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Lattice Boltzmann for flow and transport phenomena
2. The lattice Boltzmann for porous flow and
transport
Li Chen
Mail: [email protected]
http://gr.xjtu.edu.cn/web/lichennht08
XJTU, 2016
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o 2.1 Examples of porous media
o 2.2 Structural characteristics of porous media
Content
o 2.3 Reconstruction of porous media
o 2.4 LBM simulation of transport in porous media
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2.1 Examples of porous media
o Transport processes in porous media are widely encountered in scientific and engineering
problems
o Natural porous systems: enhanced hydrocarbon and geothermal energy recovery, CO2
geological sequestration, groundwater contaminant transport and bioremediation, nuclear
waste disposal…
o Artificial porous systems: fuel cell, reactor, catalysts, building material…
CO2
Porous media
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o 2.1 Examples of porous media
o 2.2 Structural characteristics of porous media
Content
o 2.3 Reconstruction of porous media
o 2.4 LBM simulation of transport in porous media
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2.2 Structural characteristics of porous media
A material that contains plenty of pores (or voids) between solid
skeleton through which fluid can transport.
Two necessary elements: skeleton and pores;
Skeleton: maintain the shape
Pores: provide pathway for fluid flow through
Stone Gas diffusion layer 13/40
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2.2.1 Porosity
The volume ratio between pore volume and total volume
Shale: <10% GDL: >70%
Porosity maybe vary from near zero to almost unity. Shale has
low porosity, around 5%. Fiber-based porous media can have a
porosity as high as 90%.
ε =𝑉pore
𝑉total
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2.2.1 Porosity
Direct method: measure the bulk volume of a porous sample
and then somehow destroying all the voids and measuring the
volume of the only solids.
Optical methods: if the porous structure is random, then the
porosity equals the “areal porosity”. The areal porosity is
determined based on the polished sections of the sample
1
2
To make the pores more visible, pores
are often impregnated with wax, plastic
or Wood’s metal. Connected and isolated
pores can be distinguished.
Depending the resolution of the
experimental techniques. Pores under the
resolution can not be identified.15/40
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Imbibition method :
a) immersing the porous sample in a perfectly wetting fluid for
a sufficiently long time, the wetting fluid will imbibe into the
pores.;
b) weight the sample before and after the imbibition, with the
density of the fluid, the volume of the pores can be
determined;
c) or displace the wetting fluid out of the porous medium and
measure the volume of the fluid.
3
Immersing
Volume of liquid
2.2.1 Porosity
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Mercury injection method:
Most of the materials are not wetted by mercury
4
Hydraulic pressure in the chamber containing both porous
medium and the mercury. As a result, the mercury will enter the
pores. If the pressure is sufficiently high, mercury will penetrate
into very small pores.
p
p1 p2
The penetration is never quite
complete as it requires infinite
pressure o perfectly fill all the
edges and corners of the pores.
High pressure will change or
even damage the structures of
the porous medium
2.2.1 Porosity
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P1(Va-(VB-Vp))=P2(Va+Vb-(VB-Vp))
Vp=VB-Va-Vb[P2/(P2-P1)]
P1
Va
VB
Vb
2.2.1 Porosity
Gas expansion method:
Calculate according to the Equation of State
5
Gas in one chamber with a porous medium is expanded to
another chamber. When equilibrium state is reached, the
pressure is measured. With known volume of the two chambers,
the volume of the porous medium can be determined.
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2.2.2 Pore size
Micropores, <2nm
Mesopores, 2~50nm
Macropores, >50nm
Rouquerol et al.
(1994) Membrane
Choquette and Pray
(1970)Carbonate rock
Micropores, <62.5µm
Mesopores, 62.5µm~4mm
Macropores, 4mm~256mm
Pore size terminology of IUPAC
International Union of Pure and Applied Chemistry
Rouck et al. 2012, further added picopore and nanopore for study of shale.
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2.2.2 Pore size
How to define pore size?
d
Klaver, Desbois et al. 2015
pore area
pore perimeter
pore long/short axis length
orientation
circularity
convexity and elongation
Theoretically, one can divide the connected pores into infinite
slice and determine the averaged diameter of each slice (volume
averaged, area averaged…). However, such theoretical definition
can not be practically implemented.
Mercury invasion method, Adsorption method, Optical method
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2.2.2 Specific surface area
Defined as the ratio between total surface area to the total
volume.
An important parameter for porous media as one of the
important type of porous media is catalyst, which requires high
specific surface area for reaction.
Catalyst of Fuel cell Packed-bed reactor21/40
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2.2.4 Tortuosity
Tortuosity: defined as the actual length traveled by a particle to
the length of the media
𝜏 =𝐿𝑖𝐿 𝐿
𝐿𝑖
Tortuosity is thus transport dependent, including flow,
diffusion, heat transfer, electrical conduct, acoustic transport.
For fluid flow it is “hydraulic tortuosity”
For diffusion it is “diffusivity tortuosity”
For electron transport, it is “conductivity tortuosity”
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o 2.1 Examples of porous media
o 2.2 Structural characteristics of porous media
Content
o 2.3 Reconstruction of porous media
o 2.4 LBM simulation of transport in porous media
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o 2.1 Examples of porous media
o 2.2 Structural characteristics of porous media
o 2.4 LBM simulation of transport in porous media
Content
o 2.3 Reconstruction of porous media
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2.4.1 Permeability
o Permeability, an indictor of the capacity of a porous medium
for fluid flow through
o In 1856, Darcy noted that for laminar
flow through porous media, the flow
rate <u> is linearly proportional to
the applied pressure gradient Δp, he
introduced permeability to describe
the conductivity of the porous media.
