2. The lattice Boltzmann for porous flow and transport

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1 Lattice Boltzmann for flow and transort henomena 2. The lattice Boltzmann for orous flow and transort Li Chen XJTU, 2018/05/30 Mail: htt://gr.xjtu.edu.cn/web/lichennht08 1/66

2 Content o 2.1 Background o 2.2 Structural characteristics of orous media o 2.3 Continuum-scale governing equation for orous media o 2.4 Pore-scale simulation: reconstruction of orous media o 2.5 Pore-scale simulation: LB for fluid flow o 2.6 Pore-scale simulation: LB for heat transfer 2/66

3 Stone Carbon fiber Metal foam Catalyst 3/66

4 Content o 2.1 Examles of orous media o 2.2 Structural characteristics of orous media o 2.3 Continuum-scale governing equation for orous media o 2.4 Pore-scale simulation: reconstruction of orous media o 2.5 Pore-scale simulation: LB for fluid flow o 2.6 Pore-scale simulation: LB for heat transfer 4/66

5 2.2 Structural characteristics of orous media o A material that contains lenty of ores (or voids) between solid skeleton through which fluid can transort. o Two necessary elements: skeleton and ores; Skeleton: maintain the shae; Pores: rovide athway for fluid flow through Black: solid White: ores 5/66

6 2.2.1 Porosity o The volume ratio between ore volume and total volume ε = V ore V total o Porosity maybe vary from near zero to almost unity. Shale has low orosity, around 5%. Fiber-based orous media can have a orosity as high as 90%. Shale: <10% GDL: >70% 6/66

7 2.2.2 Pore size Pore size terminology of IUPAC International Union of Pure and Alied Chemistry Rouquerol et al. (1994) Membrane Microores, <2nm Mesoores, 2~50nm Macroores, >50nm Rouck et al. 2012, further added icoore and nanoore for study of shale. 7/66

8 Pore size distribution 8/66

An imortant arameter for orous media as one of the imortant tye of

9 2.2.3 Secific surface area Defined as the ratio between total surface area to the total volume. An imortant arameter for orous media as one of the imortant tye of orous media is catalyst, which requires high secific surface area for reaction. Catalyst of Fuel cell Packed-bed reactor 9/66

10 2.2.4 Tortuosity Tortuosity: defined as the actual length traveled by a article to the length of the media τ = L i L L i L Tortuosity is thus transort deendent, including flow, diffusion, heat transfer, electrical conduct, acoustic transort. For fluid flow it is hydraulic tortuosity For diffusion it is diffusivity tortuosity For electron transort, it is conductivity tortuosity Excet for some very simle orous structures, there is no clear consensus on the relation between these definitions. 10/66

11 Content o 2.1 Examles of orous media o 2.2 Structural characteristics of orous media o 2.3 Continuum-scale governing equation for orous media o 2.4 Pore-scale simulation: reconstruction of orous media o 2.5 Pore-scale simulation: LB for fluid flow o 2.6 Pore-scale simulation: LB for heat transfer 11/66

12 Two velocity definition in a orous medium: v suerficial v hysical Porosity v hysical V hysical : the actual flow velocity in the ores. V suerficial ( 表观速度 ):the averaged velocity in the entire domain. V suerficial < V hysical Fluent uses suerficial velocity as the default velocity. 12/66

13 Original continuity and momentum equation ( u) 0 t ( u) t ( )( 2 u u) u Continuity equation for orous media: ( ) ( u hysical ) 0 t As the total mass of fluid is V f = V total = x y z Fluent uses suerficial velocity as the default velocity. ( ) t ( u ) 0 suerficial 13/66

14 Momentum equation for orous media: ( u ) ( u t ( u )( u ) ( ) u F hysical 2 hysical hysical hysical suerficial suerficial 2 suerficial suerficial t ) u u ( )( u ) ( ) ( ) F For incomressible steady state roblem: u Total force due to orous media suerficial 0 u 1 u u F suerficial 2 ( )( suerficial ) ( ) ( suerficial) 14/66

15 The fluid-solid interaction is strong in orous media. Porous media are modeled by adding a momentum source term: F F - u - u u k k The first term is the viscous loss term ( 黏性项 )or the Darcy term. The second term is inertial loss term ( 惯性项 )or the Forchheimer term. k is the ermeability ( 渗透率 ) of a orous media, one of the most imortant arameter of a orous media 15/66

16 Permeability ( 渗透率 ) In 1856, Darcy ( 法国工程师 ) noted that for laminar flow through orous media, the flow rate <u> is linearly roortional to the alied ressure gradient Δ, thus he introduced ermeability to describe the conductivity of the orous media. The Darcy law is as follows u k µ l μu k is ermeability with unit of Δ m 2 16/66

17 In Fluent, this force source term is exressed as 1 F - u - C 2 u u k 2 k: ermeability; C 2 : inertial resistance factor The second term can be canceled if the fluid flow is slow F - u k 1 - C 2 u 2 u u is small, thus u*u is smaller. 17/66

There have been lots of exeriments in the literature to determine the relationshi between ressure dro and velocity of different kinds of orous media, and thus to determine ermeability.

