Colloids transport in porous media: analysis and applications.

Transcription

1 Colloids transport in porous media: analysis and applications. Oleh Krehel joint work with Adrian Muntean and Peter Knabner CASA, Department of Mathematics and Computer Science. Eindhoven University of Technology. April 9, 2014

2 What are colloids? Colloid aggregation is important in: Cheese production Colloid filtration Carbon transport in seawater Nanoparticles engineering Water treatment Our goal: see the effects of aggregation on colloidal transport in porous media. Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

3 Soret effect: the heat makes the particles move Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

4 Dufour effect: the particles carry the heat Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

5 Model assumptions Figure : View of a real (a) and a model (b) aggregate u 1 u 2 u 3 u 4 u 5 size class 1 size class 2 size class 3 Figure : Each aggregate is uniquely identified by the number of monomers it s composed of. Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

6 Population balance (Smoluchowski) equation Assuming presently a well-mixed solution, we have: du k dt = R(u) := 1 2 α ij β ij u i u j u k i+j=k N α ij β ki u i b k u k + i=1 i=k N d ki b i u i u k concentration of species composed of k monomers a k the effective diameter for u k β ij = β ij (a i, a j ) aggregation rate α ij = α ij (a i, a j ) collision efficiency b k breakage coefficient d ki breakage distribution Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

7 Porous model geometry Γ Γ ε ε Y ij ε ε Y ij Figure : Microstructure of Ω ε (isotropic case on the left, anisotropic case on the right). Y ij is the periodic cell. Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

8 Microscopic model equations System for P ε : t θ ε + ( κ ε θ ε ) τ ε t u ε i + ( d ε i u ε i ) f ε t v ε i N i=1 δ u ε i θ ε = 0, in Ω ε, i δ θ ε ui ε = R i (u ε ), in Ω ε, = ai ε ui ε bi ε vi ε, on (0, T ) Γ ε, with boundary conditions: κ ε θ ε n = 0, on (0, T ) Γ ε, di ε ui ε n = ε(ai ε ui ε bi ε vi ε ), on (0, T ) Γ ε, θ ε (x, 0) = θ ε (x, 1) x [0, 1], θ ε (0, y) = θ ε (1, y) y [0, 1], u ε i (x, 0) = u ε i (x, 1) x [0, 1], i {1,..., N} u ε i (0, y) = u ε i (1, y) y [0, 1], i {1,..., N} Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

9 Model challenges t θ ε + ( κ ε θ ε ) τ ε t u ε i + ( d ε i u ε i ) f ε t v ε i N i=1 δ u ε i θ ε = 0, in Ω ε, i δ θ ε ui ε = R i (u ε ), in Ω ε, = ai ε ui ε bi ε vi ε, on (0, T ) Γ ε, with boundary conditions: κ ε θ ε n = 0, on (0, T ) Γ ε, di ε ui ε n = ε(ai ε ui ε bi ε vi ε ), on (0, T ) Γ ε. Non-linear cross-diffusion terms with a gradient Non-linear coupling via the aggregation reaction PDE-ODE coupling via the deposition condition Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

10 Model challenges t θ ε + ( κ ε θ ε ) τ ε t u ε i + ( d ε i u ε i ) f ε t v ε i N i=1 δ u ε i θ ε = 0, in Ω ε, i δ θ ε ui ε = R i (u ε ), in Ω ε, = ai ε ui ε bi ε vi ε, on (0, T ) Γ ε, with boundary conditions: κ ε θ ε n = 0, on (0, T ) Γ ε, di ε ui ε n = ε(ai ε ui ε bi ε vi ε ), on (0, T ) Γ ε. Non-linear cross-diffusion terms with a gradient Non-linear coupling via the aggregation reaction PDE-ODE coupling via the deposition condition Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

11 Model challenges t θ ε + ( κ ε θ ε ) τ ε t u ε i + ( d ε i u ε i ) f ε t v ε i N i=1 δ u ε i θ ε = 0, in Ω ε, i δ θ ε ui ε = R i (u ε ), in Ω ε, = ai ε ui ε bi ε vi ε, on (0, T ) Γ ε, with boundary conditions: κ ε θ ε n = 0, on (0, T ) Γ ε, di ε ui ε n = ε(ai ε ui ε bi ε vi ε ), on (0, T ) Γ ε. Non-linear cross-diffusion terms with a gradient Non-linear coupling via the aggregation reaction PDE-ODE coupling via the deposition condition Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

12 Model well-posedness For P ε, under the assumptions: Positivity of initial data di ε > d ε,0 i > 0, κ ε > κ ε,0 > 0 ai ε > 0, bi ε > 0 we show (for weak solutions): Existence Uniqueness Positivity Boundedness Proof idea: Galerkin method is used to prove existence of two auxiliary problems Banach fixed point theorem is used to prove the existence and uniqueness of the full problem Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

13 Rigorous upscaling Definition (Two-scale convergence) Let (u ε ) L 2 (0, T ; L 2 (Ω)), Ω R n. (u ε ) two-scale converges to a unique function u 0 (t, x, y) L 2 ((0, T ) Ω Y ) φ C0 ((0, T ) Ω, C # (Y )): T lim ε 0 0 Ω u ε φ(t, x, x ε )dxdt = 1 Y T 0 Ω Y u 0 (t, x, y)φ(t, x, y)dydxdt Denote as u ε 2 u 0. Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

