Multivariable optimization studies of cathode catalyst layer of a polymer electrolyte membrane fuel cell

chemical engineering research and design 8 9 ( 2 0 1 1 ) 10–22

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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Multivariable optimization studies of cathode catalyst layer of a polymer electrolyte membrane fuel cell M. Srinivasarao a , D. Bhattacharyya b , R. Rengaswamy c,d,∗ , S. Narasimhan a a

Department of Chemical Engineering, IIT Madras, India Department of Chemical Engineering, West Virginia University, Morgantown, USA c Department of Chemical Engineering, Texas Tech University, Lubbock, USA d Department of Chemical & Bio-Molecular Engineering, Clarkson University, Potsdam, NY 13699, USA b

a b s t r a c t The amount of current generated in a polymer electrolyte membrane fuel cell (PEMFC) depends strongly on the local conditions in a cathode such as available oxygen, surface area available for the reactions, amount of ionomer, and amount of electro-catalyst. In the present work, design parameters of a cathode catalyst layer are optimized to achieve the maximum current density at a given operating voltage. The decision variables are chosen such that they can be realized experimentally. To understand the effect of the model fidelity on the decision variables, optimization is performed with a single phase model and a two-phase model with and without membrane. Other objective functions such as maximization of current generation per catalyst loading, minimization of catalyst layer cost per power and minimization of cell cost per power are also considered to study the effects of the objective functions on the decision variables. © 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: PEMFC; Two-phase modeling; Multivariable optimization; Platinum utilization

1.

Introduction

Extensive research and development efforts are being undertaken in recent years in the field of PEM fuel cell (PEMFC) systems to make them commercially viable. However, further cost reduction and performance improvement are required to make PEMFCs a commercial success. The major cost component of a PEMFC is the platinum catalyst. The platinum loading is typically about 0.05–0.2 mg cm−2 on the anode side and 0.2–0.6 mg cm−2 on the cathode side (Gasteiger et al., 2004, 2005). More platinum is used on the cathode side because of the sluggish oxygen reduction reaction. As many limiting processes take place in the reaction layer, it is important to model this layer. Most of the models developed earlier (Springer et al., 1991; Bernardi and Verbrugee, 1991; Bernardi and Verbrugge, 1992; Berning et al., 2002; Pasaogullari and Wang, 2004) have considered the catalyst layer (CL) as very thin and ignored its effect on the performance of the cell. Even though these layers are 5–20 ␮m thick, neglecting the effect of this finite thickness can lead to errors in predicting the performance of the fuel cell.

∗

The catalyst layer can be modeled using a macro-homogenous approach or agglomerate characterization (Rao et al., 2007). By using advanced microscopy instruments like scanning electron microscope (SEM) and transmission electron microscope (TEM), various researchers have studied the morphology of the catalyst layer. Middleman (2002) has reported that the catalyst layer consists of random distribution of particles and pores. The tiny particles form agglomerates with a thin film of ionomer. These agglomerates are either spherical or cylindrical in shape. Lin et al. (2004, 2006) have used cylindrical agglomerate characterization, whereas Sun et al. (2005) and Rao et al. (2007) have used spherical agglomerate characterization for modeling the catalyst layer. Another major cost component of a PEMFC is the polymer membrane. Nafion is the commonly used membrane in a PEMFC. These membranes are expensive and can give poor performance under low humidity and high temperature conditions (Viswanathan and Helen, 2007). The performance of the cell depends mainly on the operating conditions, design of the flow field, composition and

Corresponding author at: Department of Chemical Engineering, Texas Tech University, Lubbock, TX, USA. Tel.: +1 806 742 1765. E-mail address: [email protected] (R. Rengaswamy). Received 2 January 2010; Received in revised form 12 April 2010; Accepted 22 April 2010 0263-8762/$ – see front matter © 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2010.04.020

