A. Bertei GLOSSARY
GLOSSARY
Abbreviations
ACP anion-conducting phase
BCY15 yttria doped barium cerate (BaCe 0.85 Y 0.15 O 2.925 ) DGM Dusty Gas model
CM central membrane OCV open circuit voltage
PCFC proton conducting fuel cell PCP proton-conducting phase ppm parts per million
SOFC solid oxide fuel cell TPB three phase boundary
YDC15 yttria doped ceria (Ce 0.85 Y 0.15 O 1.925 )
Symbols
a dl v double layer specific area [m 2 /m 3 ]
a v k total surface area of k-particles exposed to gas per unit volume [m 2 /m 3 ] a v l total surface area of l-particles exposed to gas per unit volume [m 2 /m 3 ] a v PCP total surface area of PCP exposed to gas per unit volume [m 2 /m 3 ]
B permeability [m 2 /s]
c dl double layer capacitance [C/(m 2 ·V)]
C j molar concentration of j in phase P (eq. II.7.1) [mol/m 3 ] c O concentration of oxygen ions in lattice per unit of perovskite-cell [-]
C O molar concentration of oxygen ions [mol/m 3 ]
c OH concentration of protonic defects per unit of perovskite-cell [-]
C OH molar concentration of protonic defects [mol/m 3 ] c VO concentration of oxide ion vacancies per unit of perovskite-cell [-]
C VO molar concentration of oxide ion vacancies [mol/m 3 ] C w,PCP concentration of water adsorbed in proton-conducting phase [mol/m 3 ] D B K
Knudsen diffusion coefficient for inert B (nitrogen) [m 2 /s]
d m,B mean diameter of gaseous inert B (nitrogen) molecule [m]
d m,w mean diameter of gaseous water molecule [m]
A. Bertei GLOSSARY
D O diffusivity of oxygen ions in lattice [m 2 /s]
D OH diffusivity of protonic defects [m 2 /s]
d p mean diameter of pores [m]
D p mean diameter of k and l-particles [m]
D s OH self-diffusivity of protonic defects [m 2 /s]
D s VO self-diffusivity of oxide ion vacancies [m 2 /s]
D wB ordinary diffusion coefficient for water-inert B (nitrogen) [m 2 /s]
D w K
Knudsen diffusion coefficient for water [m 2 /s]
D w,PCP diffusivity of water adsorbed in proton-conducting phase [m 2 /s]
PCP ,
D
waverage diffusivity of water adsorbed between CM and anodic side [m 2 /s]
E difference of potential between cathode and anode [V]
f frequency [Hz]
F Faraday constant [C/mol]
F w (des) molar flow of water desorbed through boundary (6) [mol/s]
F w (g) molar flow of water evacuated by gas phase through boundary (6) [mol/s]
F w (PCE) molar flow of water evacuated by PCP through boundary (3) [mol/s]
F w (r)
molar flow of water produced in water recombination reaction [mol/s]
G Gibbs free energy [J/mol]
g j v
generation term (eq. II.7.1) [mol/(m 3 ·s)]
i density of current normalized on electrode area [A/m 2 ]
I current [A]
i ACP density of current in anion-conducting phase (eq. II.7.2-3) [A/m 2 ] i s current per unit of length of three phase boundary [A/m]
I S(2) current calculated as surface integral on boundary (2) [A]
I S(3) current calculated as surface integral on boundary (3) [A]
I
υcurrent calculated as volume integral [A]
i 0 exchange current per unit of length of three phase boundary [A/m]
j imaginary unit [-]
J B,g diffusive molar flux of inert B (nitrogen) in gas phase [mol/(m 2 ·s)]
J w,g diffusive molar flux of water in gas phase [mol/(m 2 ·s)]
K equilibrium constant of water adsorption [Pa -1 ]
k d kinetic constant of water adsorption in proton-conducting phase [mol/(m 2 ·Pa·s)]
Kn B Knudsen number for inert B (nitrogen) [-]
Kn w Knudsen number for water [-]
A. Bertei GLOSSARY
M B molar weight of inert B (nitrogen) [kg/mol]
M w molar weight of water [kg/mol]
n number of oscillation [-]
n ˆ normal unitary vector [-]
N B,g molar flux of water in gas phase [mol/(m 2 ·s)]
N c coordination number in a uniform-sized random particle bed [-]
N hi,hj limit average number of contacts of a hi-particle with hj-particles [-]
n hi v number of hi-particles per unit volume (voids included) [m -3 ]
N H + molar flux of protons [mol/(m 2 ·s)]
N j molar flux of j (eq. II.7.1) [mol/(m 2 ·s)]
N k,k limit average number of contacts of a k-particle with k-particles [-]
N k,l limit average number of contacts of a k-particle with l-particles [-]
n k v
number of k-particles per unit volume (voids included) [m -3 ] N l,k limit average number of contacts of a l-particle with k-particles [-]
