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4

Chapter Outline 4.1 Introduction 51 4.2 Literature review 52 4.3 Governing equations 56 4.4 Case study 57 4.5 Summary 62 References 64

4.1

Introduction

In this chapter, we discuss the modeling aspects associated with portable fuel cells. However, before moving to the crux of the subject, we need to explain what we mean by “portable” as the selection of some of the governing equations, which are used to model the fuel cell, largely relies on what is meant by this term. Hoogers (2002) attempted to define portable as “a small grid-independent electric power unit ranging from a few watts to roughly 1 kW, which serves mainly a purpose of convenience.” Under this definition, the portable fuel cell units could be divided into two groups: battery replacements (typically well under 100 W) and portable power generators (up to 1 kW) (Barbir, 2012). The fuel cell units in the first group are meant to compete with batteries in powering small portable devices such as smartphones and laptops. On the other hand, the fuel cell units in the second group have to compete with conventional power generators such as gasoline or diesel generators (Hoogers, 2002). Clearly, the fuel cell units in the first group need to be significantly simplified and/or miniaturized to compete with the ever-improving batteries. However, this requirement is much less strict for the fuel cell units belonging to the second group. One effective way to significantly simplify the battery-replacement fuel cell is to make its cathode open to the ambient. As such, the oxygen (required for the completion of the overall reaction) and water vapor (required for the initial humidification of the polymeric membrane) available in the surrounding ambient are directly supplied to the fuel cell through natural (or free) convection. This mode of operation is normally described as “air-breathing” or “passive.” The supply of air to the powergenerator fuel cells is often supplied by fans installed in the housing of the fuel cell system; this can be seen, for example, from the commercially available products listed Portable Hydrogen Energy Systems. https://doi.org/10.1016/B978-0-12-813128-2.00004-8 © 2018 Elsevier Inc. All rights reserved.

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Portable Hydrogen Energy Systems

in EFOY (2017) and HORIZON (2017). With the air-breathing mode of operation, the need to have devices to store, humidify, and force air/oxygen into the fuel cell is eliminated, thus substantially simplifying the fuel cell system. In this chapter, we refer to this type of fuel cells, that is, air-breathing fuel cells, when talking about portable fuel cells. The fuels that are normally used in air-breathing fuel cells are either methanol or hydrogen. If methanol is used, the fuel cells are normally called direct methanol fuel cells (DMFCs). To comply with the scope of the book, this chapter is only concerned with the air-breathing polymer-electrolyte-membrane (PEM) fuel cells fueled with hydrogen. A good review on active and passive DMFCs is available in Zhao et al. (2009). Mathematical models for fuel cells are normally used to better understand the transport phenomena that take place within the components of the fuel cells that are of the microscale lengths or to theoretically optimize the geometric and/or operational parameters of the fuel cell to maximize its performance (or, e.g., its portability in the case of the fuel cells in question) before manufacturing and/or operating it. The literature is rich with hundreds, if not thousands, of mathematical models on PEM fuel cells. However, the models on air-breathing PEM fuel cells form a small fraction of the mass models. This is obviously due to the fact that the conventional PEM fuel cell system is easily scalable and can be used to power a multitude of portable, automotive, and stationary applications. In contrast, due to their relatively small power output, the use of air-breathing fuel cells is typically limited to small portable devices (Ismail et al., 2014). The layout of this chapter is as follows: we review the models available in the literature for the air-breathing PEM fuel cells in the following section. In Section 4.3, the equations that are normally employed in the above models are presented and explained. A case study, representing a simple model, is presented in Section 4.4. The last section summarizes the main points of the chapter.

