47
A two-dimensional simulation of developing laminar
heat and mass transfer in fuel cell channels with
uniform suction of O2 and H2
H Hassanzadeh, S H Mansouri*, M A Mehrabian, and A Sarrafi
Department of Mechanical Engineering, University of Birjand, Birjand, Iran
The manuscript was received on 14 December 2006 and was accepted after revision for publication on 8 October 2007.
DOI: 10.1243/09576509JPE401
Abstract: A numerical model was developed to simulate the momentum, heat, and mass
transfer in the fuel cell channels at steady state and developing laminar flow with constant fluid
properties. The continuity, momentum, energy, and species equations were solved for humidified
air and humidified H2 with constant mass flux suction boundary condition for O2 in the cathode
and H2 in the anode channels. A finite-volume method was used to solve the equations in twodimensional cases. Friction coefficient was related to the flowrate, whereas Nusselt and Sherwood
numbers were, respectively, related to the heat and mass transfer in the flow channels. The distribution of local Nu and Sh numbers and their average values were obtained on the porous walls of
the channels (electrodes). The local Nu and Sh numbers were increased with increasing the
pressure and Reynolds numbers, whereas the coefficient of friction and the average Nu and Sh
numbers were decreased when the pressure and Re number are increased. The mass fraction of
O2 and H2 were decreased along the channel, except at the entrance regions, whereas the mass
fraction of water vapour increased along the channel, especially on the porous wall. Solving the
species equations gave: (a) the distribution of mass concentration of O2 and H2 and other species
in the channels and on the electrode surfaces for several stochiometric ratios and different current
densities; (b) the local and average Sherwood numbers in the cathode channels for O2 at different
stochiometric ratios, different inlet wall Reynolds numbers, and several inlet pressures; and (c) the
water vapour mass fraction in both channels and the mass of liquid water if condensation occurs.
Keywords: fuel cell, modelling, laminar, mass and heat transfer, hydrogen, oxygen, channel
1
INTRODUCTION
Fuel cells are highly efficient electrochemical devices
converting chemical energy of the reactants directly
into electricity. Electrochemical processes in proton
exchange membrane (PEM) fuel cells can be performed at temperatures close to room temperature;
therefore, unlike heat engines high operating
temperatures are not necessary for achieving high efficiency. In the case of hydrogen/oxygen fuel cells, which
are the focus of most research activities today, the only
*Corresponding author: Department of Mechanical Engineering,
Shahid Bahonar University of Kerman, Islamic Republic
Boulevard, P.O. Box 76175-133, Kerman, Iran. email: mansouri@
alum.mit.edu
JPE401 # IMechE 2008
by-products are water and heat. The high efficiency of
fuel cells and the prospects of generating electricity
with no pollutions have made them a serious candidate
to power the next generation of vehicles. However, the
costs of fuel cell systems are still high to become viable
commercial products. In the past decades, substantial
efforts have been devoted to reducing the cost as well
as increasing the efficiency of the fuel cell. In this
respect, the analysis of channel flow becomes one of
the necessities in fuel cell design [1–12].
To date, there have been two major approaches for
the analysis of flow in the channels of fuel cells. The
first approach uses theoretical techniques to investigate the variation of flow structure including the
fuel concentration as well as the current generated
along the channel [1 – 4], but the analytical solutions
are only available in limited cases whenever some
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48
H Hassanzadeh, S H Mansouri, M A Mehrabian, and A Sarrafi
simplifications are feasible due to strongly non-linear
nature of the governing equations. The second
approach uses computational fluid dynamics to
examine the two- or three-dimensional flow fields
in the flow channels [5 – 12].
Doughty and Perkins [5] have investigated the
hydrodynamic entry length for the two-dimensional
laminar flow between very long parallel porous
plates in variety of cases such as uniform suction or
injection from a wall, with uniform or parabolic
entry velocity profiles and constant or variable properties. They solved the continuity and axial momentum equations using finite-difference technique and
gave two correlations for the hydrodynamic entry
length with injection or suction. They have also simulated the thermal entry problem for laminar flow
between parallel porous plates assuming constantproperties with constant and equal wall temperatures
[6]. Doughty and Perkins have also simulated the
thermal and combined entry problem for laminar
flow with constant properties between parallel
porous plates for uniform suction and injection at
constant and equal heat fluxes [7]. They revealed
that the Nusselt number (Nu) for porous plates is
higher for the constant heat flux cases than for the
constant temperature wall cases for the same wall
Peclet number (Pe) similar to the usual results for
these two boundary conditions for non-porous
walls. The effect of injection is that the thermal
entry length is continuously differentiable in x, an
effect similar to the constant temperature wall case
[6]. Rhee and Edwards [8] obtained a numerical
solution for thermally and hydro-dynamically
developing laminar flow in a flat plate duct with
uniform suction on one wall and uniform temperature or heat flux independently prescribed at
each wall.
The frictional and heat transfer characteristics of fully
developed laminar flow in a porous circular tube with
constant wall temperature was simulated by Kinney
[9] for both cases with suction and injection. The effects
of mass injection, increasing the wall friction and
decreasing the wall heat transfer were verified. Hwang
and Cheng [10] simulated a square duct with one
porous wall subject to a constant heat flux, whereas
the other three walls were adiabatic and non-porous.
