CHAPTER 2 MATHEMATICAL MODELING
2.5 Numerical Procedures
2.5.3 Model Validation
In the simulation, a uniform grids distribution is used to calculate the complex
electrochemical reaction and physical phenomena in the fuel cell. Three mesh
systems- 41 x 13 x 47, 51 x 16 x 58, and 61 x 21 x 67 are constructed to explore
numerical result dependence on computational cell numbers. Table 2.1 presents
geometrical and operating parameters of the base model in the PEM fuel cell. The
results of the polarization curve by the base model under different grid systems are
shown in Fig. 2.3. Considering both accuracy and economics, the grid system of 51 in
the z-direction, 16 in the x-direction, and 58 in the y-direction was selected for present
research.
To further check the adequacy of the numerical scheme, it is clearly seen from
Fig. 2.4 that the present predictions agree reasonably with the experimental data of
Squadrito et al. [44]. The above preliminary runs confirm that the present model and
the numerical method used are generally appropriate in analysis of the present
problem.
3 [14]
Tortuosity of the diffusion and catalyst layers
0.5 / 1.5 [14] Relative humidity of anode/cathode inlet
1.5 / 3 [44]
Stoichiometry, at / at 1.0
0.28 [45]
Porosity of membrane
0.4 / 0.4 [45]
Porosity of diffusion and catalyst layers
[14]
Permeability of membrane
/ [14]
Permeability of diffusion and catalyst layers
3 [14]
Tortuosity of the diffusion and catalyst layers
0.5 / 1.5 [14] Relative humidity of anode/cathode inlet
1.5 / 3 [44]
Stoichiometry, at / at 1.0
0.28 [45]
Porosity of membrane
0.4 / 0.4 [45]
Porosity of diffusion and catalyst layers
[14]
Permeability of membrane
/ [14]
Permeability of diffusion and catalyst layers
Table 2.1. Geometrical and operating parameters
Quantity Value
Cathode BP Anode BP
Cathode Channel
Anode Channel
Cathode GDL Cathode CL
Membrane Anode CL Anode GDL
X Y
Z
Cathode BP Anode BP
Cathode Channel
Anode Channel
Cathode GDL Cathode CL
Membrane Anode CL Anode GDL
X Y
Z
Figure 2.1 Physical and computational domains considered in this study
Begin
Solve momentum eq. obtain u*, v*, w*
Obtain pressure and velocity correction Guess P*
Set P*=P Stop
Solve for scalar equations of species, enthalpy, potential field Obtain mass imbalances
Correct pressure and velocity fields
Converged ?
Solve momentum eq. obtain u*, v*, w*
Obtain pressure and velocity correction Guess P*
Set P*=P Stop
Stop
Solve for scalar equations of species, enthalpy, potential field Obtain mass imbalances
Correct pressure and velocity fields
Converged ?
Figure 2.2 Numerical flow diagram of the solution procedure.
0 0.2 0.4 0.6 0.8 1 1.2
0 0.2 0.4 0.6 0.8 1 1.2
41 x 13 x 47 51 x 16 x 58 61 x 21 x 67
Vo lt a g e ( V )
Current Density ( A/cm
2)
Figure 2.3 Comparison of predictions on the three different grid systems.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Squadrito et al. [44]
Present result
Vo lt a ge ( V )
Current Density ( A/cm
2)
Figure 2.4 Comparison of the predicted I–V curve and the experimental data of Squadrito et al. [44].
CHAPTER 3
EFFECTS OF CATHODE HUMIDIFICATION AND POROSITY OF THE GDL ON THE GAS–LIQUID INTERFACE LOCATION IN A PEM FUEL CELL
3.1 Introduction
A PEM fuel cell is prone to gas-liquid two-phase formation due to its low
operating temperature, particularly under highly humidified or high current density
conditions. When the gas diffusion layer and the catalyst layer become saturated with
water vapor, the product water starts to condense and block open pores, reducing the
available paths for oxygen transport. This phenomenon is termed “flooding” and
becomes a major limiting factor of PEM fuel cell performance. Hence, it is critical to
understand the two-phase flow and transport in a PEM fuel cell, and a mathematical
model is useful to improve this understanding. In practice, humidification of anode
fuels and/or cathode oxidants is often used to provide sufficient membrane hydration.
