New upscaled charge transport equations for highly heterogeneous multiphase materials Markus Schmuck Maxwell Institute for Mathematical Sciences and Department of Mathematics Heriot-Watt University MIR@W 2015, Warwick November 30, 2015 Outline I) Applications and basic upscaling idea II) Upscaling of charge transport equations in heterogeneous media III) Effective macroscopic catalyst layer formulations IV) Control of Macroscopic Transport Characteristics? M. Schmuck Effective transport equations: Reliable upscaling Part I): Applications and basic upscaling idea M. Schmuck Effective transport equations: Reliable upscaling Example I: Catalyst layer in PEM fuel cell M. Schmuck Effective transport equations: Reliable upscaling Example II: Batteries M. Schmuck Effective transport equations: Reliable upscaling Effective equations for multiscale problems: Idea M. Schmuck Effective transport equations: Reliable upscaling Effective equations for multiscale problems: Idea M. Schmuck Effective transport equations: Reliable upscaling Effective equations for multiscale problems: Idea M. Schmuck Effective transport equations: Reliable upscaling Effective equations for multiscale problems: Idea M. Schmuck Effective transport equations: Reliable upscaling Upscaling: The asymptotic expansion method 1) Make the Ansatz: us (t, x) ≈ u0 (t, x, y ) + su1 (t, x, y ) + s2 u2 (t, x, y ) + . . . , where y := x/s is the microscale. 2) Insert 1) in the periodic model: (formal method by assuming differentiability) Collect terms of the same order in s. 3) “Take the leading order terms”: Solvability requirements provide the upscaled system for u0 . References: - [Bensoussan, Lions, Papanicolaou (78)], - [Cioranescu, Donato (99)], - [Chechkin, Piatnitski, Shamaev (07)], - [G.A. Pavliotis, A.M. Stuart (08)] M. Schmuck Effective transport equations: Reliable upscaling Upscaling: The asymptotic expansion method 1) Make the Ansatz: us (t, x) ≈ u0 (t, x, y ) + su1 (t, x, y ) + s2 u2 (t, x, y ) + . . . , where y := x/s is the microscale. 2) Insert 1) in the periodic model: (formal method by assuming differentiability) Collect terms of the same order in s. 3) “Take the leading order terms”: Solvability requirements provide the upscaled system for u0 . References: - [Bensoussan, Lions, Papanicolaou (78)], - [Cioranescu, Donato (99)], - [Chechkin, Piatnitski, Shamaev (07)], - [G.A. Pavliotis, A.M. Stuart (08)] M. Schmuck Effective transport equations: Reliable upscaling Upscaling: The asymptotic expansion method 1) Make the Ansatz: us (t, x) ≈ u0 (t, x, y ) + su1 (t, x, y ) + s2 u2 (t, x, y ) + . . . , where y := x/s is the microscale. 2) Insert 1) in the periodic model: (formal method by assuming differentiability) Collect terms of the same order in s. 3) “Take the leading order terms”: Solvability requirements provide the upscaled system for u0 . References: - [Bensoussan, Lions, Papanicolaou (78)], - [Cioranescu, Donato (99)], - [Chechkin, Piatnitski, Shamaev (07)], - [G.A. Pavliotis, A.M. Stuart (08)] M. Schmuck Effective transport equations: Reliable upscaling Upscaling: The asymptotic expansion method 1) Make the Ansatz: us (t, x) ≈ u0 (t, x, y ) + su1 (t, x, y ) + s2 u2 (t, x, y ) + . . . , where y := x/s is the microscale. 