Electrochimica Acta 50 (2004) 677–683
Two-phase flow analysis in a cathode duct of PEFCs
Jinliang Yuan∗ , Bengt Sundén
Division of Heat Transfer, Lund Institute of Technology, P. O. Box 118, S-22100 Lund, Sweden
Received 5 June 2003; received in revised form 27 November 2003; accepted 10 January 2004
Available online 20 August 2004
Abstract
A three-dimensional calculation method has been further developed to include two-phase, multi-component gas and heat transport processes.
A set of momentum, heat transport and gas species equations have been solved for the whole duct by coupled source terms and variable thermophysical properties. The effects of the electrochemical reactions on the heat generation and mass consumption/generation have been taken
into account as well. Liquid water saturation and its effects on the local current density has been predicted based on the calculated values of
water vapor partial pressure and temperature. The unique fuel cell conditions, such as the combined thermal boundary conditions and mass
transfer, have been employed as well.
© 2004 Elsevier Ltd. All rights reserved.
Keywords: Two-phase flow; Current density; Modeling; PEFC; Cathode
1. Introduction
Electrochemical reactions, current flow, hydro-dynamics,
multi-component transport, phase change and heat transfer
are simultaneously involved in polymer electrolyte fuel cells
(PEFCs). During operation of the PEFCs, the membrane electrode assembly (MEA) should be well humidified at all circumstances to reduce the ohmic resistance and additional
losses in the activation processes. However, if too much water is accumulated in the cathode duct, water flooding may
appear. In this case, liquid water blocks the oxidant (oxygen)
flow to the reaction sites and consequently affects the cell
performance.
It is a widely accepted fact that the PEFCs cathode is the
performance-limiting component. It is so because the potential water flooding as mentioned above and occurrence of
slower kinetics of the oxygen reduction reaction affect the
cell performance. In addition, mass transfer limitations may
happen due to the nitrogen barrier layer effects in the porous
layer, see [1]. Two-phase flow and its effects on the heat transfer, concentration variation and PEFCs performance are very
∗
Corresponding author. Tel.: +46 46 222 4813; fax: +46 46 222 8612.
E-mail address: [email protected] (J. Yuan).
0013-4686/$ – see front matter © 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.electacta.2004.01.118
important. Due to several limitations and difficulties, experimental investigations on PEFCs are still limited so far. For
the purpose of supplying the guidance for the design and optimization, comprehensive studies have been undertaken over
decades to simulate and analyze water transport, gas utilization, produced power, electrical current density distribution in
both unit and cell levels of PEFCs, see [1–5]. It is worthwhile
to note that most of the models are one- or two-dimensional
ones, and the isothermal and/or isobaric assumptions are involved, which may not be true in PEFCs as discussed in this
study.
Based on “multiple-phase and -component” mixture approach, a three-dimensional computational method has been
further developed in this study to predict water phase change,
and analyse its effects on the gas flow, heat transfer and cell
performance (current density) for a cathode duct. Momentum, heat transport and species equations have been solved by
coupled source terms and variable thermo-physical properties
of a multi-component mixture. The duct under consideration
consists of a flow duct, porous layer and solid structure. Advanced boundary conditions are applied in the analysis, such
as combined thermal boundary conditions of heat flux on the
active surface and thermal insulation on the remaining surface. Moreover, mass consumption and generation appearing
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J. Yuan, B. Sundén / Electrochimica Acta 50 (2004) 677–683
on the active surface are considered, together with interfacial
conditions of velocity, temperature and species concentration
between the flow duct and the porous layer, etc. It has been
found that two-phase flow and current density are sensitive
to the operating and configuration parameters.
2. Problem statement and formulations
Fig. 1 shows a schematic drawing of a PEFC cathode duct,
which includes the flow duct, porous layer and solid current collector. Many essential processes are involved, such as
transport of reactants (oxygen from air in the cathode, and
hydrogen in the anode) and products (water and heat if pure
hydrogen is used as the fuel), and electrons as well. The focus
of this study is to predict the two-phase flow in the composite
duct, and then assess effects of the liquid water saturation
level on the current density distribution. The following assumptions are employed in this study: the axial-velocity and
temperature distribution of the species at the inlet of flow duct
are uniform; a steady laminar flow is considered; species in
the duct are perfect and saturated with water vapor; negligible effect of the dissolved gases on water balance and no
interaction between liquid water and other gas species; the
interfacial shear force and surface tension force between the
liquid water and the gas-phase are cancelled out; liquid water has the same pressure field as the gas species; species
change and heat generation associated with the reaction are
appear on the active surface (bottom surface in Fig. 1); porous
layer is homogeneous, and both gas- and liquid-phase are in
thermal equilibrium with solid matrix, i.e., sharing the same
temperature field; liquid water appears in the form of small
droplets in the species, and multiple-phase mixture model is
employed to describe two-phase flow and heat transfer in the
composite duct.