Since water is generated in the cathode catalyst layer from electrochemical reaction
and it also tends to migrate from the anode side to the cathode side by the
electro-osmotic drag, it becomes a key issue in the design and operation of PEM fuel
cells to avoid the flooding phenomenon in the cathode. In this chapter, effects of
cathode humidification and gas diffusion layer porosity on the gas-liquid interface
location in a PEM fuel cell are investigated and described in detail. For all of the
calculations carried out in this study, the fuel flows were co-flows and inlet
stoichiometric ratios of 1.5 and 3 are used for the anode and cathode sides based on a
reference current density of 1 A/cm2. The inlet fuel, hydrogen, is assumed to be fully
humidified on the anode-side; the cell temperature and humidification temperature are
343 K and the operating pressure is 1 atm.
3.2 Effects of cathode humidification scheme
In this study, the location of the gas-liquid interface along the flow channel
direction at various cathode humidity conditions and its effect on cell performance are
elucidated by modeling a three-dimensional PEM fuel cell system using CFD. Figure
3.1 shows the effect of the relative humidity of the cathode on the location of the
interface where the liquid water begins to condense along flow channel at a cell
operating voltage of 0.7 V. The interface is defined as the location where liquid water
begins to condense. The horizontal dotted line indicates the interface between the flow
channel and the gas diffusion layer. A higher cathode relative humidity corresponds to
a smaller distance between the gas-liquid interface and the gas inlet, as clearly
displayed in Fig. 3.1 (a). Furthermore, the gas-liquid interface location moves to the
flow channel inlet region as the relative humidity of the cathode increases. This
phenomenon is caused by: (i) the decrease in the amount of evaporated water through
the flowing gas stream; (ii) the increase in the partial pressure of water and the ability
to reach the saturation pressure of the vapor water relatively quickly to form liquid
water earlier as the relative humidity of the cathode increases. On the contrary, the
gas-liquid interface location moves closer to the cathode catalyst layer when the
relative humidity of the cathode is less than 60%, as shown in Fig. 3.1 (b). In
conclusion, increasing the relative humidity of the cathode can reposition the
gas-liquid interface and cause liquid water to appear, affecting the performance of the
cell, because liquid water may occupy the pores in the porous media, reducing the
amount of fuel gas that can reach the cathode catalyst layer. Figures 3.2 (a) and 3.2 (b)
plot I–V and I–P curves at various relative humidity of the cathode with conventional
flow fields. The results in Fig. 3.2 (a) reveal that the cell performance decreases as the
relative humidity of the cathode increases, because the amount of liquid water
increases with the relative humidity. Accordingly, the pores in the porous media were
obstructed by liquid water at the cathode-side, reducing the amount of reaction gas to
the cathode catalyst layer. Therefore, the performance of the cell gradually decreases.
Also, the polarization curves do not seem to vary as the relative humidity of the
cathode increases, at an operating voltage that exceeds ~0.65 V. A large voltage results
in a small current density, and therefore, a relatively small electrochemistry reaction
rate. However, the performance of the cell decreases significantly as the operating
voltage declines below the ~0.65 V operating voltage, since the large current density
accelerates the rate of the electrochemical reaction. In particularly, in Fig. 3.2 (a), the
higher cathode relative humidity (100%) offers a better cell performance than the
lower one (20%) at an operating voltage of over ~0.65 V, because the rate of the
electrochemical reaction decreases as the operating voltage increases. Therefore, the
water content of the membrane in the initial stage increases with the relative humidity.
Thus, the cell performance is improved as the relative humidity of the cathode
increases.
By contrast, at operating voltages of below ~0.65 V, the cell performance
improves as the relative humidity of the cathode decreases, because accelerating the
electrochemical reaction produces more water, increasing the water content in the
membrane at a low cathode relative humidity, such as 20%. However, at high relative
humidity, flooding may occur at the cathode-side. Additionally, the power density
increases as the relative humidity of the cathode decreases, as shown in Fig. 3.2 (b).
3.3 Effects of cell operating voltage
Figure 3.3 shows the effects of various operating voltages on the location of
the gas-liquid interface along the flow channel at a relative humidity of the cathode of
80%. Figure 3.3 reveals that the gas-liquid interface location is close to the cathode
catalyst layer at high operating voltage, indicating that less water was generated at a
lower electrochemical reaction rate. However, since more water is generated at a high
electrochemical reaction rate, the gas-liquid interface location is close to the gas
diffusion layer and the flow channel at a low operating voltage. In closing, the
gas-liquid interface location gradually moves to the gas channel inlet region as the
operating voltage decreases, because reducing the operating voltage increases the
current density. Therefore, the electrochemical reaction rate increases.