Yuan et al. [11] simulated the fully developed laminar
flow in rectangular ducts with combined thermal
boundary conditions, i.e. constant heat flux at one
wall and constant temperature at the other walls. It
was revealed that the Nu number is sensitive to
the boundary conditions. They also investigated the
fully developed laminar flow in ducts of rectangular
and trapezoidal cross-sections when one porous
wall having mass transfer whereas the previous
combined thermal boundary conditions prevail [12].
It was revealed that mass injection through one wall
increases the friction factor and decreases the Nusselt
number.
All the above models considered air as working
fluid and treated only suction or injection. These
models solved numerically either momentum, or
momentum and energy equations, but did not solve
the species equations. Without solving the species
equations, it’s not possible to obtain the rate at
which the species transfer and their concentration
values on porous electrodes. This information is
necessary to model the fuel cells. Hence, in the current study, a two-dimensional model is used to simulate the developing laminar flow in the channels of an
air – H2 fuel cell with uniform O2 suction at the cathode or H2 suction at the anode with constant mass
flux and combined thermal boundary conditions,
i.e. constant heat flux at porous wall and constant
temperature at non-porous wall. Therefore, in
addition to the momentum and energy equations,
the species equations are also solved for O2, H2, and
water vapour.
2
2.1
ANALYSIS
Problem statement
Figure 1 shows a schematic diagram of a PEM fuel
cell. It is made of two porous electrodes, a polymer
electrolyte, two very thin catalyst layers, and two
bipolar plates. Assuming the channel walls are
straight, humidified oxidant gases enter the cathode
channel whereas humidified fuel enters the anode
channel. The oxidant and fuel diffuse through the
porous gas diffusion layers and reach the catalyst
layers where the electrochemical reactions occur.
The concentration of fuel and oxidant varies along
the channel due to consumption in electrochemical
reaction. Both gaseous reactant flows in the cathode
and anode channels are subject to fluid injection of
H2O – vapour on cathode and suction of reactants
over the porous electrode surface. Thus, they may
be simulated as they flow in one porous wall duct
with constant heat flux and constant mass flux
boundary conditions.
Fig. 1
Schematic diagram of a PEM fuel cell
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Two-dimensional simulation of developing laminar heat and mass transfer
49
2
V rYi ¼ Deff
d;i r Yi
Fig. 2
2.2
Schematic diagram of flow and coordinate used
in this model for the cathode and anode
channels
A horizontal flow channel is shown schematically in
Fig. 2. The upper wall of the channel is porous,
whereas the other wall is non-porous. The gases,
either in the anode or in the cathode channels, are
assumed to be ideal. The gas flows in both channels
are considered two-dimensional and steady; the
flows in both channels have small Reynolds numbers,
therefore being laminar and considered to be incompressible, the compression work and viscous dissipation effects are neglected.
The temperature changes within the fuel cell channels are typically small; therefore, the thermodynamic properties of the fluids in the channels are
considered constant.
The suction flow through the porous walls are
considered uniform and are described by the wall
Reynolds number
Rew ¼
vw Dh
q
In these equations, u and v are the x and y components of velocity, respectively, T the temperature,
and Yi the mass fraction of species i. The symbols r,
m, cp, k, and Deff
d,i are density, viscosity, specific heat
at constant pressure, thermal conductivity, and effective diffusion coefficient of species i in the mixture,
respectively.
2.4
Assumptions
ð1Þ
where, vw is the mass transfer velocity, Dh the hydraulic diameter which is twice the plate spacing; and q
the kinematic viscosity. In the current study, Rew is
positive for injection at the anode channel and
negative for suction at cathode. The fluids in suction
have the same temperature as the porous wall.
ð6Þ
Boundary conditions
Equations (2) to (6) form a complete set of governing
equations for five unknowns, namely u, v, p, T, and Yi.
The inlet boundary conditions are
x ¼ 0: u ¼ u0 ; v ¼ 0; T ¼ T0 ; and Yi ¼ Yi;0
The boundary conditions at the cathode channel
(non-porous and porous walls) are
y ¼ H: u ¼ 0; v ¼ 0; T ¼ Tw ; and
dYi
¼0
dy
y ¼ 0: u ¼ 0; v ¼ vw ; q ¼ qcell ¼ const:;
_ 00O2 ¼ const:
m
and the boundary conditions at the anode channel
(non-porous and porous walls) are
y ¼ 0: u ¼ 0; v ¼ 0; T ¼ Tw ; and
dYi
¼0
dy
y ¼ H: u ¼ 0; v ¼ þvw ; q ¼ qcell ¼ const:;
_ 00vw ¼ 0
_ 00H2 ¼ const: and m
m
The outlet boundary conditions are
du
dT
dYO2
¼ 0;
¼ 0;
¼ 0;
dx
dx
dx
dYH2
dYvw
¼ 0; and
¼0
dx
dx
x ¼ L:
2.3
Governing equations
The governing equations are the conservation of
mass, momentum, energy, and species for an incompressible Newtonian fluid at steady-state conditions
and they can be written as
r ðrV Þ ¼ 0
Because, it is assumed that the channel is sufficiently
long so the velocity, temperature, and species concentration fields are fully developed.
