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Modeling Studies of a Cylindrical Polymer Electrolyte Membrane Fuel Cell Cathode Srinivasarao Modekurti,† Brian Bullecks,‡ Debangsu Bhattacharyya,*,§ and Raghunathan Rengaswamy‡ †

Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India Department of Chemical Engineering, Texas Tech University, Lubbock, Texas 79409, United States § Department of Chemical Engineering, West Virginia University, Morgantown, West Virginia 26506, United States ‡

ABSTRACT: Traditional polymer electrolyte membrane fuel cells (PEMFCs) are planar. High cost and low gravimetric and volumetric power densities are two major issues with the planar design. To improve the gravimetric and volumetric power densities of the PEMFCs and to reduce the cost, a novel cylindrical PEMFC design has been developed. The performance of the air-breathing cylindrical PEMFC is found to be superior to a state-of-the-art planar cell in the high current density region. To understand the effect of various design parameters and operating conditions on the performance of the cylindrical PEMFC, twodimensional, two-phase, steady-state models of the cylindrical cell for both air-breathing and pressurized conditions have been developed in this work. The developed model of the air-breathing cylindrical PEMFC is validated with in-house experimental data. Experiments were conducted with hydrogen on the anode side and air on the cathode side. The cathode catalyst layer is modeled using spherical agglomerate characterization. With the developed model, the effects of various operating and design parameters on the performance of the cell are studied. These studies show that the performance of the cylindrical cell can be further improved by optimizing these parameters.

1. INTRODUCTION Polymer electrolyte membrane fuel cells (PEMFCs) are very promising for power generation in the low-temperature fuel cell category. To make PEMFCs commercially viable, their gravimetric power density (power density per unit weight of the cell) and the volumetric power density (power density per unit volume of the cell) should be improved, especially for manportable and automotive applications. In addition, the manufacturing cost of the cell per unit power should be decreased. The bipolar plates used in the conventional planar PEMFCs can contribute as high as 42% of the cost and 62% of the weight of a 1 kW fuel cell stack.1 According to Murphy et al.,2 the bipolar plates contribute to over 80% of the stack weight and almost all of the volume. Replacing the traditional graphite bipolar plates with a low-cost, lightweight, noncorrosive, less bulky material, or elimination of it, can reduce some of the afore-mentioned problems. In a recent publication, our group has presented a novel cylindrical design of a PEMFC that eliminates some of the preceding problems. The cell uses a wire mesh for reactant distribution and current collection.3 The cell is found to have considerably higher volumetric and gravimetric power densities in addition to lower cost when compared to a state-of-the-art planar cell. Since the cylindrical cell could be arranged to exploit the gravitational force for water removal, flooding was not observed at high current density even by squirting liquid water on the MEA. A number of patents have described the fabrication of cylindrical PEM fuel cells.4−7 Green et al.8 have described the development of a low-weight tubular PEMFC considering still or slowly moving air. The authors had used an acid-etched, roughened, stainless steel shim for current collection as well as reactant distribution in the cathode side. The authors have © 2012 American Chemical Society

highlighted the importance of the shape and size of the holes, and distribution of acid-etched stainless steel foils for the performance and water management of a tubular PEMFC. Lee et al.9 have developed an air-breathing PEM fuel cell with cylindrical configuration. Even though there are some experimental studies on the cylindrical PEMFCs, very few theoretical studies on the tubular PEMFCs can be found in the existing literature. Coursange et al.10 have compared the cell performance of both tubular and planar PEMFCs by three-dimensional (3-D) numerical simulations. Three-dimensional computational fluid dynamics (CFD) models for both planar and tubular PEM fuel cells have been presented by Sadiq Al-Baghdadi.11,12 Models of various levels of complexities, ranging from simple one-dimensional (1-D) single-phase to complex 3-D two-phase models are available in the open literature for the planar PEMFCs.13−15 Our group has previously published papers on modeling of the planar PEMFCs as well as tubular solid oxide fuel cells (SOFCs).16−19 In the present work, we have developed a two-dimensional (2-D) steady-state model of a cylindrical PEMFC cathode under air-breathing and pressurized conditions. The model for the air-breathing condition is validated with the experimental data from a tubular cell that was fabricated. The fabrication and the experimental details of this cell have been presented in our previous publications and presentations.3,20,21 Even though there are numerous studies on model-based optimization of planar PEMFCs,22−25 optimization studies of the tubular PEMFCs are rare in the open Received: Revised: Accepted: Published: 5003