chemical engineering research and design 8 9 ( 2 0 1 1 ) 10–22

Nomenclature cf Cki Cmem W eff,k

Di

Dmem W E fionomer fPt F ia icell io ref

io

Iw Jik kc kv Kwo,k mPt n nd NW,k ragg R RO2 S tCL Tcell Vcell wc wionomer wPt

fixed charge site concentration concentration of species i in region k (mol m−3 ) concentration of liquid water in the membrane (mol m−3 ) effective diffusivity of the species i in region k (m2 s−1 ) diffusivity of liquid water in the membrane (m2 s−1 ) activation energy (J mol−1 ) weight fraction of ionomer in the catalyst layer weight fraction of platinum on carbon Faraday’s constant (C mol−1 ) local current density (A (m2 Pt)−1 ) cell current density (A (m2 Pt)−1 ) exchange current density for oxygen reduction on platinum (A m−2 ) reference exchange current density for oxygen reduction on platinum (A m−2 ) interfacial transfer of water between liquid and vapor (mol m−3 s−1 ) local flux due to diffusion of species i in region k (mol m−2 s−1 ) condensation constant (s−1 ) evaporation constant (atm−1 s−1 ) permeability of liquid water inside porous region k at 100% saturation (m2 ) platinum loading inside the catalyst layer (kg Pt (m2 CL)−1 ) number of electrons taking part in the oxygen reduction reaction net electro-osmotic drag coefficient flux of liquid water in region k (mol m−2 s−1 ) agglomerate radius (m) universal gas constant (J mol−1 K−1 ) rate of oxygen reduction reaction per unit volume of the catalyst layer (mol m−3 s−1 ) source term thickness of the catalyst layer (m) cell temperature (K) cell voltage (V) mass of carbon inside the agglomerate (kg) mass of ionomer inside the agglomerate (kg) mass of platinum inside the catalyst layer (kg)

Greek letters ˇ cathode transfer coefficient εk void fraction inside region k εionomer fraction of volume occupied by the ionomer inside the catalyst layer eff,c effective proton conductivity in the catalyst layer (mho m−1 ) eff,mem  effective proton conductivity in the membrane (mho m−1 ) k ele electric conductivity in region k (S m−1 ) eff,k

ele w

c ionomer

Pt w

(mol m−3 )

effective electric conductivity in region k (S m−1 ) water content in the membrane (mol H2 O (mol SO3 )−1 ) density of carbon (kg m−3 ) density of ionomer (kg m−3 )

11

density of platinum (kg m−3 ) density of water (kg m−3 )

design parameters of the membrane electrode assembly (MEA), and MEA preparation techniques. Optimization of cell performance will require a fundamental understanding of the various processes taking place inside the cell and their influence on the cell performance. However, it is difficult to study all these processes through experiments. Hence, a detailed mathematical model is useful in simulating the effect of various model parameters on the performance of the cell. For this purpose, a two-dimensional, two-phase model has been developed. The developed model is useful for studying the effects of various operating, design, and model parameters on the cell performance. In a typical fuel cell, the concentration of the reactants decreases from inlet of the gas flow channel to the outlet. Spatial variations exist within the catalyst layer as well. The amount of current generated in the catalyst layer depends on the local conditions such as available oxygen, surface area available for reaction, the amount of ionomer, and the amount of catalyst. These local conditions in turn depend on the volume fractions of voids, ionomer, and the solids. For a given thickness of the catalyst layer, the void fraction (εr ) in the catalyst layer depends on design parameters such as, catalyst loading (mPt ), weight fraction of Platinum on Carbon (fPt ), weight fraction of ionomer inside the catalyst layer (fionomer ) and density of ionomer (ionomer ). In the optimization studies presented in this work, all the above parameters are incorporated in the steady state model so that the interactions between them are captured in the predictions. Gasteiger et al. (2003) have experimentally shown the importance of catalyst layer design parameters in reducing the MEA losses. Catalyst layers are usually prepared using a mixture of carbon supported platinum particles, ionomer solution and solvent. After evaporation of the solvent, the final CL contains solids (carbon and platinum), ionomer and voids. Hence, the decision variables used in this optimization study are platinum loading, ionomer loading, weight fraction of platinum on carbon and CL thickness. Optimization is performed throughout the polarization range using a single phase and a two phase model to show the importance of liquid water modeling on the optimization results. Dependence of the optimization results on the initial guess is also studied. A number of different objective functions are considered. It is observed that the values of the decision variables strongly depend on the objective function considered and the constraints imposed.

2.

Background

A typical objective of a cell level optimization study is to maximize the current density at a given operating voltage. A number of publications can be found in this area. Secanell et al. (2007a,b) have optimized the PEMFC cathode by numerical optimization techniques. In their earlier work (Secanell et al., 2007a), the authors have optimized the composition of the catalyst layer and showed that current density can be increased by more than 50%. This increase is due to substantial reduction in the porosity and an increase in platinum loading and volume fraction of the ionomer. In their recent work, Secanell et al. (2007b) have performed multivariable optimization studies of the PEMFC cathode. The optimization variables are

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chemical engineering research and design 8 9 ( 2 0 1 1 ) 10–22

Fig. 1 – Schematic of PEMFC cathode.

platinum loading, volume fraction of ionomer in the agglomerate, fraction of platinum on carbon, and porosity of the Gas Diffusion Layer (GDL). The authors have suggested that the performance at medium and high current densities can be improved by increasing the ionomer content and reducing the carbon and platinum loading. However, both these works did not account for the effect of liquid water, which is of primary importance for an accurate representation of the cell performance. Lin et al. (2006) have optimized the channel width ratio (), porosity of GDL (εGDL ), and porosity of the catalyst layer (εCat ). They have showed that the optimum combination of these parameters is:  = 0.55, εGDL = 0.5913, and εCat = 0.307. Wang et al. (2004a) have optimized the cathode catalyst layer structure in PEM fuel cells. They have studied three different regions namely oxygen diffusion control, proton conductivity control, and an intermediate region. In these regions they have studied the effect of structure and material properties such as porosity, agglomerate size, catalyst layer thickness, and proton conductivity on the cell performance. Diwakar and Subramanian (2005) have shown the effect of electrolyte conductivity on the electrochemical behavior of porous electrodes by modifying the existing mathematical models. Rao and Rengaswamy (2006) performed two different studies namely minimization of platinum loading and maximization of current for a single agglomerate. Jain et al. (2008) reformulated the agglomerate model of a CL into a condensed