ð2Þ
2.5
@p
V rðrvÞ ¼ þ mr2 u
@x
ð3Þ
@p
þ mr 2 v
@y
ð4Þ
V rðrvÞ ¼ V rðrcp T Þ ¼ kr2 T
JPE401 # IMechE 2008
ð5Þ
Additional equations
The friction coefficient (Cf), Nusselt number (Nu),
and Sherwood number (Sh) are the important parameters to be calculated in the fuel cell flow channels.
Friction factor is related to the flowrate, whereas
Nusselt number and Sherwood number are, respectively, related to heat and mass transfer in the flow
channels.
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H Hassanzadeh, S H Mansouri, M A Mehrabian, and A Sarrafi
The fanning friction coefficient Cf is defined as the
ratio of the wall shear stress tw ¼ m@u/@yjw to the flow
kinetic energy per unit volume [13]
tw
Cf ¼
1=2ru2m
Ð
ru dA
um ¼ Ð
r dA
ð8Þ
Whenever mass transfer is considered, the value of
um is not constant along the channel. Mean value of
u can be derived from mass balance in the duct.
Assuming that the suction is constant, the mean
value of velocity changes linearly along the channel,
and can be written as umH ¼ u0H þ xvw where vw is
positive for injection and negative for suction.
Rearranging for um we obtain
x vw
or
um ¼ u0 1 þ
H u0
x Rew
um ¼ u0 1 þ
H Re0
and Re0 ¼
u0 Dh
q
For a given geometry, local Cf in laminar flow is
independent of the surface roughness, and depends
on the flow condition as represented by Re and wall
Reynolds number as
Cf ¼ f1 ðRe; Rew Þ
ð10Þ
The Nusselt number as a dimensionless representation of the convective heat transfer is commonly
defined as [14]
hH Dh
k
ð11Þ
where hH is the convection heat transfer coefficient.
The convection heat transfer coefficient hH, is
equal to
hH ¼
For a given geometry, Nu depends on the flow
condition as represented by Re and relative effectiveness of momentum and heat transfer by diffusion in
the velocity and thermal boundary layers, as represented by Pr and suction wall Reynolds number,
Rew. Therefore, the Nusselt number has a functional
form as
Nu ¼ f2 ðRe; Pr; Rew Þ
ð14Þ
The Nusselt number on porous wall can be derived
from an energy balance on the wall surface as
dT q ¼ k dy w
ð15Þ
k@T =@yjw
Tw Tm
Nu ¼
q
Dh dT =dðy=Dh Þjw
¼
Tw Tm
Tw Tm k
and
ð16Þ
The Sherwood number or the dimensionless convection mass transfer coefficient (mass-based
analog to Nu) is commonly defined as
where
Nu ;
ð13Þ
dividing this equation by (Tw 2 Tm)k/Dh
rearranging, the Nusselt number is equal
ð9Þ
vw Dh
q
Ð
rcp Tu dA
Tm ¼ Ð
rcp u dA
ð7Þ
where um is the mean flow velocity and defined as
Rew ¼
which is defined as [15]
ð12Þ
where, Tw is the wall surface temperature and Tm the
mean flow temperature in the channel cross-section
Sh ;
hm Dh
Dd;i
ð17Þ
where, hm is the convection mass transfer coefficient,
Dh the hydraulic diameter, and Dd,i the diffusion
coefficient of species i in the media.
The convection mass transfer coefficient hm when
there is no suction or injection is equal to
hm ¼
Dd;i @Yi =@yjw
Yw;i Ym;i
ð18Þ
where, Yw,i is the surface mass concentration and Ym,i
the mean concentration of species i in the crosssection and is defined as [14]
Ð
Yi u dA
Ym;i ; Ð
u dA
ð19Þ
For a given geometry, Sh depends on: (a) the flow
conditions (as represented by Re), (b) the diffusion
of momentum in the velocity boundary layer, (c)
the diffusion of mass in the concentration boundary
layer (represented by Schmidt number), and (d)
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Two-dimensional simulation of developing laminar heat and mass transfer
Equations (2) to (6) have the general form of
wall suction Reynolds number (as represented by
Rew)
Sh ¼ f3 ðRe; Sc; Rew Þ
The diffusion of momentum
n
in the velocity boundary layer
¼
Sc ¼
The diffusion of mass in the
Dd
concentration boundary layer
When there is mass transfer through suction or injection, the total Sherwood number is made up of two
parts and the Sherwood number on porous wall can
be derived from a mass balance on the wall surface as
_ 00 ¼ rDd;i
m
dYi þrvw Yw;i
dy w
ð21Þ
Dividing this equation by (Yw,i 2 Ym,i) Dd,i/Dh and
rearranging becomes
_ 00
m
Dh
dYi =dðy=Dh Þjw
¼
rðYw;i Ym;i Þ Dd;i
Yw;i Ym;i
þ
Yw;i
Rew Sc
Yw;i Ym;i
ð22Þ
The left-hand side is the definition for Sh, whereas the
first part on the right-hand side is the definition of the
Sherwood number due to concentration gradient, Shc,
and the second term is the Sherwood number due to
mass transfer, Shm. Therefore, the total Sherwood
number can be written as
Sh ¼ Shc þ Shm
ð23Þ
where, Shm for suction flow is negative and for injection flow is positive.