December 4, 2011 February 16, 2012 March 12, 2012 March 12, 2012 dx.doi.org/10.1021/ie2028359 | Ind. Eng. Chem. Res. 2012, 51, 5003−5010

Industrial & Engineering Chemistry Research

Article

of water vapor, respectively. The switching function (q) between condensation and evaporation is given by26

literature. In this paper, a number of sensitivity studies are presented to motivate possibilities for optimization of the cylindrical PEMFCs.

q=

2. MODEL DEVELOPMENT A schematic of the cylindrical PEMFC is shown in Figure 1. The modeling domain consists of the cathode gas diffusion

1 + (|pw − pwsat. |)/(pw − pwsat. ) 2

(5)

If the partial pressure of water vapor is less than the saturation pressure (pw < pwsat.), the switching function becomes zero and the interfacial process is evaporation. On the other hand, if pw ≥ pwsat., q becomes unity and the interfacial process is condensation. The conservation equation for the liquid water can be written as 1 ⎛ ∂(rNwr ) ⎞ ⎛ ∂Nwz ⎞ ⎟ + Iw = 0 − ⎜ ⎟−⎜ r ⎝ ∂r ⎠ ⎝ ∂z ⎠

(6)

where Nwr and Nwz are liquid water flux in r- and z-directions, respectively, and are given by

Figure 1. Schematic of a cylindrical PEM fuel cell.

layer (GDL) and cathode catalyst layer (CL) for the air-breathing condition. Under the air-breathing conditions, the GDL is open to the atmosphere. In developing the half-cell model, the following assumptions are made: isothermal and steady-state operation; no variation of the concentrations/potentials in the θ-direction; ideal gas behavior of the gas mixture; consideration of Butler−Volmer kinetics for the oxygen reduction reaction; liquid form for water generated due to the electrochemical reaction; istotropic gas diffusivities and water permeability; and negligible anode overpotential. 2.1. Conservation Equations. Gas Diffusion Layer. The gaseous transport and the liquid water transport through the GDL are described by Fick’s first law and Darcy’s law, respectively. The conservation equations for the gaseous components can be written as follows: 1 ⎛ ∂(rJa, r ) ⎞⎟ ⎛ ∂Ja, z ⎞ ⎟⎟ − Iw = 0 − ⎜⎜ − ⎜⎜ r ⎝ ∂r ⎟⎠ ⎝ ∂z ⎠

(7)

ρ K wK rl ⎛ dPc ⎞ ∂s ⎜− ⎟ Nwz = − w Mw μ w ⎝ ds ⎠ ∂z

(8)

In eqs 7 and 8, Pc is the capillary pressure which is calculated by Pc = σ cos θc

ε J (s ) Kw

(9)

where ⎧1.417(1 − s) − 2.120 for θcontact < 90 ⎫ ⎪ ⎪ 2 ⎪ (1 − s) + 1.263 ⎪ ⎬ J(s) = ⎨ (1 − s)3 ⎪ ⎪ ⎪ ⎪ 2 3 ⎩1.417s − 2.120s + 1.263s for θcontact > 90 ⎭ (10)

Catalyst Layer. The conservation equations of the species along with the electrochemical reaction can be written as for O2:

(1)

⎛ ⎞ 1 ⎜ ∂(rJO2, r ) ⎟ ⎛⎜ ∂JO2, z ⎞⎟ − ⎟ + ⎜ − ∂z ⎟ + R O2 = 0 r ⎜⎝ ∂r ⎠ ⎠ ⎝

where Ja,r and Ja,z are molar fluxes of the component “a” in r- and z-directions, respectively, and are given by

(11)

for N2:

Ja, r = −Daeff ∂Ca /∂r

(2)

Ja, z = −Daeff ∂Ca /∂z

(3)

⎛ ⎞ 1 ⎜ ∂(rJN2, r ) ⎟ ⎛⎜ ∂JN2, z ⎞⎟ − ⎟ + ⎜ − ∂z ⎟ = 0 r ⎜⎝ ∂r ⎠ ⎠ ⎝

(12)

for H2O(v):