3.4 Three-dimensional species field
Figures 3.4 and 3.5 plot the oxygen and water fractions in the gas flow channel
and the gas diffusion layer, respectively, of the cathode-side in the direction of the
flow channel at an operating voltage of 0.7 V and relative humidity of the cathode at
80%. Larger oxygen fraction appears in the gas channel inlet region, and then
gradually decreases in the flow channel direction, as shown in Fig. 3.4. The results
reveal that the decrease in the oxygen fraction is caused by the catalyst layer
consuming more oxygen on the cathode-side. Hence, the oxygen fraction is lowest
near the cathode catalyst layer. By contrast, Fig. 3.5 reveals that the water fraction
gradually increases along the flow channel direction, because of the production of
water in the catalyst layer of the cathode by the electrochemical reaction, as well as
the transport of water from the anode-side to the cathode-side by electro-osmotic drag.
Accordingly, the water fraction is the highest near the cathode catalyst layer. Since the
water mass fraction is higher near the cathode catalyst layer, where the corresponding
water partial pressure exceeds the saturated vapor pressure, causing the formation of
liquid water. Therefore, Fig. 3.6 shows the liquid water saturation field in the gas
channel and the gas diffusion layer of the cathode-side at an operating voltage of 0.7
V and a relative humidity of the cathode of 80%. The figure shows that saturation of
the liquid water increases along the flow channel direction, because the
electrochemical reaction causes the partial pressure of the water to exceed the
saturated vapor pressure, causing liquid water to condense. The capillary force also
causes liquid water to move toward the gas diffusion layer. Hence, this interface is the
single-phase region when the saturation of the liquid water is zero. It becomes a
two-phase region when the saturation of the liquid water exceeds zero. According to
Fig. 3.1, to the right of the point where water condensation starts, a two-phase region
is present along the flow channel direction. A single-phase region exists on the other
side.
Figures 3.7 show that the oxygen mass fraction contours at the cathode gas
diffusion layer as the cathode humidification of (a) 20% (b) 60% (c) 100% for cell
voltage of 0.4 V. It is obvious from these plots that the oxygen mass fraction gradually
decreased along the flow channel direction from the inlet toward outlet owing to the
consumption of oxygen by the electrochemical reaction in the cathode catalyst layer.
This figure also shows that the oxygen mass fraction decreases as the cathode
humidification increases from the 20% to 100%. This is attributed to the fact that a
higher relative humidity of cathode causes greater amount of water vapor enters the
electrode; therefore more liquid water is apt to condense in the pore space. Hence, the
amount of the oxygen through the gas diffusion layer is reduced and the cell
performance is decreased. Figures 3.8 presents the mass fraction contours of water
vapor at the cathode gas diffusion layer as the cathode humidification of (a) 20% (b)
60% (c) 100% for cell voltage of 0.4 V. The results reveal the fact that water fraction
gradually increases along the flow channel direction owning to the production of
water in the cathode catalyst layer by the electrochemical reaction. Also the water
concentration increases in the gas diffusion layer as the cathode humidification
increases. The distributions of the liquid water saturation for these cases are showed
in Fig 3.9. From these figures, it is found that when the cathode humidification is 20%,
the amount of the liquid water in the gas diffusion layer is almost very small.
However, when the cathode humidification is 100%, the there is some liquid water
appears in the gas diffusion layer. Hence, the pores in the gas diffusion layer were
obstructed by liquid water, reducing the amount of oxygen to the cathode catalyst
layer. Therefore, it supports the argument of previous plot, namely, the oxygen mass
fraction decreased owing to the blocking of liquid water in the pore of the porous
media as the cathode humidification increases.
3.5 Two-phase mixture velocity field
Figure 3.10 shows the velocity field of the two-phase mixture in the gas
diffusion layer and flow channel of the cathode at the cell voltage of 0.7 V and
cathode relative humidity of 80%. As expected, there is a large difference in the
velocity scale between the gas diffusion layer and the flow channel. The mixture
velocity in the gas diffusion layer is at least two orders of magnitude smaller than that
in the flow channel, indicating that gas diffusion is the dominant transport mechanism
in the porous media. The flow field in the flow channel is fully developed in view of
the large aspect ratio of the channel length to height (equal to 93 in the present study),
as can be seen from Fig. 3.7 where the channel length is, however, not drawn to scale
for better view.