3
SOLUTION PROCEDURE
The conservation equations (equations (2) to (6))
were discretized using a finite-volume method [16]
and solved using an in-house written code based on
methodology discussed in reference [17]. This code
is designed for two-dimensional convectiondiffusion problems in x –y, r– u, and r –z coordinates.
The pressure – velocity coupling is treated by the
simple method with the incompressible form of the
pressure – correction equation. The convection –
diffusion terms are treated by power-law scheme
[17].
JPE401 # IMechE 2008
!
@
ðrwÞ þ r ðr V wÞ ¼ r ðGrwÞ þ S
@t
ð20Þ
The Schmidt number is defined as
51
ð24Þ
where w stands for generalized dependent variable, G
for generalized diffusion coefficient, and S for the
source term of w. Those terms in the transport
equation not compatible with the terms in the
above equation, are treated as the source terms
[17]. The source terms for momentum, energy and
mass transfer equations are usual source terms in
these equations [17]. For example, for incompressible
flow, the source term in momentum equation in
x-direction is 2dp/dx and for energy equation, the
source term is F, heat dissipation and so on.
A staggered grid system is introduced in order to
eliminate numerical checker board oscillations
encountered with the single grid system. In this
arrangement, species concentration, temperature,
and pressure are discretized and computed on the
main grid, whereas velocities are computed at the
staggered nodes located mid-way between the main
grid nodes. The velocity, temperature, pressure, and
species concentrations are solved using the tridiagonal matrix algorithm (TDMA) [17]. A 30120
non-uniform and symmetrical grid in x and y directions with expansion factor of 1.05 in both directions
was used. A typical simulation involving approximately 21 600 unknowns required about 60 min computation time on a Pentium 4, 500 MHz PC for the
cathode channel.
4
VALIDATION OF NUMERICAL METHOD
In order to validate the performance of the numerical
program and code, it has been applied to a duct with
thermal and concentration boundary conditions with
and without mass transfer. The simulation results are
in full agreement with established results in the literature [5, 13, 18] as shown in Figs 3 to 5.
In Fig. 3, the numerical results for Nusselt number
at constant temperature are compared with those of
Kakac et al. [18]. The data were obtained from an
empirical equation with +3 per cent error as compared with theoretical results for the thermal entry
problem in parallel plate channels with constant
temperature walls. The present results are in good
agreement with the empirical data.
In Fig. 4, the numerical results for the thermal entry
flow are compared with those of Shah and London
[13]. Shah and London developed a correlation
based on the original results of reference [19], with
3 per cent error.
For the combined entry flow, where both velocity
and temperature at the inlet are uniform, the
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52
H Hassanzadeh, S H Mansouri, M A Mehrabian, and A Sarrafi
Fig. 3
Comparison of the numerical results for Nusselt
number when constant surface temperature is
applied on both walls with those from Kakac
et al. [18]
numerical results are compared with the results
obtained by Hawg and Fan [19]. Both sets of results
were obtained using the finite-difference method
for various Prandtl numbers. The numerical results
show good agreement with those released in reference [19].
In Fig. 5, the numerical results for Nusselt number
at constant heat flux are compared with the numerical results for a thermal entry problem in a channel
with parallel porous walls with suction or injection
on both walls from Doughty and Perkins [5]. They
are in good agreement.
5
Fig. 5
Comparison of the numerical results for Nusselt
number with those from Doughty and Perkins
[5] for parallel porous walls at constant heat
flux on both walls
surface) and the other is non-porous (bipolar plate
surface). Oxygen in the cathode channel constitutes
only a small portion of flow (20 per cent at 3 atm),
the suction is weak in the cathode channel, with a suction Reynolds number Rew 20.1 at stoichiometric
ratio ¼ 3, J ¼ 1 A/cm2. The stoichiometric ratio (z ) is
defined as the ratio of the inlet mass flowrate of
oxygen to the rate of oxygen consumption in the cathode (or hydrogen in anode). The stoichiometric ratio is
related to the wall Reynolds number Rew, the inlet Reynolds number, and the geometry of the channel
RESULTS AND DISCUSSION
zjRew j ¼
As mentioned before, in the current study the PEM
fuel cell channels are considered as two parallel
plate channels. One of them is porous (electrode
Fig. 4
Comparison of the present numerical results for
Nusselt number with those from Shah and
London [13] when constant surface heat flux is
applied on both walls
H
Re0
L
ð25Þ
At the anode channel, however, hydrogen is the
major species in the anode stream (84 per cent at
3 atm) but the velocity of hydrogen at the inlet of channel is small (u0 0.1 m/s for z ¼ 1.3, J ¼ 1 A/cm2) so in
both cases the wall Reynolds numbers are small. The
wall Reynolds number is small for almost all practical
cases.