18 There have been lots of exeriments in the literature to determine the relationshi between ressure dro and velocity of different kinds of orous media, and thus to determine ermeability. Ergun equation is one of the most adoted emirical equations ( 经验公式 ) for acked bed orous media. ΔP u l D D Diameter of solid article 1 F - u - C 2 u u k 2 u 2 18/66

19 ΔP u l D D u 2 F - u k 1 - C 2 u 2 u Comaring the two equations, you can obtain C 2. k 2 3 D C D 19/66

20 Original energy equation: ( CT) t ( )( 2 u CT ) T + S For orous media: Heat transfer in fluid hase as well as in solid hase. There are two models for heat transfer: Equilibrium thermal model Non-Equilibrium thermal model 20/66

21 Equilibrium thermal model Assume solid hase and fluid hase are in thermal equilibrium. At the fluid-solid hase, temerature and heat flux are continuous. Original For the first term: ( CT) t ( u )( ) + 2 C T T S C TV (1 ) V ( C ) T V ( C ) T solid solid fluid fluid (1 )( C ) solid ( C ) fluid VT C T (1 )( C ) solid ( C ) fluid T 21/66

22 For the second term: ( u )( CT ) As convective term is only for fluid hase! For the diffusion term: For the source term TV = V (1 ) T V T s s f f 2 V (1 ) s V f T T (1 ) s f 2 2 T SV (1 ) V S s VS f 22/66

23 ( (1 )( C ) solid ( C ) t fluid T ) 2 (1 ) s f T + (1 ) Ss S f ( u ) ( CT) ( C ) (1 )( C ) ( C ) eff solid fluid eff (1 ) s f Seff (1 ) S s S f The final energy equation for orous media ( ( C ) t eff T ) ( u )( C T suerficial 2 ) eff T + S eff 23/66

24 u Continuum-scale equations for orous flow u suerficial 0 1 u u F suerficial 2 ( )( suerficial ) ( ) ( suerficial) F F - u - u u k k ( ( C ) t eff T ) ( u )( C T suerficial 2 ) eff T + S eff eff (1 ) s f 24/66

25 Content o 2.1 Examles of orous media o 2.2 Structural characteristics of orous media o 2.3 Continuum-scale governing equation for orous media o 2.4 Pore-scale simulation: reconstruction of orous media o 2.5 Pore-scale simulation: LB for fluid flow o 2.6 Pore-scale simulation: LB for heat transfer 25/66

26 2.4 Reconstruction Fiber orous media in PEM fuel cell 1. L Chen, YL He, WQ Tao, P Zelenay, R Mukundan, Q Kang, Pore-scale study of multihase reactive transort in fibrous electrodes of vanadium redox flow batteries,, Electrochimica Acta 248, ; 2. L. Chen*, H.B. Luan, Y.-L. He, W.-Q. Tao, Pore-scale flow and mass transort in gas diffusion layer of roton exchange membrane fuel cell with interdigitated flow fields, 2012, 51, , International journal of thermal science 26/66

Catalyst layer in PEM fuel cell L. Chen, G. Wu, E.

and Non-Precious Metal Cathode Catalyst Layers in

27 Catalyst layer in PEM fuel cell L. Chen, G. Wu, E. Holby, P. Zelenay, W.Q. Tao, Q. Kang, Lattice Boltzmann Pore-Scale Investigation of Couled Physical-electrochemical Processes in C/Pt and Non-Precious Metal Cathode Catalyst Layers in Proton Exchange Membrane Fuel Cell, 2015, 158, , Electrochimica Acta 27/66

28 Organic matter in shale gas L. Chen, L Zhang, Q Kang, W.Q. Tao, Nanoscale simulation of shale transort roerties using the lattice Boltzmann method: ermeability and diffusivity, 2015, 5, 8089, Scientific Reorts 28/66

