José S. Andrade Jr. Universidade Federal do Ceará Departamento de Física Flow and heat transport...
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![Page 1: José S. Andrade Jr. Universidade Federal do Ceará Departamento de Física Flow and heat transport in irregular channels Collaborators: Ascânio Dias Araújo.](https://reader036.fdocumentos.com/reader036/viewer/2022062519/5697bff31a28abf838cbc3fb/html5/thumbnails/1.jpg)
José S. Andrade Jr.
Universidade Federal do CearáDepartamento de Física
Flow and heat transport in irregular channels
Collaborators:
Ascânio Dias Araújo (UFC)
Raimundo N. Costa Filho (UFC)
Murilo P. Almeida (UFC)
Marcel Filoche (Ecole Polytechnique, France)
Bernard Sapoval (Ecole Polytechnique, France)
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Screening EffectsScreening Effects
Sapoval, Phys. Rev. Lett. (1994)Filoche and Sapoval, Phys. Rev. Lett. (2000)Andrade, Filoche and Sapoval, Chem. Eng. Sci. (2001)
temperature heat flux
Laplacian system
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Makarov TheoremMakarov Theorem
Makarov theorem (1985): “The information dimension of the harmonic measure is equal to 1 in d=2.”
Meaning: The set where the activity takes place has a dimension equal to 1. The length of the active zone is proportional to the system size.
2) Laplace equation
1) Dirichlet BC
Diffusion Cell:
.1 withLLa
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Makarov TheoremMakarov Theoremsubstrate
“alive” interface
active interface
SLL pa /
is the “screening efficiency”1S
L
LLS p /
pL
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Makarov TheoremMakarov Theorem
1
1
2
PL
iiaL ),1( Pat LL
Active Length Square Koch Curve
02 C
0C
SCC
./ jii qq
pa LL equal partition of fluxes
1aL
with
strongly “localized”
),/1( iLpi
The value La=22.9 is compatible with the prediction of the Makarov theorem, La ≈ L=27.
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Screening in flow through fractal Screening in flow through fractal channels channels Laplace & Stokes
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Screening in flow through fractal Screening in flow through fractal channels channels
Laplace & Stokes
LS
highly heterogeneous!!
Evertsz & Mandelbrot,J. Phys. A (1992)
Andrade, Araújo, Filoche & Sapovalaccepted PRL (2007)
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Screening - Inertial Effects Screening - Inertial Effects
High Reynolds
Low Reynolds
0 u
uuρpuμ
2continuity
Navier-Stokes
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Permeability & Active LengthPermeability & Active Length
Permeability
w
PKV
12/20
hK smooth channel
Vh
Re Reynolds number
Darcy`s Law
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Screening in flow through fractal Screening in flow through fractal channels channels
Random Fractal Wall active length & position in the channel
u
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Convective Heat Transport Convective Heat Transport
Heat transport between (self-similar) rough
walls: Constant properties μ, ρ, and α. Steady state, ∂T/∂t=0. Diffusion-convection equation,TTu 2.
0T),( yxu
),( yxT
wT
y
x
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Convective Heat Transport Convective Heat Transport
Temperature fields
Temperature increases from
blue to red.
25.0Pe
200Pe
510Pe
710Pe10Pe
D
VPe
Péclet number
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Convective Heat Transport Convective Heat Transport
collapse
roughness effect
200Pe
0PeHeat Flux & Péclet
smooth
g3g2
g1
Andrade et al., Physica A (2004)
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Convective Heat Transport Convective Heat Transport
rough and smooth showthe same behavior
roughness effect
Activity length & Péclet
smooth
g2
g1
g3
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Recent papers on the subject:
[1] B. Sapoval, J.S. Andrade Jr. and M. Filoche, Chem. Engng. Sci. 56, 5011 (2001). (catalysis)
[2] J. S. Andrade Jr., M. Filoche and B. Sapoval, Europhys. Lett. 55, 573 (2001). (catalysis)
[3] M. Filoche, J. S. Andrade Jr. and B. Sapoval, Physica A 342, 395 (2004). (catalysis)
[4] J. S. Andrade Jr., H. F. da Silva, M. B. da Silva and B. Sapoval, Phys. Rev. E 68, 049802 (2004). (catalysis, Knudsen diffusion)
[5] B. Sapoval, M. H. A. S. Costa, J. S. Andrade Jr. E M. Filoche, Fractals 12, 381 (2004). (catalysis)
[6] J. S. Andrade Jr., E. A. A. Henrique, M. P. Almeida e M. H. A. S. Costa, Physica A 339, 296 (2004). (heat transport)
[7] M. Filoche, J. S. Andrade Jr. and B. Sapoval, AIChE Journal 51, 998 (2005). (catalysis)
[8] B. Sapoval et al., Physica A 357, 1 (2005). (review)
[9] J. S. Andrade Jr., A. D. Araújo, M. Filoche e B. Sapoval, accepted PRL (2006). (screening)
[10] M. Filoche, D. Grebenkov, J. S. Andrade Jr. E B. Sapoval, submitted (2006). (catalysis)
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NanopercolationAndrade, Azevedo, Costa Filho and Correa
Filho, Nano Letters (2005)
Motivation
Nanotechnology
Material design
Functional polymers:Drug delivery Improved catalyst supportsSupramolecular structures
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Dendrimers
Branching molecules
Functional polymers
Fractal dimension ≈ 2.5
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Dendrimers are real molecules!!
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SFM images of (1) individual molecules, and (2) thin films.
[Frauenrath H., Prog. Polym. Sci. (2005)]
Dendrimers
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Percolating Molecules: Generation
Square or honeycomb lattices of size L
Spanning cluster at p=pc
Sites are carbon atoms connected by single bonds
The valence is adjusted to 4 with hydrogen atoms
Fractional stoichiometry → CxHy
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Molecular Mechanics
Vbnd
Force fields from Classical Mechanics → potential energy V
Vang
MM+ force field → optimized geometry
Comparison with semi-empirical PM3 → 3% difference
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Simulations: square lattice
C
H
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Simulations
carbon nanosheet
square lattice
honeycomb lattice
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Simulations Size, Sampling and
Properties
Critical square and honeycomb lattices
Square lattice
Honeycomb lattices
Radius of Gyration (Rg)
L 12 15 18 21 24 27 30
Nrea 300 300 150 150 80 80 50
L 10 14 18 22 26 30
Nrea 300 150 150 150 80 80
a
a
N
ii
a
N
ii
g rrwithN
rrR
10
20
1
||
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ResultsFractal Dimension
fdg MR /~ 1
)(50.2 squared f )(63.2 honeycombd f
WdW LV ~
)(84.1 squaredW )(87.1 honeycombdW
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Nanopercolation: Is it possible?
SEM of carbon nanosheets grown on Si substrate [Wang et al., Carbon (2004)]
Carbon Nanosheets Self-Organized Percolation (SOP)
Typical cluster grown under an SOP rule [Andrade et al., Physica A (1997), Alencar et al., PRE (1997)]
( 1) ( ) [ ( )]Tp t p t k N N t