Iw represents the interfacial transfer of water between liquid and vapor states and can be written as26 ε (1 − s) Iw = kcond k y (p − pwsat. )q RTcell w w εk sρw (p − pwsat. )(1 − q) + kevap Mw w

ρ K wK rl ⎛ dPc ⎞ ∂s ⎜− ⎟ Nwr = − w Mw μ w ⎝ ds ⎠ ∂r

⎛ ⎞ 1 ⎜ ∂(rJH2O, r ) ⎟ ⎛⎜ ∂JH2O, z ⎞⎟ − ⎟ + ⎜ − ∂z ⎟ − Iw = 0 r ⎜⎝ ∂r ⎠ ⎠ ⎝

(13)

for H2O(l): 1 ⎛ ∂(rNwr ) ⎞ ⎛ ∂Nwz ⎞ ⎟ − 2R O2 + Iw = 0 ⎜− ⎟ + ⎜− r⎝ ∂r ⎠ ⎝ ∂z ⎠

(4)

In the preceding equation, kcond and kevap are condensation and evaporation constants. s is the saturation level (defined as the fraction of void volume occupied by the liquid water), and εk is the void fraction. pw and pwsat. are partial and saturation pressures

(14)

where the term RO2 is the consumption of oxygen per unit volume of the catalyst layer. In this work, the catalyst layer is modeled using spherical agglomerate characterization as shown 5004

dx.doi.org/10.1021/ie2028359 | Ind. Eng. Chem. Res. 2012, 51, 5003−5010


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in Figure 2. This characterization assumes that the catalyst particles are spherical agglomerates with a thin film of ionomer

Considering the species transport and ohmic losses in the cathode, the overpotential within the CL can be calculated using the following expression:27 ηr = Vcell − Voc + 8.453 × 10−4(Tcell − 298.15) −

RTcell ln(pO ) + ΩI 2 nF

(19)

where pO2 is the partial pressure of oxygen, Tcell is the operating temperature of the cell in Kelvin, Ω is the ohmic resistance of the cell, and I is the current generated that is calculated by I =

z=L ⎛

∫z = 0

⎜ −ε mem ⎝

3/2κ

mem

⎞ dz⎟wcell dz ∂r r = CL − mem ⎠

∂ϕr

(20)

Figure 2. Schematic of spherical agglomerate characterization (redrawn from Rao et al.16).

The boundary conditions for the species at various domains are shown in Table 1.

around them. It is also assumed that the water generated due to the electrochemical reaction forms an ultrathin layer around the ionomer film before participating in condensation/evaporation. It is assumed that the diffused oxygen from the manifold to the catalyst layer dissolves and diffuses through both water and ionomer films before reaching the reaction sites. The concentration of the oxygen at the ionomer/spherical agglomerate interface is16,17

3. MODEL VALIDATION To solve the system of equations, MAPLE and MATLAB software have been used. The equations along with the boundary conditions were written and discretized in MAPLE. This generates a system of nonlinear algebraic equations (AEs) that are exported to and solved in MATLAB. A grid sensitivity study is carried out for deciding the final grid size so that an acceptable numerical accuracy is achieved, keeping the computational burden tractable in the MATLAB environment. At the final grid size, the change in the current density is lesser than 0.1% from the previous coarser grid size. Figure 3 shows

⎧ ⎫ ⎪ ⎪ RTcell ⎬ CO2|ns = ⎨ ⎪ ⎪ ⎩ HO2,memCO2, r ⎭ ⎧ ⎞ ⎛⎛ ⎞⎛ ⎞ ⎪ δ 1 ⎨1 + ⎜⎜⎜⎜ mem ⎟⎟⎜ ⎟ζk reacn⎟⎟ ⎪ ⎠ ⎝⎝ DO2,mem ⎠⎝ a1 ⎠ ⎩ ⎫ ⎞⎪ ⎛⎛ ⎞⎛ HO ,w ⎞⎛ ⎞ δ 2 ⎟⎜ 1 ⎟ζk reacn⎟⎬ + ⎜⎜⎜⎜ w ⎟⎟⎜⎜ ⎟ ⎟⎪ ⎠⎭ ⎝⎝ DO2,w ⎠⎝ HO2,mem ⎠⎝ a1 ⎠

(15)

The conservation equation for the protons can be written as ⎛ ⎞ 1 ∂(rJϕr ) ⎟ ⎛ ∂Jϕz ⎞ ⎟ + nFR O = 0 − ⎜⎜ −⎜ 2 r ⎝ ∂r ⎟⎠ ⎜⎝ ∂z ⎟⎠