4.2

Literature review

Generally speaking, the mathematical models vary according to the basis of the classification. Based on the spatial dimensionality, they could be 0-, 1-, 2-, or 3-D models. Based on the temporal dimensionality, they are classified as transient or steady state. Based on how they are solved, the models are either analytic or numerical. Based on whether or not the governing equations of the model are derived from theory, the models are classified as theoretical, semiempirical, or empirical. Further, the models could be classified as single-phase or multiphase, isothermal or nonisothermal, etc. based on whether or not some of the physics are solved. The models available in literature for the air-breathing PEM fuel cells almost cover all the combinations of the elements of the above classifications. One more classification that is specific to the air-breathing PEM fuel cells is based on whether the cathode current collector, also known as the cathode flow distributor, is window-based (also known as ribbed) or channel-based (also known as ducted) (Ismail et al., 2013); see Fig. 4.1. In the following paragraphs, we present the models of the air-breathing PEM fuel cells

Modeling of portable fuel cells

53

Fig. 4.1 Examples of (A) window-based and (B) channel-based current collectors (Ismail et al., 2013).

that have been reported in the literature. As a note for the reader, if only the spatial dimensionality has been mentioned, then the model is numerical and steady state. Schmitz et al. (2004) developed a 2-D model for an air-breathing PEM fuel cell with rectangular vertical openings in the cathode current collector. The model developed showed that the cell performance improves with increasing the opening ratio of the current collector from 33% to 80%. A 3-D model for an air-breathing PEM fuel cell with an array of circular holes in the cathode current collector was created by Hwang (2006). The model suggests that the fuel cell with the staggered arrangement of the holes performs slightly better than that with the in-line arrangement of the holes. Also, for both arrangements, the optimum opening ratio was found to be about 30%, which ensures both a good supply of oxygen and good electric contact with the adjacent cathode gas diffusion layer (GDL). There was a slight discrepancy between the experimental and the modeling data; for a given current, the model slightly overpredicts the cell potential. The most probable reason for this discrepancy is that the natural convection over the cathode surface was not taken into account. If taken into account, the formed boundary layers over the cathode surface would impose mass and thermal resistances, and therefore, the predicted performance would be closer to the actual one. Zhang and Pitchumani (2007) built a 2-D model for a window-based air-breathing PEM fuel cell. They found that the fuel cell performs better with shorter heights due to better utilization of the active area caused by better exposure to the ambient air. They also investigated the effect of the cell orientation and found that the fuel cell gives the best performance if it is oriented vertically. On the other hand, the worst performance was found to be obtained when oriented horizontally with the open cathode facing upward. Further, they found that the pressure and relative humidity of the anode side have a significant effect on the performance of the fuel cell.

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Portable Hydrogen Energy Systems

Ying and coworkers developed 2-D (Ying et al., 2005a) and 3-D (Wang and Ouyang, 2007; Ying et al., 2005b,c) models for a channel-based air-breathing PEM fuel cell. They showed that there exists an optimum channel width (Ying et al., 2005c). A small channel width results in a small hydrodynamic entry length and as a consequence a poor heat and mass exchange. On the other hand, the authors suggested that a large channel width causes the two formed boundary layers to be distant from each other and consequently the heat exchange with the central region of the channel becomes poor. The effect of the channel depth was investigated by Tabe et al. (2006) who developed a 3-D model. For a given channel width, they found that the deeper is the channel, the greater is the concentration gradient at the cathode surface, and accordingly the higher is the oxygen supply to the cathode electrode. Matamoros and Br€ uggemann (2007) developed a 3-D model for a channel-based air-breathing PEM fuel cell. They found that the cell performance improves with increasing ambient temperature and this is due to enhanced natural convection driven by temperature difference. However, for a fully humidified fuel, the effect of the ambient humidity was found to be negligible. Also, the authors showed that the current density is a maximum near the ends of the channel and diminishes toward its central regions; this is due to the inefficiency of natural convection to drive air toward the latter regions. To this end, the shorter is the fuel cell, the more uniform is the current density, and the better is the cell performance. Further, for a fuel cell with a realistic channel length, only a small amount of platinum is required—the main rate limiting is the concentration losses caused by the insufficiency of oxygen supplied by natural convection. Any increase in the platinum loading would only result in a minor improvement in the local current densities near the ends of the channels. Zhang et al. (2007) built a three-dimensional model to investigate the effects of some geometric parameters on the performance of an air-breathing PEM fuel cell stack. The stack consists of a number of fuel cell cartridges; each cartridge consists of two cells sharing a hydrogen chamber. They found that there is a need to have a minimum vertical gap between the fuel cell and the bottom substrate and a minimum spacing between the cartridges in order to maximize the supply of fresh air to the cathode and, consequently, enhance the stack performance. Any increase in the vertical gap and the spacing between the cartridges beyond these minimum values would have an insignificant impact on the stack performance and lead to an undesirable increase in the size of the fuel cell stack. Litster et al. (2006) proposed a two-dimensional model for an air-breathing PEM fuel cell stack with nanopore GDLs, which can be used to increase the active area of the deposited catalyst. In their model, they only considered the cathode electrode, and the rest of the components were represented by a single solid component. They showed that the proposed GDL sufficiently exchanges the reactant and the product gases between the catalyst layer and the ambient. In other words, the natural convection induced by the heat of reaction was illustrated to provide a sufficient amount of oxygen to the reactive areas through the proposed GDL. As one would expect, the main focus of all the above models is on the design of the open cathode. However, there have been a few models whose main focus was to