A constant flow rate U = Uin for the saturated gas mixture
at pressure Pin = 105 Pa is specified at the inlet of the flow
duct, while U = 0 is specified at the inlet for the solid layer
and porous layer. Only the right half of the duct is considered
by imposing symmetry conditions at the mid-plane.
The following transport processes and parameters are considered in the present model, namely multi-component mixture flow based on convection and diffusion in the flow
duct and porous layer; species mass fraction for O2 , H2 O(v)
and H2 O(l) ; mass generation and consumption and effects
on the species composition change by the reaction on the
bottom surface in Fig. 1; heat generation by the reaction;
heat generation/absorption caused by water phase change
(condensation/vaporization) and heat transfer (convection
and/or conduction in all the components of the duct). Moreover, non-uniform gas pressure and temperature distributions
in the duct have been considered, together with variable
thermal–physical properties based on the gas species composition and/or temperature.
Due to space limitation, the governing equations, such
as the continuity, momentum, energy and species equations,
together with boundary and interfacial conditions are complemented with a detailed description of the mathematical
model in Appendix A.
Local current density I is an important parameter of the
cell, and essential for source term calculations related to mass
injection/suction and heat generation by the electrochemical
reaction. It should be noted that a constant value is typically
prescribed for the current density on the active surface in
the literature and in our previous work as well [6,7]. In this
study, this limitation is released. The local current density
is calculated by considering the effects of the liquid water
saturation when liquid water appears. It reads [1]
(1 − s)φO2
O2,trans F
I = Io
(1)
exp −
Vover
φO2 ,ref
RT
b
where I is the local current density based on the Tafel equation
along the active surface, Io exchange current density per real
catalyst area, s liquid water saturation, φO2 oxygen species
mass concentration, O2,trans the transfer coefficient for the
oxygen reaction, Vover cathode over-potential, φO2 ,ref the oxygen reference concentration (e.g., P = 1 atm) for the oxygen
reaction; other symbols can be found in the nomenclature.
It should be mentioned that, by employing (1−s), the effect
of liquid water saturation on the surface availability of the
reaction site is accounted for in Eq. (1). The liquid-phase saturation s describes the liquid water volume fraction in the
species mixture. It reads [2]
s=
ρφw − ρg φwv
ρwl − ρg φwv
(2)
in which, ρg is gas-phase density, ρwl liquid-phase density.
The density of the two-phase species mixture is
ρ = ρg (1 − s) + ρwl s
(3)
3. Numerical method
Fig. 1. Schematic drawing of a PEFC cathode duct.
A finite-volume method is employed to solve the governing equations and the boundary conditions, the interfacial
J. Yuan, B. Sundén / Electrochimica Acta 50 (2004) 677–683
conditions as presented in Appendix A. The solution domain
is discretized with a uniform grid size in the cross-section. A
non-uniform distribution of grid points is applied in the main
flow direction with an expansion factor to get finer meshes in
the entrance region of the duct. An in-house computational
fluid dynamics (CFD) code is further developed to include
variable thermo-physical properties and the liquid and vapor
water two-phase flow.
The important feature of this model is based on the approach of the multi-component two-phase mixture. The phase
change and its effects on the gas flow and heat transfer are
considered. The amount of water undergoing phase change
is calculated based on the partial pressure of water vapor
and the saturation pressure. It is worthwhile to note that twophase flow and heat transfer have been concerned and implemented to get local pressure, temperature and species component composition; this model is therefore considered as a
non-isothermal and -isobaric model. It is also worthwhile to
note that the source term is added to the species conservation equations for water phase change [7]. When the partial
pressure of water vapor is greater than the saturated pressure,
water vapor will condense to liquid water. Consequently, its
mass fraction will be reduced together with a release of water
latent heat until the partial pressure equals the local saturation
pressure. On the other hand, if the partial pressure is lower
than the saturation pressure, the liquid water will evaporate
if liquid water is available. It should be mentioned that the
source term concerning the water phase change and the associated heat source term correspond to the control volumes
where two-phase water appears, and these are not treated as
the boundary conditions.