(16)

where Jϕr and Jϕz are the ionic fluxes in r- and z-directions written as Jϕr = −κ eff ϕ ∂ϕr /∂r

(17)

Jϕz = −κ eff ϕ ∂ϕr /∂z

(18)

Figure 3. Performance comparison between tubular and planar cells.

the model validation for both of the cells with our in-house experimental data. The planar cell model has been presented elsewhere.16 The model parameters and the correlations used in the model development are presented in Table 2 and Table 3, respectively. The model parameters corresponding to the MEA

Table 1. Boundary Conditions (C = Concentration, J = Gaseous Flux, N = Liquid Flux)a variable

a

entrance

CO2

CO2 = CO2,o

CN2

CN2 = CN2,o

CH2O

CH2O = CH2O,o

s ϕr

s=0

GC/DL

DL/CL

CL/mem

COGC2 = CODL2 CNGC2 = CNDL2 CHGC2O = CHDL2O GC DL

CODL2 = COCL2 , JODL2 = COCL2 CNDL2 = CNCL2 , JNDL2 = JNCL2 CHDL2O = CHCL2O , JHDL2O = JHCL2O PcDL = PcCL, NwDL = NwCL

∇COCL2 = 0

s

=s

∇ϕr = 0

∇CNCL2 = 0 ∇CHCL2O = 0 ∇sCL = 0 ϕr = 0

Boundary conditions along z-direction: Fluxes of all the variables are zero at the channel entrance and exit. 5005



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Table 2. Design and Model Parameters Design Parameters outer radius of channel, mm 13 platinum (for pressurized conditions) loading, mg cm−2 outer radius of GDL, mm 5 membrane thickness, μm GDL thickness, μm 200 fraction of Pt on carbon in CL GDL porosity 0.7 fraction of ionomer in CL CL thickness, μm 15 active area, cm2 CL porosity 0.42 Model Parameters Kwo,GDL, m2 8.7 × 10−12 ρPt, kg m−3 Kwo,CL, m2 3 × 10−15 ρw, kg m−3 α 1 σ, N m−1 ragg, μm (tuning parameter) 0.29 θcontact(GDL), deg io, A m−2 (tuning parameter) 0.84 × 10−6 θcontact(CL), deg kcond, s−1 100 Ωplanar3, Ω kevap, atm−1 s−1 100 Ωtubular3, Ω −3 ρc, kg m 1800

0.5 50 0.4 0.3 10

21450 977.3 0.0625 110

Figure 4. Concentration profile of O2 in the catalyst layer (CL) of the air-breathing cylindrical cell.

100 0.011 0.023

Table 3. Correlations Used in Model Development16 Di,eff

εk 3/2(1 − s)3/2 Di m

Dim

−1 ⎛ N yj ⎞ ⎜ ⎟ (1 − yi )⎜ ∑ ⎜ D ⎟ ⎝ j = 1; i ≠ j ij ⎠

HOmem 2

⎛ 666 ⎞ 1.33 exp⎜− ⎟ ⎝ Tcell ⎠

HOw2

⎛ 498 ⎞ 5.08 exp⎜− ⎟ ⎝ Tcell ⎠

DOmem 2

⎛ 2768 ⎞ 3.1 × 10−7 exp⎜− ⎟ ⎝ Tcell ⎠

κeff

⎛ ⎛ 1 1 ⎞⎞ 100(0.005139λ w − 0.00326) exp⎜⎜1268⎜ − ⎟⎟⎟ 303 T ⎝ ⎝ cell ⎠⎠

Figure 5. Liquid water saturation profile in the catalyst layer (CL) of the air-breathing cylindrical cell.

are the same in both models. The ohmic resistance used in this model is based on the EIS data from our in-house cell.3 Figure 3 shows that the errors between the experimental data and model results are within acceptable limits. The scaled relative errors in current density between the experimental data and model results are 0.0915 and 0.1043 for the cylindrical and planar cells, respectively. The concentration profile of O2, the liquid water saturation profile, and the ionomer potential profile in the CL of the air-breathing cylindrical cell are shown in Figures 4, 5, and 6, respectively. As expected, the figures show that there is no change in the concentration of the gaseous species in the in-plane direction (z-direction) as the cell is the airbreathing type.

