2. The lattice Boltzmann for porous flow and transport
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1 Lattice Boltzmann for flow and transport phenomena 2. The lattice Boltzmann for porous flow and transport Li Chen XJTU, 2016 Mail:
2 Content o 2.1 Examples of porous media o 2.2 Structural characteristics of porous media o 2.3 Reconstruction of porous media o 2.4 LBM simulation of transport in porous media 2/40
3 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, CO 2 geological sequestration, groundwater contaminant transport and bioremediation, nuclear waste disposal o Artificial porous systems: fuel cell, reactor, catalysts, building material CO 2 Porous media 3/40
4 2.1 Examples of porous media Stone 4/40
5 2.1 Examples of porous media Soil 5/40
6 2.1 Examples of porous media Dried up river beds 6/40
7 2.1 Examples of porous media Wood 7/40
8 2.1 Examples of porous media Lung 8/40
9 2.1 Examples of porous media Metal foam 9/40
10 2.1 Examples of porous media Pack-bed reactor 10/40
11 2.1 Examples of porous media Road 11/40
12 Content o 2.1 Examples of porous media o 2.2 Structural characteristics of porous media o 2.3 Reconstruction of porous media o 2.4 LBM simulation of transport in porous media 12/40
13 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
14 2.2.1 Porosity The volume ratio between pore volume and total volume ε = V pore V total 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%. Shale: <10% GDL: >70% 14/40
15 2.2.1 Porosity 1 2 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 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
16 2.2.1 Porosity 3 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. Volume of liquid Immersing 16/40
17 2.2.1 Porosity 4 Mercury injection method: Most of the materials are not wetted by mercury 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. The penetration is never quite complete as it requires infinite pressure o perfectly fill all the edges and corners of the pores. p High pressure will change or even damage the structures of the porous medium p 1 p 2 17/40
18 2.2.1 Porosity 5 Gas expansion method: Calculate according to the Equation of State 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. P 1 V a V b V B P 1 (V a -(V B -V p ))=P 2 (V a +V b -(V B -V p )) V p =V B -V a -V b [P 2 /(P 2 -P 1 )] 18/40
19 2.2.2 Pore size Pore size terminology of IUPAC International Union of Pure and Applied Chemistry Rouquerol et al. (1994) Membrane Choquette and Pray (1970)Carbonate rock Micropores, <2nm Mesopores, 2~50nm Macropores, >50nm Micropores, <62.5µm Mesopores, 62.5µm~4mm Macropores, 4mm~256mm Rouck et al. 2012, further added picopore and nanopore for study of shale. 19/40
20 2.2.2 Pore size How to define pore size? d Klaver, Desbois et al 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 20/40
21 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 reactor 21/40
22 2.2.4 Tortuosity Tortuosity: defined as the actual length traveled by a particle to the length of the media τ = L i L L i L 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 22/40
23 Content o 2.1 Examples of porous media o 2.2 Structural characteristics of porous media o 2.3 Reconstruction of porous media o 2.4 LBM simulation of transport in porous media 23/40
24 2.3 Reconstruction θ Gas diffusion layer 24/40
25 Organic matter in shale 25/40
26 Dried bed------vonoroi mesh 26/40
27 Content o 2.1 Examples of porous media o 2.2 Structural characteristics of porous media o 2.3 Reconstruction of porous media o 2.4 LBM simulation of transport in porous media 27/40
28 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 μu o o o leakage in the test too high Re number fluid is not viscous. < u >= k µ Δp l Δp o q flow rate (m/s), μ the viscosity, pressure drop Δp, length of the porous domain l, k is the permeability. 28/40
29 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 29/40
30 Bounce-back at the solid surface 1 f t t t f t f t f t eq i ( x ci, ) i ( x, ) ( i ( x, ) i ( x, )) Collision f ' 1 eq ( x, t) ( f (, ) (, )) i i x t fi x t Streaming f t t t f t ' i( x ci, ) ( x, ) Macroscopic variables calculation i 30/40
31 Bounce-back at the solid surface Pressure drop across x direction Periodic boundary condition y Analyze the detailed flow field Calculate permeability based on Darcy equation. < u >= k µ Δp l 31/40
32 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 k 2 3 d 180 (1 ) 2 32/40
33 Permeability o Fibrous beds have received special attention for its wide applications such as filter which can form stable structures of very high porosity. k c r A( 1) B 33/40
34 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): /40
35 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 35/40
36 1/k Forchheimer equation o For flow with very high rate, inertial effects can also become significant. Therefore an inertial term is added to the Darcy s equation. Δp = μ < u > ρβ < u >< u > l k 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. ( Δp ) /(μ < u >) = 1 β ρ<u> l k µ 1 k = 1 k + βre pseudo Re 1 4 k = 1 k + βre pseudo Re pseudo 36/40
37 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. C J D x C H C L J ' D eff C x Deff D Bruggeman equation: α =1.5 37/40
38 Evolution equation 1 g t t t g t t g t eq i ( x ci, ) i ( x, ) (g i ( x, ) i ( x, )) Collision g ' 1 eq ( x, t) (g (, ) (, )) i i x t gi x t Streaming g t t t g t ' i( x ci, ) ( x, ) Macroscopic variables calculation C g i Diffusivity and collision time i 1 D (1 J0 )( 0.5) 2 38/40
39 After the concentrationfield is obtained, the effective diffusivity is calculated by Bruggeman equation: α=-1.48 Chen et al. Electrochemica Acta, /40
40 Effective Knudsen diffusivity Concentration Deff 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, /40
41 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 T q ' x T H T L eff 41/40
42 2.5. Heat transfer 硬质聚氨酯发泡材料 密度低 孔隙率极高, 因而导热系数极低, 常作为保温隔热材料使用 42/40
43 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
44 二氧化硅气凝胶 等效结构单元体数值重构 (a) 颗粒型微观结构 气凝胶电镜扫描图 (b) 开孔型微观结构 44/40
45 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 : 1 gi i t t t gi t gi t gi t eq r e, r, r, r, 单松弛的 BGK 模型 其中, 是温度分布函数, 是无量纲松弛时间, g eq i 是平衡态温度分布函数 g i eq i g T / 7, i 0 6 e i 是格子离散速度 : e i c D3Q7 模型 45/40
46 ±10% 误差 开孔型微观结构, 颗粒骨架连续 图不同压力下气凝胶导热系数实验值与模拟值的对比 颗粒型微观结构, 颗粒骨架不连续 46/40
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