The in-house CFD code has been further developed to
handle variable properties of gas species. These properties are
calculated as functions of pure component properties and the
species concentration, and are updated for every new iteration
[6,7].
679
oxygen reaction O2,trans = 0.5, over-potential Vover = 0.3 V.
It should be noted that all the results presented hereafter are
for the base case condition unless otherwise stated.
As revealed in [6,7], a parabolic velocity profile is clearly
observed in the flow direction for the fluid duct. On the other
hand, the velocity in the porous layer is very small except in
the region close to the flow duct. It should be noted that the
temperature increases monotonically along the main flow direction. A variation in temperature is also found in the crosssection with a slightly larger value close to the bottom surface
[7], due to the heat generation by both the reaction close to the
active surface, and the latent heat release by water condensation in the two-phase region. By considering the local value,
the effects of the temperature distribution on the saturation
pressure can be obtained.
It is revealed from the calculated results that oxygen mass
concentration decreases along the main flow direction in both
the flow duct and the porous layer, see Fig. 2a. As revealed
in [6,7], the partial pressure of water vapor is bigger in the
regions close to the active site, and smaller at the interfacial
region and in the flow duct. Based on the calculated partial
pressure and the saturation one, the liquid water is predicted.
It is found that the liquid water appears in both the flow duct
and the porous layer, with the biggest mass fraction (around
4. Results and discussion
Parameters of a typical PEFC cathode duct and its porous
layer, presented in the common literature, are applied as a
base case in this study. Duct geometries are employed as follows [3,4]: overall channel is 10 cm × 0.20 cm × 0.16 cm (x
× y × z), gas flow duct is 10 cm × 0.12 cm × 0.08 cm (x × y
× z), while the diffusion layer is 10 cm × 0.04 cm × 0.16 cm
(x × y × z). Oxidant inlet temperature Tin = 70 ◦ C, inlet
relative humidity η = 100%; oxygen binary diffusion coefficient DO2 ,f = 2.84 × 10−5 m2 /s, water vapor diffusion co(v)
efficient DH2 O,f = 1.25 × 10−5 m2 /s and liquid water diffu-
sion coefficient DH2 O,f = 1.0 × 10−5 m2 /s; For the porous
layer, thermal conductivity ratio θ k = 1.0, porosity ε = 0.5
and permeability β = 2 × 10−10 m2 ; inlet Rein = 50, thermal
conductivity ks = 5.7 W/mK for the solid layer; the exchange
current density Io = 0.01 A/m2 , the transfer coefficient for the
(l)
Fig. 2. Mass concentration profiles for: (a) oxygen; and (b) liquid water
saturation along main flow direction of a PEFC cathode duct at the base
case.
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J. Yuan, B. Sundén / Electrochimica Acta 50 (2004) 677–683
10%) in the porous layer close to the exit (not shown here).
Because the saturation pressure is proportional to the local
temperature, smaller saturated pressures can be expected for
the flow duct compared to the porous layer. This is the reason
why the liquid water can appear in the flow duct as well, but
with smaller mass fractions (less than 5%).
As shown in Fig. 2b, the liquid saturation s is zero at
the inlet region and increases along the flow direction. It is
also true that the liquid saturation s decreases from the active
site to the flow duct at the same x position due to the liquid
water transport, as discussed in [7]. The liquid water occupies
more volume in the porous layer with the largest value of s
appearing in the corner close to the active surface at the exit
of the duct, while the single-phase gas species occupy most
of the flow duct except that small values of s can be observed
in the interface region after a certain distance downstream the
inlet.
As mentioned earlier, the local current density I is one of
the most critical parameters for fuel cell performance, and for
modeling as well. It is possible to calculate the distribution of
the local current density once the detailed distribution of the
oxygen at the active site is available. It is assumed that the
catalyst load is evenly distributed at the active site, and the
activation over-potential therefore is evaluated by a constant
value.