Figure 6. Ionomer potential profile in the catalyst layer (CL) of the air-breathing cylindrical cell.

4. MODEL ENHANCEMENT FOR THE PRESSURIZED CELL In this model, the cathode flow channel is included in the modeling domain. The conservation equations for the gaseous components can be written as 2RAD1Ja, r ∂(uCa) − − − Iw = 0 ∂z RAD2 2 − RAD12 (21)

where RAD1 and RAD2 are the inner and outer radii of the gas channel. The term Iw is zero for nonwater components. For liquid water, the conservation equation can be written as ρ ∂(sgcu) 2RAD1Nw − w − + Iw = 0 Mw ∂z RAD2 2 − RAD12 5006

(22)



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6. PARAMETRIC STUDIES Though the air-breathing cell performs reasonably well, it is primarily limited in oxygen supply at high current density. A model of a pressurized cell is developed by enhancing the model of the air-breathing cell. Then this model is utilized to carry out a number of parametric studies to study the effect of the cell temperature, platinum loading, and ionomer loading on the cell performance. We believe that results from such studies can reduce the experimental trial-and-error cycle time that inevitably accompanies the development of any new technology. First, we present a model enhancement for a pressurized cell before these parametric studies are undertaken. 6.1. Temperature. An increase in temperature enhances the reaction rate and improves the mass-transfer rate. On the other hand, an increase in temperature decreases the molar concentration of oxygen at a given pressure and also increases the ohmic loss of the current collector. At a given operating pressure, an increase in temperature reduces the proton conductivity due to dehydration of membrane at higher temperatures. The effect of operating temperature on cell performance is shown in Figure 9. With the increase in temperature, a negligible

The polarization curve under pressurized conditions is shown in Figure 7. As the operating pressure is increased, the cell

Figure 7. Performance of the pressurized cylindrical cell for various pressures.

performance improves in the entire polarization range. In the polarization range studied, the cell does not reach its limiting current density when the operating pressure is increased from the atmospheric pressure to 1.5 atm.

5. PERFORMANCE COMPARISON BETWEEN THE CYLINDRICAL AND PLANAR PEMFCS The planar and tubular cells vary in their concentration and ohmic polarizations. A comparison of polarization curves with and without concentration losses is shown in Figure 8

Figure 9. Effect of temperature on cell performance.

increase in the performance is observed in the 40−50 °C temperature range (not shown in the figure). Further rise in temperature (60 °C) results in relatively low performance. Even though temperature has a modest effect in the range studied, it can have strong effect when it exceeds some higher/lower values due to drying/freezing. 6.2. Flow Rate. An increase in the cathode air flow rate reduces the concentration losses and hence increases the cell performance. Since less amount of oxygen is required at low current densities, the increase in the flow rate has a negligible effect on the performance at high voltages. However, as the cell produces more current, it requires a larger amount of oxygen. The increase in the flow rate not only supplies the required amount of oxygen for the cathode reduction reaction but also increases the catalyst utilization. The polarization curves for various air flow rates are shown in Figure 10. As the flow rate is increased from 0.25 to 1 LPM, a significant increase in the performance improvement is observed. Further rise in the flow rate gave a slight increase in the current in the mass-transfer limiting region. 6.3. Platinum Loading. A greater amount of platinum reduces activation losses. However, this effect is dominant

Figure 8. Polarization curves of the cylindrical (air-breathing) and planar cells with and without concentration polarization.

for both types of cells. The operating conditions and model parameters used in this study are the same as those used in the model validation. Figure 8 shows that the concentration losses are more in the planar geometry in the entire polarization range. Mainly due to this, the performance of the cylindrical cell is better than the planar cell at high current densities where the concentration polarization is the dominating loss. As shown in Figure 3, this higher performance is achieved at high current densities even though the ohmic resistance of the cylindrical cell is more than the planar cell. 5007



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Figure 10. Polarization curves at various flow rates of air. Figure 12. Spatial distribution of liquid saturation inside the catalyst layer at 0.3 V.

mainly in the low current density region where the activation loss plays the key role. At high current densities, concentration and ohmic losses dominate. An increase in the platinum loading will in fact lead to an increase in the concentration overpotential due to reduction in the pore volume. The effect of platinum loading throughout the polarization range is shown in Figure 11. Figure 11 shows that as the platinum loading is

6.4. Ionomer Loading. An increase in the ionomer loading results in a decrease in the ionic losses. On the other hand, the void volume decreases causing higher concentration losses. Polarization curves for different ionomer loadings are shown in Figure 13. Figure 13 shows that when

Figure 11. Effect of platinum loading on the cell performance (f pt = 0.2, f ionomer = 0.35, tCL = 20 μm).