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55

understand the transport phenomena in air-breathing PEM fuel cells (Hamel and Frechette, 2011; O’Hayre et al., 2007; Rajani and Kolar, 2007; Ismail et al., 2013, 2014). Hamel and Frechette (2011) developed a 1-D analytic model to study water transport across the membrane in a micro air-breathing PEM fuel cell. They showed that, beyond a certain current, the membrane suffers from dry-out at the anode side. This phenomenon arises because after a certain optimal current, the electroosmotic drag, which is proportional to the current drawn from the fuel cell, dominates over water back diffusion, thus resulting in a membrane dry-out at the anode side. Based on these findings, the authors suggested that the anode is humidified to delay the onset of the limiting current. O’Hayre et al. (2007) developed a 1-D analytic model to investigate the factors that limit the performance of window-based air-breathing fuel cells. They found that there exists an optimum thickness for the cathode GDL; a thin GDL tends to reject water easily from the cathode, thus causing a membrane dry-out; a thick GDL features a high thermal resistance that results in an increased cell temperature and, consequently, a membrane dry-out. Also, they reported that there was no performance gain if the GDL thermal conductivity was increased beyond 30 W m1 K1. It was shown that the heat and mass transfer coefficients associated with the natural convection are the rate limiting factors for air-breathing PEM fuel cells. If the values of these coefficients are significantly increased, the cell temperature at which the membrane dry-out occurs would never be reached, and the cell performance would be only limited by the ohmic losses. Further, the authors found that, for typical ambient conditions of 20°C and 40% relative humidity, the thermal gradient, compared with the concentration gradients, is the main driving force of the natural convection. Rajani and Kolar (2007) built a 2-D model for a window-based air-breathing PEM fuel cell. They found that the thicknesses of the boundary layers associated with natural convection have a significant impact on the performance of the fuel cell. The shorter is the height of the fuel cell, the lower are the overall thicknesses of the velocity boundary layer and the thermal and the concentration boundary layers and, therefore, the higher are the temperature and concentration gradients at the cathode surface—this leads to increased heat and mass transfer coefficients. Consequently, the shorter fuel cells feature better performance due to lower concentration losses due to better oxygen supply; lower ohmic losses due to better heat dissipation, which delays the membrane dry-out; and higher activation overpotential, which is inversely proportional to the temperature. Ismail et al. (2013) developed a thermal 2-D model for an air-breathing PEM fuel cell. They reported that heat was dissipated more effectively if the fuel cell was oriented vertically or horizontally facing upward than if it was oriented horizontally facing downward. In a subsequent work, Ismail et al. (2014) developed a 0-D numerical model for an air-breathing PEM fuel cell. They showed that, at intermediate cell potentials, mild ambient temperature and low humidity are favored to maintain high membrane conductivity and mitigate water flooding. However, at low cell potentials, low ambient temperature and high humidity are favored to avoid the dry-out of the polymeric membrane.