form for minimization of platinum loading at various values of current density. They also obtained optimum platinum distribution along the CL width by dividing the CL into various zones. However, this study neither considered the other layers of cathode in detail nor takes liquid water into account. Song et al. (2005) obtained optimal distributions of platinum and Nafion in the catalyst layer. They studied the distributions of each variable by keeping the other constant along with a study wherein both the variables are changed simultaneously. The authors concluded that optimal distribution of Nafion content is a linearly increasing function and the optimal distribution of platinum is a convex increasing function across the thickness of the catalyst layer. Song et al. (2004), in another work, optimized the current density at a given electrode potential. The decision variables are Nafion loading, platinum loading, catalyst layer thickness, and porosity. The authors have performed optimization studies considering single and two parameters. Results from single or two parameter optimization may be less accurate in comparison with multi-parameter optimization. Further, all these studies have assumed that water exists only in vapor form. In addition, all these studies excluded the membrane and the anode from the modeling domain. Due to fast reaction kinetics and comparatively small overpotential, anode may not have much influence on the cell performance. However, the effect of membrane (as water and proton transport takes place through

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chemical engineering research and design 8 9 ( 2 0 1 1 ) 10–22

the membrane) on the cell performance is significant. Furthermore, because of the typical operating condition in a PEMFC, water is expected to be in liquid form and hence neglecting the liquid water in the model can reduce the model accuracy significantly. Therefore, for the optimization study, a two phase cathode model including the membrane is developed. Optimization studies are performed throughout the polarization range for three cases: single phase modeling excluding the membrane, two-phase modeling excluding the membrane and, two-phase modeling including the membrane. The results are then analyzed. Further, various objective functions such as maximization of performance, minimization of CL cost per power and the cell cost per power are also considered in this paper.

Table 1 – Source terms. Component/variable

Layer

O2 O2 N2 H2 O (v) H2 O (l) H2 O (l) s s r r

S

GDL, MPL CL GDL, MPL, CL GDL, MPL, CL GDL, MPL CL GDL, MPL CL CL Membrane

0 RO2 0 −Iw Iw −2RO2 + Iw 0 −nFRO2 nFRO2 0

Table 2 – Operating conditions.

3.

Model development

The schematic of the PEMFC cathode studied here is shown in Fig. 1. The detailed model development is described in our earlier papers (Rao et al., 2007; Srinivasarao et al., 2010). The experimental design parameters are taken from Wang et al. (2004b). A thin micro-porous layer (MPL) is considered in between the GDL and CL. The CL is modeled using spherical agglomerate characterization with thin films of ionomer surrounding the agglomerate. It is assumed that the water produced at the reaction sites diffuse through the ionomer film and reaches the agglomerate surface. There it forms a thin layer before moving out (Lin et al., 2004; Rao et al., 2007). A three phase contact is required in the CL for the transportation of reactant gases, electron and protons. The following assumptions are also made in developing the models:

Fuel Oxidant Temperature Pressure Air flow rate Relative humidity

are given below: eff

− ∇ · (−Di,k ∇Ci,k ) + S = 0

Gaseous species

:

Liquid water

:

− ∇ · Nw,k + S = 0

Electrons

ele ∇ 2 s,k + S = 0

Protons 1. The cell is at steady state and in isothermal condition. 2. The contribution due to convection of species is negligible. 3. The GDL and the MPL are composed of carbon fibers and void space. 4. The CL consists of particles of platinum on carbon, ionomer, and void space. 5. Water generated due to the electrochemical reaction is in vapor form (single phase model) or in the liquid form (twophase model). 6. Water transport through the polymer membrane is only by electro-osmotic drag and back diffusion (in case of twophase model with membrane). 7. Gas diffusivity and electric conductivity are isotropic in all the layers. 8. Liquid water permeability is isotropic (for two-phase model).

Pure hydrogen Air (21% O2 , 79% N2 ) 353.15 K 2 atm. 25 × 10−6 m3 /s 100%

:

:

(1) (2)

eff,k

(3)

eff,l

ion ∇ 2 ϕr,l + S = 0

Water through membrane : ∇ ·

(4)

i

a

F

mem nd − Dmem w ∇Cw



=0

(5)

In the above equations, S represents the corresponding source terms. The source terms of various components in various layers are given in Table 1. The boundary conditions used to solve these equations are continuity of flux and concentration in the direction of diffusion. The equations with the specified boundary conditions are discretized and solved in MAPLE-MATLAB environment. The operating conditions, design parameters, and the model parameters are given in Tables 2–4 respectively.