2) Insert 1) in the periodic model: (formal method by assuming differentiability) Collect terms of the same order in s. 3) “Take the leading order terms”: Solvability requirements provide the upscaled system for u0 . References: - [Bensoussan, Lions, Papanicolaou (78)], - [Cioranescu, Donato (99)], - [Chechkin, Piatnitski, Shamaev (07)], - [G.A. Pavliotis, A.M. Stuart (08)] M. Schmuck Effective transport equations: Reliable upscaling Part II): Upscaling of charge transport equations in heterogeneous media M. Schmuck Effective transport equations: Reliable upscaling Charged porous media [M.S. & M.Z. Bazant, SIAM J. Appl. Math., 75(3), 1369-1401 (2015).] Reference configuration: Solid-electrolyte composite Y \ Ys Y := I Ys Extension: Presence of a surface charge density σs on I Replace (continuity of fluxes) −λ2 ∇n φs ∂Y s = −α∇n φs ∂Y s on I , by −κ(x/s)∇n φs ∂Y s = sσs (x/s)∂Y s on I , for the Debey length λ and dielectric permittivity α := that κ̂(x) := λ2 χΩs (x) + αχΩ\Ωs (x) M. Schmuck Effective transport equations: Reliable upscaling m f such 6 Y Reference cell Y 2 := Y \ Y 1 Periodic covering by cells Y B~ B~ B~ B~ B~ B~ B~ B~ B~ B~ B~ B~ ? Homogenous approximation B~ B~ B~ B~ B~ B~ B~ B~ B~ B~ B~ B~ ` 6 ( → 0) - Ω 6 ` B~ B~ B~ B~ B~ B~ ? M. Schmuck Effective transport equations: Reliable upscaling L ? Local Thermodynamic Equilibrium (LTE) Definition: (Scale separation) We say that the chemical potential µ is scale separated if and only if ( 0 on the reference cell Y , ∂µ(u0 (x, t)) = ∂µ(u0 (x,t)) ∂xk on the macroscale Ω , ∂xk where u0 (x) is the upscaled/slow variable solving the corresponding upscaled equation. M. Schmuck Effective transport equations: Reliable upscaling Charged porous media [M.S. & M.Z. Bazant, SIAM J. Appl. Math., 75(3), 1369-1401 (2015).] + + + ∂t ns = div ∇ns + ns ∇φs Micro: ∂t ns− = div ∇ns− − ns− ∇φs −div (κ̂(x/s)∇φs ) = ns+ − ns− in Ωs in Ωs in Ω + + ∇n ns + ns ∇n φs = 0 Micro interface: ∇n ns− − ns− ∇n φs = 0 −κ̂(x/s)∇n φs = sσs (x/s) becomes under local thermodynamic equilibrium + + + p∂t n0 = div D̂∇n0 + n0 M̂∇φ0 Macro: p∂t n0− = div D̂∇n0− − n0− M̂∇φ0 −div (κ̂eff ∇φ0 ) = p(n0+ − n0− ) + ρs where ρs := M. Schmuck 1 |Y | R I on Is on Is on Is in Ω in Ω in Ω σs (x, y ) do(y ). Effective transport equations: Reliable upscaling Charged porous media [M.S. & M.Z. Bazant, SIAM J. Appl. Math., 75(3), 1369-1401 (2015).] + + + ∂t ns = div ∇ns + ns ∇φs Micro: ∂t ns− = div ∇ns− − ns− ∇φs −div (κ̂(x/s)∇φs ) = ns+ − ns− in Ωs in Ωs in Ω + + ∇n ns + ns ∇n φs = 0 Micro interface: ∇n ns− − ns− ∇n φs = 0 −κ̂(x/s)∇n φs = sσs (x/s) becomes under local thermodynamic equilibrium + + + p∂t n0 = div D̂∇n0 + n0 M̂∇φ0 Macro: p∂t n0− = div D̂∇n0− − n0− M̂∇φ0 −div (κ̂eff ∇φ0 ) = p(n0+ − n0− ) + ρs where ρs := M. Schmuck 1 |Y | R I on Is on Is on Is in Ω in Ω in Ω σs (x, y ) do(y ). Effective transport equations: Reliable upscaling Rigorous error bounds / Convergence rates: Theorem: [M.S. 2012, ZAMM 92:304–319 (2012)] Let ∂Ω be of class C ∞ and ξ k , ζ kl ∈ W 1,∞ (Y ). Then, for e1s := ns+ − K1s n0+ , e2s := ns− − K1s