The Darcy’ law is as follows
o q flow rate (m/s), μ the viscosity, pressure drop Δp, length of
the porous domain l, k is the permeability.
< 𝑢 >= −𝑘
µ
Δp𝑙
Δp
𝜇u
o leakage in the test
o too high Re number
o fluid is not viscous.
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Schematic of Darcy’s experiment Simulation mimic the experiment
Δp
Δp
NS equation is solved at the pore scale. Non-slip
boundary condition for the fluid-solid interface
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eq1( , ) ( , ) ( ( , ) ( , ))i i i i if t t t f t f t f t
x c x x x
Collisioneq' 1
( , ) ( ( , ) ( , ))i i it f t f tf
x x x
'( , ) ( , )ii if t t t f t x c x
Streaming
Macroscopic variables calculation
Bounce-back at
the solid surface
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Analyze the detailed flow field
Calculate permeability based on
Darcy equation.
< 𝑢 >= −𝑘
µ
Δp𝑙
Bounce-back at the solid surface
Pressure drop across x direction
Periodic boundary condition y
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2.4.1. Permeability
o One of the most famous empirical relationship between
permeability and statistical structural parameters is proposed
by Kozeny and Carman (KC) equation for beds of particle
2 3
2180 (1 )
dk
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2.4.1. Permeability
o Fibrous beds have received special attention for its wide
applications such as filter which can form stable structures of
very high porosity.
2 1( 1)
1
Bck r A
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Close relation between flow and porous structure
Two-point correlation function Eulerian correlation function
Jin, C., et al. (2016). "Statistics of highly heterogeneous flow fields confined to three-dimensional random porous media." Physical Review E 93(1): 013122.34/40
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Validation of Darcy’s law
o Darcy’s law is only valid for incompressible, slow and viscous
flow (creeping flow), where fluid flow is dominated by viscous
force.
o Typically any flow with Re lower than one is clearly laminar.
Experimental results show that porous flow with Re up to 10
can be described by Darcy’s law.
𝑅𝑒 =ρ𝑢𝑑
μ< 10
o Most of fluid flow in porous media cases fall in this category.
Example: fluid flow in gas diffusion layer of a PEMFC;
ground water under subsurface…
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Forchheimer equation
Δp
𝑙= −
μ
𝑘< 𝑢 > −ρβ < 𝑢 >< 𝑢 >
o β is referred to as the beta factor (non-Darcy coefficient). The
first term account for the viscous contribution to the pressure
drop, where the second accounts for the inertial effect.
o For flow with very high rate, inertial effects can also become significant.
Therefore an inertial term is added to the Darcy’s equation.
(Δp𝑙
) /(μ < 𝑢 >) = −1
𝑘− β
ρ<𝑢>
µ1
𝑘′ =1
𝑘+ β𝑅𝑒pseudo
1
𝑘′=1
𝑘+ β𝑅𝑒pseudo
𝑅𝑒pseudo
Re
1/𝑘
4
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2.4.2. Effective diffusivity
Mass transfer is an important processes in many engineering
and scientific problems. Diffusivity D is adopted to describe the
capacity of a porous medium for mass transport.
CH
CLC
J Dx
eff'C
J Dx
effD D
Bruggeman equation: α =1.5
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eq1( , ) ( , ) (g ( , ) ( , ))i i i i ig t t t g t t g t
x c x x x
Collision
eq' 1( , ) (g ( , ) ( , ))
i i it t g tg
x x x
'( , ) ( , )ii ig t t t g t x c x
Streaming
Macroscopic variables calculation
iC g
Evolution equation
Diffusivity and collision time
0
1(1 )( 0.5)
2D J
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Bruggeman equation:
α=-1.48
After the concentrationfield is
obtained, the effective diffusivity
is calculated by
Chen et al. Electrochemica Acta, 2015 39/40
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Concentration
Effective Knudsen diffusivity
effD D
Bruggeman equation: α =1.5
Our prediction: α=2.8~4.0
Void space are very tortuous
Chen et al. Scientific reports, 2015, Chen et al. Fuel, 2015 40/40
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2.4.3. Effective thermal conductivity
Estimating effective thermal conductivity of porous media is
thus important.
硬质聚氨酯发泡材料 C/C-SiC复合材料 气凝胶
TH TL eff'T
qx
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2.5. Heat transfer
二氧化硅气凝胶
以纳米微粒相互聚集构成多孔网络结构，并在网络孔隙中充满气态介质的超轻纳米多孔材料
在航空航天、保温节能等领域具有广泛的应用前景!43/40
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eq1, , , ,i i i i ig t t t g t g t g t
r e r r r
其中， 是温度分布函数， 是无量纲松弛时间， 是平衡态温度分布函数
ig eqig
eq / 7, 0 6ig T i
是格子离散速度：ie
0 1 1 0 0 0 0
0 0 0 1 1 0 0
0 0 0 0 0 1 1
ei c
单松弛的BGK模型
D3Q7模型
数值模拟方法
采用D3Q7格子Boltzmann模型，对于材料中的不同相组分，其对应于温度控制方程的温度分布函数演化方程：
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