Table 3 – Design parameters.

3.1.

Governing equations

The transport of gaseous species is completely governed by diffusion due to concentration gradients in all the porous layers. The liquid water transport is described by capillary pressure gradients. Electro-osmotic drag and back diffusion are considered for the transport of water in the polymer membrane. Bruggeman relations are used in the calculation of effective gas diffusivities, electron, and proton conductivities. The effects of liquid water are also taken in the calculation of effective gas diffusivities in the GDL, MPL and the CL. The governing equations for the two-phase model with membrane

Channel length Channel width Channel height No. of channels Land width GDL thickness GDL porosity MPL thickness MPL porosity CL thickness CL porosity Platinum loading Membrane thickness Active area of the cell

100 mm 1 mm 1 mm 7 1 mm 280 ␮m 0.7 40 ␮m 0.5 20 ␮m 0.112 0.4 mg cm−2 45 ␮m 14 cm2

Wang et al. (2004b) Wang et al. (2004b) Wang et al. (2004b) Wang et al. (2004b) Wang et al. (2004b) Wang et al. (2004b)

Wang et al. (2004b) Wang et al. (2004b) Wang et al. (2004b)

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chemical engineering research and design 8 9 ( 2 0 1 1 ) 10–22

Table 4 – Model parameters. Model constants F n R c Pt w

96485.3 4 8.314 1800 21450 977.3



Model parameters



1 Tcell

1 Tref



C mol−1

Rao et al. (2007)

J mol−1 K−1 kg m−3 kg m−3 kg m−3

Rao et al. (2007) Rao et al. (2007) Rao et al. (2007)

A m−2

io

iref exp − RE 0

E

If Vcell > 0.79, 76,500 Else, 27,700

J mol−1 J mol−1

Sun et al. (2005)

If Vcell > 0.79, 3.85 × 10−4 Else, 1.50 × 10−2 If Vcell > 0.79, 1 Else, 0.495 + 0.0023(Tcell − 300) 0.1 8.7 × 10−12 3 × 10−14 9 ×10−15 −28.42 −56.84 −113.68 100 100 1000 1000 1200 1200

A m−2 A m−2

Sun et al. (2005)

ref

io ␤

ragg Kwo,GDL Kwo,MPL Kwo,CL −(dPc /ds) in GDL −(dPc /ds) in MPL −(dPc /ds) in CL kc kv GDL ele MPL ele CL ele cf

4.

Results and discussion

4.1.

Maximization of current density

−

Sun et al. (2005)

Sun et al. (2005) ␮m m2 m2 m2 N m−2 N m−2 N m−2 s−1 atm−1 s−1 S m−1 S m−1 S m−1 mol m−3

Rao et al. (2007)

Rao et al. (2007) Rao et al. (2007) Rao et al. (2007) Rao et al. (2007) Rao et al. (2007)

Lin and Nguyen (2006a)

The objective here is to maximize the cell current density at a given operating cell voltage subject to some equality and/or inequality constraints. Typically there are two approaches to solve this optimization problem. The first approach is a simultaneous approach where the model equations – including the discretized species conservation equations and the boundary conditions – are the equality constraints and nonnegative void fraction in the reaction layer is the inequality constraint. In the other approach, known as sequential approach, there are no equality constraints. The inequality constraint is the same as in the previous case. For every combination of the decision variables (reaction layer design parameters), the steady state model is solved to calculate current density. In the sequential approach, the decision variables are catalyst layer design parameters alone. In the simultaneous approach, both design parameters and the species concentrations constitute decision variables. Since discretized equations are used to solve the model equations, the decision variables will be more in simultaneous approach. In this work, sequential optimization is considered due to ease of formulation/solution. The optimization problem is solved using constrained nonlinear optimization solver, fmincon, available in MATLAB’s optimization tool box. To verify the robustness of the solution, different sets of initial guesses are considered. The procedure for sequential optimization is shown in Fig. 2. In the present study, three different cases are considered. Case 1: Optimization of a PEMFC cathode without water and without membrane.

Fig. 2 – Schematic for sequential approach of optimization.

