Multi species Lattice Boltzmann Models and Applications to Sustainable Energy Systems

Multi species Lattice Boltzmann Models and Applications to Sustainable Energy Systems Pietro Asinari, PhD Dipartimento di Energetica (DENER), Politecnico di Torino (POLITO), Torino 10129, Italy e-mail: pietro.asinari@polito.it, home page: http://staff.polito.it/pietro.asinari SIAM Conference on Computational Science and Engineering March 2 6, 2009, Miami Hilton Hotel, Miami, FL (USA) Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 1 / 29

Outline of this talk 1 Technological application Solid Oxide Fuel Cells (SOFC) 2 Lattice Boltzmann scheme Simplified AAP model (1) Equilibrium, (2) collisional matrix and (3) forcing Implementation by variable transformation 3 Model analysis Diffusive scaling Grad moment expansion 4 Numerical validation (1) Non-Fickian test case: Stefan tube (2) Solvent test case in a Poiseuille flow Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 2 / 29

Technological application Outline Compass 1 Technological application Solid Oxide Fuel Cells (SOFC) 2 Lattice Boltzmann scheme Simplified AAP model (1) Equilibrium, (2) collisional matrix and (3) forcing Implementation by variable transformation 3 Model analysis Diffusive scaling Grad moment expansion 4 Numerical validation (1) Non-Fickian test case: Stefan tube (2) Solvent test case in a Poiseuille flow Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 3 / 29

Technological application Solid Oxide Fuel Cells (SOFC) Solid Oxide Fuel Cells (SOFC) Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 4 / 29

Technological application Solid Oxide Fuel Cells (SOFC) Electron Microscopy for analyzing material structure Scanning Electron Microscopy (SEM) together with Energy Dispersion Spectrometry (EDS) for catching different solid phases in SOFC electrodes (anode is showed) Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 5 / 29

Technological application Solid Oxide Fuel Cells (SOFC) 3D Reconstruction by CHIMERA R Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 6 / 29

Technological application Solid Oxide Fuel Cells (SOFC) Spatial dependence of effective diffusivity Direct numerical calculation obtained by Lattice Boltzmann scheme Additional details are reported in Asinari et al., J. Power Sources, 2007 Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 7 / 29

Lattice Boltzmann scheme Outline Compass 1 Technological application Solid Oxide Fuel Cells (SOFC) 2 Lattice Boltzmann scheme Simplified AAP model (1) Equilibrium, (2) collisional matrix and (3) forcing Implementation by variable transformation 3 Model analysis Diffusive scaling Grad moment expansion 4 Numerical validation (1) Non-Fickian test case: Stefan tube (2) Solvent test case in a Poiseuille flow Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 8 / 29

Lattice Boltzmann scheme Simplified AAP model Multiple relaxation time AAP model on D2Q9 lattice Let us consider the LBM AAP model [Andries, Aoki, and Perthame, JSP 106, 2002; Asinari, PRE 77, 056706, 2008] for mixture modeling, namely f σ ˆt + V f σ [ i = A σ fσ( ) (ρ σ, u ] ˆx σ) f σ + dσ, (1) i where V i is a list of i-th components of the velocities in the considered lattice (for simplicity, let us consider D2Q9) V 1 = [ 0 1 0 1 0 1 1 1 1 ] T, (2) V 2 = [ 0 0 1 0 1 1 1 1 1 ] T, (3) f = f σ( ), f σ is a list of discrete distribution functions and d σ is a proper forcing term. Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 9 / 29

Lattice Boltzmann scheme Simplified AAP model Properties of target velocity in AAP model The target velocity can be expressed as u σ = u + ( ) m 2 B σς 1 x ς (u ς u σ ). (4) m ς σ m ς B mm If m σ = m for any σ, then (Property 1, which is the key for satisfying the Indifferentiability Principle) u σ = u + ( ) m 2 B mm 1 x ς (u ς u σ ) = u. (5) mm B ς mm Multiplying Eq. (4) by mass concentration x σ and summing over all the component yields (Property 2) x σ u σ = u + ( ) m 2 B σς 1 x σ x ς (u ς u σ ) = u. m σ σ ς σ m ς B m m (6) Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 10 / 29

