − M (1 − X m ) < c A1 >V 2 ⎥ , < cB > 0 =< cB 2 >V 1 (C6)
V1,m+1 ⎣ ⎦
1 ⎡ V2 ⎤
< cC > m+1 = ⎢V1,m < cC > m − M X m < c A1 >V 2 ⎥ , < cC > 0 =< cC 2 >V 1 (C7)
V1,m+1 ⎣ ⎦
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The volume of the “major” liquid component increases as follows
V2
V1,m+1 = V1,m + , V1, 0 = V1 . (C8)
M
Finally the molar amounts of the reactants consumed in the second stage of mixing read:
∆N A 2 = V1 < c A1 >V 1 +V2 < c A1 >V 2 (C9)
∆N B 2 = V1 < cB1 >V 1 − V1,M < cB > M (C10)
∆N C 2 = V1 < cC1 >V 1 − V1,M < cC > M (C11)
ACKNOWLEDGEMENTS
This work was supported financially by the Polish State Committee for Scientific Research (grant No
7 T09C 056 21). The author wishes to thank a diploma student Mrs. A. Krasińska for help in conducting
the selectivity measurements.
SYMBOLS – OZNACZENIA
ci – concentration of ith reactant, mol/m3
stężenie i reagenta
cP – mass fraction of the polymer, kg/kg
ułamek masowy polimeru
Vj – mean concentration of ith reactant in jth volume after 1st stage of mixing, mol/m3
średnie stężenie i reagenta w j objętości po 1 etapie mieszania
Vj – corrected mean concentration of ith reactant in jth volume, mol/m3
skorygowane średnie stężenie i reagenta w j objętości
m – mean concentration of ith reactant after mixing of mth portion of base, mol/m3
średnie stężenie i reagenta po zmieszaniu m porcji zasady
Di – coefficient of molecular diffusivity of ith reactant, m2/s
współczynnik dyfuzji molekularnej i reagenta
G – shear rate, 1/s
szybkość ścinania
ki – reaction rate constant, m3/(s·mol)
stała szybkości reakcji
M – volume partition of the base solution
podział objętości roztworu zasady
Ni – number of moles of ith reactant, mol
liczba moli i-tego reagenta
n – rotational speed of the inner cylinder, 1/s
szybkość obrotowa wewnętrznego cylindra
r – radial coordinate, m
promień
~
ri – reaction rate, mol/(m3·s)
szybkość reakcji
s – striation thickness, m
grubość warstewki
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t – time, s
czas
V – solution volume, m3
objętość roztworu
V1,m – volume of acid and ester solution after adding mth portion of base, m3
objętość roztworu kwasu i estru po dodaniu m porcji zasady
vθ – tangential liquid velocity, m/s
obwodowa prędkość płynu
X – selectivity
selektywność
Xm – selectivity referring to mixing of m-th portion of the base solution
selektywność odnosząca się do mieszania m porcji zasady
x′,y′ – local coordinates, m
współrzędne lokalne
δ – segregation scale, m
skala segregacji
γcrit – critical amount of shear
krytyczna ilość ścinania
µ – dynamic viscosity, Pa·s
lepkość dynamiczna
ρ – density, kg/m3
gęstość
ωi – angular velocity, rad/s
prędkość kątowa
SUBSCRIPTS - INDEKSY DOLNE
A – NaOH
B – HCl
C – CH2ClCOOC2H5
H – high viscosity liquid
ciecz o dużej lepkości
L – low viscosity liquid
ciecz o małej lepkości
P – polyethylenepolypropylene glycol
glikol polietylenopolipropylenowy
0 – initial value
wartość początkowa
1 – “major” component
składnik większościowy
2 – “minor” component
składnik mniejszościowy
∞ – far field value
wartość w głębi roztworu
REFERENCES
[1] RANZ W.E., AIChEJ, 1979, 25, 41.
[2] BAŁDYGA J., ROŻEŃ A., MOSTERT F., Chem. Eng. J., 1998, 69, 7.