N l,l limit average number of contacts of a l-particle with l-particles [-]
n l v
number of l-particles per unit volume (voids included) [m -3 ]
N O molar flux of oxygen ions in lattice [mol/(m 2 ·s)]
N OH molar flux of protonic defects [mol/(m 2 ·s)]
N O -2
molar flux of oxygen ions [mol/(m 2 ·s)]
n v total number of particles per unit volume (voids included) [m -3 ] n v,as total number of particle per unit volume after sintering [m -3 ] n v,bs total number of particle per unit volume before sintering [m -3 ]
N w,g molar flux of water in gas phase [mol/(m 2 ·s)]
N w,PCP molar flux of water adsorbed in proton-conducting phase [mol/(m 2 ·s)]
OCV open circuit voltage [V]
P ratio of radii of particles r k /r l (only in chap. III) [-]
P total pressure [Pa]
p B partial pressure of inert B (nitrogen) [Pa]
P ex total pressure of external atmosphere [Pa]
P hi,hj ratio of radii of particles r hi /r hj [-]
p H2 partial pressure of hydrogen [Pa]
p k percolation probability of k-particles [-]
p l percolation probability of l-particles [-]
P max maximum ratio of radii to avoid segregation (only in chap. III) [-]
A. Bertei GLOSSARY
P max maximum pressure inside central membrane [Pa]
P min minimum ratio of radii to avoid segregation (only chap. III) [-]
p O2 partial pressure of oxygen [Pa]
p w partial pressure of water [Pa]
p w an
partial pressure of water at the anodic side [Pa]
p w ex
partial pressure of water in the external atmosphere [Pa]
R ACE ohmic resistance of anionic electrolyte [Ω]
r ACPi radius of a generic i anion-conducting particle [m]
r CM radius of central membrane [m]
r E radius of electrodes [m]
R ex ohmic resistance of the external circuit [Ω]
r fi radius of a generic fi pore former-particle [m]
R g universal gas constant [J/(mol·K)]
R h hydraulic radius [m]
r hi radius of hi-particle [m]
r k radius of k-particles [m]
r l radius of l-particles [m]
R p polarization resistance of central membrane [Ω ·m 2 ]
R p,an polarization resistance of anode [Ω ·m 2 ]
R p,cat polarization resistance of cathode [Ω ·m 2 ]
R PCE ohmic resistance of protonic electrolyte [Ω]
r PCPi radius of a generic i proton-conducting particle [m]
S dopant level [-]
S fi surface area fraction of generic fi pore former-particles [-]
S hi surface area fraction of hi-particles [-]
S k surface area fraction of all k-particles [-]
S l surface area fraction of all l-particles [-]
t time [s]
T absolute temperature [K]
t ACE thickness of anionic electrolyte [m]
t CM thickness of central membrane [m]
t PCE thickness of protonic electrolyte [m]
V ACP electric potential of anion-conducting phase [V]
v ads rate of water adsorption in proton-conducting phase [mol/(m 2 ·s)]
A. Bertei GLOSSARY
V an electric potential at the anode [V]
V cat electric potential at the cathode [V]
V el,an electric potential of the electronic conductor at the anode [V]
V el,cat electric potential of the electronic conductor at the cathode [V]
v m molar average velocity [m/s]
V PCP electric potential of proton-conducting phase [V]
x radial coordinate [m]
x w molar fraction of water in gas phase [-]
x w ex
molar fraction of water in external atmosphere [-]
y axial coordinate [m]
z impedance normalized on section of electrodes [Ω ·m 2 ] Z overall average coordination number (only in chap III) [-]
Z impedance [Ω]
z’ real component of impedance z [ Ω·m 2 ]
Z’ real component of impedance Z [Ω]
z’’ imaginary component of impedance z [ Ω ·m 2 ]
Z’’ imaginary component of impedance Z [Ω]
Z hi average coordination number of hi-particles [-]
Z hi,hj average number of contacts between a hi-particle and hj-particles [-]
Z hi,k average number of contacts between a hi-particle and k-particles [-]
Z hi,l average number of contacts between a hi-particle and l-particles [-]
Z k average coordination number of k-particles [-]
Z k,k average number of contacts between a k-particle and k-particles [-]
Z k,k-th average number of k-k contacts at percolation threshold [-]
Z k,l average number of contacts between a k-particle and l-particles [-]
Z l average coordination number of l-particles [-]
Z l,k average number of contacts between a l-particle and k-particles [-]
Z l,l average number of contacts between a l-particle and l-particles [-]