Lattice Boltzmann scheme (1) Discrete local equilibrium (1) Equilibrium, (2) collisional matrix and (3) forcing Let us consider a matrix M = [1; V 1 ; V 2 ; V 2 1 ; V 2 2 ; V 1 V 2 ; V 1 (V 2 ) 2 ; (V 1 ) 2 V 2 ; (V 1 ) 2 (V 2 ) 2 ] T, then the discrete local equilibrium is defined as Mf σ( ) = Π 0 Π 1 Π 2 Π 11 Π 22 Π 12 Π 221 Π 112 Π 1122 = ρ σ ρ σ u σ1 ρ σ u σ2 p σ + ρ σ (u σ1 )2 p σ + ρ σ (u σ2 )2 ρ σ u σ1 u σ2 ρ σ u σ1 /3 ρ σ u σ2 /3 p σ /3 + ρ σ (u σ1 )2 /3 + ρ σ (u σ2 )2 /3. (7) Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 11 / 29

Lattice Boltzmann scheme (1) Equilibrium, (2) collisional matrix and (3) forcing (2) Multiple relaxation time collisional matrix The collisional matrix is A σ = M 1 Λ σ M and Λ σ = 0 0 0 0 0 0 0 0 0 0 λ δ σ 0 0 0 0 0 0 0 0 0 λ δ σ 0 0 0 0 0 0 0 0 0 + 0 0 0 0 0 0 0 + 0 0 0 0 0 0 0 0 0 λ ν σ 0 0 0 0 0 0 0 0 0 λ δ σ 0 0 0 0 0 0 0 0 0 λ δ σ 0 0 0 0 0 0 0 0 0 + where λ δ σ = p B mm /ρ = p B(m, m)/ρ, + = (λ ξ σ + λ ν σ)/2, = (λ ξ σ λ ν σ)/2, λ ν σ = λ ν = 1/(3 ν) and λ ξ σ = λ ξ (2 ϕ σ ) = (2 ϕ σ )/(3 ξ) (where p σ = ϕ σ ρ σ /3)., (8) Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 12 / 29

(3) External forcing Lattice Boltzmann scheme (1) Equilibrium, (2) collisional matrix and (3) forcing The external source d σ is designed in moment space as [ d σ = M 1 0, ˆρ σ ĉ σ1, ˆρ σ ĉ σ2, (ˆρ σĉ σ1 ), (ˆρ ] σĉ σ2 ) T, 0, 0, 0, 0, ˆx 1 ˆx 2 (9) where ĉ σi = â i + ˆb σi and â i is the acceleration due to an external field acting on all the components in same way (for example, the gravitational acceleration), while ˆb σi is the acceleration due to a second external field discriminating the nature of the component particles (for example, the electrical acceleration). As it will be clarified later on by the asymptotic expansion, the additional terms affecting the stress tensor components must be considered in order to compensate the deficiencies (in terms of symmetry properties) of the considered lattice. Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 13 / 29

Lattice Boltzmann scheme Discrete operative formula Implementation by variable transformation Eq. (1) is formulated for discrete velocities, but it is still continuous in both space and time. Since the streaming velocities are constant, the Method of Characteristics is the most convenient way to discretize space and time (simplest formulation of the LBM scheme). Applying the second-order Crank Nicolson yields (let us consider the BGK case for sake of simplicity) f σ + [ ] ] = f σ + (1 θ) λ σ fσ( ) f σ + θ λ + σ [f + σ( ) f σ +, (10) where θ = 1/2. The previous formula would force one to consider quite complicated integration procedures [Asinari, PRE 2006]. A simple variable transformation has been already proposed in order to simplify this task [He et al., JCP 1998]. Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 14 / 29

