CHAPTER 8 Modeling of PEM fuel cell stacks with hydraulic ... · PDF fileModeling of PEM fuel...

CHAPTER 8

Modeling of PEM fuel cell stacks withhydraulic network approach

J.J. Baschuk & X. LiDepartment of Mechanical Engineering, University ofWaterloo, Canada.

Abstract

Polymer electrolyte membrane (PEM) fuel cells convert the chemical energy ofhydrogen and oxygen directly into electrical energy. Waste heat and water are thereaction by-products, making PEM fuel cells a promising zero-emission powersource for transportation and stationary co-generation applications. In this study, amathematical model of a PEM fuel cell stack is formulated. The distributions of thepressure and mass flow rate for the fuel and oxidant streams in the stack are deter-mined with a hydraulic network analysis. Using these distributions as operatingconditions, the performance of each cell in the stack is determined with a mathe-matical, single cell model that has been developed previously. The stack model hasbeen applied to PEM fuel cell stacks with two common stack configurations: theU and Z stack design. The former is designed such that the reactant streams enterand exit the stack on the same end, while the latter has reactant streams enteringand exiting on opposite ends. The stack analyzed consists of 50 individual activecells with fully humidified H2 or reformate as fuel and humidified O2 or air as theoxidant. It is found that the average voltage of the cells in the stack is lower thanthe voltage of the cell operating individually, and this difference in the cell per-formance is significantly larger for reformate/air reactants when compared to theH2/O2 reactants. It is observed that the performance degradation for cells operatingwithin a stack results from the unequal distribution of reactant mass flow among thecells in the stack. It is shown that strategies for performance improvement rely onobtaining a uniform reactant distribution within the stack, and include increasingstack manifold size, decreasing the number of gas flow channels per bipolar plate,and judicially varying the resistance to mass flow in the gas flow channels fromcell to cell.

www.witpress.com, ISSN 1755-8336 (on-line) WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press

doi:10.2495/1-85312-840-6/08

284 Transport Phenomena in Fuel Cells

1 Introduction

Polymer electrolyte membrane (PEM) fuel cells convert the chemical energy ofhydrogen and oxygen directly and efficiently into electrical energy with by-productsof heat and liquid water. PEM fuel cells also have a high power density, quickstart-up and load following characteristics, making them attractive zero emissionpower sources [1]. Before PEM fuel cells can be successfully commercialized,the production cost must be reduced from the current estimate of approximately$200/kW to $30/kW [2]. Increasing the energy conversion efficiency and poweroutput of the PEM fuel cells could decrease the cost per kW, and thus severalempirical and mathematical modeling studies have been undertaken for the purposeof understanding and predicting PEM fuel cell performance.

In order to satisfy the power demand of most applications, several PEM fuel cellsmust be connected in series to form a PEM fuel cell stack. However, heat and watermanagement strategies, which are successful for single PEM fuel cells, are difficultto implement in a stack environment; the efficiency and power output of a PEMfuel cell operating within a stack are lower than the performance of a PEM fuelcell operating independently [3]. Thus, single cell PEM fuel cell models cannot bedirectly applied for PEM fuel cell stack optimization. This is because the operatingconditions for each cell in a stack are typically not the same as the conditions at thestack inlet, and are different among the cells themselves due to the non-uniformreactant flow distribution among the cells, influenced by the pressure loss associatedwith each flow passage.

Several modeling studies of PEM fuel cell stacks exist in the published litera-ture. Empirical models, originally developed for a single PEM fuel cell, have beenextended to model PEM fuel cell stacks. The empirical, single cell model of Kimet al. [4] was applied to a stack by Chu et al. [5]. The stack voltage was char-acterized as a function of current density using terms that represented activationoverpotential, ohmic overpotential, and mass transport limitations. The general-ized steady state electro-chemical model (GSSEM) [6] has been applied to bothsingle PEM fuel cells and stacks. Stack voltage has a functional dependence onthe partial pressure of the reactants, current density and temperature through termsaccounting for activation, concentration and ohmic overpotential, CO poisoning,and performance degradation due to aging [7]. Mathematical PEM fuel cell modelshave also been extended to simulate stack performance. The single cell model ofNguyen and White [8] was isothermal, two-dimensional, steady state, and incorpo-rated mass transport in the electrode backing, the electro-chemical reaction of thecathode, and proton migration in the polymer electrolyte. By modeling the reactantflow in the gas flow channels and stack manifold as a pipe network, Thirumulai andWhite [9] extended the single cell model to simulate stack performance.

Due to the exothermic nature of the electro-chemical reactions occurring withina PEM fuel cell, thermal management within a stack is a significant considera-tion for stack design. Maggio et al. [10] investigated the temperature and currentdensity distribution in a PEM fuel cell stack using a three-dimensional model.The temperature distribution in the cooling plate, cooling water and membrane


Modeling of PEM fuel cell stacks with hydraulic network approach 285

electrode assembly was found through application of conservation of energy, whilethe electro-chemical performance of the PEM fuel cell was determined with theempirical relationship of Patel et al. [11]. The model of Maggio et al. [10] wasdeveloped for a stack operating in steady state, but the model of Lee and Lalk [12]allowed for a non-steady state simulation. As with the model of Maggio et al. [10],the model of Lee and Lalk [12] determined the temperature distribution within thestack using conservation of energy and the voltage of each cell in the stack wasfound with the empirical model of Kim et al. [4].

When used as a power source for stationary or transportation applications, a PEMfuel cell stack requires auxiliary equipment for providing fuel, oxidant and heatremoval; the stack operates as a component in a larger energy conversion system.Barbir et al. [13] developed a model of a PEM fuel cell stack system consisting ofa stack, air compressor subsystem for the stack oxidant supply, gasoline reformersubsystem for the stack fuel supply, and cooling subsystem. The electro-chemicalperformance of the stack was modeled using an empirical, linear current-voltagerelationship and the system water balance and efficiency was investigated at variousoperating pressures and temperatures. A PEM fuel cell stack system consisting ofa stack, air compression subsystem, compressed hydrogen supply subsystem, andcooling subsystem was modeled by Cownden et al. [14]. The stack voltage andpower output were determined with the GSSEM of [6] and the efficiencies of boththe stack and system were examined.

This study formulates a PEM fuel cell stack model. The reactant distributionwithin the stack is modeled by treating the stack manifold and gas flow channelsas a pipe network, and the voltages of the cells in the stack are determined withthe single cell, steady state, isothermal model developed previously by the presentauthors [15]. U and Z configuration stacks operating with humidified hydrogen orreformate as the fuel and humidified oxygen or air as the oxidant are simulated,strategies for reducing the unequal distribution of reactants within the stack areexamined, and methods for the improvement of stack performance are describedbased on the results of the present study.

2 Model formulation

In general, a PEM fuel cell stack consists of several PEM fuel cells connected inseries, as illustrated in Fig. 1. The cathode side of the PEM fuel cell is exposed tothe oxidant, while fuel is introduced to the anode side of the cell. Each cell in thestack consists of several components. The bipolar plates conduct electrons and havegrooves, referred to as gas flow channels, that supply fuel or oxidant to the PEM fuelcell. The anode and cathode electrode backings also conduct electrons and allowreactants to access the catalyst layer. In the catalyst layer, electro-chemical reactionsconvert the chemical energy of the fuel and oxidant to electrical energy. Reductionof oxygen occurs in the cathode catalyst layer and the oxidation of hydrogen occursin the anode catalyst layer. The polymer electrolyte membrane conducts the protonproduced by hydrogen oxidation to the cathode for participation in the reductionof oxygen.