To understand the behaviour of humidified air flow
in the cathode channel and humidified hydrogen
flow in the anode channel, the continuity, momentum, energy, and species equations are solved
numerically. Parameters such as coefficient of friction, Nusselt number, Sherwood number, mole fraction of oxygen, hydrogen, and water vapour at
different stoichiometric ratios are calculated.
5.1
Flow in fuel cell channels with suction of O2
and H2
The behaviour of humidified air with suction of O2 in
cathode channel and humidified hydrogen with
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Two-dimensional simulation of developing laminar heat and mass transfer
53
suction of H2 in anode channel are analysed in this
section. In order to accomplish this, the species
equation was solved in addition to the continuity,
momentum, and energy equations. The velocity,
temperature, and concentration were considered
uniform at the inlet of the channel. With the assumption of constant properties, humidified air enters the
cathode channel at a pressure of 3 atm (Pr ¼ 0.775) or
5 atm (Pr ¼ 0.7575), and humidified hydrogen at a
pressure of 3 atm (Pr ¼ 0.956) enters the anode channel. oxygen in the cathode channel and hydrogen in
the anode channel are removed from the channels
through the porous wall. The relative amount of
hydrogen or oxygen removed from the channel is
determined by the stoichiometric ratio z.
5.2
Cathode channel
Figure 6 shows the mass fraction contours of oxygen
and water vapour along the channel at a current density of J ¼ 1 A/cm2 and z ¼ 2. The mass fraction of
oxygen decreases along the cathode channel because
of O2 suction whereas the mass fraction of water
vapour increases. The inlet air is fully humidified
(saturated) at inlet temperature (80 8C), so that
addition of water vapour causes condensation to
occur along the channel. In the current study, generation of liquid water in the cathode has not been
considered. In practice, liquid water formation and
flooding can be a significant problem.
In Fig. 7, the distribution of oxygen mass fraction
along the y-direction is shown at the inlet, middle,
and outlet plains of the channel. As the stoichiometric ratio is assumed to be constant, the mean
value of concentration at every cross-section will be
constant. Constant mass flux is also assumed for
oxygen on the cathode (or hydrogen on the anode)
electrode surface. Increasing the current density will
increase the inlet velocity, therefore, the mass fraction concentration is increased near the non-porous
wall due to both convection, and decreasing the
mass fraction concentration on porous wall.
Fig. 7
Therefore, at any cross-section the lateral concentration gradient is increased as shown in Fig. 7.
Figure 8 displays the O2 mass fraction distribution
along the cathode electrode (porous wall) at different
current densities and stoichiometric ratios. If the stoichiometric ratio is kept constant and the current density is increased, the mass fraction concentration of
oxygen decreases on the porous wall (electrode surface) due to increasing the oxygen consumption.
Higher stoichiometric ratios will lead to a more uniform mass fraction concentration distribution but
Fig. 8
Fig. 6
Contours of mass fraction for oxygen and water
vapour along the channel
JPE401 # IMechE 2008
Distribution of oxygen mass fraction in the
y-direction at the inlet, the middle, and the
end of channel
Distribution of oxygen mass fraction on
electrode surface at different current densities
and stoichiometric ratios
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H Hassanzadeh, S H Mansouri, M A Mehrabian, and A Sarrafi
this has the consequence of reducing the overall efficiency of the system. This is because more un-reacted
oxygen and hydrogen escape from the system.
Figure 9 shows the average friction coefficient on
the porous wall of cathode fuel cell channel at different inlet Reynolds numbers. Increasing the stoichiometric ratio at a specified inlet Reynolds number,
changes the average friction factor slightly such that
as z ! 1 (i.e. no O2 suction) the average friction
factor remains nearly constant. At a constant Reynolds number, increasing the inlet pressure,
decreases the coefficient of friction, because the density of humidified gas is increased.
Figure 10 shows the average Nusselt number on the
porous and non-porous walls at different values of
stoichiometric ratios. The average values were
obtained with a uniform distribution of temperature
and velocity at channel inlet. At a constant stoichiometric ratio, increasing the inlet Reynolds number
increases both the current density and the production
of heat. Therefore, the average Nusselt number on the
porous wall will increase. At a constant inlet Reynolds
number, increasing the stoichiometric ratio decreases
slightly the average Nusselt number.
Figure 11 shows the local Sherwood number on
porous wall of cathode channel. The local values
were obtained assuming the uniform distribution of
concentration and velocity at the channel inlet. At a
constant stoichiometric ratio, increasing the inlet
Reynolds number increases the gradient of mass fraction on porous wall and therefore the local Sherwood
number is increased. Increasing the inlet Reynolds
number causes the concentration entry length to
increase.
Fig. 10
Figure 12 shows the average Sherwood number in
cathode channel at different inlet Reynolds numbers.
