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!