Fig. 3 shows a local current density distribution on the active surface (in the x-z plane). It can be found that the current
density is high near the entrance, and then decreases along
the main flow direction. It is so because the oxygen transfer
to the reaction site is larger near the entrance region, which is
dominated by the oxygen convection (more detailed discussion can be found in [6]). The reduced current density downstream is due to the small oxygen transport rate controlled
by the diffusion [6]. It is also true that the current density is
non-uniformly distributed in the cross-section with a smaller
value in the corner region beneath the solid current collector. This is due to a longer transport distance from flow duct
to the active site. From Fig. 4, a similar conclusion can be
drawn, i.e., the current density is non-uniform along the main
flow direction and in the cross-section as well. It should be
Fig. 3. Local current density distribution in a cathode duct of a PEFC at the
base case.
Fig. 4. Current density distribution for the cross-sections along a PEFC
cathode duct at the base case.
noted that the liquid water saturation affects this non-uniform
distribution as indicated by Eq. (1).
Fig. 5 shows the current density distribution along the
main flow direction and reveals the effects of the liquid water saturation. As discussed earlier, liquid water saturation
level is low in the entrance region, and consequently almost
identical current density can be observed for both cases in
Fig. 5. While after a certain distance downstream the inlet,
the effects of the liquid water saturation becomes more clear,
about 20% lower current densities are predicted at the exit
if two-phase effects are considered. As liquid water appears,
the oxygen supply to the active surface is impeded and subsequently the local current density is reduced, which depend
on the liquid water saturation level.
Sensitivity studies for the effects of the liquid water saturation on the current density by varying some of the parameters from the base condition are conducted and presented
hereafter. The effect of the inlet temperature on the current
density is shown in Fig. 6. At a high temperature (80 ◦ C),
it was revealed in [7] that liquid water saturation is smaller
compared to the case at a low temperature (60 ◦ C) in both the
porous layer and the flow duct. This indicates that a high inlet
temperature contributes to a smaller liquid water saturation
level, and less effect on the decrease in the current density
that can be observed in Fig. 6. It is known that humidifying
of the inlet species can increase the water vapor concentration. Consequently, it is easier to form two-phase flow in the
duct, particularly in the porous layer where the water vapor
is generated as well. This is because water condensation will
Fig. 5. Liquid water effects on the current density distribution along a PEFC
cathode flow direction.
J. Yuan, B. Sundén / Electrochimica Acta 50 (2004) 677–683
681
the current density distribution appearing in a PEFC cathode
duct. It has been revealed that the non-uniformly distributed
current density is due to the non-uniformity of the oxygen
distribution and transport rate to the active surface, as well
as the liquid water saturation effects. It is found that a high
saturation level reduces the current density, as in the cases of
low inlet temperature and high inlet humidity of gas species.
Acknowledgement
Fig. 6. Effects of the inlet temperature of the gas species on the current
density distribution along a PEFC cathode flow direction.
The National Fuel Cell Programme by the Swedish Energy
Agency financially supports the current research.
Appendix A. Governing equations and other
equations implemented
The mass continuity equation is written as,
∇ × (ρeff v) = Sm
(A.1)
The source term Sm in the above equation accounts for the
mass balance caused by the reaction from/to the active surface Aactive (bottom surface in Fig. 1). It corresponds to the
consumption of oxygen consumption in cathode side and generation of water in the duct, respectively.
Fig. 7. Effects of the inlet relative humidity on the current density distribution along the flow direction.
occur only when the water vapor pressure exceeds the saturated pressure. In general, the fuel cell performance increases
with the increase of the cell operating temperature, because
the exchange current density and the membrane conductivity increase for a higher temperature. However, it has been
revealed by the experimental data in [8] that there is no apparent change in the fuel cell performance with the change in
the cathode humidification temperature, except the decreasing trend of the limit current density can be observed for the
increase of the cathode humidification temperature due to the
decrease of the effective porosity of the gas diffusion layers
and the decrease of the oxygen concentration, as discussed
in [8].
As found in [7], a decrease in the inlet humidity alone
would lead to a small liquid water saturation in the porous
layer and the flow duct, and consequently less effects on the
decrease in the current density. It can be found in Fig. 7 that
the dry inlet species (relative humidity is 0) will contribute to
a smaller current density decrease, compared with the base
case (relative humidity is 100%).