Figure 1: Schematic of a polymer electrolyte membrane fuel cell stack.

The fuel and oxidant for each PEM fuel cell are supplied by the stack mani-fold, with the anode manifold supplying fuel and the cathode manifold supplyingoxidant. The major component of the fuel for a PEM fuel cell is hydrogen, withcarbon dioxide and carbon monoxide being present if reformate fuel is utilized. Thepresence of carbon monoxide severely degrades the performance of a PEM fuelcell through the mechanism of CO poisoning [16]. Mitigation of CO poisoning ispossible with oxygen or air bleeding, whereby 1 to 4% oxygen is added to the fuel;thus oxygen and nitrogen can also be present in the fuel stream. The oxidant used ina PEM fuel cell is oxygen, with nitrogen being present if air is used as the oxygensupply. The gas flow channels remove the water produced by the electro-chemicalreactions within the MEA and supply the humidity required to avoid polymer elec-trolyte membrane dehydration; thus liquid and vapor phase water are present inboth the oxidant and fuel streams. In addition to fuel and oxidant, water is circu-lated through cooling plates in order to remove the heat produced by the PEM fuelcells and maintain a constant stack temperature.

The stack performance, often measured in terms of the stack voltage, can bedetermined by:

Estack =Ncell∑

1

Ecell −Ncell∑

1

ηcp, (1)

where Ncell is the total number of fuel cells in the stack, Ecell is the voltage of eachcell (from bipolar plate to bipolar plate), and ηcp is the ohmic loss due to a coolingplate. In this study, the voltage of each cell is found with the single cell model of



Baschuck and Li [15]. The single cell model is one-dimensional and assumes thatthe cell is isothermal and operating in steady-state with fully humidified reactants.Cell voltage is calculated with:

Ecell = Erev − ηa − |ηc| − 2ηbp − 2ηe − ηm, (2)

where Erev is the reversible cell voltage, ηa and ηc are the overpotentials attributedto the anode and cathode catalyst layers, respectively. The voltage losses caused bythe bipolar plate, electrode backing and polymer electrolyte membrane are denotedby ηbp, ηe, and ηm, respectively.

The reversible cell voltage is the cell potential obtained at thermodynamic equi-librium. It is a function of temperature and reactant concentration through a modifiedversion of the Nernst equation. The cell voltage is reduced from the reversible cellvoltage by the overpotentials associated with the various components of the PEMfuel cell. The voltage losses attributed to the bipolar plate and electrode backingare the result of electron migration; the overpotential is calculated by consideringthe electrode backing and bipolar plate as electrical resistances. Proton migrationis responsible for the voltage loss in the polymer electrolyte membrane and thusthe voltage loss is determined by the Nernst-Planck equation. The conductivity ofthe polymer electrolyte is a function of hydration, but the single cell model [15]assumes that the reactants and the polymer electrolyte are fully humidified; thus theconductivity is constant. Therefore, the polymer electrolyte membrane overpoten-tial is a function of the membrane properties, such as conductivity and thickness,and current density.

For a PEM fuel cell operating with CO-free fuel, the cathode catalyst layer over-potential is the major voltage loss. The anode and cathode catalyst layer overpoten-tials are found by considering species conservation, proton and electron migrationwithin the catalyst layers. Proton and electron migration within the catalyst layersare related to the protonic and ionic current through Ohm’s law.

Species conservation requires modeling of reaction kinetics and mass transport.Oxygen reduction is modeled with the Butler-Volmer equation in the cathode cata-lyst layer, while in the anode catalyst layer the adsorption and desorption of H2, COand O2, the electro-oxidation of the adsorbed hydrogen and carbon monoxide, andthe heterogeneous oxidation of H2 and CO by O2 are included in the reactionkinetics. The reaction rates in the catalyst layers are functions of overpotential andreactant concentrations; the functional dependency on concentration necessitatesconsideration of mass transport. The concentrations within the catalyst layers areinfluenced by resistance to mass transport from the gas flow channels to the electrodebacking, within the electrode backing, and within the catalyst layer. The mass trans-fer from the gas flow channels to the gas flow channel/electrode backing interfaceis calculated using a logarithmic mean concentration relationship. Mass transportwithin the electrode backing and catalyst layers is assumed to be through diffusiononly and the diffusion coefficients are formulated such that a variable amount of liq-uid water can exist within the pore space of the electrode backing and catalyst layers;thus the PEM fuel cell can be simulated with a variable degree of water flooding.



Through consideration of species conservation, proton and electron migration,ordinary differential equations are generated which can be solved for the catalystlayer overpotential. Therefore, the catalyst layer overpotential depends on reactantconcentration, current density, and the properties of the gas flow channels, electrodebacking and catalyst layer, such as porosity, thickness, and conductivity.

Determination of the reversible cell potential and overpotentials requires severalinput parameters, which can be classified as operating or design parameters. Designparameters depend on the manufacture of the PEM fuel cell and include properties,such as conductivity and porosity, and geometric dimensions. Design parameterscan be further classified according to the components of a PEM fuel cell; thusthere are bipolar plate, electrode backing, catalyst layer, and polymer electrolytemembrane design parameters. The operating parameters include current density,temperature, pressure, reactant composition and stoichiometry.

The PEM fuel cells in a stack will have the same design parameters. Due to theseries connection, the current density in each cell will be equal. As well, the circu-lation of cooling water allows each cell to have the same temperature. However,the pressure, reactant composition and stoichiometry can vary from cell to cell ifthe mass flow rate and pressure distributions within the stack are unequal.

Therefore, the stack model presented here consists of two parts: the single cellmodel and the stack flow model. The single cell model, as described above, deter-mines the voltage of each cell in the stack based on the cell inlet pressure, tem-perature, stoichiometry, and reactant composition in the gas flow channels, as wellas the current density and design parameters. In order to find the cell inlet pres-sure, temperature, stoichiometry and reactant composition, the mass flow rate andpressure distributions among each cell within the stack must be determined; thisconstitutes the stack flow model. The voltage loss attributed to the cooling plate ineqn (1) is determined by assuming that the cooling plate has the same overpotentialas the bipolar plate in the single cell model. The single cell model is described indetail elsewhere [15] and only the stack flow model will be presented here.

2.1 Stack flow model

The mass flow rate and pressure distributions within the stack are coupled; thusthey must be solved simultaneously. The fuel or oxidant flow within the stack ismodeled as a pipe network, and is illustrated in Fig. 2. Two stack configurationsare considered in this study: the U and Z configurations. Other stack configurationscan be treated similarly. The U configuration is illustrated in Fig. 2(a) and is char-acterized by the stack inlet and outlet being on the same end of the stack. The Zconfiguration, which is shown in Fig. 2(b), has the flow inlet and outlet on oppositeends of the stack. For both the U and Z configuration, the section of stack manifoldthat supplies the reactants to the gas flow channels of the PEM fuel cells is referredto as the top section, while the gas flow channels exit into the bottom section ofstack manifold.

