Disperse Microstructuring in Continuous...

INTRODUCTION Micro-structuring in shear- and elongational dispersing flow fields links the areas of process engineering and material engineering. The latter includes the structure-rheology relationships. Rheologically the viscous behaviour of multiphase food systems is in general non-Newtonian, mostly shear-thinning and time-dependent. In many cases elastic properties are also not negligible. Dispersing as a flow structuring unit operation regarded here, generates the product microstructure due to shear- and normal stresses which act in the dispersing flow velocity field. A detailed description of the velocity field can be received from flow visualization experiments and from Computational Fluid Dynamics (CFD).

The coupling of local velocity field information and rheological material functions which are received from rheometric shear and elongation experiments, allow for determining the related local shear and normal stress distributions. Micro-structuring is triggered by the stresses acting locally in the flow field and occurs in general if critical stresses of the “structuring units” like macromolecular networks, droplets, particle aggregates are exceeded.

In order to generate a flow-induced structure the structuring units have to experience along their flow tracks a certain total deformation which requires sufficient shear-/elongational rates and residence times. Such shear induced structural changes can be reversible like for macromolecule or drop deformation/ relaxation or irreversible like for network or drop break-up.

STRUCTURE RHEOLOGY RELATIONSHIPS Laminar flow fields are of uni-/multiaxial shear or elongational nature or a mixture of both. Within such flows, the rheological behaviour mirrors the dynamic behaviour of structural units which can change by orientation, deformation and aggregation or de-aggregation (= structuring mechanisms). The latter is synonymous to dispersing. The flow induced structuring generates non-Newtonian behaviour even for

Disperse Microstructuring in Continuous Narrow Annular Gap Reactors

E.J. Windhab, M. Stranzinger, S.F.M. Kaufmann, P. Fischer, K.

Feigl

Laboratory of Food Process Engineering Institute of Food Science

Swiss Federal Institute of Technology Zürich (ETH Zürich) CH - 8092 Zürich, Switzerland

In reactors for multiphase fluid processing the coupling of velocity field and rheological material functions determines heat and mass transfer, residence time distribution as well as shear and normal stresses acting on disperse structuring units (SUs) like particle or macromolecular aggregates, droplets and bubbles within a continuous fluid phase. Such stresses cause structural changes of the SUs thus leading to new flow induced microstructures. Main flow structuring mechanisms for disperse SUs are deformation, orientation, aggregation and dispersing. The generation of flow induced microstructures can be reversible or irreversible. The time-dependant structure formation can generally be described by structuring-restructuring kinetics. Disperse structuring means that SU specific critical stresses or related deformations and energies have to be exceeded in order to apply structural changes. Narrow angular gap reactors considered here are formed by a rotating inner cylinder on which tools like wall scrapers, pins or screw flights are mounted, and a concentric outer cylinder. In such annular flow gaps under rotational and superimposed axial flow, narrow distributions of acting flow stresses and residence time as well as good heat transfer conditions are set up. To get information about acting stresses and residence time within different sections of the narrow angular gap reactor flow field; a coupled experimental and numerical approach was set up. Experimentally Particle Imaging Velocimetry (PIV) was used to characterise the flow field. Applying Computational Fluid Dynamics (CFD), local flow fields for non-Newtonian disperse fluid systems were also calculated and compared to related experimental PIV results. Furthermore so-called particle tracking calculation was developed which allows for quantifying shear-/elongation rates and related stresses the SUs experience moving along their flow pads and based on a coupled Boundary Integral Method (BIM) also allows to calculate the transient deformation of model SUs (fluid drops). From this, criteria for micro-structuring process optimisation where derived.

6th World Congress of Chemical Engineering Melbourne, Australia 23-27 September 2001

dispersions with Newtonian continuous fluid phases. The shear rate-, disperse phase concentration- and time dependent viscous behaviour can be described by the structure-related fluid immobilization behaviour, which depends strongly on the structuring mechanisms described before and investigated in previous work [1]. Elongational flow fields induce micro-structural changes most efficiently due to the fact that there is no rotational contribution of the velocity field which, like in shear flows, causes structuring units to rotate. Rotation prevents structuring units from being strongly deformed, because the acting stresses may not act long enough to exceed a critical deformation. This is the more the case if the structuring units are highly viscous or of solid-like nature. This is demonstrated in figure 1 for the break-up of droplets of diameter x in uniaxial shear- and elongation flows by the well known critical Weber number Wec (also denoted as capillary number Ca) as a function of the viscosity ratio λ (= ηdisperse / ηcontinuous)[2].