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4.3

Portable Hydrogen Energy Systems

Governing equations

The governing equations used in the models for air-breathing PEFCs are similar to those used in the models for the conventional PEFCs. As will be shown later, the difference between the two groups of models is only on how the boundary conditions for the temperature and the concentrations of chemical species at the cathode are treated. Therefore, in this section, we present the conservation equations that are typically solved in the conventionally modeled PEFCs, and in the next section, we highlight the boundary conditions that are specific to the open cathode in the air-breathing PEFC. The transport of any physical quantity may be cast in the following generic conservation equation (Cheddie and Munroe, 2005): ∂ ð φÞ + r:ðuφÞ ¼ r:ðΓrφÞ + S ∂t

(4.1)

where φ is the property to be solved for, u is the velocity of the fluid, and Γ is the diffusion coefficient. The first and second terms on the left-hand side of Eq. (4.1) are the transient and convection terms, respectively. The first and second terms on the right-hand side of the equation are the diffusion and source/sink terms. The source/sink term, S, represents the generation or consumption of the property φ. A full set of equations include the conservation equations of mass, momentum, species, charge, and energy. For completeness, we briefly list these equations. The conservation of mass equation could take the following form: ∂ðερÞ + r:ðερuÞ ¼ 0 ∂t

(4.2)

where ε is the bulk porosity and ρ is the density of the fluid (e.g., air). The conservation of momentum could be expressed as follows: ∂ðερuÞ + r:ðερuuÞ ¼ r:ðεμruÞ + Sm ∂t

(4.3)

where μ is the dynamic viscosity and Sm is the momentum source term. The conservation of the chemical species equation is given by. ∂ðερYk Þ + r:ðερuY k Þ ¼ r: ρDeff rY + Sk k k ∂t

(4.4)

where Yk is the mass fraction of species k, Deff k is the effective diffusion coefficient of species k, and Sk is the source term for species k. Due to their negligible contribution, the conservation of charge equation is normally expressed without the transient and convective terms: r: σ eff c rφc + Sc ¼ 0

(4.5)

Modeling of portable fuel cells

57

where σ c, φc, and Sc are the effective electric conductivity, the electric potential, and the source term for the phase c, respectively (i.e., solid or electrolyte phase). The conservation of energy equation may be described as follows: ∂ ερCp T + r: ερCp uT ¼ r: κeff rT + ST ∂t

(4.6)

where T is the temperature, κ eff is the effective thermal conductivity of the medium, Cp is the specific heat capacity of the fluid, and ST is the energy source term. It should be noted that, when modeling the fuel cell, not all the equations are applicable to the various components of the fuel cells. For example, Eq. (4.5) is not applicable to the flow channels as there is no transport of electrons/protons in such regions. Also, the transient terms in all the conservation equations vanish when ignoring the transient operation of the fuel cells that has been the case for most of the reported PEFC models. Further, some equations may not be solved due to the simplifications and/or assumptions made to the model. For example, the conservation of mass and momentum equations, that is, Eqs. (4.2), (4.3), is not normally solved for if the flow channels are not included into the model based on the assumption that the effects of the convective flow in the porous media on the fuel cell performance are negligible. It should be also noted that the source term for a given conservation equation may be different from one component to another. For example, the source term for the oxygen species in the cathode catalyst layer is its consumption rate, whereas the same source term is zero in the cathode GDL as there is no chemical reaction. Listing all the closure relations and defining all the involved physical parameters will distract us from the main aim of this chapter that is to introduce to the reader a brief account on the modeling of air-breathing PEM fuel cells. However, for more details on the mathematical formulation of the models, the reader is referred to, for example, Ismail et al. (2015) and Weber and Newman (2004).

4.4

Case study

For the sake of illustration, we present in this section a simple model. It is 1-D, single-phase, and for a fuel cell under steady-state operation. Fig. 4.2 shows the schematic representation of the computational domain. Table 4.1 shows the set of the governing equations used in the model, and Table 4.2 shows the boundary conditions used to solve these equations. Note that the equations listed in Table 4.1 are

Anode GDL

Anode catalyst layer

Membrane

Cathode catalyst layer

Cathode GDL

Fig. 4.2 A schematic representation of the modeled air-breathing PEFC.