If two PEM fuel cells and the connecting manifold sections are considered, asillustrated in Fig. 2, then the pressure losses in the gas flow channels and manifold



Figure 2: Schematic of (a) U and (b) Z stack manifold configurations.

sections are related:

�Picell

(mi

cell,in

) − �Pitop

(mi

top

) − �Pi+1cell

(mi+1

cell,in

) + α�Pibot

(mi

bot

) = 0,

(3)

where α equals −1 for the U configuration, 1 for the Z configuration, and i = 1 atthe stack inlet. Each pressure loss is a function of the mass flow rate in the manifoldsection or gas flow channel; thus the pressure and mass flow rate distributions mustbe solved simultaneously. The mass flow rate in the top sections of the manifoldcan be related to the mass flow rate entering the gas flow channels of each PEMfuel cell:

mitop = mstack

in −i∑

j=1

m jcell,in, (4)



where mstackin is the mass flow rate at the inlet of the stack. The mass flow rate in the

bottom sections of the stack manifold depends on the manifold configuration:

mibot =

i∑j=1

m jcell,out Z configuration

Ncell∑j=i+1

m jcell,out U configuration,

(5)

where micell,out is the mass flow rate exiting the gas flow channels. Because of the

reactions occurring within the anode and cathode catalyst layers, the mass flow rateat the inlet of the gas flow channels is not equal to the value at the outlet. The inletand outlet mass flow rates are related:

micell,out = mi

cell,in − �mr , (6)

where �mr is the mass consumed in the catalyst layers. Hence, the mass generatedwill be written with a minus sign. The anode and cathode values of �mr differ andare functions of current density.

Equation (3), in conjunction with eqns (4) and (5), can be applied to generateNcell −1 equations with Ncell unknown mi

cell,in values. One final equation is required

to solve for the unknown micell,in’s, and this equation is the mass conservation for

the stack as a whole:

mstackin =

Ncell∑i=1

micell,in. (7)

With the addition of eqn (7), determination of the mass flow rate and pressuredistributions within the stack is possible. However, the relationship between thepressure loss and mass flow rate must be determined first, as well as the value of�mr for the anode and cathode gas flow channels. These are presented in the nextsections.

2.2 Manifold pressure loss

The pressure loss in either the top or bottom section of stack manifold is foundwith:

�P = �Pm + �Pf , (8)

where �Pm is the pressure loss due to the change in momentum of the fluid and�Pf is the pressure loss due to the wall friction. The losses due to the branchingof the flow, or minor losses, are not included in this formulation because appropri-ate coefficients are not available for the flow conditions encountered. However, aseparate experimental investigation is under way to develop empirical correlationfor the minor loss coefficient associated with the branching/confluence flow, and



therefore, minor loss will be integrated in future studies. The pressure loss due toa change in momentum is given by:

�Pm = m

Am(Vout − Vin), (9)

where m is the mass flow rate in the manifold section, Am is the manifold cross-sectional area and V is the velocity. The velocity within the manifold section canbe found with:

V = m

ρAm, (10)

where ρ is the density of the fluid. The fluid in both the manifold and gas flowchannels consists of a mixture of a multi-component gaseous phase and liquidwater. The density of this mixture depends on pressure, temperature and the massfraction of the gaseous phase; Appendix A describes the calculation of density.

The pressure loss due to friction in the manifold sections is given by:

�Pf = 2Cf Lsρave(Vave)2

d stackh

, (11)

where Cf is the friction coefficient, Ls is the length of the manifold section betweenthe gas flow channels of the PEM fuel cells, and d stack

h is the hydraulic diameterof the stack manifold. The subscript “ave” refers to the arithmetic average of theinlet and outlet values; because density is a function of pressure and mass fractionof the gas phase, the inlet and outlet values will differ. The distance Ls depends onthe thickness of the PEM fuel cell and is equal to the thickness of two bipolar plates,one MEA and one cooling plate. In this study, the cooling plate is assumed to havethe same dimensions as the bipolar plate. The friction coefficient is a function ofthe Reynolds number [17]:

Cf ={

16(Redh )−1 Redh ≤ 2000

0.079(Redh )−1/4 Redh ≥ 4000,(12)

where Redh is the Reynolds number of the flow based on the hydraulic diameter:

Redh = ρaveVaved stackh

µave. (13)

The viscosity of the fluid is denoted by µ and is calculated with the equationsdescribed in Appendix A. As in eqn (11), the subscript “ave” denotes the arithmeticaverage of the inlet and outlet values. For Reynolds numbers between 2000 and4000, a linear relationship is used for the friction coefficient:

Cf = Cf |Redh= 2000 +

(Redh − 2000

2000

)(Cf |Redh

= 4000 − Cf |Redh= 2000

), (14)

where Cf |Redh= 2000 and Cf |Redh

= 4000 are the friction coefficient values at Reynoldsnumbers of 2000 and 4000, respectively.



2.3 Cell pressure loss

The pressure loss calculation for the PEM fuel cell gas flow channels is similar tothe manifold sections, with the total pressure loss being calculated with eqn (8).However, due to the change in mass flow rate between the inlet and outlet of the gasflow channels, the formulation of �Pm and �Pf differ from the manifold sectionformulation. The pressure loss due to momentum change is:

�Pm = 1

Afc(moutVout − minVin) , (15)

where Afc is the cross-sectional area of a gas flow channel. The bipolar plate, withthe manifold and gas flow channels, is illustrated in Fig. 3. Two flow channelconfigurations are shown: serpentine and parallel. Both configurations allow forseveral gas flow channels to exist on a single bipolar plate. In this study, the gasflow channels on a bipolar plate are assumed to have the same resistance to massflow; hence all of the gas flow channels on a single bipolar plate have the samemass flow rate. Therefore, the pressure losses of all the gas flow channels on thebipolar plate are found by considering only the flow in one gas flow channel. Themass entering and exiting one gas flow channel can be found with:

min = mcell,in

nc, (16)

mout = mcell,out

nc, (17)

where nc is the number of gas flow channels per bipolar plate.

Figure 3: Illustration of the bipolar plate with serpentine and parallel gas flowchannel configurations.



The pressure loss due to friction is determined with:

�Pf = 2Cf Lfcρave (Vave)2

d fch

, (18)

where Lfc is the length of a gas flow channel and d fch is the gas flow channel

hydraulic diameter. The determination of the friction coefficient, average densityand average velocity in eqn (18) is the same as for the stack manifold sections.This implies that the effect on the friction coefficient of wall suction due to thereactants flowing into the catalyst layers for electrochemical reactions and the wallblowing due to the reaction products coming out of the cathode catalyst layer are notaccounted for. The wall suction/blowing might in reality have significant impact onthe friction coefficient, their impact on the transport of momentum, heat and massis being investigated numerically and will be incorporated later once the relevantinformation is available.

Further, in the above approach for the pressure loss calculation associated withreactant stream in the flow channels built on bipolar plates, the minor loss associatedwith the bend or turn of the flow direction, as mandated by the serpentine flowchannel design, has not been included due to the lack of relevant information. Anexperimental study is currently under way to measure the associated pressure loss inthe serpentine flow channels, and correlations for the minor loss coefficient will bedeveloped and included to improve the model formulated here. However, it mightbe pointed out that the minor loss for the branching/confluence flow associated withthe flow in the manifolds and in the serpentine flow channels will affect the totalpressure loss within the stack, but is relatively small in its impact on the reactantmass distribution among the cells in the stack. Therefore, their effect on the finalresults shown in this study is small as well, and can be neglected.