At a specified inlet Reynolds number, increasing the
stoichiometric ratio reduces the relative consumption of oxygen in the fuel cell so that the gradient of
concentration near the wall is decreased. Therefore,
the Sherwood number and its average value on
porous wall are decreased. At a constant value of stoichiometric ratio, increasing the inlet Reynolds
number also increases the average Sherwood
Fig. 11
Fig. 9
Distribution of average Cf in cathode channel at
different stoichiometric ratios and inlet
pressures
Distribution of average Nusselt number on
porous wall in the cathode channel at
different stoichiometric ratios and inlet
pressures
Distribution of average Sherwood number on
porous wall in the cathode channel at
different stoichiometric ratios, and inlet
pressures
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Two-dimensional simulation of developing laminar heat and mass transfer
Fig. 12
Distribution of local Sherwood number on
porous wall in the cathode channel at
different Reynolds numbers
number because the concentration gradient near the
porous wall is increased.
5.3
Anode channel
The practical stoichiometric ratio of hydrogen in the
anode channel is 1.2 [14]; therefore, most of the
hydrogen is consumed in the anode channel (or suction boundary condition in this model). The results
are obtained assuming uniform velocity, temperature, and concentration at inlet and no water
vapour removed from or added to the channel. However, in PEM fuel cells, water is generated on the cathode side.
Figure 13 shows the mass fraction contours of
hydrogen and water vapour in the anode channel at
J ¼ 1 A/cm2. Concentration of hydrogen is reduced
along the channel due to reaction (or hydrogen suction) and concentration of water vapour is increased
due to decreasing hydrogen concentration. The lateral concentration gradient of hydrogen in the
anode channel is smaller than the cathode channel,
because the effective mass transfer coefficient of
hydrogen in water vapour is nearly three times that
of oxygen in water vapour; additionally, the inlet velocity of the mixture in the anode channel is smaller
than that of the cathode channel and therefore the
convection effect is small.
The mass concentration of hydrogen on the electrode surface (porous wall) is reduced due to suction.
The mass concentration of water vapour is increased
on this surface, because the water vapour is forced to
the porous wall but then diffused back to the channel
due to suction. The two effects counterbalance each
other and therefore the mass flux of species becomes
zero.
Mass fraction concentration of hydrogen on porous
wall at constant inlet wall Reynolds number is shown
in Fig. 14. Same as the cathode channel, increasing
the stoichiometric ratio makes the reaction distribution more uniform but this has the consequence
of reducing the overall efficiency of the cell. The
behaviour of H2 and O2 mass fraction concentration
is similar except at the inlet of the cathode channel
where the effect of convection due to large Reynolds
number is higher than that the anode channel.
Figure 14 shows the H2 concentration curves at stoichiometric ratios up to 2. This is similar to Fig. 8
showing similar results for O2.
Figure 15 shows the average friction coefficient on
the porous wall of the anode channel at different inlet
Reynolds numbers. Increasing the Reynolds number
reduces the average friction factor. Similar behaviour
is expected for laminar flow in non-porous channels,
but for the same Reynolds number, the friction
Fig. 14
Fig. 13
Mass fraction contours of hydrogen and water
vapour into the anode channel
JPE401 # IMechE 2008
55
Mass fraction of hydrogen on porous wall at
J ¼ 1 A/cm2 with different stoichiometric
ratios
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56
H Hassanzadeh, S H Mansouri, M A Mehrabian, and A Sarrafi
Distribution of average Cf in the anode channel
at different stoichiometric ratios and 3 atm
Fig. 17
Distribution of local Nusselt number on porous
wall in the anode channel at different inlet
Reynolds numbers
coefficient on the porous wall is greater than the nonporous wall.
Local and mean Nu and Sh numbers on porous
wall are shown in Figs 16 to 18. As mentioned
before, this result was obtained at uniform distribution for velocity, temperature, and concentration
at the channel inlet. Increasing the inlet Reynolds
number increases the temperature and concentration
entry length and local Nu and Sh numbers. The behaviour of local and mean Nu and Sh numbers in the
anode channel is similar to the cathode channel,
but their changes in the anode channel is smaller
than the cathode channel due to smaller changes of
Reynolds number.
Modelling the flow in membrane electrode assembly (MEA) of PEM fuel cell, it is necessary to have the
mass fractions of O2 and H2 and local or average Sherwood number, temperature distribution, and local or
average Nusselt number on electrodes surface. When
calculating these parameters, one can model the MEA
of PEM fuel cell without having to model the flow in
fuel cell channels. As previously mentioned, this is a
single-phase study and thus, the presence of liquid
water on the cathode has been neglected.
Fig. 16
Fig. 18
Fig. 15
Distribution of local Sherwood number on
porous wall in the anode channel at different
inlet Reynolds numbers
Distribution of average Sherwood and Nusselt
numbers on porous wall in the anode
channel at different stoichiometric ratios
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JPE401 # IMechE 2008
Two-dimensional simulation of developing laminar heat and mass transfer
Fig. 19
Comparing the results of this study with
experimental data from reference [19] and
linear distribution of species in channels
In the current study, the results obtained from the
simulation program for the flow channels using a
two-dimensional model for MEA of PEM fuel cell
have been compared with the related experimental
data. This comparison is shown in Fig. 19, where
the E– I curve from the present study is compared
with the corresponding experimental data from
Ticianelli et al. [20] and also the linear distribution
of species in the channels. A full detail of this comparison and PEM fuel cell modelling is given in
reference [21].
6
at the staggered nodes and the variables were
solved using TDMA algorithm.