5. Conclusions
This study concerns three-dimensional predictions of simultaneous species component distribution and heat transfer, with main focus on the two-phase flow and its effects on
Sm = Sm,O2 + Sm,H2 O
I
Aactive
(1 + 2α)I
= − MO2 +
M H2 O
4F
V
2F
(A.2)
in which, V is the volume of control volumes at active site.
The momentum equation reads
∇ × (ρeff vv) = −∇P + ∇ × (µeff ∇v) + Sdi
(A.3)
Eq. (A.3) is valid for both the porous layer and the flow duct,
by including a source term Sdi
µeff v
Sdi = −
− ρeff BVi |V |
(A.4)
β
The first term on the right hand side accounts for the linear
relationship between the pressure gradient and flow rate by
the Darcy law, while the second term is the Forchheimer term
taking into account the inertial force effects, i.e., the nonlinear relationship between pressure drop and flow rate. In
Eq. (A.4), β is the porous layer permeability, and V represents
the volume-averaged velocity vector of species mixture. For
example, the volume-averaged velocity component U in the
x direction is equal to εUp , in which ε is the porosity, Up the
average pore velocity (or interstitial velocity in the literature).
This source term accounts for the linear relationship between
the pressure gradient and flow rate by the Darcy law. It should
be noted that Eq. (A.3) is formulated to be generally valid for
both the flow duct and the porous layer. The source term is
zero in the flow duct, because the permeability β is infinite.
Eq. (A.3) then reduces to the regular Navier–Stokes equation.
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J. Yuan, B. Sundén / Electrochimica Acta 50 (2004) 677–683
For the porous layer, the source term, Eq. (A.4) is not zero,
and the momentum Eq. (A.3) with the non-zero source term
in Eq. (A.4) can be regarded as a generalized Darcy model.
The energy equation reads
keff
∇ × (ρeff vT ) = ∇ ×
∇T + Swp
(A.5)
cp,eff
in which, Swp is the heat source associated with the water
phase change (condensation/vaporization) when two-phase
flow is considered as in this study.
Swp = Jwl × hwl
∇ × (ρeff vφ) = ∇ × (ρeff Dφ,eff ∇φ) + Sφ
(A.7)
where φ is mass fraction. The above equation is solved for the
mass fraction of O2 , H2 O(v) and H2 O(l) . The concentration
of inert species, nitrogen, is determined from a summation
of the mass fractions of the other species. The source term
in Eq. (A.7) includes water vapor and liquid water caused by
phase change. It is written as follows
Pw,sat − Pwv massi
Swv = −Swl = MH2 O
(A.8)
P − Pw,sat
Mi
i
In Eq. (A.8), massi is mass of species i; Pwv is partial
pressure of water vapor, Pw, sat saturation pressure at the local
temperature, while P is local pressure. If the partial pressure
of water vapor is greater than the saturation pressure, water
vapor will condense, and a corresponding amount of liquid
water is formed. Water vapor partial pressure Pwv in the above
equations is calculated based on its concentration and local
pressure of the gas mixture, while its saturated pressure Pw, sat
at local temperature reads
+ 1.445 × 10−7 T 3
(A.9)
The boundary conditions on the solid active surfaces can be
written as
U = V − Vm = W = 0,
−keff
∂φ
= Jφ
∂y
∂T
= qb ,
∂y
at bottom wall (y = 0)
(A.10)
Jφ = 0
q = 0 (or T = Tw ),
at top and side walls
∂V
∂U
∂T
∂φ
=
=W =
=
=0
∂z
∂z
∂z
∂z
at mid-plane (z = a/2)
(A.13)
The detailed procedure to obtain this value was discussed in
[6], and the final form is as follows
Sm = ρeff Rem
ν a
Dh A
(A.14)
where Rem = Vm Dh /ν is wall Reynolds number caused by the
electrochemical reaction. The other variables can be found in
the nomenclature list. qb in Eq. (A.10) is the heat source
caused by the reaction
qb = −
I
(HH2 O MH2 O − I × Vcell
2F
(A.15)
where (H is the enthalpy change of water formation. The first
term in the above equation accounts for the quantity of water,
and the second one takes care of the current density generated
by the electrochemical reaction. When the inhomogeneous
current density is concerned, the local heat generation should
be defined for various zones and locations.