2.4 Mass consumed in the catalyst layers

The amount of mass consumed in the anode and cathode catalyst layers differ, butin general can be written as:

�mr =∑

i=species

�mir , (19)

where the summation is over all of the species present in the catalyst layers. For theanode catalyst layer, the species present are H2, CO, CO2, O2, N2 and H2O. Theamount of H2, CO and O2 consumed in the anode catalyst layer can be calculatedusing Faraday’s law:

mH2r

M H2

+ �mCOr

M CO− 2

�mO2r

M O2

= IδAcell

2F, (20)

where Iδ is the current density, Acell is the active area of a PEM fuel cell, F is theFaraday constant, and M is the molecular weight. The concentration of CO is



typically at the ppm level, while the concentration of O2 is 1 to 4 percent; as aresult the amount of O2 and CO removed from an anode gas flow channel will bemuch less than the amount of H2 removed. Thus, in order to simplify the solutionprocedure, the values of �mCO

r and �mO2r are neglected. Since the amount of CO

consumed is negligible, the mass flow rate of CO2 will not change between theinlet and outlet of a gas flow channel. Nitrogen, if present, does not react in theanode catalyst layer. The water produced in the cathode catalyst layer is removedby the cathode gas stream in practical PEM fuel cell stack operation, and thus thisstudy assumes that no water is added or removed from the anode gas flow channels.Therefore, the only species that is consumed in the anode catalyst layer is H2, andthe mass consumed for the anode is:

�mr = �mH2r . (21)

In the cathode catalyst layer, the species present are O2, N2, and H2O. Nitrogendoes not react in the cathode catalyst layer and the amount of oxygen consumedcan be calculated with Faraday’s law:

�mO2r = IδAcell

4FM O2 . (22)

All of the water produced by the PEM fuel cell is assumed to enter the cathode gasflow channels; thus the amount of water consumed (actually generated, hence thenegative sign in the equation below) in the cathode catalyst layer becomes:

�mH2Or = − IδAcell

2FM H2O. (23)

The negative sign denotes that the water produced by the cathode catalyst layer isadded, not removed from, the cathode gas flow channels. Therefore, the amount ofmass consumed by the cathode catalyst layer becomes:

�mr = �mO2r + �mH2O

r . (24)

2.5 Boundary conditions

The boundary conditions for the above model formulation are specified at the stackinlet, that include the temperature, pressure, reactant composition. The reactantflow rate is determined based on the stack current density specified and the stoi-chiometry of the reactant desired. Then the stack performance and the reactant atthe stack outlet are determined by the model presented earlier. The following sec-tion on numerical procedure provides the details on how the boundary conditionsare implemented for the model formulated.

3 Numerical procedure

As with the single cell model, the input parameters for the stack flow model areclassified as operating and design parameters. The design parameters are the stack



manifold dimensions and stack configuration. Operating parameters include thestack current density, temperature, and the pressure, stoichiometry and reactantcomposition at the stack inlet. The stack stoichiometry is defined as:

S stacka = 2FN stack

H2

NcellIδAcell, (25)

S stackc = 4FN stack

H2

NcellIδAcell, (26)

where the inlet molar flow rates of hydrogen and oxygen to the anode and cathodesides of the stack, respectively, are denoted by N stack

H2and N stack

O2. Using stoichiome-

try and current density, the inlet molar flow rates of hydrogen in the anode manifoldand oxygen in the cathode manifold can be calculated. The mass flow rate at thestack inlet can be determined with the molar flow rates and reactant composition.

The relationship between pressure and mass flow rate in the manifold sectionsand gas flow channels is non-linear; therefore the mass flow rate and pressure dis-tributions must be calculated using an iterative procedure. The numerical solutionbegins with assumed values of mi

cell,in and two levels of iteration are then required.The outer level of iteration solves for the pressure and mass flow rate distributionswithin the stack. For a given mass flow rate, the procedure for determining thepressure loss in the gas flow channels or stack manifold is iterative; thus an innerlevel of iteration is required to find the values of �Pi

cell, �Pitop and �Pi

bot for theestimate of mass flow rate.

3.1 Outer iteration

Using estimated values of micell,in, values for mi

top and mibot can be calculated using

eqns (4) and (5). Applying these values of mass flow rate to eqn (3) results in apressure residual:

rip = �Pi

cell

(mi

cell,in

) − �Pitop

(mi

top

) − �Pi+1cell

(mi+1

cell,in

) + α�Pibot

(mi

bot

). (27)

The residual can be set to zero by changing the assumed mass flow rates:

0 = �Picell

(mi

cell,in + �micell,in

) − �Pitop

(mi

top + �mitop

)− �Pi+1

cell

(mi+1

cell,in + �mi+1cell,in

) + α�Pibot

(mi

bot + �mibot

). (28)

Combining eqns (27), (28) and using a Taylor series expansion yields a relationshipbetween the �mi

cell,in, �mitop and �mi

bot:

−rip = d�Pi

cell

dmicell,in

�micell,in − d�Pi

top

dmitop

�mitop

− d�Pi+1cell

dmi+1cell,in

�mi+1cell,in + α

d�Pibot

dmibot

�mibot, (29)



where the derivatives are calculated numerically [18]. From eqns (4) and (5), �mitop

and �mibot can be rewritten in terms of �mi

cell,in:

�mitop = −

i∑j=1

�micell,in, (30)

�mibot =

i∑j=1

�m jcell,in Z configuration

Ncell∑j=i+1

�m jcell,in U configuration.

(31)

Thus, eqn (29) can be used to make Ncell − 1 equations for the Ncell values of�mi

cell,in; it becomes for the Z configuration stack:

−rip = d�Pi

cell

dmicell,in

�micell,in + d�Pi

top

dmitop

i∑j=1

�m jcell,in

− d�Pi+1cell

dmi+1cell,in

�mi+1cell,in + d�Pi

bot

dmibot

i∑j=1

�m jcell,in. (32)

and for the U configuration stack:

−rip = d�Pi

cell

dmicell,in

�micell,in + d�Pi

top

dmitop

i∑j=1

�m jcell,in

− d�Pi+1cell

dmi+1cell,in

�mi+1cell,in − d�Pi

bot

dmibot

i∑j=1

�m jcell,in. (33)

The final equation needed to solve for the �micell,in comes from the overall mass

conservation eqn (7):Ncell∑i=1

�micell,in = 0. (34)

Equations (34) and either (32) or (33) can be solved for the corrections to theassumed mass flow rate distribution with a linear equation solver, such as LUdecomposition [18]. The mass flow rate distribution estimation is then updated:

micell,in|new = mi

cell,in|old + �micell,in. (35)

Using the updated values of micell,in, new values of �mi

cell,in are found until con-vergence, which is achieved when:

Ncell∑i=1

∣∣∣rip

∣∣∣ ≤ 1 × 10−3 Pa. (36)



3.2 Inner iteration

The pressure loss in the gas flow channels and stack manifold are functions of theinlet and outlet mass flow rate, density, and viscosity. Density and viscosity arefunctions of the mass fraction of the gas phase (χ), the species mole fractionsin the gas phase (xi), and the pressure. However, the outlet pressure is not knownuntil the pressure loss is calculated; hence an iterative procedure is required todetermine the pressure loss in the gas flow channels or stack manifold.

