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Application of chaotic advection to thermal mixing
1 Physics of Mixing, Leiden 2011
Application of Chaotic Advection to Thermal
Mixing
January 23th 2011
Kamal El [email protected]
Yves Le [email protected]
Mechanical and Electrical Engineering Laboratory (SIAME)
University of Pau, France
Application of chaotic advection to thermal mixing
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Chaotic mixing in Pau: a few examples
● Reactive mixing in chaotic Dean flows● Mapping method:
optimization of mixing – scalar statistics
● Two rod mixers for concentrated emulsions
● Thermal mixing in a two rod mixer:● Stirring protocols● Thermal wall BC
● Fluid rheology● Temperaturedependence
Application of chaotic advection to thermal mixing
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Helicoidal reactor
Alternated angles between bends: +90°, -90°, +90°,
-90°...
« Chaotic » reactor
Diffusive and reactive chaotic mixing in alternated Dean flows Open flows Diffusive mixing: conductivity profiles Reactive mixing: concentration profiles
Experimental pilot
80 bends
Boesinger, Le Guer, Mory, AIChE J. 2005
Application of chaotic advection to thermal mixing
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Concentration profiles
Reactive mixing
Re = 335
Helicoidal reactor Chaotic reactor
Bend 22
Bend 76
Minimal conversion
Maximal conversion
Re = 335
Taken samples
-0,9 -0,7 -0,5 -0,3 -0,1 0,1 0,3 0,5 0,7 0,9
Y (cm)
Sampling probe
Bend 55
Y
Application of chaotic advection to thermal mixing
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Mapping method: optimization of mixing and scalar statistics
T1 T2 T3 T4
R
R
a
a
Bend outlet
T1 transformation
M1 matrix
• Anisotropic unstructured mesh mapping method
• Flow in a 100 bend sequence• Four plane orientations considered: k x 90° with k = 0, 1, 2 or 3
Bend inlet
Cn=M i . Cn−1 with i=1,4
Final concentration distribution:
Cn = Mi . Cn-1
with i = 1, 2, 3 or 4
Le Guer et al., Chem. Eng. Sci. 2004, Comm. Nonlin. Sci. Num. Sim. 2006
Application of chaotic advection to thermal mixing
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Pattern recurrence: 20 bends∆Cfinal = 0,002
10 20 50 100
Chaotic stirring protocol: +90°,+90°,+90°,90°,90°, ...
nonGaussian PDF
Intermittency Rare but important events for large fluctuations (tails)
- Selfsimilar PDFs - « Gaussian hat » for small fluctuations- Asymetric tails for large fluctuations
Concentration fluctuations
Application of chaotic advection to thermal mixing
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Two rod mixers for concentrated emulsions
0
10
20
30
40
50
60
70
80
90
100
0 5 10 15 20 25 30 35
Emulsification time (min)
d32 (µm)d32 (µm)
Emulsification time (min)
0
2
4
6
8
10
12
3 6 9 12 15 18 21 24 27 30 33 36 39
%
d (µm)
0
2
4
6
8
10
12
3 6 9 12 15 18 21 24 27 30 33 36 39
%
d (µm)
0
2
4
6
8
10
12
3 6 9 12 15 18 21 24 27 30 33 36 39
%
d (µm)
Highly concentrated O/W emulsion
Low energy mixing
Emulsification
Chaotic mixing strategy to obtain a narrow distribution of droplets size
90% oil
10% water+ surf.
Caubet et al., AIChE J., 2011 - S. Caubet PhD thesis 2010
Two rod mixer with continuous flow
Application of chaotic advection to thermal mixing
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Thermal mixing● Highly viscous fluids with high Pr number
like viscous oils or polymeric solutions● classical mixing with turbulent flows are energy
consuming● delicate fluids may be damaged by stirring
● High Pr: heat transfer by thermal conduction is weak compared to convective transport
Need to effective advection flows with creation of striations in the temperature field.
Application of chaotic advection to thermal mixing
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Fluid heating objectives
● Using chaotic advection for thermal mixing in the case of heating walls ● Two simultaneous objectives:
● Enhancement of parietal heat transfer: energy extraction from the walls.
● Homogenization of the temperature in the whole fluid to avoid hot or cold spots: rapid distribution of the extracted energy.