Among various interfacial conditions between the porous
layer and gas flow region, the continuity of velocity, shear
stress, temperature, heat flux, mass fraction and flux of
species (for the oxygen, water vapor and liquid water, respectively) are adopted
U− = U+ ,
(µeff ∂U/∂y)− = (µf ∂U/∂y)+
(A.16)
T− = T+ ,
(keff ∂T/∂y)− = (kf ∂T/∂y)+
φ− = φ+ ,
(ρeff Dφ,eff ∂φ/∂y)− = (ρeff Dφ,eff ∂φ/∂y)+
(A.17)
(A.18)
log10 Pw,sat = −2.179 + 0.029T − 9.183 × 10−5 T 2
U = V = W = 0,
O2 + 4e− + 4H+ → 2H2 O
(A.6)
where Jwl is mass flux of liquid water by phase change, hwl
water latent heat. The species conservation equations are formulated into a general one,
− ρeff Dφ,eff
It should be noted that all the walls for the above boundary
conditions are on the external surfaces of the solid layer and
porous layer. In Eq. (A.10), Vm is the wall velocity of mass
transfer caused by the electrochemical reaction
(A.11)
(A.12)
here subscript + (plus) is for fluid side, while − (minus) for
porous layer side. Moreover, the thermal interfacial condition
Eq. (A.17) is also applied at an interface between the porous
layer and solid layer with ks instead of keff . It should be noted
that the properties in the above equations with subscript eff
are effective ones. For the flow duct, the effective properties
are reduced to regular values of the species mixture based
on the species composition, or regarded as constant values in
some cases; while in the porous layer, there are many factors
affecting the effective properties, such as microstructure of
the porous layer, species composition and local temperature
etc. It is not easy to obtain more accurate values so far because
the available data of the porous layer structure are still limited.
It has been found that setting µeff = µf and ρeff = ρf provides
good agreement. For the sake of simplicity, this approach
is adopted here as well. To reveal the porous layer effects,
on the other hand, parameter studies are carried out for the
conductivity keff and species diffusion coefficients D␾, eff by
J. Yuan, B. Sundén / Electrochimica Acta 50 (2004) 677–683
employing the ratios θ
θk =
keff
kf
(A.19)
θD =
Dφ,eff
Dφ
(A.20)
In Eq. (A.19), kf is the species mixture conductivity in the
porous layer, and estimated by a typical method
1
xi −1
kf = ×
(A.21)
xi kfi +
2
kfi
i
where xi is the mole fraction, and kfi conductivity of the
species component. The diffusion coefficients Dφ in Eq.
(A.20) are the values of the species components in the species
(v)
(l)
mixture, i.e., DO2 , DH2 O and DH2 O for oxygen, water vapor
and liquid water, respectively. However, the binary diffusion
coefficients of the components in pure air are used as estimation of Dφ .
The effective diffusivity ratios are corrected by applying
the so-called Bruggemann correction to account for the effects of porosity in the porous layer
θD = ε1.5
(A.22)
It should be noted that the thermal–physical properties of the
species mixture, such as the density ρf , viscosity νf are estimated as functions of the local concentration as well. According to Dalton’s law, relative humidity of the species mixture
is defined as
Pwv
P
η=
= xwv
(A.23)
Pw,sat
Pw,sat
where P is the pressure, Pwv the water vapor partial pressure, Pw, sat the saturation pressure identified in Eq. (A.9),
xwv water vapor molar fraction.
Appendix B. Nomenclature
a
b
D
F
hduct
width of porous layer (m)
width of flow duct (m)
diffusion coefficient (m2 /s)
Faraday constant (96487 C/mol)
height of the duct (m)
hporous
I
k
O2, trans
P
Re
s
T
Ui
x,y,z
683
thickness of porous layer (m)
local current density (A/m2 )
thermal conductivity (W/mK)
transfer coefficient for the oxygen reaction
pressure (Pa)
Reynolds number (UDh /ν), dimensionless
water saturation (dimensionless)
temperature (◦ C)
velocity components in x, y and z directions, respectively (m/s)
Cartesian coordinates
Greek symbols
β
permeability of porous layer (m2 )
ε
porosity (dimensionless)
φ
mass fraction (dimensionless)
η
relative humidity (%)
ρ
density (kg/m3 )
Subscripts
b
bottom wall
H2 O
water vapor
in
inlet
O2
oxygen
s
solid layer
wl
water liquid
wp
water phase change
wv
water vapor
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