From the estimated mass flow rates in the outer iteration, the pressure, massfraction of gas phase and species mole fractions in the gas phase are known at theinlet of the gas flow channel or stack manifold. In order to obtain values of χ andxi at the outlet, the mass flow rate of each species at the outlet must be known. Forthe species other than water, calculation of the mass flow rate is straightforward.The mass flow rates of the species other than water at the inlet and outlet of thestack manifold sections are equal; the mass flow rates of H2 in the anode and O2 inthe cathode gas flow channels are reduced by �mH2

r and �mO2r , respectively, and

all other non-water mass flow rates are the same at the inlet and outlet of the gasflow channels.

The amount of water exiting in liquid or vapor phase depends on the outletpressure of the gas flow channels or stack manifold section. The maximum molefraction of water in the gaseous phase at the outlet is:

xmaxH2O,out = PH2O

sat

Pkout

, (37)

where PH2Osat is the saturation pressure of water and Pk

out is the estimated value ofoutlet pressure. With this mole fraction, the maximum mass flow rate of water vaporcan be found:

mmaxH2O(g),out =

xmaxH2O(g),outM H2O(g)

1 − xmaxH2O(g),out

∑i �=H2O(g);H2O(�)

mi,out

M i. (38)

The total mass flow rate of water (liquid and vapor) is:

mtotalH2O,out = mH2O(g),in + mH2O(�),in − �mH2O

r , (39)

where �mH2Or is only non-zero for the cathode gas flow channels. Using eqns (38)

and (39), the mass flow rates of the liquid and vapor water can be found:

mH2O(g),out =

mtotalH2O,out mmax

H2O(g),out ≥ mtotalH2O,out

mmaxH2O(g),out mmax

H2O(g),out < mtotalH2O,out,

(40)

mH2O(�),out =

0 mmaxH2O(g),out ≥ mtotal

H2O,out

mtotalH2O,out − mmax

H2O(g),out mmaxH2O(g),out < mtotal

H2O,out.(41)



Knowledge of the outlet mass flow rates allows for the calculation of the outletmole fractions in the gas phase and the mass fraction of the gas phase:

xi,out = mi,out/M i∑j �=H2O(�)

mj,out/M j, (42)

χout = 1 − mH2O(�),out∑mi,out

. (43)

With the values of pressure, mole fraction, mass fraction and mass flow rate atthe inlet and outlet, the pressure loss in either the gas flow channel or stack manifoldcan be determined with eqn (8). This value of �P can then be used to generate anew estimate for the outlet pressure:

Pk+1out = Pin − �P. (44)

Iteration continues until: ∣∣∣∣∣Pk+1out − Pk

out

Pkout

∣∣∣∣∣ ≤ 1 × 10−6. (45)

3.3 Numerical procedure summary

Using the inner and outer iterations, the mass flow rate and pressure distributionsin the stack are solved with the following procedure, for a given current densityoutput from the stack:

1. An initial estimate for micell,in is made.

2. Using the inner iteration and the estimated value of micell,in, �Pi

cell, �Pitop, and

�Pibot are calculated.

3. The values of �micell,in are calculated and used to generate new values for

micell,in.

4. Steps 2 to 3 (the outer iterations) are repeated until convergence.5. The current density output can be varied, and the above procedures repeated

in order to determine the stack performance for a range of loading conditions.

The solution of the stack flow model provides the mass flow rate, composition andpressure at the inlet of each PEM fuel cell in the stack. These values are used by thesingle cell model of [15] to calculate values of Ecell for each cell. The cell voltages,along with the value of ηcp also calculated by the model of [15], are used in eqn (1)to determine the stack voltage.

4 Results and discussion

Figure 4 compares the performance of a single PEM fuel cell operating indepen-dently with U and Z configuration stacks consisting of 50 cells. This comparison



Figure 4: Polarization curves comparing the performance of a single PEM fuelcell with 50 cell, U and Z configuration stacks operating with (a) H2/air,(b) reformate/air and (c) H2/O2 reactants. The stack polarization curvesare plotted by dividing the stack voltage by the number of cells in thestack.

is accomplished by comparing the single cell voltage to the stack voltage dividedby the number of cells in the stack (average cell voltage). Both the single cell andthe cells of the stacks have an active area of 240 cm2 and a serpentine gas flowchannel configuration with three gas flow channels per bipolar plate. The relevantdesign parameters for various cell and stack components, such as the bipolar plate,electrode backing, catalyst layer and polymer electrolyte membrane are given inTable 1. The anode and cathode sides of the cells and stacks are assumed to bethe same. The cross-sectional areas of both the anode and cathode stack manifoldsare equal to 1.54 cm2, and the design parameters for the anode and cathode sidesof the cell or stack are considered to be equal. The single cell and stacks operate



Table 1: Design parameters for various cell and stack components.

Parameter Value

Bipolar Plate ρbp 6 × 10−5 � · mW 0.155 mL 0.155 mhp 0.002 mhc 0.002 mws 0.00262 mwc 0.002 mnc 3ng 11

Electrode Backing ρbulke 6 × 10−5 � · mδe 2.5 × 10−4 mφe 0.4

Catalyst Layer δc 2.0465 × 10−5 mmPt 0.004 kg/m2

fPt 0.2�m 0.9κs 72700 S/m

Polymer Electrolyte δm 1.64 × 10−4 mMembrane KE 7.18 × 10−20 m2

Kp 1.8 × 10−18 m2

CH+ 1200 mole/m3

DH+ 4.5 × 10−9 m2/s

Stack wm 0.0124 mhm 0.0124 m

Ncell 50

with anode and cathode inlet pressures of 250 kPa, stoichiometries of 1.1 and 2,respectively, and a temperature of 358 K.

The performance of a single cell and stack operating with fully humidified hydro-gen as the fuel and fully humidified air, consisting of 21% O2 and 79% N2, as theoxidant (H2/air reactants) is compared in Fig. 4(a). Figure 4(b) compares single celland stack performance with fully humidified reformate, consisting of 75% H2 and25% CO2, as the fuel and fully humidified air as the oxidant (reformate/air reac-tants), while single cell and stack performance with fully humidified hydrogen asthe fuel and fully humidified oxygen as the oxidant (H2/O2 reactants) is comparedin Fig. 4(c). For operation with H2/air and reformate/air reactants, the average cellvoltage in the stack is less than the voltage of a single cell operating independently.



Figure 5: Voltage of each cell in 50 cell U and Z configuration stacks operatingwith (a) H2/air reactants and at the current density of 0.41 A/cm2 and(b) reformate/air reactants and at the current density of 0.32 A/cm2.

The difference between cell and stack performance is greater when reformate/airreactants are utilized. Operation with H2/O2 reactants results in the average cellvoltage in the stack being almost the same as the single cell voltage; this agreeswith the experience that a stack designed for H2/O2 reactants, in general, will notoperate properly for H2/air or reformate/air reactants. From Fig. 4, it is also apparentthat the Z configuration stacks have a better performance than the U configurationstacks.