We =0.25τx

σ (1)

Droplets are preferably used as “model structuring units” due to their isotropy, homogeneity and

symmetry. In uniaxial shear the drop break-up is limited to the domain λ ≤ 4, in uniaxial elongational flow fields, this limitation does not exist (Fig. 1)

10-4 10110-010-110-210-3 102

1.0

0.1

0.2

0.5

-0.4

pure shear

pure uniaxial elongation

criti

cal W

eber

num

ber W

e c

viscosity ratio λ = ηdispers / ηcontinuous

��

��

superimposed shear and elongation

-0.12

Figure 1: Critical Weber number Wec as a function of the viscosity ratio λ = ηdisperse / ηcontinuous for uniaxial

shear- and elongational flows [2]. EXPERIMENTS IN DISPERSING FLOW FIELDS Model Flow Fields

In order to get experimental results from laminar dispersing flow fields which are close to real flow fields in dispersing apparatus, the experimental set up should allow to investigate pure uni- or biaxial shear- and elongational flows but also their well defined superposition. Such conditions are locally fulfilled in a Four-Roller Apparatus (FRA) and Cross-Slot Channels (CSC) [3-5]. These flow apparatus were constructed in such a way, that laser optics for optical rheometry, small angle light scattering (SALS) and microscopic / image analysis tools (e.g. CSLM) can easily be adapted. Within the FRA the velocity ratio α of the two diagonal roller pairs determines whether pure shear (α = 0), pure elongation (α = 1) or mixed flows of defined elongation to shear ratio (0 < α ≤ 1) are generated. From systematic drop break-up visualization experiments the functions Wec = f(λ, α) as shown in figure 1 for α = 0.1 were confirmed. Single Drop Rheology From the dynamic drop behaviour during deformation and relaxation, interfacial elasticity and viscosity as well as interfacial tension can be estimated under small drop deformation conditions (interface ratio ξ = A/A0 ≤ 1.05) as investigated in [6] and [7]. Equation 3 is derived for the drop relaxation from a mechanical spring/dashpot model of the drop interface [8].


σ =

Ý γ x ηc

2 D 19λ +1616λ +16

(2)

A / A0 = exp{τ2 (GG1 + GG2 )2

2GG12GG 2

2 exp(−2 GG1GG2

(GG1 + GG 2 )(ηd + ηc)t)} (3)

Emulsion Rheology Within multiphase fluid systems containing higher disperse fractions of SUs (e.g. emulsion droplets) the disperse structure is determined by a dynamic equilibrium between SU de-aggregation (e.g. drop break-up) and SU re-aggregation (e.g. drop re-coalescence). Experimentally equilibrium drop size distributions were measured in pure shear flow fields (concentric cylinder gaps) as well as in superimposed shear and elongational flow fields within ex-centric cylinders for diluted (φV ≤ 0.05) and concentrated (φV = 0.3) model O/W emulsions. Figure 2 shows equilibrium drop size distributions from pure shear flow experiments. The related equilibrium viscosity function η( Ýγ ) as well as so-called structure viscosity functions η( Ý γ * , Ýγ ) (= viscosity functions at constant drop size, measured by step-shear experiments) are described elsewhere [9].

droplet size x / µm

0

0.05

0.1

0.15

0.2

0.25

0.3

1 10 100 1000

50 1/s

300 1/s

500 1/s

800 1/s

num

ber

vol.

dist

ribut

ion

q3(

x)

dispersing

coalescencePv

hc

Figure 2: Equilibrium drop size distributions after shear at various constant shear rates (pure uniaxial

shear in concentric cylinder gap) [9]. Modelling and Flow Simulation In order to get information about the locally acting shear- and normal stresses as well as the related shear- and elongational rates, Computational Fluid Dynamics (CFD) was applied to model flows in the geometries described before. Using a partly modified finite element code (SEPRAN; TU Delft, NL [10]), the shear- and elongation rate fields were calculated from the velocity field. Results are shown exemplary for the FRA in Fig. 3 [11]. Fig. 4 demonstrates additionally elongation rate fields in three different experimentally investigated cross-slot flow channels [12].

Calculations of the stress/strain - history (SSH) of droplets or other SUs along flow tracks (particle tracks) within the apparatus flow, were also performed. The application of such a particle tracking calculation is demonstrated in the next main chapter where the application of this methodology to a continuous rotor/stator-dispersing reactor is described.