The friction coefficient, Nusselt number, and
Sherwood number were the important parameters
to be calculated in the fuel cell flow channels. Friction coefficient was related to the flowrate, whereas
Nusselt number and Sherwood number were related
to heat and mass transfer in the flow channels,
respectively. The distribution of local and mean
Nu and Sh numbers were obtained on the porous
walls of the channels (electrodes). These parameters
were increased when increasing the pressure and
Reynolds number whereas the coefficient of friction
and its average were decreased. The mass fraction
of O2 and H2 decreased along the channel, except
for the entrance regions, whereas the mass fraction
of water vapour increased along the channel
especially on the porous wall. If the cathode gas
had 100 per cent relative humidity at inlet and
Px ¼ P0, then from thermodynamics, PH2O 2
vapour ¼ PH2O 2saturated. Thus, when water is
added, it will liquefy and mass fraction of N2 will
increase. Solving the species equations gave: (a)
the distribution of mass concentration of O2 and
H2 and other species in channels and on the surface
of electrodes for several stochiometric ratios and
different current densities; (b) the local and average
Sherwood number in cathode channels for O2 at
different stochiometric ratios, different inlet wall
Reynolds numbers and several inlet pressures; and
(c) the water vapour mass fraction in both channels
and the mass of liquid water when condensation
occurs.
REFERENCES
CONCLUSION
A numerical model is developed to simulate the
momentum, heat, and mass transfer in the air–H2
fuel cell channels at steady state and developing laminar flow with constant fluid properties. The channel is
composed of a porous and a non-porous (impermeable) wall. The continuity, momentum, energy, and
species equations were solved for humidified air and
humidified H2 with constant mass flux suction boundary condition for O2 in the cathode and H2 in the
anode channels. Obtaining such parameters as mass
fraction concentration of O2 and H2 and their average
Sherwood number in channel flow are very important
for modelling the fuel cell processes.
A finite-volume method was used to solve the governing equations in two-dimensional cases. A staggered non-uniform grid system was introduced, and
the equations for species concentration, temperature, and pressure were discretized and computed
on the main grid, whereas velocities were computed
JPE401 # IMechE 2008
57
1 Berman, A. Laminar flow in channels with porous walls.
J. Appl. Phys., 1953, 24, 1232.
2 White, F. M., Baefield, B. F., and Goglia, M. J. Laminar
flow in a uniformly porous channel. J. Appl. Mech., 1958,
25, 613.
3 Raithby, G. D. and Knudsen, D. C. Hydrodynamic
development in a duct with suction and blowing.
Trans. ASME, J. Appl. Mech., 1970, 92, 548 –550.
4 Lessner, P. and Newman, J. Hydrodynamics and mass
transfer in a porous-wall channel. J. Electrochem. Soc.,
1984, 131, 1828–1831.
5 Doughty, J. R. and Perkins Jr, H. C. Hydrodynamic
entry length for laminar flow between parallel
porous plates. Trans. ASME, J. Appl. Mech., 1970, 92,
548– 550.
6 Doughty, J. R. and Perkins Jr, H. C. The thermal entry
problem for laminar flow between parallel porous
plates. J. Heat Transf., 1971, 93, 476– 478.
7 Doughty, J. R. and Perkins Jr, H. C. The thermal and
combined entry problem for laminar flow between parallel porous plates. J. Heat Transf., 1972, 94, 233 –234.
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H Hassanzadeh, S H Mansouri, M A Mehrabian, and A Sarrafi
8 Rhee, S. J. and Edwards, D. K. Laminar entrance flow in
a flat plate duct with asymmetric suction and heating.
Numer. Heat Transf., 1981, 4, 85 –100.
9 Kinney, R. B. Fully developed frictional and heat transfer characteristics of laminar flow in porous tubes.
Int. J. Heat mass Transf., 1968, 11, 1393–1401.
10 Hwang, G. J. and Cheng, Y. C. Developing laminar flow
and heat transfer in square duct with one-walled injection and suction. Int. J. Heat Mass Transf., 1993, 36,
2429–2440.
11 Yuan, J., Rokni, M., and Sunden, B. The development
of heat transfer and gas flow modeling in solid oxide
fuel cells. In Solid oxide fuel cells (SOFC VI) (Eds
S. C. Singhal and M. Dokiya) 1999, pp. 1099–1108 (The
Electrochemical Society, Pennington).
12 Yuan, J., Rokni, M., and Sunden, B. Simulation of fully
developed heat and mass transfer in fuel cell ducts with
different cross-sections. Int. J. Heat Mass Transf., 2001,
44, 4047–4058.
13 Shah, R. K. and London, A. L. Laminar flow forced convection in ducts, 1978 (Academic Press, New York).
14 Li, X. Principles of fuel cell, 2005 (Taylor & Francis,
New York).
15 Incropera, F. P. and Dewitt, D. P. Fundamentals of heat
and mass transfer, 1990 (John Wiley & Sons, New York).
16 Versteeg, H. K. and Malalasekera, W. An introduction to
computational fluid dynamics. In The finite volume
method, 1995 (Longman Scientific & Technical, England).