Table 4.1 The set of conservation equations solved in the model Equation Conservation of chemical species

Conservation of electronic charge Conservation of protonic charge Conservation of energy

Mathematical expression ∂ ∂Yk + Sk ¼ 0 : ρDeff k ∂x ∂x

∂ eff ∂φs + Ss ¼ 0 : σ s ∂x ∂x ∂ ∂φm + Sp ¼ 0 : σ eff m ∂x ∂x r. (keff r T) + ST ¼ 0

Source/sink terms 8 Ic > > MO2 in CCL > > 4F > > > < I Ic c + nd MH2 O in CCL Sk ¼ > 2F F > > > > > > : Ia MH in ACL 2 2F Ic in CCL Ss ¼ Ia in ACL Sm ¼

Ic in CCL Ia in ACL

8 2 is > > in GDLs > eff > > σ > s > > > > > i2m i2s TΔS > > + + I η in CCL > c < σ eff σ eff c 2F m s ST ¼ > > i2m i2s > > + eff + Ia ηa in ACL > eff > > σm σs > > > > 2 > > i > : m in membrane σm

Table 4.2 The boundary conditions used in the model Boundary AMB jCGDL

Boundary condition keff

∂T ¼ ht ðT∞ Ts Þ ∂x

Deff O2

∂YO2 ¼ hO2 ðYO2 , ∞ YO2 , s Þ ∂x

Deff H2 O

∂YH2 O ¼ hH2 O ðYH2 O, ∞ YH2 O, s Þ ∂x

φs ¼ Vcell CGDLj CCL

∂φm ¼0 ∂x

CCL jMEM

∂YO2 ∂YH2 O ∂φ ¼ 0; ¼ 0; s ¼ 0 ∂x ∂x ∂x

MEM jACL

∂YH2 ∂φ ¼ 0; s ¼ 0 ∂x ∂x

ACL jAGDL

∂φm ¼0 ∂x

AGDL jACH

YH2 ¼ YH2O; T ¼ To; φs ¼ 0

Modeling of portable fuel cells

59

the conservation equations presented in the last section but after removing the transient and convective terms, assuming steady-state condition and the dominance of diffusion transport. Also, it should be noted that the conservation equations of mass and momentum, that is, Eqs. (4.1), (4.2), have not been solved because the flow channels were not included into the model. What matters most for us in the model are the boundary conditions of the temperature and the concentrations of the chemical species used for the open cathode (i.e., the interface between the cathode GDL and the ambient) because they differentiate the air-breathing PEM fuel cell from the normal PEM fuel cell. As shown in Table 4.2, the boundary conditions used for the temperature and the concentration of the chemical species at the interface between the cathode GDL and the ambient are the “convection surface condition” rather than “constant surface temperature or concentration.” With such a boundary condition, the temperature or the concentration of oxygen or water at the open cathode is determined by the heat/mass flux into (or out of ) the fuel cell and the heat/mass transfer coefficients. The amount and the direction of the flux are determined by the source/sink terms of the transported property. The heat (ht) and mass (hm) transfer coefficients can be determined using the following expressions: ht ¼

Nu κ Lt

hm ¼

Sh Dij Lm

(4.7)

(4.8)

where Nu is the average Nusselt number, Sh is the average Sherwood number, κ is the thermal conductivity of the fluid in the surrounding ambient that is air in our case, Dij is the diffusion coefficient of species i in j (in our case, i is either oxygen or water, and j is nitrogen), and Lt and Lm are the characteristic lengths associated with heat transfer and mass transfer, respectively. Note that the Nusselt and Sherwood numbers are described as average and so are the heat and mass transfer coefficients; they are averaged over the characteristic lengths Lt and Lm, respectively. Assuming that the fuel cell resembles a homogeneous isothermal plate, Nu can be calculated using some empirical correlations. The selection of the correlation depends on how the fuel cell is oriented (Bergman et al., 2011):

Nu ¼

8 1=4 > 0:67Ra > > 0:68 + > h i4=9 ðfor vertical plateÞ > > < 1 + ð0:492=Pr Þ9=16 > > > 0:54Ra1=4 ðfor horizontal heated plate facing upwardÞ > > > : 0:52Ra1=5 ðfor horizontal heated plate facing downwardÞ

(4.9)

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Portable Hydrogen Energy Systems

where Ra is the average Rayleigh number and Pr is the Prandtl number, which both can be obtained as follows: Ra ¼

gβðTs T∞ Þ L3t να

(4.10)

Pr ¼

ν α

(4.11)

where g is the magnitude of the acceleration due to gravity (9.81 m s2), ν is the kinematic viscosity, α is the thermal diffusivity, and β is the thermal expansion coefficient that can be approximated for ideal gases as follows: β¼