Simulation of Drop Deformation and Break-up Using a boundary integral element method (BIM) developed by Loewenberg et al. [13, 14], the drop deformation in superimposed shear-/elongation flows was calculated, in a first step assuming steady state flow conditions, clean interfaces (no surfactants) and Newtonian fluids. As the only interfacial property a constant equilibrium interfacial tension σ was taken into account. If critical Weber numbers Wec are derived from such calculations, the expected increase from pure elongation to pure shear is found as shown in Fig. 5. The calculated results for Wec in pure shear and pure elongation flow are in very good agreement with the


experimental results demonstrated in Fig. 1 (comparison for λ = 1). The calculation provides also reliable data for the mixed flow domain [15].

In our recent work we coupled the FEM based particle tracking calculation with the BIM calculation of drop deformation thus simulating the drop deformation under transient mixed shear/elongation flow conditions [11, 15]. This is demonstrated in Fig. 6 for the flow of a droplet along a particle track in an eccentric cylinder/cylinder gap.

Figure 3: CFD simulation of elongation (left) and shear rate fields (right) for a Four Roller Apparatus

(program SEPRAN) [11].

Figure 4: CFD simulation of streamlines (top) and elongation rate fields for different cross slot flow

channels (program SEPRAN) [12].


elongational flow

simple shear flow

0.5

0.1

0.2

0.3

0.4

0.00.40.20.0 0.80.6 1.0

criti

cal W

e -

nu

mb

er

We

c

f low type factor α

Figure 5: Critical Wec for drop break-up in steady shear/elongation flow defined by parameter α from BIM

simulation for λ = 1 [15].

0 100 200 300 400 500 600Time [-]

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Def

orm

atio

n R

ates

[-]

A

B

C

D

E

F

G

A

B

C

D

E

F

G

D A

B

F

C

E G

Figure 6: Transient drop deformation in mixed shear elongation flow field (eccentric cylinder gap, inner

cylinder rotating), λ = 2, streamline / particle track r = 4.6 [15]. Processing in continuous Narrow Annular Gap Reactors (NAGR) In narrow annular gap reactors (NAGR) which consist basically of a concentric cylinder gap geometry with rotating inner cylinder, the treated fluid systems experience a rather narrow range of shear stresses. Commonly dispersing/mixing elements are adapted to the rotating inner and /or the outer cylinder wall. In the case of in-built wall-scraping rotor- or stator blades or pins, which are in general inclined relative to a tangential plane to the cylinder wall (see Fig. 7), higher elongation rates act locally. This has been proved experimentally using Particle Imaging Velocimetry (PIV) as well as by CFD calculations which applied a particle tracking method as shown before in the previous chapter for model flow cells [16]. For the investigated flow gap between a rotating dispersing blade which is mounted to the inner cylinder scraping the surface of the outer cylinder, and the inner cylinder surface, three particle tracks are shown in Fig. 8. For various flow gap ratios of rS = 0.25, 0.4, 0.6 and a constant blade angle of β = 30° the calculated shear and elongation rates ( Ýγ , Ý ε 1) along the three selected particle tracks are shown in Fig. 9.


x

y

moving walls

trailing edge scraping edge

leading edgegap edge

ωrotor

stator

scraper blade

ro

ri

rotorω rotor

stator

ω rotor

βh0hgap

bb

Figure 7: Cross section of narrow annular gap reactor (NAGR) with wall-scraping dispersing /mixing blade

(inner cylinder rotating, blades fixed at inner cylinder, vcirc >> vaxial) [16].

Beside maximum shear- and elongation rates, the shearing and elongation times and consequently the total shear- and elongational deformations are considered in order to generate a flow induced (i.e. dispersed) microstructure. Elongational and shear rate integrated over time provide the elongational and shear deformation (ε, γ). Thus, the calculated elongational rate Ýε 1 and the shear rate Ýγ with respect to the local (e1, e2) coordinate system are integrated between t = tstart and t = tend, as defined in Fig. 8. Equations 4 and 5 describe the time integrals for ε1 and γ, which are approximated numerically applying a trapezoidal rule according to Burden and Faires using a time step ∆t, according to the time increments used for the numerical particle tracking (NPT).