29 Porous rock L. Chen, Q. Kang, H. Viswanathan, W.Q. Tao, Pore-scale study of dissolution-induced changes in hydrologic roerties of rocks with binary minerals, 2014, 50 (12), WR015646, Water Resource Research 29/66

30 Fractures under subsurface L. Chen, F Wang, Q, Kang, J. Hyman, H. Viswanathan, W. Tao, Generalized lattice Boltzmann model for flow through tight orous media with Klinkenberg's effect, 2015, 91(3), , Physical Review E. 30/66

31 Random sheres with uniform ga Li Chen, et al. Chemical Engineering Journal, 2018, Pore-scale study of effects of macroscoic ores and their distributions on reactive transort in hierarchical orous media; L Chen, M Wang, Q Kang, W Tao, Pore scale study of multihase multicomonent reactive transort during CO2 dissolution traing, Advances in Water Resources, 2018, 116, /66

32 Content o 2.1 Examles of orous media o 2.2 Structural characteristics of orous media o 2.3 Continuum-scale governing equation for orous media o 2.4 Pore-scale simulation: reconstruction of orous media o 2.5 Pore-scale simulation: LB for fluid flow o 2.6 Pore-scale simulation: LB for heat transfer 32/66

33 Digitalized Structures A 2D matrix is adoted to reresent the orous media, with 1 as solid and 0 as fluid /66

34 Flow around a square 34/66

35 Flow around a circle 35/66

36 Permeability o Permeability, an indictor of the caacity of a orous medium for fluid flow through o In 1856, Darcy noted that for laminar flow through orous media, the flow rate <u> is linearly roortional to the alied ressure gradient Δ, he introduced ermeability to describe the conductivity of the orous 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 Δ l Δ o q flow rate (m/s), μ the viscosity, ressure dro Δ, length of the orous domain l, k is the ermeability. 36/66

37 Schematic of Darcy s exeriment Simulation mimic the exeriment Δ Δ NS equation is solved at the ore scale. Non-sli boundary condition for the fluid-solid interface 37/66

38 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, ) Macroscoic variables calculation i 38/66

39 Bounce-back at the solid surface Pressure dro across x direction Periodic boundary condition y Analyze the detailed flow field Calculate ermeability based on Darcy equation. < u >= k µ Δ l 39/66

40 Permeability o One of the most famous emirical relationshi between ermeability and statistical structural arameters is roosed by Kozeny and Carman (KC) equation for beds of article k 2 3 d 180 (1 ) 2 L. Chen, L Zhang, Q Kang, W.Q. Tao, Nanoscale simulation of shale transort roerties using the lattice Boltzmann method: ermeability and diffusivity, 2015, 5, 8089, Scientific Reorts 40/66

41 Permeability o Fibrous beds have received secial attention for its wide alications such as filter which can form stable structures of very high orosity. k 2 1c r A( 1) 1 B 1. L Chen, YL He, WQ Tao, P Zelenay, R Mukundan, Q Kang, Pore-scale study of multihase reactive transort in fibrous electrodes of vanadium redox flow batteries,, Electrochimica Acta 248, ; 2. L. Chen*, H.B. Luan, Y.-L. He, W.-Q. Tao, Pore-scale flow and mass transort in gas diffusion layer of roton exchange membrane fuel cell with interdigitated flow fields, 2012, 51, , International journal of thermal science 41/66

42 Content o 2.1 Examles of orous media o 2.2 Structural characteristics of orous media o 2.3 Continuum-scale governing equation for orous media o 2.4 Pore-scale simulation: reconstruction of orous media o 2.5 Pore-scale simulation: LB for fluid flow o 2.6 Pore-scale simulation: LB for heat transfer 42/66

43 LB model for heat transfer Evolution equation 1 eq gi ( x cit, t t) gi ( x, t) ( gi( x, t) gi ( x, t)) is( x, t) Equilibrium distribution function Temerature Thermal diffusivity g T(1 3 c u) eq i i i T g i i C 1 ( x) = ( 0.5) 3 t 2 43/66

44 LB model for heat transfer Standard energy governing equation ( ct) t ( c ut) ( T ) S Energy equation recovered from LB ( c T ) t ( c ut ) ( c T ) S where / c thermal diffusivity of the material. 44/66

45 At the fluid-solid interface f T ( T ) n f f f Ts ( T ) s n s s s f f f T f ( CT P ) n f T s s ( CT) n P s s Conjugate heat transfer In LB Satisfy Dirichlet- and Neumann-like boundary restriction at the same time Only when heat caacity of the two materials is the same, the conjugate heat transfer condition form the LB is equal to ractical one!!!! 45/66