Z l,l-th average number of l-l contacts at percolation threshold [-]
α transfer coefficient [-]
α B factor defined in (eq. IV.2.11) for flux of inert B in gas phase [mol/(m·s)]
α w factor defined in (eq. IV.2.9) for flux of water in gas phase [mol/(m·s)]
β B factor defined in (eq. IV.2.12) for flux of inert B in gas phase [mol·s/kg]
β w factor defined in (eq. IV.2.10) for flux of water in gas phase [mol·s/kg]
A. Bertei GLOSSARY
δ density of perovskite-cells per unit volume [mol/m 3 ]
∆V eq difference of potential between ACP and PCP at local equilibrium [V]
ε r relative permittivity of proton-conducting phase [-]
ε 0 permittivity of the void [F/m]
ζ fi number fraction of generic fi pore former-particles [-]
ζ hi number fraction of hi-particles [-]
ζ hi as
number fraction of hi-particles after sintering [-]
ζ k number fraction of k-particles [-]
ζ k as
number fraction of k-particles after sintering [-]
ζ k-th number fraction of k-particles at percolation threshold [-]
ζ l number fraction of l-particles before sintering [-]
ζ l as
number fraction of l-particles after sintering [-]
ζ l-th number fraction of l-particles at percolation threshold [-]
η overpotential between ACP and PCP (if not specified) [V]
η ACP.ohm overpotential related to ohmic losses in ACP [V]
η an anodic overpotential [V]
η cat cathodic overpotential [V]
η CM overpotential of the overall central membrane [V]
η g,evac overpotential related to energy losses in gas phase [V]
η PCP.ohm overpotential related to ohmic losses in PCP [V]
θ mean of contact among particles [rad]
θ kk angle of contact between a k-particle and a k-particle [rad]
θ kl angle of contact between a k-particle and a l-particle [rad]
θ lk angle of contact between a l-particle and a k-particle [rad]
θ ll angle of contact between a l-particle and a l-particle [rad]
κ B Boltzmann constant [J/K]
λ v TPB length of three phase boundary per unit volume [m/m 3 ] µ dynamic viscosity of the mixture water-inert B (nitrogen) [kg/(m·s)]
µ B dynamic viscosity of inert B (nitrogen) [kg/(m·s)]
µ w dynamic viscosity of water [kg/(m·s)]
ρ c charge density [C/m 3 ]
σ conductivity [S/m]
A. Bertei GLOSSARY
σ ACE conductivity of anionic electrolyte [S/m]
σ ACP conductivity of anion-conducting phase [S/m]
σ app /σ correction factor for apparent conductivity [-]
σ k conductivity of k-particles [S/m]
σ l conductivity of l-particles [S/m]
σ PCE conductivity of protonic electrolyte [S/m]
σ PCP conductivity of proton-conducting phase [S/m]
τ tortuosity (only in chap. III) [-]
τ period of oscillation [s]
τ g tortuosity of gas phase [-]
φ g initial porosity [-]
φ g f
fraction of porosity created by pore formers [-]
φ g fin
final porosity [-]
φ P volume fraction phase P referred to whole volume (eq. II.7.1) [-]
φ PCP volume fraction of PCP referred to whole volume [-]
ϕ phase of oscillation [rad]
ψ ACP volume fraction of anion-conducting phase before sintering [-]
ψ ACP as
volume fraction of anion-conducting phase after sintering [-]
ψ f volume fraction of pore formers before sintering [-]
ψ fi volume fraction of generic fi pore former-particles [-]
ψ hi volume fraction of hi-particles relative to the total solid [-]
ψ hi as
volume fraction of hi-particles after sintering [-]
ψ k volume fraction of k-particles relative to the total solid [-]
ψ k as
volume fraction of k-particles after sintering [-]
ψ k-th volume fraction of k-particles at percolation threshold [-]
ψ l volume fraction of l-particles relative to the total solid [-]
ψ l as
volume fraction of l-particles after sintering [-]
ψ l-th volume fraction of l-particles at percolation threshold [-]
ψ PCP volume fraction of proton-conducting phase before sintering [-]
ψ PCP as
volume fraction of proton-conducting phase after sintering [-]
ω pulsation [rad/s]
∇ gradient [m -1 ]
A. Bertei GLOSSARY
⋅
∇ divergence [m -1 ]
∇
2Laplace operator (i.e. ∇
2= ∇ ⋅ ∇ ) [m -2 ]
Superscripts app apparent
B Bouvard and Lange (1991) method C Chen et al. (2009) method
eff effective
eq equilibrium condition N this study
S Suzuki and Oshima (1983) method sat saturation
Z Zhu and Kee (2008) method
° standard condition
* computer simulation