Application of chaotic advection to thermal mixing
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Differences between mixing of species and thermal mixing
● Molecular diffusivity is usually 2 decades lower than thermal diffusivity: temperature striations vanish more rapidly
● For temperature: the scalar source is located at the wall (transport from/to walls must be promoted) while for species, the scalar to mix is present in the fluid and static walls reduce its mixing rate (Gouillart et al. PRL, 2010)
Different mixing strategies
Application of chaotic advection to thermal mixing
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The Two Rod Mixer● Simple geometry: two vertical
rods in a cylindrical tank or duct (batch or continuous mixer)
● Rods and tank rotate around their respective axes
● Chaotic advection induced by temporal modulation of rotation speed
● 2D study in a transversal cut plan
Application of chaotic advection to thermal mixing
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The Two Rod Mixer● R1/R3 = 1/5● ● Continuously modulated or
alternated rotation
Jana et al., JFM (269),1994. Price et al., Royal Society, 2004.
Application of chaotic advection to thermal mixing
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Numerical Modeling● Unsteady NavierStokes and
energy eqs. solver (Tamaris)● Unstructured FV method ● Second order accurate spatial
and temporal schemes● HR resolution non linear
convective scheme: low numerical diffusion
● Pressurevelocity coupling: SIMPLE algorithm
Computational mesh10,000 cells
Application of chaotic advection to thermal mixing
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Mixing indicators
● Mean temperature (energy supply)
● Standard deviation (homogenization)
● Nusselt number (parietal heat transfer)
Dirichlet Neumann
Application of chaotic advection to thermal mixing
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Steady flow topologies
Rotating tank
Non rotating tank
Non rotating rods
Stirring Config.
Rod 1
Rod 2
Tank
1 (+) () (+)
2 (+) (+) (+)
3 () () (+)
Application of chaotic advection to thermal mixing
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Effect of modulation type
Non Modulated Cont. Modulated Alternated
Application of chaotic advection to thermal mixing
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Continuous modulation Alternated modulation
t = 4 t = 4.5 t = 4 t = 4.5 Parabolic points
Application of chaotic advection to thermal mixing
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Animation
Application of chaotic advection to thermal mixing
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Temperature Probability Distribution Functions
(PDF)
Effect of stirring configuration
Rescaled temperature
Strange eigenmodes
0<T<1
1 2 3
Flow configurations
Application of chaotic advection to thermal mixing
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Fixed color map 0<T<1
Selfsimilar temperature patterns
Adjusted color map
Application of chaotic advection to thermal mixing
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Thermal mixing using a constant heat fluxconstant heat flux boundary condition
● Heating with a Constant Heat Flux (CHF) is also of common industrial usage as Fixed Wall Temperature (FWT).
Power supply to the fluid:
FWT: depends on the flow
CHF: Fixed
Fluid temperature:
CHF: linear evolution
FWT: asymptotic limit
Different mixing strategies
Application of chaotic advection to thermal mixing
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CHF
FWT
Temperature extrema evolution in the fluid
FWT decreasing heat flux in time
ALTCM
NMALT
NM
CM
Application of chaotic advection to thermal mixing
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PDF(T)CHF
FWT
ALT
ALT
Nu (Tank + rods)
CHF
Application of chaotic advection to thermal mixing
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Thermal mixing of NonNewtonian fluids Powerlaw fluids:
shearthinning (n=0.5)
Newtonian (n=1)
shearthickening (n=1.5)
BC: Constant Wall Temperature
Application of chaotic advection to thermal mixing
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Continuous Modulation Alternated rotation
t = 4
t = 4.5
n = 0.5 n = 1 n =1.5 n = 0.5 n = 1 n =1.5
Application of chaotic advection to thermal mixing
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Continuous Modulation
Alternated Rotation
Application of chaotic advection to thermal mixing
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Effect on mixing of viscosity temperaturedependence
Powerlaw nonNewtonian fluids:
with
where Pearson number
● Shearthinning (n=0.5)● Searthickening (n=1.5)
● Heating ● Cooling
Application of chaotic advection to thermal mixing
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n = 0.5 Heating
t = 5
T
V
t = 4.5 t = 5
n = 0.5 Cooling
B = 0 B = 5 B = 0 B = 5
Shear thinning fluid
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Conclusion
● Global thermal chaotic mixing is very sensitive to the wall kinematics● Alternated rotation of walls gives better mixing than continuous modulation (key role of parabolic points)● Mixing strategy should be adapted to the thermal wall boundary condition● The mixing efficiency deteriorates for shearthinning fluid and for high temperaturedependence of viscosity.
Application of chaotic advection to thermal mixing
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Thank you !
End
Application of chaotic advection to thermal mixing
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Effect of rotation direction and period size
Stirring configuration
Rod 1 Rod 2 Tank
1 (+) () (+)
2 (+) (+) (+)
3 () () (+)
global combined indicator
= 15 s
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Effect of rod Eccentricity
t=7.3
Application of chaotic advection to thermal mixing
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n = 0.5
Heating Cooling
n = 1.5