46 /66

47 LB model for redicting thermal conductivity Standard energy governing equation ( ct) t ( c ut) ( T ) S f s ( c T ) ( T ) u 0 ( T ) seudo-caacity of solid hase scheme s f s f s X. Chen, P. Han, A note on the solution of conjugate heat transfer roblems using SIMPLE-like algorithms, Int. J. Heat Fluid Flow 21 (2000) /66

48 s f Half lattice node SRT is emloyed. Thus cannot take into account anisotroic thermal conductivity Jinku Wang a, Moran Wang b, Zhixin Li, A lattice Boltzmann algorithm for fluid solid conjugate heat transfer, 2007, 46, /66

49 MRT model Yoshida, H. and M. Nagaoka, Multile-relaxation-time lattice Boltzmann model for the convection and anisotroic diffusion equation. Journal of Comutational Physics, (20): /66

50 For heterogeneous anisotroic materials. Aerogel: low density and k Different comonents with large variations of k Effective thermal conductivity WZ Fang, L. Chen, JJ Gou, WQ Tao, Predictions of effective thermal conductivities for three-dimensional four-directional braided comosites using the lattice Boltzmann method, International Journal of Heat and Mass Transfer, 2016, 92, /66

51 General case For general heat transfer rocess, the unsteady term should be considered, and thus the seudo-caacity of solid hase scheme fails. 1. Treat the interface as boundary Treat the interface as boundary. Then the roblem is changed to construct boundary condition at the hase interface. 2. Re-arrange the governing equation Re-arrange the energy equation and add additional source term into the governing equation. 51/66

52 1. Treat the interface as boundary Dirichlet condition g ( x, t t) g ( x, t) i f i f Neumann equation t gi ( x, t t) g ( x, t) J x f i f d D d x s i i x f 1 2 g ( x i f, t t) ( ) gi ( f, t) ( ) gi( s, t) 1 x 1 x 1 2 g ( x i s, t t) ( ) gi ( s, t) ( ) gi( f, t) 1 x 1 x ( C ) s ( C) f Like Li, et al. PHYSICAL REVIEW E 89, (2014) 52/66

53 2. Re-arrange the governing equation ( c T ) t ( ct) t ( c ut) ( T ) S ct ( cut ) ( c( )) S c c h 1 1 c( ) = c( h h ) c c c = h c h 1 c ( ct) t 1 ( cut ) ( h ) ( ch ) S S * c Hamid Karani et al., PRE, 2015, 91, /66

54 2. Re-arrange the governing equation ( ct) t ( c ut) ( T ) S h* ( C ) T 0 C / ( C ) 0 ( h*) h* ( h* u) ( t ( C ) ) h* h h* * ( h* u) h* u ( t t ( C ) ) 0 0 h* 1 h* h* ( h* u) ( ) u t ( C ) 0 54/66

55 h* 1 1 h* ( h* u) ( Ch*) u t ( C ) C 0 h* 1 1 h* ( h* u) ( Ch*) u t ( C ) h* C ( h*) ( C h*) ( C ) u 1 1 h* ( h*) ( Ch*) u ( C ) h* 1 1 h* ( h* u) ( h*) ( Ch* ) u t ( C ) 0 Hamid Karani et al., PRE, 2015, 91, S * 55/66

56 LB model for mass transfer Evolution equation 1 eq gi ( x cit, t t) gi ( x, t) ( gi( x, t) gi ( x, t)) is( x, t) Equilibrium distribution function Concentration Diffusivity g C(1 3 c u) eq i i i D C g i 1 ( x) ( 0.5) 3 t i 2 L. Chen, et al., Pore-scale modeling of multihase reactive transort with hase transition and dissolution-reciitation in closed systems, 84 (4), (2013), Physical Review E 56/66

57 At the fluid-solid interface s f Mass transort does not take lace inside the solid hase. Thus, only mass transort inside the void sace needs to be considered. g f C L C HC C g L Dg s n g n L The non-continuous concentration across the hase interface oses additional challenge for numerical simulations. g 57/66

58 同舟共济渡彼岸! 58/59

Modeling Volume Changes in Porous Electrodes

Modeling Volume Changes in Porous Electrodes Journal of The Electrochemical Society, 53 A79-A86 2006 003-465/2005/53/A79/8/$20.00 The Electrochemical Society, Inc. Modeling olume Changes in Porous Electrodes Parthasarathy M. Gomadam*,a,z John W.