3.6 Effects of the cathode gas diffusion layer porosity
The porosity of the cathode gas diffusion layer has two different effects on the
fuel cell performance: as the porous region provides the space for the reactants to
diffuse towards the catalyst layer region, an increase in the porosity means that the
onset of mass transport limitations occurs at higher current densities, i.e. it leads to
higher limiting currents. The adverse effect of a high porosity is an expected increase
in the contact resistance. Contact resistance occurs at all interfaces inside the fuel cell,
the most important one being the interface between the bipolar plates and the gas
diffusion layers. In this study, the solid phase conductivity is assumed constant and
does not change with the variation of GDL porosity. Hence, the location of the
gas-liquid interface along the flow channel direction at various porosities of the
cathode gas diffusion layer and its effect on cell performance were elucidated. Figure
3.11 shows the effects of the cathode gas diffusion layer porosity on the location of
the gas-liquid interface where the liquid water begins to condense along flow channel
direction at a cell voltage of 0.7 V and a relative humidity of the cathode of 80%. The
results in Fig. 3.11 show that the location of the gas-liquid interface appears early and
is closed to the flow channel inlet region in the gas diffusion layer as the gas diffusion
layer porosities decreased. These phenomena can be explained as that given smaller
porosity of the gas diffusion layer, the liquid water is more difficult to transport out of
the gas diffusion layer, hence, the liquid water will spread throughout the gas
diffusion layer. Once the liquid water block the pores of the porous media, causing the
hinder the diffusion transport of fuel gas to the cathode catalyst layer, thus limiting the
electrochemical reaction and causeing the decrease of cell performance. On the
contrary, the higher porosity of the cathode gas diffusion layer facilitates the transport
of water from the cathode catalyst layer to the flow channel. Hence, less water blocks
the pores of the gas diffusion layer, increasing the vacant pores space and helping the
reactant diffusion to the cathode catalyst layer. Nevertheless, the variation of the
gas-liquid interface location becomes small as the porosity changes from 0.7 to 0.8.
The phenomenon can be conjectured as that when the cathode gas diffusion layer
porosity is too big, the capillarity effect is not sensible, hence, the gas-liquid interface
location doesn’t seem to change obviously.
Figures 3.12 (a) and 3.12 (b) show I-V and I-P curves at various porosities of
the cathode gas diffusion layer at a relative cathode humidity of 80%. In Fig. 3.12 (a)
the I–V curves demonstrate that a better performance can be obtained by using a gas
diffusion layer of higher porosity. The remarkable drop in cell potential is caused by a
mass transfer or concentration loss, which is a consequence of shortage of the reactant
gas at high current density in which more reactant gas is required for fast reaction. A
gas diffusion layer of higher porosity has an ability of stronger diffusion transport,
which is beneficial in that it supplements the reactant gas to the cathode catalyst layer
and thus shifts the occurrence of the performance drop to a higher value of the current
density. However, Fig. 3.12 (a) also shows that when the cathode gas diffusion layer
porosity reaches the value of 0.8, the cell performance almost does not change
comparing with the value of 0.7. This phenomenon also conform the result of Fig.
3.11. The variation of power density is also shown in Fig. 3.12 (b). It reveals that the
power density increases with the cathode gas diffusion layer porosity. In closing, if the
variation of solid phase conductivity is neglected, the cell performance is enhanced as
the cathode gas diffusion layer porosity increase. But when the cathode gas diffusion
layer porosity reaches the value of 0.7, the cell performance is not enhanced.
Figures 3.13 and 3.14 illustrate the same oxygen and water mass fractions at
various porosities of the cathode gas diffusion layer, respectively, in the direction of
the flow channel at an operating voltage of 0.7 V and a cathode relative humidity of
80%. According to the model equations, a higher porosity of gas diffusion layer
accelerates the speed of oxygen diffusion to the cathode catalyst layer, as displayed in
Fig. 3.13. Because of the aperture of the pores being relatively large, meaning that it
may not block by the liquid water and the diffusion transport of reactant gas is easy to
pass through the pores of gas diffusion layer. Hence, the oxygen does not spread
throughout the gas diffusion layer, and can reach the cathode catalyst layer. On the
contrary, the reactant gas spread throughout the gas diffusion layer for a small
porosity of gas diffusion layer. The diffusion transport of the reactant gas and the
liquid water by the capillary force to pass though the gas diffusion layer is difficult.
Fig. 3.13 also shows that when the porosity value is 0.7, no matter what the porosity is
increased, the oxygen mass fraction does not change obviously because of the fuel gas
or water almost pass though the gas diffusion layer directly. Figure 3.14 shows a
similar result with Fig 3.13. That is, the water moves though the gas diffusion layer
more easily at higher porosity of gas diffusion layer and the water does not
remarkably occupy the pores and thus preventing the oxygen diffusion transport.
3.7 Summary
An isothermal, multi-dimensional, multi-component, computational fluid
An isothermal, multi-dimensional, multi-component, computational fluid