Application of Chaotic Advection to Thermal Mixing · Application of chaotic advection to thermal...

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



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



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



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



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



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



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



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.



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.



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



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



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.



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



Mixing indicators

● Mean temperature (energy supply)

● Standard deviation (homogenization)

● Nusselt number (parietal heat transfer)

Dirichlet Neumann



Steady flow topologies

Rotating tank

Non rotating tank

Non rotating rods

Stirring Config.

Rod 1

Rod 2

Tank

1 (+) () (+)

2 (+) (+) (+)

3 () () (+)



Effect of modulation type

Non Modulated Cont. Modulated Alternated



Continuous modulation Alternated modulation

t = 4 t = 4.5 t = 4 t = 4.5 Parabolic points



Animation



Temperature Probability Distribution Functions

(PDF)

Effect of stirring configuration

Rescaled temperature

Strange eigenmodes

0<T<1

1 2 3

Flow configurations



Fixed color map 0<T<1

Selfsimilar temperature patterns

Adjusted color map



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



CHF

FWT

Temperature extrema evolution in the fluid

FWT decreasing heat flux in time

ALTCM

NMALT

NM

CM



PDF(T)CHF

FWT

ALT

ALT

Nu (Tank + rods)

CHF



Thermal mixing of NonNewtonian fluids Powerlaw fluids:

shearthinning (n=0.5)

Newtonian (n=1)

shearthickening (n=1.5)

BC: Constant Wall Temperature



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



Continuous Modulation

Alternated Rotation



Effect on mixing of viscosity temperaturedependence

Powerlaw nonNewtonian fluids:

with

where Pearson number

● Shearthinning (n=0.5)● Searthickening (n=1.5)

● Heating ● Cooling



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



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.



Thank you !

End



Effect of rotation direction and period size

Stirring configuration

Rod 1 Rod 2 Tank

1 (+) () (+)

2 (+) (+) (+)

3 () () (+)

global combined indicator

= 15 s



Effect of rod Eccentricity

t=7.3



n = 0.5

Heating Cooling

n = 1.5

Application of Chaotic Advection to Thermal Mixing · Application of chaotic advection to thermal...

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