josé s. andrade jr. universidade federal do ceará departamento de física flow and heat transport...
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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)
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
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
Makarov TheoremMakarov Theoremsubstrate
“alive” interface
active interface
SLL pa /
is the “screening efficiency”1S
L
LLS p /
pL
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.
Screening in flow through fractal Screening in flow through fractal channels channels Laplace & Stokes
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)
Screening - Inertial Effects Screening - Inertial Effects
High Reynolds
Low Reynolds
0 u
uuρpuμ
2continuity
Navier-Stokes
Permeability & Active LengthPermeability & Active Length
Permeability
w
PKV
12/20
hK smooth channel
Vh
Re Reynolds number
Darcy`s Law
Screening in flow through fractal Screening in flow through fractal channels channels
Random Fractal Wall active length & position in the channel
u
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
Convective Heat Transport Convective Heat Transport
Temperature fields
Temperature increases from
blue to red.
25.0Pe
200Pe
510Pe
710Pe10Pe
D
VPe
Péclet number
Convective Heat Transport Convective Heat Transport
collapse
roughness effect
200Pe
0PeHeat Flux & Péclet
smooth
g3g2
g1
Andrade et al., Physica A (2004)
Convective Heat Transport Convective Heat Transport
rough and smooth showthe same behavior
roughness effect
Activity length & Péclet
smooth
g2
g1
g3
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)
NanopercolationAndrade, Azevedo, Costa Filho and Correa
Filho, Nano Letters (2005)
Motivation
Nanotechnology
Material design
Functional polymers:Drug delivery Improved catalyst supportsSupramolecular structures
Dendrimers
Branching molecules
Functional polymers
Fractal dimension ≈ 2.5
Dendrimers are real molecules!!
SFM images of (1) individual molecules, and (2) thin films.
[Frauenrath H., Prog. Polym. Sci. (2005)]
Dendrimers
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
Molecular Mechanics
Vbnd
Force fields from Classical Mechanics → potential energy V
Vang
MM+ force field → optimized geometry
Comparison with semi-empirical PM3 → 3% difference
Simulations: square lattice
C
H
Simulations
carbon nanosheet
square lattice
honeycomb lattice
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
||
ResultsFractal Dimension
fdg MR /~ 1
)(50.2 squared f )(63.2 honeycombd f
WdW LV ~
)(84.1 squaredW )(87.1 honeycombdW
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