25% track (dashed line)scraper blade

rotor

stator

50% track (dash dotted line)

75% track (solid line)

ω

gap edget=0

tracks start

tracks end

xw

yw

r

θ

Figure 8: Selected particle tracks around dispersing blade in Narrow Annular Gap Reactor [16].

shea

r ra

te γ

[-]

⋅

time t [-]

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

-10 -5 0 5 10

rs=0.25

rs=0.4

rs=0.6

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

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5.0

-10 -5 0 5 10

rs=0.25

rs=0.4

rs=0.6

time t [-]

elon

gatio

nal r

ate

ε 1 [-

]

Figure 9: Shear and elongational rates along selected particle tracks in Narrow Annular Gap Reactor with

blade angle β = 30°; Re = 10, various scraper blade gap size ratio rS [17].

ε1 = Ý ε 1 (t)dt

tstart

t end

∫ γ = Ý γ (t)dttstart

t end

∫ (4, 5)


The total particle deformations ε1 and γ shown in Fig. 10a are compared for the three representative

particle tracks, positioned at 25%, 50% and 75% for a shear-thinning fluid (n = 0.65), with a flow incidences of β = 150°, at Re = 80. For optimization of a structuring process the total energy density rates separately for the elongational and for the shear component are of further interest. The time integral of the elongational dissipation component in primary flow direction e1 describes the elongational energy density eE (energy per volume), given with Equation 6, whereas the shear dissipation component contributes to the integral of the shear energy density eS, as given with Equation 7. The velocity vector is formulated in the polar coordinate system (r, θ). Applying the rotational coordinate transformation to the velocity gradient ∇ v , eE and eS are found in the local coordinate system (e1, e2).

eE = 2ηs Ý γ ( )tstart

t end

∫ ∂vr∂r( )2

+ 1r

∂v θ∂θ + vr

r( )2

dt (6)

eS = ηsÝ γ ( )

tstart

t end

∫ ∂vθ∂r + 1

r∂v r∂θ − vθ

r[ ]2dt (7)

Both Equations 6 and 7 are formulated in terms of the shear viscosity ηS( Ýγ ) as a function of the shear rate Ýγ , which is transformed onto the local coordinate system (e1, e2). For the investigated non-Newtonian model fluid system with a watery 2% CMC solution as the continuous fluid phase, ηS( Ýγ ) was described by a Carreau -Yasuda model defined by Equation 8.

η −η∞

η0 −η ∞ = [1 +(λ* Ý γ )a ](n−1)/ a (8)

elon

gatio

nal e

nerg

y e E

[-]

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

-8 -4 0 4 8 12time t [-]

25% track

50% track

75% track

0

20

40

60

80

100

120

-8 -4 0 4 8 12time t [-]

25% track

50% track

75% track

shea

r en

ergy

eS

[-]

t=tend

t=tstart

ωt=0

xw

yw

r

θ

Figure 10: a) Elongational deformation ε1 and shear deformation γ along selected particle tracks from t = tstart to t = tend b.) Volumetric elongational energy eE and shear energy eS for selected particle tracks from t

= tstart to t = tend (Re = 80, n = 0.65, β = 150˚).


Approximating the time integrals between t = tstart and t = tend applying a trapezoidal rule, the expressions

eE and eS can be found numerically. The flow structuring shear- and elongation related volumetric energy inputs according to Equations 5 and 6 are also compared for the three representative particle tracks, positioned at 25%, 50% and 75% of the scraper blade gap width hgap. Fig. 10b shows the increase of eE and eS as a function of time between t = tstart and t = tend along the three representative particle tracks for a shear-thinning fluid system (n = 0.65) with β = 150° at Re = 80. All elongational and shear energies commonly develop in a step function manner. This behaviour is caused by the pronounced turning sections of the tracks around the scraper blade.

Summarizing the CFD results it can be stated that the main influence on the deformation energy depicts the flow incidence, which varies the total average elongational energy by about 95% between blade angles of β = 30° and β = 110°. The maximum change for the total average shear energy at β = 90° and β = 110° is calculated as 33%. Comparing the impact of the rotor velocity with the influence of the flow incidence, variations of the scraper blade angle cause higher structuring anisotropy with respect to elongational and shear strain, than increasing the Reynolds number between 10 and 80. All flow cases show larger total shear energies than total elongational energies. The minimum difference is found for Re = 10, β = 30°, n = 0.65 with a factor of 3.5.