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Table 5 – The effects of design variables on various losses/parameters. Decision variable (When increased with the other three decision variables being constant)

Effective mass transfer diffusivity

Platinum loading Reaction layer thickness Weight fraction of ionomer

Decreases Increases Decreases

Electronic resistance

Ionic resistance

Decreases Increases No change

Decreases Increases Decreases

Effective area of the catalyst per unit volume of reaction layer

Increases Decreases No change

Table 6 – Range of design parameters for optimization. Decision variable

Base case value

mPt (mgPt cm−2 ) fPt fionomer tCL (␮m)

Lower bound

0.40 0.20 0.25 20

0.05 0.05 0.05 10

Upper bound 1.0 1.0 1.0 20

Table 7 – Initial guess values for the design parameters. Design parameter

Initial guess 1

mPt (mgPt cm−2 ) fPt fionomer tCL (␮m)

Initial guess 2

0.45 0.25 0.25 16

0.2 0.8 0.8 10

Case 2: Optimization of a PEMFC cathode with water and without membrane. Case 3: Optimization of a PEMFC cathode with water and with membrane. Optimization formulation Objective function: Maximization of iCell at a given VCell Decision variables : fPt , fionomer , mPt , tCL Subject to

0 < εr < 1 0 < εionomer < 1 0 < εsolid < 1

The volume fractions of voids, ionomer, and solids in the catalyst layer are a function of the optimization variables. εr = 1 −

mPt tCL fPt

εionomer =

1 − f

Pt

C 1

tCL ionomer

+

mPt fPt

εsolids = 1 − εr − εionomer

fPt fionomer + Pt [1 − fionomer ] ionomer

 f  ionomer 1 − fionomer

 (6)

Initial guess 3 0.6 0.5 0.4 10

4.1.1. Single phase model without consideration of the polymer membrane (Case 1) In this case, optimization is performed for a PEMFC cathode with a model that does not consider the polymer membrane and assumes existence of a single phase. The initial guess for this problem is shown in Table 7. The maximum cell current densities predicted are compared against those from the base case model. The comparison between the base case and the optimized case are shown in Fig. 3. The maximum cell current densities obtained for each of the voltages are plotted and compared with the base case polarization curve. The optimized cell current densities are shown as dashed lines to distinguish the fact that they are not to be treated as a characteristic current–voltage curve. This is due to the fact that each point results in a different set of catalyst layer parameters; whereas, for the base case polarization curve the design parameters are constant. From the results it is observed that

(7) (8)

Non-negativity constraints are imposed on these variables. The maximum value of platinum loading is set as 1 mg cm−2 . Even though this value is more than the current state of art, it is chosen to demonstrate the effect of platinum loading at low current densities. The catalyst layer thickness is considered to be between 10 and 20 ␮m. In open literature, this variable is usually taken as a constant. However, if this value is changed, the optimum values of the decision variables can change. The effects of the chosen design variables on several factors that dictate the final performance of a fuel cell are shown in Table 5. Optimization is performed for all the points on the base case polarization curve. The base case parameters and the upper and lower bounds on the decision variables are shown in Table 6.

Fig. 3 – Comparison of base case i-v curve and optimization points without water.

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Fig. 4 – Optimum platinum loadings for 15 and 20 ␮m thick catalyst layers without water. at each operating voltage there exists an optimum combination of the catalyst layer design parameters which achieves the best performance under the same operating conditions. At low current densities, the platinum loading hits the upper bound. This is because at low current densities, activation overpotential limits the cell performance. The electrolyte and void fractions decrease in comparison with base case. At high current densities, more oxygen is required and hence more void volume is optimal. From the results, it is observed that the fraction of platinum on carbon and the fraction of ionomer increase with current density, whereas the amount of platinum loading decreases. The fraction of platinum on carbon is about 0.47 at medium and high current densities. This result matches well with the reported optimum fractions of 0.4669 and 0.467 at operating voltages of 0.676 and 0.476 V respectively by Secanell et al. (2007b). The optimum fraction of ionomer increases from low to high current densities. The catalyst layer thickness is the same for all voltages and hits the upper bound of 20 ␮m. This is due to the increase of the effective reaction area with an increase in thickness. Due to a thicker catalyst layer, the optimized platinum loading is higher than the current state of the art. When the upper bound is reduced from 20 to 15 ␮m, optimum platinum loading is reduced at the expense of performance drop. The optimized platinum loadings and the current densities with 20 and 15 ␮m are shown in Figs. 4 and 5 respectively. From the plots it is observed that when the catalyst layer thickness is increased the cell performs better but at the expense of more platinum consumption. The cell performance is slightly better for 20 ␮m thick catalyst layer than 15 ␮m, but the required amount of platinum is relatively high. It is important to note that higher the thickness of the catalyst layer, higher the transport and ionic losses. Beyond a particular thickness, the cell performance is expected to decrease.