Lattice Boltzmann scheme Variable transformation Implementation by variable transformation (Step 1) Let us apply the transformation f σ g σ defined by g σ = f σ θ λ σ [ fσ( ) f σ ]. (assuming dσ = 0) (11) (Step 2) Let us compute the collision and streaming step leading to g σ g σ + by means of the modified updating equation g σ + = g σ + λ [ ] σ fσ( ) g σ, (12) where λ σ = λ σ /(1 + θλ σ ). (Step 3) Finally let us come back to the original discrete distribution function g σ + f σ + by means of f + σ = g+ σ + θ λ + σ f + σ( ) 1 + θ λ +. (13) σ Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 15 / 29

Outline Compass Model analysis 1 Technological application Solid Oxide Fuel Cells (SOFC) 2 Lattice Boltzmann scheme Simplified AAP model (1) Equilibrium, (2) collisional matrix and (3) forcing Implementation by variable transformation 3 Model analysis Diffusive scaling Grad moment expansion 4 Numerical validation (1) Non-Fickian test case: Stefan tube (2) Solvent test case in a Poiseuille flow Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 16 / 29

Diffusive scaling Model analysis Diffusive scaling In the following asymptotic analysis [Junk et al., 2005], we introduce the dimensionless variables, defined by x i = (l c /L) ˆx i, t = (UT c /L) ˆt. (14) Defining the small parameter ɛ as ɛ = l c /L, which corresponds to the Knudsen number, we have x i = ɛ ˆx i. Furthermore, assuming U/c = ɛ, which is the key of derivation of the incompressible limit [Sone, 1971], we have t = ɛ 2 ˆt. Then, AAP model is rewritten as ɛ 2 f σ t + ɛ V f [ ] σ i = A σ f x σ( ) + A σd σ f σ. (15) i In this new scaling, we can assume α f σ = f σ / α = O(f σ ) and α M = M/ α = O(M), where α = t, x i and M = ˆρ σ, ˆq σi where ˆq σi = ˆρ σ û σi. Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 17 / 29

Model analysis Grad moment expansion Single species macroscopic equations Grad moment method can be applied to recover the single species macroscopic equations [Asinari & Ohwada, Comput. Math. Appl., in press], namely ρ σ t + (ρ σu σi ) x i = 0, (16) y σ x i = ς B σς y σ y ς (u ςi u σi ) + ρ σb σi p, ( σ ρ σ b σi = 0) (17) (ρ σ u σi ) t + x j (ρ σ u σiu σj) + p σ x i = 1 2 (ρ σ u σi ) 3 λ ν x 2 + 1 2 (ρ σ u σk ) + ρ σ a i. (18) j 3 λ ξ x i x k Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 18 / 29

Model analysis Mixture macroscopic equations Grad moment expansion Summing over the components [Asinari, PRE 77, 056706, 2008] u i t + (x σ u x σiu σj) + 1 p = ν 2 u i j ρ x i σ x 2 j + a i. (19) Clearly the previous equation is not completely consistent with the canonical Navier Stokes system of equations for the barycentric velocity u i. The same result would be obtained by using the Hilbert expansion [Asinari, PRE 2006]. If and only if the component particles have similar masses, i.e. m σ = m for any σ, then u σi = ui and u i t + u u i j + 1 p = ν 2 u i x j ρ x i x 2 j + a i. (20) Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 19 / 29

Numerical validation Outline Compass 1 Technological application Solid Oxide Fuel Cells (SOFC) 2 Lattice Boltzmann scheme Simplified AAP model (1) Equilibrium, (2) collisional matrix and (3) forcing Implementation by variable transformation 3 Model analysis Diffusive scaling Grad moment expansion 4 Numerical validation (1) Non-Fickian test case: Stefan tube (2) Solvent test case in a Poiseuille flow Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 20 / 29

Ternary mixture Numerical validation In case of ternary mixture Eq. (17) reduces to n y 1 = B 12 y 1 k 2 + B 13 y 1 k 3 (B 12 y 2 + B 13 y 3 )k 1, (21) n y 2 = B 21 y 2 k 1 + B 23 y 2 k 3 (B 21 y 1 + B 23 y 3 )k 2, (22) n y 3 = B 31 y 3 k 1 + B 32 y 3 k 2 (B 31 y 1 + B 32 y 2 )k 3. (23) The molecular weights are m σ = [1, 2, 3], the homogeneous internal energies are [e σ = 1/3, 1/6, 1/9] and consequently the corrective factors are ϕ σ = [1, 1/2, 1/3]. The theoretical Fick diffusion coefficient is D σ = α/m σ, where α [0.002, 0.8] and the theoretical Maxwell Stefan diffusion resistance is given by B σς = β ( 1 m σ + 1 m ς ) 1/2, β [5, 166]. (24) Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 21 / 29