Figure 13. Polarization curves for various ionomer loadings ( f pt = 0.2, mpt = 0.4 mg/cm2, tCL = 20 μm).

increased from 0.25 to 0.35 mg/cm2, there is an overall improvement in the performance in the entire polarization range. However, as the loading is further increased to 0.45 mg/cm2, there is an insignificant increase in the performance in the low to medium current density region. However, the cell performance is worse in the higher current density region. This suggests that there is an optimum platinum loading. The saturation profiles in the CL for different platinum loadings at 0.3 V are shown in Figure 12. The numbers shown on the X- and Y-axes represent the discretizations in the r-direction and z-direction respectively. Number 1 on the X-axis represents the GDL/CL interface and number 10 represents the CL/mem interface. Similarly, on the Y-axis, numbers 1 and 20 represent the channel entrance and exit, respectively. The saturation is found to be considerably higher for platinum loading of 0.45 mg/cm2 compared to the lower platinum loadings. This is due to a decrease in the porosity with an increase in the Pt loading for a given thickness of the CL. The higher saturation leads to higher concentration losses in the cell with higher platinum loading in the mass-transfer limited region.

the weight fraction of ionomer is increased from 20 to 30%, performance is improved throughout the operating voltages. When the weight fraction of ionomer is increased to 40%, the cell performance decreased, especially at high current densities. This suggests that there is an optimum loading of ionomer for improving the cell performance, especially at higher current densities. This happens because of the interplay between the ohmic and concentration losses as the ionomer content is changed. The spatial distribution of ionomer potential in the CL at an operating voltage of 0.4 V is shown in Figure 14. The gradient in the potential is more for the ionomer loading of 20 wt %. As the loading is increased, the overpotential due to proton transport decreases and it has a positive impact on the cell performance. On the other hand, with the increase in the ionomer loading, the void volume reduces and hence the mass-transfer losses increase. The partial pressure distribution of oxygen at the voltage of 0.4 V is shown in Figure 15. 5008



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between the change in the activation losses and concentration losses as the platinum loading is changed. The study shows that an optimum loading of ionomer content can improve the cell performance, especially at higher current densities. This happens because of the interplay between the ohmic and concentration losses as the ionomer content is changed. The parametric studies show that a multivariable optimization study can significantly improve the cell performance in the entire polarization range.

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AUTHOR INFORMATION

Corresponding Author

*Tel.:+1-3042939335. Fax: +1-3042934139. E-mail: Debangsu. [email protected]. Notes

The authors declare no competing financial interest.

■

Figure 14. Spatial variation of ionomer potential inside the catalyst layer (0.4 V).

Figure 15. Spatial profile of oxygen partial pressure inside the catalyst layer (0.4 V).

Figures 14 and 15 show that there is a strong trade-off between the ohmic and concentration losses.

7. CONCLUSIONS A two-dimensional two-phase model of a cylindrical PEM fuel cell cathode is developed. The developed model is validated with the experimental data from an in-house, air-breathing cylindrical cell. The model is extended to a pressurized cell by considering the gas flow channel. As the operating pressure is increased from atmospheric pressure to 1.5 atm, the cell does not reach the limiting current density in the entire polarization range. This shows the performance improvements that can be achieved with a pressurized cell compared to the air-breathing cell. Effects of change in the operating conditions (temperature and flow rate) and catalyst layer design parameters (platinum and ionomer loadings) on the cell performance are studied. In these studies, each parameter is varied one at a time, keeping all other parameters constant. The temperature is found to have a modest effect on the performance in the temperature range studied. As the air flow rate is increased from a low flow rate, the cell performance improves significantly. Further increase in the flow rate has negligible effect. The study suggests that there is an optimum platinum loading because of an interplay 5009