In order to express the flow structuring energy contributions of the two distinguished flow types (shear, elongation), Equations 9 and 10 define the (local) flow type contribution factors χ E , χS with regard to the investigated scraper blades for elongational flow χ E and shear flow χS , respectively. For χ E ≥ 0.14, which was the lowest contribution factor for the elongational flow type already a qualitative improvement of the dispersing efficiency (i.e. drop size reduction) at constant energy input, compared to a pure shear flow field was obtained. The subject of our ongoing research work in this field is to improve the dispersing efficiency as a function of χ E in more detail. Within the work reported here the maximum values of χ E = 0.35 was obtained for Re = 10, β = 150°, rS = 0.6, and n = 0.65 [17].

χ E =

eE

etot

χS =eS

etot

(9, 10)

ACKNOWLEDGEMENTS This research project (Project No. AiF-FV 11763 N) was partly supported by the German Ministry of Economics and Technology (BMWi) through the Arbeitsgemeinschaft industrieller Forschungs-vereinigungen "Otto von Guericke" e.V. (AiF). NOMENCLATURE β scraper blade angle e E,S energy density (due to shear- (S) or elongational deformation (E)) Ýε elongation rate ε elongational deformation Ýγ shear rate

γ shear deformation λ viscosity ratio χ energy density ratio σ interfacial tension ηS

shear viscosity

η0, ∞ lower / upper Newtonian shear viscosity

References 1 Windhab E. (2000) ‘Fluid immobilization-A structure related key mechanism for the viscous flow

behaviour of concentrated suspension systems’, Applied Rheology 10, 134-144 2 Grace, H.P. (1982) ‘Dispersion Phenomena in High Viscosity Immiscible Fluid Systems’, 3rd Eng.

Found. Res. Conference Mixing, Andover/USA and in Chem. Eng. Commun. 14, 225-277 3. Taylor G.I. (1932) ‘The viscosity of a fluid containing small drops of another fluid‘, Royal Society

London, A134 4. Bentley B.J. and Leal L.G. (1986) ‘An experimental investigation of drop deformation and break-up

in steady two-dimensional linear flows’, Journal of Fluid Mechanics 167, 241-283


5. Cox R.G. (1969) Journal of Fluid Mechanics 37, 601-623 6. Luciani A., Champagne M.F. and Utracki L.A. (1996) Polymer Network Blends 6, 41-51 7. Wolf B., Windhab E. (1995) ‘Interfacial Rheology of Deformable Droplets in Viscometric Flows’,

Rheology 5, 182-18 8. Wolf B. (1995) ´Untersuchungen zum Formverhalten mikroskopisch kleiner Fluidtropfen in

stationären und instationären Scherströmungen’, Ph.D. Thesis ETH Zürich, Nr. 11068 9. Windhab E.J. (1988) ‘Method for describing the time dependent rheological behaviour of

concentrated suspension’, Suppl. to Rheologica Acta; Progress and Trends in Rheology II, Steinkopff Verlag Darmstadt/Germany

10. Segal G. (1993) SEPRAN Code, Ingenieursbureau SEPRA, Leidschendam/The Netherlands 11. Kaufmann S.F.M., Fischer P., Windhab E.J. (2000) ’Investigation of droplet dispersing processes in

shear and elongational flow’, Proceedings Second International Symposium on Food Rheology and Structure, Zürich/Switzerland, 404-405

12. Maruyama K., Fischer P., Windhab E.J. (2000) ‘CFD calculations for various Cross Slot Channel geometries’, Internal Research Report, ETH Zürich, Zürich/Switzerland

13. Christini V., Blawzdziewicz J., Loewenberg M., (1998) ‘Drop break-up in three-dimensional viscous flow’, Physics of Fluids 10, 1781-1783

14. Loewenberg, M., Hinch E.J. (1996) ‘Numerical simulation of a concentrated emulsion in shear flow’, Journal of Fluid Mechanics 321, 395-419

15. Feigl K., Windhab E.J. (2001) ´Numerical simulation of drop break-up in shear and elongational flow’, Proceedings 3th Pacific Rim Conference on Rheology, Vancouver/Canada

16. Stranzinger M., Feigl K., Windhab E.J. (1997) ‘Shear and Secondary Flow Behaviour in Annular Gaps of Scraping’ Crystallizers, Proceedings First International Symposium on Food Rheology and Structure, Zürich/Switzerland, 333-337

17. Bieder A., Stranzinger M., Windhab E.J. (2000) ‘Einfluss der Prozessgeometrie auf das Strukturierungsverhalten in Ringspaltreaktoren’, Semesterwork ETH Zürich, Zürich/Switzerland

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