1 Tf

(4.12)

Tf is the film temperature that is the average value of both the surface temperature Ts and the ambient temperature T∞. Note that all the properties ν, α, and κ are those of air and are all estimated at Tf using, for example, the tabulated data available in Bergman et al. (2011). Given that the molecular and thermal diffusivities of the fluid are comparable (which is the case for air), Sh can be estimated using correlations that are analogous to those listed in Eqs. (4.9)–(4.12). The difference between the two sets of equations is that, for the average Sherwood number, the thermal diffusivity is replaced by the mass diffusivity Dij; the characteristic length associated with heat transfer is replaced by that associated with the mass transfer Lm; and the temperatures of the surface and ambient are replaced by the corresponding densities, that is, ρs and ρ∞. Note that the subscript m refers to either oxygen (O2) or water vapor (H2O). Employing the boundary conditions stated in Table 4.2, the conservation equations in Table 4.1 have been solved using a numerical solver, COMSOL Multiphysics®. The computational domain was discretized and refined especially at the interfaces until a mesh-independent solution is obtained. Table 4.3 shows the values of all the key parameters used in the model; they were mainly taken from (Ismail et al., 2017) and (Bernardi and Verbrugge, 1992). The meanings of all the symbols shown in the text, including those in the tables, are listed in the nomenclature at the end of the chapter. Having solved the model, Fig. 4.3 presents the polarization curves of the modeled air-breathing fuel cell as the orientation changes. The figure shows that the fuel cell performance is not very sensitive to the orientation of the fuel cell; a slight performance gain is obtained when the fuel cell is oriented vertically or horizontally with the open cathode facing upward. These results are in accordance with the experimental data presented in Fig. 13 in Kim et al. (2009). Compared with those of the horizontal orientation with the open cathode facing downward, the mass transfer coefficients of

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Table 4.3 List of the constants and physical parameters used in the model Parameter

Value

Ambient temperature (T∞) Pressure ( p) Oxygen/nitrogen molar ratio Ambient relative humidity Thickness of GDL Thickness of catalyst layer Thickness of membrane Porosity (ε) of GDL Porosity (ε) of catalyst layer Henry’s constant (H) for oxygen in the ionomer Henry’s constant (H) for hydrogen in the ionomer Effective electric conductivity (σ eff s ) of GDL Effective electric conductivity (σ eff s ) of catalyst layer Effective ionic conductivity (σ eff m ) of catalyst layer electrolyte Ionic conductivity (σ m) of membrane electrolyte Thermal conductivity (keff) of GDL Thermal conductivity (keff) of catalyst layer Thermal conductivity (keff) of membrane Reference concentration (Cref) of dissolved oxygen Reference concentration (Cref) of dissolved hydrogen Binary diffusivity (Dij) of oxygen in nitrogen Binary diffusivity (Dij) of water vapor in nitrogen Binary diffusivity (Dij) of oxygen in water vapor Binary diffusivity (Dij) of hydrogen in water vapor Reference exchange current density (Io) at cathode Reference exchange current density (Io) at anode Charge transfer coefficient (α) at cathode Charge transfer coefficient (α) at anode Cell voltage (Vcell) Net drag coefficient (nd) Emissivity Entropy change (ΔS) Characteristic length associated with heat transfer (Lt) Characteristic length associated with mass transfer (Lm)

293.15 K 1 atm 0.21:0.79 50% 250 μm 15 μm 100 μm 0.6 0.48 0.3125 atm m3 mol1 0.0456 atm m3 mol1 100 S m1 13 S m1 0.15 S m1 1.2 S m1 1.0 W m1 K1 0.2 W m1 K1 0.2 W m1 K1 0.85 mol m3 41.08 mol m3 1.86 105 m2 s1 2.58 105 m2 s1 2.47 105 m2 s1 8.54 105 m2 s1 1.6 105 A m3 1.4 1011 A m3 1 0.5 1.17 V 1 0.9 163.2 J mol1 K1 0.07 m 0.03 m

the oxygen for the other two orientations (i.e., the vertical and the horizontal with the open cathode facing upward) are significantly higher (see Fig. 4.4A), supplying more oxygen to the fuel cell especially at low cell potentials; see Fig. 4.5. Note that the heat transfer coefficient behaves similarly to the mass transfer coefficient of oxygen (see Fig. 4.4B); however, the effect of the former coefficient was found to be almost