NOMENCLATURE PEMFC = polymer electrolyte membrane fuel cell CFD = computational fluid dynamics GC = gas channel GDL = gas diffusion layer CL = catalyst layer mem = membrane MEA = membrane electrode assembly Ca = concentration of species a (mol m−3) Da eff = effective diffusivity of species a (m2 s−1) DOmem = diffusivity of oxygen in ionomer (m2 s−1) 2 Dwmem = diffusivity of liquid water in the membrane (m2 s−1) E = activation energy (J mol−1) f pt = weight fraction of platinum on carbon f ionomer = weight fraction of ionomer F = Faraday’s constant (C g−1 equiv−1) = Henry’s constant for air−ionomer interface (atm m3 HOmem 2 mol−1) HOw2 = Henry’s constant for air−water interface (atm m3 mol−1) icell = cell current density (A m−2 Pt−1) Iw = interfacial transfer of water between liquid and vapor (mol m−3 s−1) Ja,r = local flux due to diffusion of species a in r-direction (mol m−2 s−1) Ja,z = local flux due to diffusion of species a in z-direction (mol m−2 s−1) Kw = absolute permeability of liquid water inside porous region (m2) Krl = relative permeability mpt = platinum loading inside the catalyst layer (kg of Pt m−2 CL−1) Mw = molecular weight of water (g mol−1) n = number of electrons taking part in the oxygen reduction reaction Nw = flux of liquid water (mol m−2 s−1) Pc = capillary pressure (atm) R = universal gas constant (J mol−1 K−1) RAD1 = inner radius of the annulus (m) RAD2 = outer radius of the annulus (m) RO2 = rate of oxygen reduction reaction per unit volume of the catalyst layer (mol m−3 s−1) s = liquid water saturation tCL = thickness of the catalyst layer (m) Tcell = cell temperature (K) u = air velocity at the cathode inter (m s−1) dx.doi.org/10.1021/ie2028359 | Ind. Eng. Chem. Res. 2012, 51, 5003−5010


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(18) Bhattacharyya, D.; Rengaswamy, R.; Caine, F. Isothermal models for anode-supported tubular solid oxide fuel cells. Chem. Eng. Sci. 2007, 62, 4250−4267. (19) Bhattacharyya, D.; Rengaswamy, R. A review of solid oxide fuel cell (SOFC) dynamic models. Ind. Eng. Chem. Res. 2009, 48, 6068− 6086. (20) Bullecks, B.; Bhattacharyya, D.; Rengaswamy, R.; Campbell, G. Development of a tubular proton exchange membrane fuel cell Experimental and modeling studies. AIChE Annual Meeting, Philadelphia, PA, USA, 2008 (21) Bullecks, B.; Bhattacharyya, D.; Rengaswamy, R. Steady-state modeling and optimization of a tubular proton exchange membrane fuel cell. AIChE Annual meeting, Nashville, TN, USA, 2009 (22) Rao, R. M.; Rengaswamy, R. Optimization study of an agglomerate model for platinum reduction and performance in PEM fuel cell cathode. Chem. Eng. Res. Des. 2006, 84, 952−964. (23) Jain, P.; Biegler, L. T.; Jhon, M. S. Parametric study and estimation in CFD-based PEM fuel cell model. AIChE J. 2008, 54 (8), 2089−2100. (24) Jain, P.; Jhon, M. S.; Biegler, L. T. Optimization of polymer electrolyte fuel cell cathodes. Electrochem. Solid-State Lett. 2008, 11 (10), B193−B196. (25) Srinivasarao, M.; Bhattacharyya, D.; Rengaswamy, R.; Narasimhan, S. Multivariable optimization studies of cathode catalyst layer of a polymer electrolyte membrane fuel cell. Chem. Eng. Res. Des. 2011, 89, 10−22. (26) He, W.; Yi, J. S.; Nguyen, T. V. Two-phase flow model of the cathode of PEM fuel cells using interdigitated flow fields. AIChE J. 2000, 46 (10), 2053−2064. (27) Barbir, F. PEM Fuel Cells−Theory and Practice; Elsevier: Amsterdam, 2005.

Vcell = cell voltage (V) Greek Letters

δmem = thickness of the ionomer film covering the agglomerate (m) δw = thickness of the water layer on top of the agglomerate (m) θcontact = contact angle μw = viscosity of water (kg m−1 s−1) ρw = density of water (kg m3) σ = surface tension (N m−1) Subscript

a = index for the species O2, N2, and H2O (v)

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REFERENCES

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