n0− , and e3s := φs − K1s φ0 , there holds 2 1 es (T ) ≤ CT s , 2 2 es (T ) ≤ CT s , 3 2 es 1 (T ) ≤ C (T + 1) s , H (Ω) where K1s u0 (t, x) := 1−s N X ! ξ k (x/s)∂xk u0 (t, x) . k =1 Related Reference: Two-scale converge result - [M. S., COMMUN MATH SCI, 9(3):685-710 (2011)] M. Schmuck Effective transport equations: Reliable upscaling Rigorous error bounds / Convergence rates: Theorem: [M.S. 2012, ZAMM 92:304–319 (2012)] Let ∂Ω be of class C ∞ and ξ k , ζ kl ∈ W 1,∞ (Y ). Then, for e1s := ns+ − K1s n0+ , e2s := ns− − K1s n0− , and e3s := φs − K1s φ0 , there holds 2 1 es (T ) ≤ CT s , 2 2 es (T ) ≤ CT s , 3 2 es 1 (T ) ≤ C (T + 1) s , H (Ω) where K1s u0 (t, x) := 1−s N X ! ξ k (x/s)∂xk u0 (t, x) . k =1 Related Reference: Two-scale converge result - [M. S., COMMUN MATH SCI, 9(3):685-710 (2011)] M. Schmuck Effective transport equations: Reliable upscaling 2) Key ideas in the proof: Step 1: Define error variables eis := usi − s d X r =1 ξ ir (y ) ∂u0i ∂xr for i = 1, 2, 3. Step 2: Determine equations for eis based on ∂u(x, y ) 1 = ∂x u(x, y ) + ∂y u(x, y ) , ∂x s such that for r = 1, 2 r ∂es r r 3 r ∂t = div ∇es + zr es ∇es + sFs er = sGr s s 3 = e1 − e2 + sF3 −div κ(x/s)∇e s s s s e3 = sG3 s s M. Schmuck in Ωs ×]0, T [ , on ∂Ωs ×]0, T [ , in ΩT , on ∂Ω×]0, T [ . Effective transport equations: Reliable upscaling 3) Charged porous media [M.S. & M.Z. Bazant, SIAM J. Appl. Math., 75(3), 1369-1401 (2015).] D̂ M̂ 6= kT Einstein relations: Consequence: Seem to loose Boltzmann distribution in equilibrium and Einstein relations ! But: Introduce the mean field approximations ∇n± := D̂∇n± , ∇φ := M̂∇φ , then again the Boltzmann distribution is obtained, i.e., kTn± M± ∇µ̃ = D± ∇n± + z± n± kTM± ∇φ , where µ = M. Schmuck µ̃ kT and φ = eΦ kT . Effective transport equations: Reliable upscaling 3) Charged porous media [M.S. & M.Z. Bazant, SIAM J. Appl. Math., 75(3), 1369-1401 (2015).] D̂ M̂ 6= kT Einstein relations: Consequence: Seem to loose Boltzmann distribution in equilibrium and Einstein relations ! But: Introduce the mean field approximations ∇n± := D̂∇n± , ∇φ := M̂∇φ , then again the Boltzmann distribution is obtained, i.e., kTn± M± ∇µ̃ = D± ∇n± + z± n± kTM± ∇φ , where µ = M. Schmuck µ̃ kT and φ = eΦ kT . Effective transport equations: Reliable upscaling 3) Charged porous media [M.S. & M.Z. Bazant, SIAM J. Appl. Math., 75(3), 1369-1401 (2015).] D̂ M̂ 6= kT Einstein relations: Consequence: Seem to loose Boltzmann distribution in equilibrium and Einstein relations ! But: Introduce the mean field approximations ∇n± := D̂∇n± , ∇φ := M̂∇φ , then again the Boltzmann distribution is obtained, i.e., kTn± M± ∇µ̃ = D± ∇n± + z± n± kTM± ∇φ , where µ = M. Schmuck µ̃ kT and φ = eΦ kT . Effective transport equations: Reliable upscaling 3) Charged porous media [M.S. & M.Z. Bazant, SIAM J. Appl. Math., 75(3), 1369-1401 (2015).] D̂ M̂ 6= kT Einstein relations: Consequence: Seem to loose Boltzmann distribution in equilibrium and Einstein relations ! But: Introduce the mean field approximations ∇n± := D̂∇n± , ∇φ := M̂∇φ , then again the Boltzmann distribution is obtained, i.e., kTn± M± ∇µ̃ = D± ∇n± + z± n± kTM± ∇φ , where µ = M. Schmuck µ̃ kT and φ = eΦ kT . Effective transport equations: Reliable upscaling Example I: Straight channels Assumption: Insulating porous matrix, i.e., α → 0 The correction tensor D̂ = M̂ becomes 1 0 0 D̂ := p 0 0 0 . 