Fig. 5 – Optimum cell performance for 15 and 20 ␮m thick catalyst layers without water. formance compared to ionic and electronic resistance. Due to accumulation of liquid water, effective porosity and hence the diffusivity of gases decrease. To account for mass transfer losses due to liquid water, a saturation term is incorporated in the calculation of effective gas diffusivity. The effective gas diffusivity is calculated by Eq. (9): eff,k

Di

= (εk )

3/2

(1 − sk )

3/2

Dki

(9)

In Eq. (9), s represents the liquid saturation in the porous layer. The effective gas diffusivity for single phase can be calculated by eff,k,g

Di

= (εk )

3/2

Dki

The loss of gas diffusivity defined as

(10) eff,k eff,k,g −D ) i i eff,k,g D i

(D

× 100 due

to saturation is shown in Fig. 6. The loss is about 50% at a saturation of 0.4. Optimization is performed throughout the polarization range using the liquid water model under the same conditions. The comparison between the base case and the optimized current densities are shown in Fig. 7. When compared to the previous case, the base case and optimized current densities

4.1.2. Two phase model without the polymer membrane (Case 2) Modeling of liquid water is necessary due to typical operating temperature of a PEMFC. The generated water should be removed from the reaction sites to provide more active area for reaction and also to reduce mass transport losses. From experimental measurements, Suzuki et al. (2004) found that gas diffusion in the CL has more influence on the cell per-

Fig. 6 – Reduction in gas diffusivity (%) due to the liquid water.

chemical engineering research and design 8 9 ( 2 0 1 1 ) 10–22

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Fig. 7 – Comparison of base case i–v curve and optimization points with water and without membrane.

Fig. 8 – Comparison of base case i–v curve and optimization points with water.

are more in this case. This is due to consideration of another mechanism for removal of water from the cell as this model considers convective transport of liquid water by capillary gradient from the CL to the gas flow channel along with diffusive transport of water in the gas phase that is considered in the earlier model. It is observed that there is a reduction in platinum loading. This is due to the availability of more active area for reaction due to efficient removal of water from the reaction sites. Similarly, as more liquid saturation provides high ionic conductivity, the optimum membrane fractions are less than the previous case. In this study, the proton transport and water transport through the membrane are not considered. The ionic and mass transfer resistances depend strongly on the water content in the CL. Usually, there is net liquid water flow from the anode to the cathode due to dominant electroosmotic drag which results in more saturation in the CL. As a result of this, mass diffusivity decreases and ionic conductivity increases. At low current densities, these aspects may not play a major role. However, at medium and high current densities, their effect can be significant.

the mass transfer losses due to liquid water are significant in this scenario, high void volume is required in addition to high platinum amount. On the other hand, higher liquid saturation decreases ionic losses. From Fig. 10a, it is observed that optimum ionomer loadings and hence ionomer volumes (Fig. 11b) are significantly less than Cases 1 and 2. With decrease in weight fraction of platinum, the specific surface area available for the reaction increases (Rao and Rengaswamy, 2006). The usage of more platinum enhances the extent of reaction. Due to these simultaneous effects, more platinum loadings with less weight fraction of platinum on carbon are optimal design parameters at higher voltages. Similar results are observed from the plots in Figs. 10b and c. The optimized catalyst layer thickness is same at all operating voltages similar to the previous cases (Fig. 10d). The optimum volumes of voids, ionomer and solids are shown in Fig. 11. The major difference observed is in the optimum ionomer volume. A major reduction in the ionomer volume is observed from Fig. 11a. This volume is utilized by the voids (to reduce the transport loss) and solids (more platinum). The optimum void and solid fractions are shown in Figs. 11b and c respectively. Optimization is performed with three different initial guesses throughout the polarization range. The initial guesses are selected such that they cover the entire range of design parameters. The

4.1.3. Two phase model with the polymer membrane (Case 3) Water management is a very critical issue in PEMFC. Excessive water causes flooding that causes an increase in mass transfer resistance. Lack of adequate water results in drying that increases the ionic resistance. To minimize these losses, proper water content has to be maintained. The generated protons at the anode CL are transferred to cathode CL through polymer membrane along with water (electro-osmotic drag). Though there is back diffusion from cathode to anode due to the water formed at the cathode CL, the drag dominates. Hence, the liquid saturations in the CL are more than Case 2. As mentioned earlier, saturation favors proton transport, but diminishes gas transport. As a result, the cell performance in both the base and optimized cases are lower than the previous two cases. From the results, it is observed that there is a considerable change in the optimized design parameters compared to Cases 1 and 2 though the qualitative trend is similar. Current and power densities for both the base and optimized CL are shown in Figs. 8 and 9 respectively. The optimized design parameters of the CL for all the cases are shown in Figs. 10 and 11. At low current densities, high catalyst loadings are required as explained before due to activation losses. Since

Fig. 9 – Comparison of base case and optimized power densities with water.

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Fig. 10 – Optimum values of: (a) weight fraction of ionomer, (b) weight fraction of platinum on carbon, (c) platinum loading, and (d) catalyst layer thickness.

Fig. 11 – Optimum values of: (a) void fraction, (b) ionomer fraction, and (c) solid fractions.