The performance differences between the stacks and single cells are caused byvoltage variations among the cells in the stack. This cell-to-cell voltage variationis illustrated for U and Z configuration stacks in Fig. 5, with Figs 5(a) and 5(b)showing cell-to-cell variation for H2/air and reformate/air reactants, respectively.The average cell voltage in the stack of Fig. 5 is approximately 0.6, making thecurrent density used for the H2/air simulation 0.41 A/cm2 and 0.32 A/cm2 for thereformate/air simulation. The voltage of each cell in the stack is compared to thevoltage of a single cell operating independently, with the cells of the stack beingnumbered starting at the stack inlet. Near the stack inlet and outlet, the cells ofthe Z configuration stacks have a higher voltage than the single cell operatingindependently, while only the cells near the stack inlet have a higher performancethan the single cell for the U configuration stacks. However, the majority of the cellsin the stacks have voltages less than the single cell, which results in the averagecell voltage of the stack being less than the single cell voltage.



The cell-to-cell voltage variation can be quantified by the voltage spread of thestack:

SE = Emaxcell − Emin

cell1

Ncell

∑Ncell1 Ecell

× 100%, (46)

where Emaxcell and Emin

cell are the maximum and minimum cell voltages, respectively,within the stack. For the H2/air reactants of Fig. 5(a), the voltage spread for theU configuration stack is 9.8% and 5.0% for the Z configuration stack. Operationwith reformate/air reactants yields voltage spreads of 15% and 6.2% for the U andZ configuration stacks, respectively. Although not illustrated in Fig. 5, operationwith H2/O2 reactants and at the current density of 0.88 A/cm2, which correspondsto the average cell voltage of 0.6 V, results in voltage spreads of only 0.0021% and0.0018% for the U and Z configuration stacks, respectively. These voltage spreadvalues show that a high voltage spread corresponds to a poor stack performancewhen compared to a single cell. Z configuration stacks have a better performanceand lower voltage spread than U configuration stacks, while operation with refor-mate/air reactants results in the lowest performance and the highest voltage spread.Therefore, in practice, a low voltage spread and uniform cell-to-cell performanceis desirable in order to maximize cell performance.

The cell-to-cell voltage variations shown in Fig. 5 are the result of the pressureloss distribution, leading to non-uniform distribution of mass flow rate among thecells within the stack. This unequal distribution of mass flow rate among the cellsin the stack creates cell-to-cell variations in the anode and cathode stoichiometry.To illustrate how the mass flow rate distribution affects the cell voltage distributionin a stack, the anode and cathode cell stoichiometry in a Z configuration stack ispresented in Fig. 6. Operation with H2/air reactants and at the current density of0.41 A/cm2 results in little anode stoichiometry variation among the cells in thestack, as illustrated in Fig. 6(a). However, the cathode stoichiometry variation issimilar to the cell voltage distribution shown in Fig. 5(a). The cathode stoichiometryvaries more than the anode stoichiometry due to the larger mass flow rate in thecathode gas flow channels. The average Reynolds number in the anode gas flowchannels is around 32, but due to the presence of inert N2 gas, the Reynolds numberis approximately 640 in the cathode gas flow channels.

The use of reformate/air reactants and a current density of 0.32 A/cm2 results invariation in both the anode and cathode stoichiometry, as illustrated in Fig. 6(b).The anode stoichiometry shows more variation for reformate fuel than for hydrogenfuel due to the higher flow rate caused by the presence of the inert CO2; the averageReynolds number in the corresponding gas flow channels is about 112 for thereformate fuel.

In a manner similar to the cell voltage variation, the cell-to-cell variation instoichiometry can be quantified by the stoichiometry spread of the stack:

SS = S maxcell − S min

cell1

Ncell

∑Ncell1 Scell

× 100%, (47)



Figure 6: Anode and cathode stoichiometry of each cell in a 50 cell, Z configura-tion stack operating with (a) H2/air reactants and at the current densityof 0.41 A/cm2 and (b) reformate/air reactants and at the current densityof 0.31 A/cm2.

where S maxcell and S min

cell are the maximum and minimum values, respectively, ofcell stoichiometry within the stack. For the H2/air reactants of Fig. 6(a), the anodestoichiometry spread is only about 1%, while the cathode stoichiometry spreadis 50%. The use of reformate/air reactants, as illustrated in Fig. 6(b), results inan anode stoichioimetry spread of 10% and a cathode stoichiometry spread of42%. The cathode stoichiometry spread is greater for the case with H2/air reac-tants than for reformate/air reactants because the H2/air case uses a larger currentdensity. With a larger current density, the mass flow rate increases and generates alarger stoichiometry spread; the average Reynolds number in the cathode gas flowchannels, when reformate/air reactants and a current density of 0.32 A/cm2 areused, is approximately 501 while, as mentioned previously, the average Reynoldsnumber in the cathode gas flow channels is about 640 for the H2/air case. However,the larger anode stoichiometry spread results in a larger cell voltage spread whenreformate/air reactants are used.

Obtaining a uniform mass flow rate distribution within a PEM fuel cell stackreduces the voltage spread and improves stack performance. The degree of flowuniformity through the side branches of a manifold system depends on the ratio ofcross-sectional area between the side branches and manifold, and the resistance tomass flow in the side branches and manifold [19]. The ratio of cross-sectional areabetween the side branches and manifold is defined as:

f = nbAb

Am, (48)



Figure 7: Effect of stack manifold cross-sectional area on voltage spread for (a) Zand (b) U configuration stack with 3 gas flow channels per bipolar plate.The current density for H2/air reactants is 0.41 A/cm2, reformate/air reac-tants is 0.32 A/cm2, and H2/O2 reactants is 0.88 A/cm2.

where nb is the number of side branches, Ab is the cross-sectional area of each sidebranch, and Am is the cross-sectional area of the manifold. For the PEM fuel cellstacks considered in this study, eqn (48) becomes:

f = ncNcellAfc

Am. (49)

In principle, a uniform mass flow rate distribution is achieved when f approacheszero. However, nearly uniform flow can be obtained in reality at a finite value of f .From eqn (49), f can be reduced by increasing the cross-sectional area of the stackmanifold.

The effect of stack manifold cross-sectional area, or manifold area, on volt-age spread for stacks operating with H2/air reactants and a current density of0.41 A/cm2, reformate/air reactants and a current density of 0.32 A/cm2, andH2/O2 reactants and a current density of 0.88 A/cm2 is illustrated in Fig. 7.Althoughthe different mass flow rates in the anode and cathode manifold could warrant dif-ferent manifold areas, the results shown in Fig. 7 are for anode and cathode manifoldareas that are equal. It is seen that the voltage spread decreases as the manifold areais increased if H2/air or reformate/air reactants are used. Within the range of mani-fold areas considered in Fig. 7, the voltage spread is almost zero and independent of



Table 2: Critical stack manifold areas and f values, corresponding to a voltagespread of 1%, for stacks using three gas flow channels per bipolar plate.

Reactants Configuration Current Density Manifold Area f

H2/air Z 0.41A/cm2 3.3 cm2 1.8reformate/air Z 0.32A/cm2 3.8 cm2 1.6H2/air U 0.41A/cm2 4.9 cm2 1.2reformate/air U 0.32A/cm2 5.6 cm2 1.1

manifold area if H2/O2 reactants are utilized. In reality, the critical manifold area,below which significant cell-to-cell voltage variations occur, is much smaller forthe H2/O2 reactants.