0 0 1 Literature: Auriault & Lewandowska 1997 M. Schmuck Effective transport equations: Reliable upscaling Example I: Straight channels Assumption: Insulating porous matrix, i.e., α → 0 The correction tensor D̂ = M̂ becomes 1 0 0 D̂ := p 0 0 0 . 0 0 1 Literature: Auriault & Lewandowska 1997 M. Schmuck Effective transport equations: Reliable upscaling Example II: Perturbed straight channels Assumption: Insulating porous matrix, i.e., α → 0 The correction tensor D̂ = M̂ becomes 0.3833 0 0 0 0 0 . D̂ := p 0 0 1 Literature: Auriault & Lewandowska 1997 M. Schmuck Effective transport equations: Reliable upscaling Example II: Perturbed straight channels Assumption: Insulating porous matrix, i.e., α → 0 The correction tensor D̂ = M̂ becomes 0.3833 0 0 0 0 0 . D̂ := p 0 0 1 Literature: Auriault & Lewandowska 1997 M. Schmuck Effective transport equations: Reliable upscaling Part III): Effective Proton Transport in Catalyst Layers M. Schmuck Effective transport equations: Reliable upscaling 1) Effective catalyst layer in PEMFC: [with P. Berg (NTNU)] Periodic catalyst layer: Microscopic scenario ρs and Butler-Volmer H2 O r PEM H + , c+ O2 , cO e− x2 x1 Butler-Volmer: M. Schmuck GDL O2 + 4H + + 4e− 2H2 O Effective transport equations: Reliable upscaling Result: [M.S. & P. Berg, Appl. Math. Res. Express. 2013(1):57-78 (2013)] Under large overpotential, i.e., Φs − Φeq 0, we have s −∆cO = 0 in Ωs Micro bulk: −div (∇c+s + c+s ∇φs ) = 0 in Ωs −div (κ̂(x/s)∇φs ) = c+s in Ω Interface: s s n s n s −∇n cO = sβO (cO ) O (c+ ) + exp [−αc (Φ − Φ0 )] −∇n c+s − c+s ∇n Φs = sβ+ (cOs )nO (c+s )n+ exp [−αc (Φs − Φ0 )] −κ(x/s)∇n Φs = sσs (x, x/s) on I on I on I turns after upscaling under scale separation into O n n −div D̂ ∇CO = β O (C+ ) +(CO ) O exp (−αc (Φ − Φ0 )) , −div D̂+ ∇C+ + C+ M̂+ ∇Φ = β + (C+ )n+ (CO )nO exp (−αc (Φ − Φ0 )) , −div ε̂(λ2 , γ)∇Φ = pC+ + ρs , where β O := βO Λ = M. Schmuck i0 Λ 4eDO and β + := βO Λ = i0 Λ 4eD+ with Λ := |I|. Effective transport equations: Reliable upscaling in Ω in Ω in Ω Result: [M.S. & P. Berg, Appl. Math. Res. Express. 2013(1):57-78 (2013)] Under large overpotential, i.e., Φs − Φeq 0, we have s −∆cO = 0 in Ωs Micro bulk: −div (∇c+s + c+s ∇φs ) = 0 in Ωs −div (κ̂(x/s)∇φs ) = c+s in Ω Interface: s s n s n s −∇n cO = sβO (cO ) O (c+ ) + exp [−αc (Φ − Φ0 )] −∇n c+s − c+s ∇n Φs = sβ+ (cOs )nO (c+s )n+ exp [−αc (Φs − Φ0 )] −κ(x/s)∇n Φs = sσs (x, x/s) on I on I on I turns after upscaling under scale separation into O n n −div D̂ ∇CO = β O (C+ ) +(CO ) O exp (−αc (Φ − Φ0 )) , −div D̂+ ∇C+ + C+ M̂+ ∇Φ = β + (C+ )n+ (CO )nO exp (−αc (Φ − Φ0 )) , −div ε̂(λ2 , γ)∇Φ = pC+ + ρs , where β O := βO Λ = M. Schmuck i0 Λ 4eDO and β + := βO Λ = i0 Λ 4eD+ with Λ := |I|. Effective transport equations: Reliable upscaling in Ω in Ω in Ω I-V curves: Circle shaped pores (R = `/a, a = 3, 6, 12) M. Schmuck Effective transport equations: Reliable upscaling I-V curves: Square with (R = `/a, a = 3, 6, 12) M. Schmuck Effective