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Table 8 – Optimization results for maximization of current density per mg of catalyst loading. Cell voltage (V) 1 0.9 0.8 0.7 0.6

icell /mPt (A/mg) 0.0041 0.0929 0.7577 1.8253 3.0258

icellopt /mptopt (A/mg) 0.0048 0.1008 0.7849 1.9555 4.1945

Fig. 12 – Sensitivity of optimized current density to initial guesses. optimized polarization curves for different initial guesses are shown in Fig. 12. At low current densities, activation losses are completely dominant compared to other losses and hence higher platinum loadings result in best performance irrespective of the model used for optimization. At medium and high current densities, as other losses become important, the model plays a key role in finding the optimized parameters. From Fig. 10c, it is observed that there is a significant difference in optimized platinum loadings at a voltage of 0.5 V. The optimization considering liquid water suggests more void fraction in the CL to minimize the mass transport losses at the expense of a reduced ionomer content. As a result, the optimized ionomer loadings are less than the single phase case (Fig. 10a). The results are demonstrated at each voltage to show that the optimum combination is different. However, in making the CL the optimum design parameters can be chosen from a typical operating voltage (0.5–0.65 V) or nearest commercially available parameters. The polarization curves for the base case design parameters and the optimum design parameters at 0.5 V are shown in Fig. 13. From Fig. 13 it is observed that the new combination of design parameters provides better performance throughout the polarization range.

Fig. 14 – Comparison of optimized and base case platinum loadings for maximization of icell /mPt .

4.2. Maximization of current density per mg of catalyst loading From the above studies it is observed that though there is a significant improvement in the performance this comes at the cost of higher platinum loadings than the current state of the art. The optimum platinum loadings obtained in the maximization of cell performance are 1 mg/cm2 and 0.6 mg/cm2 at low and high current densities respectively. However the commercial MEAs are made up of low platinum loadings such as 0.4–0.5 mg/cm2 . In order to reduce the catalyst loading, another study is performed with the two-phase model considering the membrane. The objective function for this optimization problem is maximization of cell performance per mg of catalyst loading. In addition to the inequality constraints of the above problem, an additional constraint is considered to ensure that the optimum current density is greater than that of the base case. This constraint is imposed only to make sure that there is no compromise in the cell performance. Optimization formulation Objective function: Maximization of icell /mpt at a given VCell Subject to 0 < εr < 1, 0 < eionomer < 1, 0 < εsolid < 1 and icellopti ≥ icellbase

Fig. 13 – Performance curves for the base case and for the optimal design parameters at 0.5 V.

Optimization is performed at various operating voltages and the currents generated per mg of platinum for both the base and optimized cases are shown in Table 8. The optimized platinum loadings are shown in Fig. 14. From the results, it can be observed that there is a significant increase in the current generation per mg of catalyst loading. It is also important to mention that there is a significant platinum reduction. There is about 25% reduction in the platinum loading for the same

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chemical engineering research and design 8 9 ( 2 0 1 1 ) 10–22

Table 9 – Optimization results for minimization of catalyst layer cost per unit power. Voltage (V)

Current density (mA/cm2 ) Base case

0.8 0.6 0.4

265.47 1081.7 1895.3

Optimized case 292.02 1189.9 2084.8

Cost function ($/W) Base case 0.61182 0.2002 0.17139

% Gain in performance

% Gain in cost

10 10 10

14.81 33.12 40.43

Optimized case 0.5212 0.1339 0.1021

base performance at 0.6 V. This is due to significant reduction of mass transfer resistance which provides more active surface for the reaction.

4.3.

Minimization of catalyst layer cost per unit power

In the conventional method of CL fabrication, catalyst ink can be prepared by synthesizing a uniform mixture of carbon supported particles of platinum with Nafion solution. The catalyst ink is coated on the GDL or membrane. In this study, the cost of carbon supported platinum (f (Pt/c)) and the ionomer solution are considered. The objective of the problem is minimization of the CL cost ($) per power (W) generated. There can be many solutions to the multi-objective problem of cost minimization depending on the cell performance constraints. In this study, a minimum of 10% increase in the cell performance compared to the base case is considered as a constraint. Since 10% increment could be achieved at all voltages, this particular constraint is considered. Optimization formulation Objective function: Minimization of (C1 (w(pt) + w(c)) + C2 wionomer )/Power $/W Subject to 0 < εr < 1, 0 < εionomer < 1, 0 < εsolid < 1 and icellopti ≥ 1.1(icellbase ) w(Pt) + w(c) =

(mPt × 10−3 )area (g) f (Pt/c)

C1 = cost of platinum and carbon = 251.7f(Pt/c) + 6.6092 $/g. The above expression is obtained from curve fitting of f (Pt/c) and its corresponding cost data (www. fuelcellstore.com). wionomer =

(mPt × 10−3 )area fionomer (g) 1 − fionomer f (Pt/c)

C2 = cost of ionomer = 24.0566 $/g (the cost is calculated from the cost of Nafion solution DE1021 10 wt.% 100 ml) (www. fuelcellstore.com). Optimization is performed at one voltage of each limiting zones and the results are shown in Table 9. The comparison between the base case and optimized platinum loadings are shown in Fig. 15. From the results it is observed that there is a drastic reduction in the cost of the CL with about 10% more performance than the base case. The gains in the costs are 14%, 33%, and 40% and reductions in the platinum amounts are 8.5%, 33.4% and 50.2% at 0.8, 0.6 and 0.4 V respectively. At higher voltages lot more platinum is needed if one were max-

Fig. 15 – Comparison of optimized and base case platinum loadings for minimization of catalyst layer cost per unit power. imizing performance because the potential driving force for the electrochemical reactions is less. Hence there is not much margin to decrease the cost at higher voltages. This is reflected in the optimization solution. The optimum platinum loading from this study is about 0.26642 mg cm−2 at 0.6 V whereas the optimum amount is 0.2895 mg cm−2 from the previous study.