The U configuration stacks exhibit greater cell-to-cell variation than the Z con-figuration stacks, while operation with reformate/air reactants results in a largervoltage spread than operation with H2/air reactants. This is evident in Table 2,which lists the critical manifold areas and f values corresponding to a voltagespread of 1%. The U configuration stacks require a larger manifold area than theZ configuration stacks in order to obtain a voltage spread of 1%. For a given stackconfiguration, reformate/air reactants require a larger manifold area than H2/airreactants; thus stacks designed for H2/air reactants may not operate effectively ifswitched to reformate/air reactants.

Decreasing the number of gas flow channels per bipolar plate (nc) can alsoreduce the value of f in eqn (49), resulting in less cell-to-cell voltage variation.To illustrate this, Fig. 8 shows the effect of manifold area on voltage spread forU and Z configuration stacks employing a bipolar plate with one serpentine gasflow channel. Table 3 tabulates the bipolar plate design parameters used for thesimulation results of Fig. 8. Other than the value of nc used (3 in Fig. 7 and 1in Fig. 8), all other parameters used in the simulations of Fig. 8 are the same asin Fig. 7, such as stack configuration, reactant composition and current density.The trends illustrated in Figs 7 and 8 are also similar. Increasing the manifold areadecreases the voltage spread if H2/air or reformate/air reactants are used, while themanifold area does not affect the voltage spread when H2/O2 reactants are used.For a given manifold area, the Z configuration stack has a smaller voltage spreadthan the U configuration stack.

Although reducing the number of gas flow channels does not affect the generalrelationship between voltage spread and manifold area, a reduction in nc signifi-cantly decreases the magnitude of the voltage spread. Therefore, the manifold areacorresponding to a voltage spread of 1% would be expected to be smaller if one,rather than three, gas flow channel grooves exist on each bipolar plate. Table 4 liststhe critical manifold areas and f values corresponding to a voltage spread of 1%for stacks utilizing one gas flow channel per bipolar plate. Comparing the entriesof Tables 2 and 4, it is evident that the critical manifold areas for the nc = 1 stacksare approximately one-third of those when nc = 3. However, less variation occurs



Figure 8: Effect of stack manifold cross-sectional area on the voltage spread for(a) Z and (b) U configuration stack with 1 gas flow channel per bipo-lar plate. The current density for H2/air reactants is 0.41 A/cm2, refor-mate/air reactants is 0.32 A/cm2, and H2/O2 reactants is 0.88 A/cm2.

Table 3: Bipolar plate design parameters utilizing one gas flowchannel per bipolar plate.

Parameter Value

ρbp 6 × 10−5 � · mW 0.155 mL 0.155 mhp 0.002 mhc 0.002 mws 0.00213 mwc 0.002 mnc 1ng 37

between the f values, with the f values for stacks with nc = 1 being approximately20% larger than the values with nc = 3.

Both the manifold cross-sectional area and the number of gas flow channels caninfluence the pressure loss of the stack. The pressure loss for the cathode stream in aZ configuration stack is illustrated in Fig. 9. Air is used as the cathode reactant and



Table 4: Critical stack manifold areas and f values, corresponding to a voltagespread of 1%, for stacks using one gas flow channels per bipolar plate.

Reactants Configuration Current Density Manifold Area f

H2/air Z 0.41A/cm2 1.1 cm2 2.2reformate/air Z 0.32A/cm2 1.4 cm2 1.8H2/air U 0.41A/cm2 1.6 cm2 1.6reformate/air U 0.32A/cm2 2.0 cm2 1.3

Figure 9: Cathode stream stack pressure loss for a Z configuration stack operatingwith H2/air reactants, a current density of 0.41 A/cm2, and 1 or 3 gasflow channels per bipolar plate (nc).

the current density is 0.41 A/cm2. The pressure loss increases by approximately 9times if the number of gas flow channels per bipolar plate is reduced from threeto one, caused by the combined effects of a lengthened flow path and higher flowvelocity in the gas flow channel. However, the general effect of manifold area onthe stack pressure loss is the same regardless of the number of gas flow channels.As the manifold area is increased, the pressure loss decreases initially, but thenbecomes independent of manifold area. This plateau in the pressure loss/manifoldarea plot indicates that the manifold area is sufficiently large such that the frictionloss attributed to the manifold does not contribute to the overall stack pressureloss; the stack pressure loss depends almost solely on the gas flow channels. Inpractice, nc = 1 is the minimum that can be used. However, depending on the



cell size, the flow path and the momentum loss in the gas flow channel may betoo large, resulting in an excessive pressure loss; this large pressure loss increasesthe parasitic power required for gas compression and decreases the overall stackeffeciency. Thus, nc = 3 is often used for large stacks consisting of large cells.

In a pipe network, the distribution of mass flow rate is influenced by the flowresistance in each pipe; pipes with a high resistance to mass flow have smaller massflow rates than pipes with a low resistance to mass flow. Therefore, one method ofreducing the cell-to-cell variation of mass flow rate and cell voltage in a PEM fuelcell stack is to alter the resistance to mass flow in the gas flow channels. In thisstudy, the resistance to mass flow in the gas flow channels is altered through theaddition of a term in the pressure loss due to friction, eqn (18):

�Pf = ζ2Cf LfcρaveV 2

ave

d fch

, (50)

where ζ is the flow resistance parameter that, if greater than one, increases theresistance to mass flow in the gas flow channel. Practically, the resistance to massflow can be increased by either decreasing the gas flow channel hydraulic diameter,increasing the length of the gas flow channel, or installing a flow obstruction in thegas flow channel.

Reducing voltage spread by varying the flow resistance parameter among thecells in the stack can be illustrated by considering a Z configuration stack operatingwith H2/air reactants and a current density of 0.41 A/cm2. In order to reduce cell-to-cell cathode stoichiometry variation, the flow resistance parameter for the cathodegas flow channels is set according to:

ζi = S ic

S minc

, (51)

where S ic is the cathode stoichiometry of cell i in Fig. 6(a) and S min

c is theminimum cathode stoichiometry in Fig. 6(a). Using these values of ζi, the cell-to-cell variation of cathode stoichiometry is greatly reduced when compared tothe variation resulting from using a constant ζ = 1, as illustrated in Fig. 10. Thevariable values of ζi are shown in the inset of Fig. 10. The reduction in cathodestoichiometry variation results in a 0.5% voltage spread, which is much smallerthan the 5% voltage spread achieved by using ζ = 1 for all cells in the stack.Significantly, the use of the variable flow resistance parameter only increases thecathode stack pressure loss from 1.8 kPa to 2.0 kPa; this increase of 11% is muchsmaller than the pressure loss increase of over 800% incurred if the voltage spreadreduction is achieved by reducing the number of gas flow channels per bipolarplate from three to one. Thus, the energy required to overcome the stack pressureloss would be less if a variable ζi is used, allowing for greater system efficiency.However, varying the flow resistance from cell to cell may be difficult to implementsince it requires the customization of each bipolar plate.