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Portable Hydrogen Energy Systems

1.2 Vertical Horizontal facing upward Horizontal facing downward

1

Potential (V)

0.8

0.6

0.4

0.2

0

0

100

200

300

400

500

600

700

800

2

Current density (mA/cm )

Fig. 4.3 The polarization curves of the air-breathing PEFC for the three different investigated orientations.

negligible as the air-breathing fuel cell, under the current operating conditions (i.e., 20°C and 1 atm), is mainly limited by the mass transfer of oxygen than by the transfer of heat. Before we conclude the chapter, it should be noted that, for models with higher dimensionality (i.e., 2- or 3-D), the boundary conditions do not need to be necessarily at the interface between the open cathode and the ambient. An ambient region surrounding the open cathode could be modeled, and the boundary conditions are prescribed at the borders of this region. The mass and momentum conservation equations, Eqs. (4.1), (4.2), are solved in such models to take into account the buoyancy effects taking place in the ambient region, eliminating the need to use empirical correlations to estimate heat and mass transfer coefficient. Such models have been presented in, for example, Zhang and Pitchumani (2007), Zhang et al. (2007), and Ismail et al. (2013).

4.5

Summary

The literature review shows that there exist a limited number of models of airbreathing PEM fuel cells. This is attributed to the fact that, compared with the conventional PEM fuel cells, the use of air-breathing PEM fuel cells is, due to their low output power, mainly limited to small portable applications. The main findings of the models reported in the literature for the air-breathing PEM fuel cells have been

Modeling of portable fuel cells

9

63

× 10−3 Vertical Horizontal facing upward Horizontal facing downward

Mass transfer coefficient (m/s)

8 7 6 5 4 3 2 1 0

0.2

0.4

(A)

0.6

0.8

1

1.2

Potential (V) 4.5 Vertical Horizontal facing upward Horizontal facing downward

Heat transfer coefficient (W/(m2 K))

4 3.5 3 2.5 2 1.5 1 0.5 0

(B)

0

0.2

0.4

0.6

0.8

1

1.2

Potential (V)

Fig. 4.4 (A) The mass transfer coefficient of oxygen and (B) the heat transfer coefficient as a function of cell potential for the three different investigated orientations.

64

Portable Hydrogen Energy Systems

9 Vertical Horizontal facing upward Horizontal facing downward

8.5

Oxygen concentration (mol/m3)

8 7.5 7 6.5 6 5.5 5 4.5 4

0

0.2

0.4

0.6

0.8

1

1.2

Potential (V)

Fig. 4.5 The concentration of oxygen at the surface of the cathode as a function of cell potential for the three different investigated orientations.

presented. The mathematical formulation of the models of the air-breathing PEM fuel cells is similar to that of the conventional fuel cells. The only difference between the two formulations lies in how the boundary conditions at the cathode side are treated. If the ambient region surrounding the open cathode is not incorporated into the model, convection surface conditions are prescribed at the interface between the open cathode and the ambient to compute the heat and mass transfer coefficients and subsequently the temperature and the concentrations at the above interface. A 1-D, single-phase, and steady-state model for an air-breathing PEM fuel cell has been presented, and the results show that, for the given operating conditions, the performance of the air-breathing fuel cell is slightly sensitive to its orientation. Finally, if the ambient region surrounding the open cathode is considered, then the boundary conditions are prescribed at the borders of that ambient region.

References Barbir, F., 2012. Fuel cell applications. In: PEM Fuel Cells: Theory and Practice. second ed. Elsevier, Waltham. Bergman, T.L., Lavine, A.S., Incropera, F.P., Dewitt, D.P., 2011. Fundamentals of Heat and Mass Transfer, seventh ed. John Wiley & Sons, Hoboken, New Jersey. Bernardi, D.M., Verbrugge, M.W., 1992. A mathematical model of the solid-polymerelectrolyte fuel cell. J. Electrochem. Soc. 139, 2477–2491.

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