4.4.

Minimization of the cell cost per unit power

In the previous study, the cost of CL alone is considered and the hardware cost of the cathode per power is kept constant irrespective of the cell performance and CL cost. In this section, to benchmark the cost reduction when factors outside of the MEA are considered, an approximate hardware cost (300$) of the cell is considered without any constraint on the cell performance. Optimization formulation Objective function: Minimization of (300 + C1 (w(Pt) + w(c)) + C2 wionomer )/Power $/W Subject to 0 < εr < 1, 0 <εionomer < 1, 0 < εsolid < 1 The optimization is performed at 0.8, 0.6 and 0.4 V and the results are given in Table 10. The comparison between the base case and the optimized case are shown in Fig. 16. In comparison to the previous results, the optimized platinum loading is significantly more than the base case. These loadings and

Table 10 – Optimization results for minimization of the cell cost per unit power. Voltage (V)

Current density (mA/cm2 ) Base case

0.8 0.6 0.4

303.46 1214.3 1920.3

Optimized case 364.7 1332.5 2147.8

Cost function ($/W) Base case 88.8436 29.6033 28.0794

% Gain in performance

% Gain in cost

Optimized case 74.4576 27.1724 25.1632

20.18 9.73 11.85

16.19 8.21 10.39

chemical engineering research and design 8 9 ( 2 0 1 1 ) 10–22

Fig. 16 – Comparison of optimized and base case platinum loadings for minimization of cell cost per unit power.

the cell performance of this study are similar to case 1 where the cell performance is maximized. These results show the influence of both the objective function and constraints on the optimized performance and the decision variables.

5.

Conclusions

To emphasize the importance of liquid water modeling and the polymer membrane on the optimized design variables, PEMFC cathode optimization is performed for three different cases: (i) two-dimensional one-phase model without considering the membrane, (ii) two-dimensional two-phase model without considering the membrane and, (iii) two-dimensional two-phase model considering the membrane. It is observed that there exists an optimum combination of parameters at each operating voltage which achieves the best performance under the same operating conditions. To find the optimum design parameters, an optimization formulation for maximization of cell performance is developed. The decision variables are platinum loading (mPt ), fraction of platinum on carbon (fPt ), fraction of ionomer (fionomer ), and the catalyst layer thickness (tCL ). To eliminate infeasible combinations, constraints are imposed on void, ionomer, and solid fractions of the catalyst layer. Optimization is performed in all the limiting resistance regions and the results are analyzed. From the results it is concluded that a two-phase model with membrane is essential for realistic results. At low current densities, optimized platinum loading is high mainly due to the limiting activation losses. As ohmic and mass transport losses become significant at medium and high current densities, optimum platinum loading is reduced and the ionomer and void fractions are increased. Similar results are also observed for other cases. Optimum platinum loadings are more in the case of two-phase model with membrane. The increase in performance compared to the base case is 10–98%. Since the chosen base case itself shows a high performance, the increase is about 10–25% at high current densities. However, at low current densities the rise in performance is about 65–98% due to more platinum loading. The predicted optimum platinum loading values are higher than the current state of the art due to more mass transfer resistance and thicker catalyst layer. When the thickness is decreased from 20–15 ␮m, optimum platinum utilization is reduced at the expense of a drop in

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the performance. In the second formulation, platinum loading is minimized. The results indicate that it is possible to get the same performance with a reduced platinum loading with optimum design parameters. There is about 38% more cell performance per mg of platinum at a voltage of 0.6 V. When the cost of the CL per unit power is minimized, the optimum platinum loadings are reduced even further. The cost of the CL per cell power is reduced by 40% along with 10% improvement in the performance. Finally, the approximate hardware cost is added with the CL cost, and the total cost is minimized per cell power. The results show improvement in the cell performance along with a significant reduction in the total cost. The optimum platinum loadings and cell performance are close to the results of case 1. From the above studies, it can be concluded that there is a strong tradeoff between the cell performance and the platinum loadings in some regions of the polarization curve. The optimized CL design parameters are strongly dependent on the objective function and also the constraints. With this type of detailed model it might be possible to perform multivariable multi-objective optimization. Future work will address these types of optimization problems.

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