Finally, it should be pointed out that the above modeling results are based on a5 kW 50 cell PEM fuel cell stack. However, we don’t have access to the test results



Figure 10: Cathode stoichiometry for a Z configuration stack using a constant flowresistance parameter of ζ = 1 and the variable flow resistance parameter(dashed curve), which is given in the inset.

for the validation of the present model. Thus the above results should be treatedas qualitative for the moment. However, the single cell model used in this studyhas been validated against the single cell test results [15], and the stack flow modelhas been validated against experimental results for a different application [23, 24].In this sense, the present model can be used as a useful tool for the design andoptimization of PEM fuel cell stacks.

5 Conclusions

The performance of 50 active cell, U and Z configuration stacks were simulatedusing a mathematical model that consisted of two parts. The first part of the modeldetermined the distribution of mass flow rate and pressure in the stack manifold andthe gas flow channels of the fuel cells through a hydraulic network analysis. Theresults of the hydraulic network analysis were used as an input parameter for thesecond part of the model, which calculated the voltage of each cell in the stack witha previously developed, mathematical model. Therefore, the distribution of massflow rate within the stack influenced the voltage of each cell in the stack; a smallmass flow rate in a cell resulted in a low cell voltage. This relationship betweenmass flow rate distribution and individual cell voltage lead to the performanceof fuel cells operating within a stack being lower when compared to a fuel celloperating independently. The magnitude of the performance difference between



single cells and cells within stacks was larger for U than for Z configuration stacks,and greater when the anode/cathode reactant compositions were fully humidifiedreformate/air, rather than H2/air. Eliminating the performance differential could beachieved by ensuring that each cell in the stack had the same mass flow rate, andthree methods of attaining a uniform mass flow rate distribution were examined.The first method was increasing the cross-sectional area of the stack manifold,while the second involved decreasing the number of gas flow channels per bipolarplate. Finally, a uniform distribution of mass flow rate within the stack could beachieved by varying, from cell to cell, the resistance to mass flow in the gas flowchannels.

Acknowledgments

This work is a part of a large research project on PEM fuel cells and related tech-nologies supported by the Auto 21 NCE, NRC Institute for Fuel Cell Innovation,Hydrogenics Corporation and PalCan Fuel Cells Ltd. Partial funding is also pro-vided by the Natural Sciences and Engineering Research Council of Canada.

Appendix A: Property determination

The fluid in the gas flow channels and stack manifold is assumed to be composed ofa multi-component gas phase with liquid water droplets. The density and viscosityof the fluid are required to calculate the pressure loss in the gas flow channels andstack manifold. The density of the fluid is given by [20]:

1

ρ= χ

ρg+ 1 − χ

ρH2O(�)

, (52)

where ρg is the density of the gas mixture and ρH2O(�) is the density of liquid water.The density of the gas phase is calculated with the ideal gas relation:

ρg =∑

i=species

xiPM i

RT, (53)

where the summation includes all gaseous species present in the gas flow channelor stack manifold. The liquid water density can be found in [21] and is equal to968 kg/m3 at a temperature of 85◦C.

The viscosity of the fluid in the gas flow channels and stack manifold can befound with [20]:

1

µ= χ

µg+ 1 − χ

µH2O(�)

, (54)



where µg is the viscosity of the gas mixture and µH2O(�) is the viscosity of liquidwater. The Wilke correlation is used to find the viscosity of the gas mixture [22]:

µg =N∑

i=1

xiµi∑Nj=1 xjφij

(55)

φij = 1√8

(1 + M i

M j

)−1/21 +

(µi

µj

)1/2(

M j

M i

)1/4

2

,

where the individual gas viscosities are found using a power law [17].

µi

µi,293 K=

(T

293 K

)n

. (56)

The viscosity of liquid water can be found with the relationship [17]:

ln

(µH2O(�)

µo

)= −1.704 − 5.306 � + 7.003 �2,

� = 273 K

T, (57)

µo = 1.788 × 10−3 kg/(m · s).

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[24] Zhang, J. & Li, X., Coolant flow distribution and pressure loss in ONANtransformer windings, Part II: Optimization of design parameters. IEEETransactions on Power Delivery, 19(1), pp. 194–199, 2004.

Nomenclature

Acell Active area of fuel cell (m2)Am Cross-sectional area of stack manifold (m2)Afc Cross-sectional area of gas flow channel (m2)Cf Wall friction coefficientCH+ Fixed charge concentration in the polymer electrolyte (mole/m3)DH+,ref Diffusion coefficient of H+ in the polymer electrolyte (m2/s)dh Hydraulic diameter (m)E Cell or stack voltage (V)fpt Mass ratio of platinum to carbon supportfw Volume fraction of electrode backing void flooded by liquid waterF Faraday constant (96495 C/mole)hc Depth of gas flow channel (m)hm Height of stack manifold cross-section (m)hp Thickness of solid portion of bipolar plate (m)Iδ Cell current density (A/m2)KE Electrokinetic permeability of polymer electrolyte membrane (m2)Kp Hydraulic permeability of polymer electrolyte membrane (m2)�m Fraction of catalyst layer void space occupied by polymer electrolyte�H2O(�) Fraction of catalyst layer void space occupied by liquid waterL Length of fuel cell active area (m)Lfc Length of gas flow channel (m)Ls Length of stack manifold between cells (m)m Mass flow rate (kg/s)mPt Platinum mass loading per unit electrode area (kg/m2)�mr Amount of mass consumed in the catalyst layer (kg/s)M i Molecular weight of species i (kg/mole)Ncell Number of cells in a stacknc Number of bipolar plate gas flow channelsng Number of times a serpentine gas flow channel traverses the

bipolar plateN stack

i Molar flow rate of species i at the stack inlet (mole/s)�P Pressure loss (Pa)�Pf Pressure loss due to friction (Pa)�Pm Pressure loss due to momentum change (Pa)P Pressure (Pa)R Universal gas constant (8.314 J/mole · K)Redh Reynolds numberri

p Pressure residual (Pa)



S Spread (%)S StoichiometryT Fuel cell or stack temperature (K)V Velocity (m/s)W Width of fuel cell active area (m)wc Width of gas flow channel (m)wm Width of stack manifold cross-section (m)ws Width of gas flow channel support (m)xi Mole fraction of species i

Greek symbols

α Equals −1 for U and 1 for Z configuration stacksδc Thickness of catalyst layer (m)δe Thickness of electrode backing (m)δm Thickness of polymer electrolyte membrane (m)ζ Flow resistance parameterη Overpotential (V)ks Conductivity of catalyst layer solid phase (S/m)µ Viscosity (N · s/m2)ρ Density (kg/m3)ρbulk

e Resistivity of electrode backing (� · m)ρbp Resistivity of bipolar plate (� · m)φe Porosity of electrode backingφc Porosity of catalyst layerχ Mass fraction of the gaseous phase

Subscripts

a Anodeave Average valuebot Stack manifold bottom sectionbp Bipolar platec Catalyst layer; cathodecell Fuel celle Electrode backingfc Gas flow channelg Gasin Inlet value� Liquidm Polymer electrolyte membraneout Outlet valuerev Reversibletop Stack manifold top section



Superscripts

cell Fuel cellk Iteration numbermax Maximum valuemin Minimum valuenew Current value of an iterative parameterold Previous value of an iterative parameterstack Stack


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