14 Two-scale compactness Theorem (Two-scale compactness on domains) (i) From each bounded sequence (u ε ) in L 2 (0, T ; L 2 (Ω)), a subsequence may be extracted which two-scale converges to u 0 (t, x, y) L 2 ((0, T ) Ω ε Y ). (ii) Let (u ε ) be a bounded sequence in H 1 (0, T, H 1 (Ω)), then there exists ũ L 2 ((0, T ) H 1 # (Y )) such that up to a subsequence (uε ) two-scale converges to u 0 L 2 (0, T ; L 2 (Ω)) and u ε 2 x u 0 + y ũ. Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

15 Two-scale convergence for the thermal colloids problem Lemma With positive initial data and coercive and bounded diffusion and heat conduction coefficients, the following holds: (i) u i u i and θ ε θ in L 2 (0, T ; H 1 (Ω)), (ii) u i ui and θ ε θ in L ((0, T ) Ω), (iii) t u i t u i and t θ ε t θ in L 2 (0, T ; H 1 (Ω)), (iv) u i u i and θ ε θ strongly in L 2 (0, T ; H β (Ω ε )) for 1 2 < β < 1 and ε ui u i L 2 ((0,T ) Γ ε) 0 as ε 0, 2 2 (v) u i ui, u i x u i + y ui 1 where ui 1 L 2 ((0, T ) Ω ε ; H# 1 (Y )), (vi) θ ε 2 θ, θ ε 2 x θ + y θ 1 where θ 1 L 2 ((0, T ) Ω ε ; H# 1 (Y )), 2 (vii) v i vi L ((0, T ) Ω ε (0, T ) Γ ε ) and 2 t v i t v i L 2 ((0, T ) Ω ε (0, T ) Γ ε ). Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

16 A model for colloids transport with deposition t u i + ( d i u i ) = R i (u) in Ω, (1) t v i = a i u i b i v i on Γ, (2) with the boundary conditions and the initial conditions d i u i n = a i u i b i v i on Γ, (3) d i u i n = 0 on Γ N, (4) u i (t, x) = u D (t, x) on Γ D, (5) u i (0, x) = u 0 i (x) in Ω, (6) v i (0, x) = v 0 i (x) on Γ. (7) Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

17 Nondimensionalized system After rescaling, we obtain this system: t u i + ( d i u i ) = ΛR i (u) in Ω, (8) t v i = Bi(a i u i b i v i ) on Γ, (9) with the boundary conditions d i u i n = ε(a i u i b i v i ) on Γ, (10) d i u i n = 0 on Γ N, (11) u i (t, x) = u D(t, x) u 0 on Γ D, (12) and the initial conditions u i (0, x) = u0 i (x) u 0 in Ω, (13) v i (0, x) = v 0 i (x) v 0 on Γ. (14) Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

18 Rigorous homogenization result The final upscaled system: t u i ( D i u i ) + A i u i B i v i = ΛR i (u) in Ω, (15) t v i = A i u i B i v i in Ω (16) with effective parameters: D ijk = di ε (y)(δ jk + y w i )dy i {1,..., N} (17) Y A i := Bi ai ε (y) dσ(y) i {1,..., N} (18) Γ ε B i := Bi bi ε (y) dσ(y) i {1,..., N} (19) Γ ε where w j (y) solve the cell problem: y (d ε i (y) w j ) = ( d ε i (y)) j j {1,..., d}, y Y (20) Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

19 Cell problem solution: isotropic case Figure : Solutions to the cell problems that correspond to isotropic periodic geometry. The resulting effective diffusion tensor: D= Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

20 Cell problem solution: anisotropic case Figure : Solutions to the cell problems that correspond to anisotropic periodic geometry. The resulting effective diffusion tensor: D= Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

21 A model with a blocking function for the deposition t n = v p n + D h n f πa 2 p t θ, (21) t θ = πa 2 pknb(θ), (22) with the boundary conditions { n 0 t [0, t 0 ] n(t, 0) =, (23) 0 t > t 0 and initial conditions n (t, L) = 0, (24) ν n(0, x) = 0, (25) θ(0, x) = 0. (26) Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

22 Simulation of the aggregation effects on deposition u1 u2 u (together) u (single species) 0.7 mass in the system time Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

23 Simulation for the Langmuirian blocking function u1 u2 v1 v mass in the system mass in the system time time Figure : The effect of the Langmuirian dynamic blocking function B(θ) = 1 βθ on the deposition (right) versus no blocking function (left). u 1 and u 2 are the breakthrough curves, while v 1 and v 2 are the concentrations of the deposited species. Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

24 Simulation for the RSA blocking function u1 u2 v1 v mass in the system mass in the system time time Figure : The effect of the RSA dynamic blocking function B(θ) = 1 4θ βθ (θ βθ) (θ βθ) 3 on the deposition (right) versus no blocking function (left). u 1 and u 2 are the breakthrough curves, while v 1 and v 2 are the concentrations of the deposited species. Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

25 Simulation of the aggregation effects on deposition u1(s) u2(s) v1(s) v2(s) u1(d) u2(d) v1(d) v2(d) mass in the system time Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

26 Implementation details cell problem solver (FEM): deal.ii - a numeric library for parallel computations spatial discretization (FEM): DUNE - a numeric library for parallel computations time discretization: CVODE library from the SUNDIALS package Fully coupled or operator splitting Code is 2D/3D Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

27 Multiscale FEM General idea: L ε u ε = f u ε = 0 in Ω on Ω Approximate u ε with u h V h = span{φ i K : K K h } s.t.: L ε φ i = 0 a(u h, v) = f (v) in K K h v V h Error estimate: u ε u h 1,Ω C 1 h f 0,Ω + C 2 ε h Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

28 Outlook Multiscale Finite Element method implementation for the system Pore clogging due to deposition Corrector estimates for the cross-diffusion problem MSFEM estimates and implementation for the cross-diffusion problem Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27

29 Last slide Thank you for your attention. Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, / 27