[3] MOHR W.D., SAXTON R.L., JEPSON C.J., Ind. Eng. Chem., 1957, 49, 1855.
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[4] MURAKAMI Y., KATSUMASA F., UOTANI S., J. Chem. Eng. of Japan, 1972, 5, 85.
[5] MIDDLEMAN S, Fundamentals of Polymer Processing, McGraw-Hill, New York, 1977.
[6] ROŻEŃ A., Investigation of Micromixing in Viscous Liquids, PhD Thesis, Warsaw University of
Technology (1995).
[7] ROŻEŃ A., BAŁDYGA J., Inż. Chem. i Proc., 2004, 25(2), 439.
[8] ROŻEŃ A., BAKKER R.A., BAŁDYGA J., Chem. Eng. J., 2001, 84, 413.
[9] BAŁDYGA J., BOURNE J.R., Chem. Eng. Commun., 1984, 28, 243.
[10] WILSON I.R., Reactions of non-metallic inorganic compounds, Bamford C.H., Tippers C.F.H. (Ed.),
Comprehensive Chemical Kinetics, Elsevier, 1972, 6, 329.
[11] KIRBY A.J., Hydrolysis and formation of esters of organic acids, Bamford C.H., Tippers C.F.H.
(Ed.), Comprehensive Chemical Kinetics, Elsevier, 1972, 10, 161.
[12] SMART J. R., LEIGHTON D. T., Phys. Fluids A, 1991, 3, 21.
[13] PATHAK J. A., DAVIS M. C., HUDSON S. D., MIGLER K. B., J. Colloid Interface Sci., 2002, 255, 391.
[14] BILBY B.A., KOLBUSZEWSKI M.L., Proc. R. Soc. Lond. 1977, A 355, 335
ANTONI ROŻEŃ
BADANIA MIKROMIESZANIA CIECZY RÓŻNIĄCYCH SIĘ LEPKOŚCIAMI
W układzie reakcji równoległych neutralizacji kwasu zasadą oraz hydrolizy estru zasadą przeprowadzo-
no badania mikromieszania cieczy różniących się lepkością w reaktorze z przepływem Couette’a.
Wykazano, że selektywność końcowa reakcji testowych zależy od różnic lepkości mieszanych roztworów
reagentów. Najmniejszą selektywność, wskazującą na dobre mieszanie w reaktorze, uzyskano dla roztwo-
rów reagentów o jednakowej lepkości. Największą selektywność uzyskano dla roztworów reagentów
najbardziej różniących się lepkością. Za pogorszenie się warunków mieszania w układzie odpowiadała
destabilizacja przepływu Couette’a, wywołana różnicami lepkości mieszanych cieczy.
Przedstawiono nowy model mikromieszania w przepływie Couette’a, bazujący na podziale procesu
mieszania na dwa etapy: szybkie mieszanie okresowe w stabilnym przepływie ścinającym i wolne mie-
szanie półokresowe po wystąpieniu destabilizacji przepływu i wtórnej segregacji mieszanych cieczy.
Model użyto do wyznaczenia zależności pomiędzy krytyczną ilością ścinania, powodującą destabilizację
przepływu, a stosunkiem lepkości mieszanych cieczy. Zależność tę wyznaczono z porównania zmierzo-
nych i obliczonych selektywności reakcji testowych. Dalsze obliczenia wykazały jednak, że
model przewiduje za niskie wartości selektywności dla innej serii pomiarów. Przyczyny tych rozbieżno-
ści zostały wyjaśnione i będą wykorzystane w dalszych pracach..
Zmierzono współczynniki dyfuzji molekularnej reagentów testowych i polimeru zwiększającego lep-
kość roztworów reagentów oraz stałej szybkości reakcji alkalicznej hydrolizy estru w układach woda
–polimer. Na tej podstawie opracowano korelacje określające współczynniki dyfuzji reagentów i polime-
ru oraz stałą szybkości reakcji hydrolizy estru obowiązujące w szerokim zakresie stężeń polimeru w ukła-
dach woda–polimer.
Wpłynęło 12 kwietnia 2005
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