17 Patankar, S. V. Numerical heat transfer and fluid flow,
1980 (Hemisphere, Washington, DC).
18 Kakac, S., Shah, R. K., and Aung, W. Handbook of singlephase convective heat transfer, 1987 (John Wiley & Sons,
New York).
19 Hwang, C. L. and Fan, L. T. Finite difference analysis
of forced convection heat transfer in entrance region
of a flat rectangular duct. Appl. Sci. Res., 1964, A13,
401– 422.
20 Ticianelli, E. A., Derouin, C. R., Redondo, A., and
Srinivasan, S. Methods to advanced technology of
proton exchange membrane fuel cell. J. Electrochem.
Soc., 1988, 135(9), 2209–2214.
21 Hassanzadeh, H. Two dimensional modeling of PEM
fuel cell. PhD Thesis, Shahid Bahonar University of
Kerman, Kerman, Iran, October 2006.
22 Poling, B. E., Prausnitz, J. M., and O’Connel, J. P. The
properties of gases and liquids, 1987 (McGraw-Hill,
New York).
23 Gurau, V., Liu, H., and Kakac, S. Two-dimentional
model for proton exchange membrane fuel cells,
AICHE J., 1998, 44, 2410–2422.
24 Mills, A. F. Mass transfer, 2001 (Prentice Hall, New York).
G
D
z
q
m
r
w
generalized diffuision coefficient
a finite difference of a property
stoichiometric ratio
kinematics viscosity (suction flow)
viscosity (pa.s)
density (kg/m3)
general variable in equation (25)
Subscripts
d
H
m
w
0
diffusion
heat transfer
mean, mass transfer
wall
inlet condition or standard temperature and
pressure condition
APPENDIX 2
The parameters used in this model are shown in
Table 1. The binary diffusivities, Dd,ij, were determined
using data in reference [22] and scaled with temperature and pressure according to the relation
APPENDIX 1
Notation
Cf
cp
Dd
Dh
hH
hm
H
J
k
L
ṁ00
Nu
Pe
P
Pr
q
Re
Rew
S
Sh
Sc
T
u
v
vw
X
Y
binary diffusion coefficient (m2/s)
effective diffusion coefficient of species
i (m2/s)
heat transfer coefficient (W/m2K)
mass transfer coefficient (kg/m2K)
height of the channel (m)
current density (A/cm2)
thermal conductivity (W/m K)
length of channel (m)
mass flux (kg/s m2)
Nusselt Number
Peclet number
pressure (Pa)
Prandtl number
heat flux (W/m2)
Reynolds number
wall Reynolds number
source term in equation (25)
Sherwood number
Schmidt number
temperature
velocity component in x-direction
velocity component in y-direction
mass transfer velocity in equation (1)
mole fraction
mole fraction
Dd,ij
Deff
d,i
friction coefficient
specific heat at constant pressure (J/kg K)
diffusion coefficient (m2/s)
hydraulic diameter (m)
P0 T 1:5
Dd;ij ðT ; PÞ ¼ Dd;ij ðT0 ; P0 Þ
P T0
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ð26Þ
JPE401 # IMechE 2008
Two-dimensional simulation of developing laminar heat and mass transfer
Table 1
Physical parameters and properties at 353 K
Gas channel length L (m)
Gas channel width H (m)
Binary oxygen diffusivity in vapour water DO22VW
(m2/s)
Binary oxygen diffusivity in nitrogen DO22N2
(m2/s)
Binary nitrogen diffusivity in vapour water
DN22VW (m2/s)
Binary hydrogen diffusivity in vapour water
DH22VW (m2/s)
Binary hydrogen diffusivity in carbon dioxide
DH22CO2 (m2/s)
Binary carbon dioxide diffusivity in vapour water
DCO22VW (m2/s)
Oxygen/nitrogen ratio
Inlet pressure at cathode cannel Pca (atm)
Inlet pressure at anode cannel Pca (atm)
7.62E 2 2
7.62E 2 4
1.153E 2 5
9.69E 2 6
mmix ¼
n
X
Xi mi
n
P
i¼1
Xj fij
ð28Þ
n
X
Xi ki
n
P
i¼1
Xj fij
ð29Þ
j¼1
3.76E 2 5
2.78E 2 5
8.83E 2 6
0.21/0.79
3
3
kmix ¼
j¼1
where mi and ki are the viscosity and thermal conductivity of species i. For both relations the value of fij is
equal
2
!1=2 32
Mj 1=4 5
1
Mi 1=2 4
mi
fij ¼ pffiffiffi 1 þ
1þ
Mj
mj
Mi
8
ð30Þ
ð27Þ
i=j
In this relation, Xj is the mole fraction of species j.
JPE401 # IMechE 2008
The viscosity and thermal conductivity of the
humidified mixture in the anode and cathode were
obtained using the following relations [24]
1.05E 2 5
The effective diffusion coefficient of species i in the
bulk mixture, Deff
d,i, is given in terms of binary diffusion
coefficient Dd,ij as [23]
1 Yi
Deff
d;i ¼ P
Xj /Dd;ij
59
M is the molecular weight. Based on this relationship,
the thermal conductivity of mixture depends on viscosity of the components.
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