transport equations: Reliable upscaling I-V curves: Cross shaped pores (R = `/a, a = 3, 6, 12) M. Schmuck Effective transport equations: Reliable upscaling I-V curves: Shamrock (R = `/a, h = R/2, a = 3, 6, 12) M. Schmuck Effective transport equations: Reliable upscaling 2) Extension towards fluid flow [M. Schmuck & P. Berg, J ELECTROCHEM SOC, 161(8):E3323-E3327 (2014)] Micro bulk: 2 s − ∆u + ∇ps = κfs := −κc+s ∇φs divu = 0 ∂t c − div DO ∇c s − Peu c s = 0 O O O ∂t c+s − div D+ (∇c+s + c+s ∇φs ) − Peus c+s = 0 s s −div (E(x/s)∇φ ) = c+ Interface: s uτ = 0 usn = − 2s Rw (c+s , cOs , η s ) −∇n c s = sβO (c s )nO (c+s )n+ exp [−αc (Φs − Φ0 )] O O −∇n c+s − c+s ∇n Φs = sβ+ (cOs )nO (c+s )n+ exp [−αc (Φs − Φ0 )] s −κ(x/s)∇n Φ = sσs (x, x/s) where s s s Rι (c+ , cO , η s ) := βι c+ and βι := M. Schmuck in Ωs in Ωs in Ωs in Ωs in Ω i0 L 4eDι n+ s cO n O for ι ∈ {+, O} and βw = exp (−αc η s ) , i0 Mw ρw F Effective transport equations: Reliable upscaling on I on I on I on I on I Scale Separation: Local Equilibrium Local Thermodynamic Equilibrium (LTE): A system depending on a flow velocity U is in local thermodynamic equilibrium if and only if 0= ∂ µι − U k , ∂xk for 1 ≤ k ≤ N , on the level of microscale Y , where ι ∈ {O, +}, Uk is the k -th velocity component of the upscaled fluid velocity U and µι denotes the electrochemical potenials w.r.t. upscaled variables, i.e., ( ln CO if ι = O , µι := ln C+ + z+ Φ if ι = + . M. Schmuck Effective transport equations: Reliable upscaling Result: Two-scale asymptotics Idea: Apply LTE and multiscale expansions of the form v s (x) = v (x, x/s) ≈ V (x, x/s) + sv1 (x, x/s) + O(s2 ) , where V denotes the upscaled variable. Result: + U = C+ M ∇Φ − K∇P , in Ω , div U = − 12 β w (C+ )n+ (CO )nO exp (−αc (Φ − Φ0 )) , O ( θ∂t CO − div D ∇CO − PeUCO in Ω , = 14 β O (C+ )n+ (CO )nO exp (−αc (Φ − Φ0 )) , in Ω , + ( + θ∂t C+ − div D ∇C+ + C+ M ∇Φ − PeUC+ = β + (C+ )n+ (CO )nO exp (−αc (Φ − Φ0 )) , n M. Schmuck −div E∇Φ = θC+ + Qs in Ω , in Ω , Effective transport equations: Reliable upscaling 3) Moving frame approach: Strongly periodic fluid flow Periodic flow problem: −µ∆y u + ∇y p = e1 divy (u) = 0 u=0 u and p are Y -periodic. M. Schmuck in Y 1 , in Y 1 , on IY , Effective transport equations: Reliable upscaling M. Schmuck Effective transport equations: Reliable upscaling Ansatz: Asymptotic expansion with drift vj := Peloc |Y 1 | R Y1 uj (y ) dy ∞ X v v u (t, x) = u t, x − t, x/ ≈ U(t, x) + i ui t, x − t, x/ i=1 Result: ( O θ∂t CO − div D (u)∇CO = 14 β O (C+ )n+ (CO )nO exp (−αc (Φ − Φ0 )) , ( + + θ∂t C+ − div D (u)∇C+ + C+ M ∇Φ in Ω , = β + (C+ )n+ (CO )nO exp (−αc (Φ − Φ0 )) , in Ω , n −div E∇Φ = θC+ + Qs . M. Schmuck Effective transport equations: Reliable upscaling Part IV): Control of Macroscopic Transport Characteristics M. Schmuck Effective transport equations: Reliable upscaling Microscopic formulation Material characteristic: Composite with high contrast in electric permittivity ⇒ strongly oscillating electric potential Y \ Ys Y := I Ys + + + in Ωs ∂t ns = div ∇ns + ns ∇φs Micro: ∂t ns− = div ∇ns− − ns− ∇φs in Ωs −div (κ̂(x/s)∇φs ) = ns+ − ns− in Ω + + ∇n ns + ns ∇n φs = 0 on Is Micro