Numerical validation (1) Non-Fickian test case: Stefan tube (1) Non-Fickian test case: Stefan tube It is essentially a vertical tube, open at one end, where the carrier flow licks orthogonally the tube opening. In the bottom of the tube is a pool of quiescent liquid. The vapor that evaporates from this pool diffuses to the top. p 1 (0, x) = p 1 (0, 0) 1 [ ( )] x L/2 1 tanh + p s, (25) 2 δx p 2 (0, x) = p 2 (0, 0) 1 [ ( )] x L/2 1 tanh + p s, (26) 2 δx p 3 (0, x) = [1 p 3 (0, 0)] 1 [ ( )] x L/2 1 + tanh + p 3 (0, 0), 2 δx (27) where the constant p s = 10 4 has been introduced for avoiding to divide per zero. Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 22 / 29

(1) Stefan tube Numerical validation (1) Non-Fickian test case: Stefan tube Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 23 / 29

Numerical validation (2) Solvent test case in a Poiseuille flow (2) Solvent test case in a Poiseuille flow Let us consider a ternary mixture in an infinitely long gap between parallel plates. The initial conditions for the partial pressures are given by [ ( p 1 (0, x 1, x 2 ) = p 1 + sin 2π x )] 1 + p s, (28) p 2 (0, x 1, x 2 ) = p [ 1 + cos L 1 ( 2π x 1 L 1 )] + p s, (29) p 3 (0, x 1, x 2 ) = 1 p 1 (0, x 1, x 2 ) p 2 (0, x 1, x 2 ), (30) where p = p s = 0.01. The mixture flow is induced by a proper external force a 1 = 0.001 (effecting only the mixture barycentric velocity). In this case, the Schmidt number is Sc 1 = ν /D 1 = ν B 13. Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 24 / 29

Numerical validation (2) Solvent test case in a Poiseuille flow (2) Solvent test case in a Poiseuille flow Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 25 / 29

Numerical validation (2) Solvent test case in a Poiseuille flow (2) Solvent test case in a Poiseuille flow Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 26 / 29

How to improve the stability of the scheme Some brand new analytical results for discrete lattices have been recently pointed out [Asinari & Karlin, PRE, in press]: in particular, the generalized Maxwellian state (with prescribed diagonal components of the pressure tensor) and the constrained Maxwellian state (with prescribed trace of the pressure tensor). All the previously introduced equilibria for LB are found as special cases of the previous results (!!). Some new LB schemes, namely Entropic Quasi Equilibrium (EQE) and Linearized Quasi Equilibrium (LQE) with both tunable bulk viscosity and H theorem have been proposed. In case of some simple preliminary tests, the LQE model was able to achieve the same accuracy of the usual BGK model with a rougher mesh (approximately half), leading to a remarkable speed up of the run time. Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 27 / 29

Conclusions In the present talk, a LBM scheme for mixture modeling, which fully recovers Maxwell Stefan diffusion model with external force in the continuum limit, without the restriction of the macroscopic mixture-averaged approximation, was discussed. As a theoretical basis for the development of the LBM scheme, a recently proposed BGK-type kinetic model for gas mixtures [Andries et al., JSP 2002] was considered. This essentially links the LBM development to the recent progresses of the BGK-type kinetic models and opens new perspectives (e.g. reactive flows). In the reported numerical tests, the proposed scheme produces good results on a wide range of relaxation frequencies. For improving the current stability region, extension of recently proposed Entropic Quasi Equilibrium idea (based on new analytical results) is currently under development. Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 28 / 29

Thank you!! Pietro Asinari, PhD (POLITO DENER) LBM for Sustainable Energy Systems SIAM Conference CSE09 29 / 29