interface: ∇n ns− − ns− ∇n φs = 0 on Is −κ̂(x/s)∇n φs continuous over Is M. Schmuck Effective transport equations: Reliable upscaling Microscopic formulation Material characteristic: Composite with high contrast in electric permittivity ⇒ strongly oscillating electric potential Y \ Ys Y := I Ys + + + in Ωs ∂t ns = div ∇ns + ns ∇φs Micro: ∂t ns− = div ∇ns− − ns− ∇φs in Ωs −div (κ̂(x/s)∇φs ) = ns+ − ns− in Ω + + ∇n ns + ns ∇n φs = 0 on Is Micro interface: ∇n ns− − ns− ∇n φs = 0 on Is −κ̂(x/s)∇n φs continuous over Is M. Schmuck Effective transport equations: Reliable upscaling Idea: Use modified expansions (u1s := ns+ , u2s := ns− ) urs = ur0 − s N X ξ rk (t, x,x/s) k =1 φs = φ0 − s N X k =1 N X ∂φ0 + s2 ζ rkl (t, x, x/s)ur0 + . . . ∂xk for r = 1, 2 , k ,l=1 ξφk (x/s) N X ∂φ0 ∂ 2 φ0 + s2 ζφkl (x/s) + ... , ∂xk ∂xk ∂xl k ,l=1 where ξ rk (·, ·, y ) ∈ V (ΩT , W] (Y s )), ξ 3k (y ) ∈ W] (Y ), ζ rkl (·, ·, y ) ∈ V (ΩT , W] (Y s )), and ζ 3kl (y ) ∈ W] (Y ) solve elliptic cell problems. Result: u0 is solution of the following upscaled system ( p∂t ur0 −p∆ur0 + div (Dr (t, x)∇φ0 ) − div zr ur0 M∇φ0 = 0 −div 0 ∇φ0 = p u10 − u20 where p :=|Y s | / |Y |is the porosity and the tensors Dr (t, x) := Drkl (t, x) 1≤k ,l≤N , M := {Mkl }1≤k ,l≤N , and 0 := 0kl 1≤k .l≤N are defined by M. Schmuck Effective transport equations: Reliable upscaling in ΩT , in ΩT , Idea: Use modified expansions (u1s := ns+ , u2s := ns− ) urs = ur0 − s N X ξ rk (t, x,x/s) k =1 φs = φ0 − s N X k =1 N X ∂φ0 + s2 ζ rkl (t, x, x/s)ur0 + . . . ∂xk for r = 1, 2 , k ,l=1 ξφk (x/s) N X ∂φ0 ∂ 2 φ0 + s2 ζφkl (x/s) + ... , ∂xk ∂xk ∂xl k ,l=1 where ξ rk (·, ·, y ) ∈ V (ΩT , W] (Y s )), ξ 3k (y ) ∈ W] (Y ), ζ rkl (·, ·, y ) ∈ V (ΩT , W] (Y s )), and ζ 3kl (y ) ∈ W] (Y ) solve elliptic cell problems. Result: u0 is solution of the following upscaled system ( p∂t ur0 −p∆ur0 + div (Dr (t, x)∇φ0 ) − div zr ur0 M∇φ0 = 0 in ΩT , −div 0 ∇φ0 = p u10 − u20 in ΩT , where p :=|Y s | / |Y |is the porosity and the tensors Dr (t, x) := Drkl (t, x) 1≤k ,l≤N , M := {Mkl }1≤k ,l≤N , and 0 := 0kl 1≤k .l≤N are defined by M. Schmuck Effective transport equations: Reliable upscaling 1 := |Y | Z 1 Mik := |Y | Z Drik (t, x) 0ik : = 1 |Y | N X Y s j=1 δij ∂yj ξ rk (t, x,y ) dy , N n o X δik − δij ∂yj ξ 3k (y ) dy , Y s j=1 Z X N Y j=1 κ̂(y ) δik − δij ∂yj ξ 3k (y ) dy . Reference: [M. Schmuck, J MATH PHYS, 54(2):21 p.021504 (2013)] M. Schmuck Effective transport equations: Reliable upscaling Conclusion: I I I I Presented formal and rigorous upscaling/homogenization methods. Systematically derived upscaled charge transport equations valid for different pore geometries Developed a framework for deriving effective macroscopic catalyst layer equations Upscaling provides means to control transport on the macroscale! Thank You for Your Attention! Conclusion: I I I I Presented formal and rigorous upscaling/homogenization methods. Systematically derived upscaled charge transport equations valid for different pore geometries Developed a framework for deriving effective macroscopic catalyst layer equations Upscaling provides means to control transport on the macroscale! Thank You for Your Attention!