THREE-DIMENSIONAL ANALYSIS OF TWO-PHASE FLOW AND ITS ... · research project at Lund Institute of Technology (LTH), while the research work at Dalian Institute of Chemical Physics

THREE-DIMENSIONAL ANALYSIS OF TWO-PHASEFLOW AND ITS EFFECTS ON THE CELL PERFORMANCEOF PEMFC

Jinliang Yuan and Bengt SundenDivision of Heat Transfer, Lund Institute of Technology,Lund, Sweden

Ming Hou and Huamin ZhangFuel Cell R&D Center, Dalian Institute of Chemical Physics, Chinese Academyof Science, Dalian, China

For a cathode duct of a proton-exchange membrane fuel cell, a three-dimensional analysis

method is further developed to include two-phase, multicomponent gas and heat transport

processes. A set of momentum, heat transport, and gas species equations is solved for the

whole duct by coupled source terms and variable thermophysical properties. The effects of

the electrochemical reactions on the heat generation and mass consumption=generation are

taken into account. The effects of liquid water on the local current density and cell per-

formance are discussed by incorporating the Tafel formula and a liquid-phase saturation

function. The numerical predictions are compared with experimental results, and good

agreement is demonstrated between the present results and the measured ones in terms of

polarization curve.

INTRODUCTION

Several types of fuel cells are currently under development. It has been foundthat proton-exchange membrane fuel cells (PEMFCs) have some advantages, such asrelative simplicity of design and operation, low-cost materials, and self-starting atlow temperatures. PEMFC systems are expected to play a significant role in the nextgeneration of primary or auxiliary power for stationary, portable, and automotivesystems. Electrochemical reactions, current flow, hydrodynamics, multicomponenttransport, phase change, and heat transfer are simultaneously involved in PEMFCs.During operation of PEMFCs, the membrane electrode assembly (MEA) should bewell humidified at all circumstances to reduce the ohmic resistance and additionallosses in the activation processes. However, if too much water is accumulated in the

Received 22 December 2003; accepted 23 April 2004.

The National Fuel Cell Programme of the Swedish Energy Agency financially supports the current

research project at Lund Institute of Technology (LTH), while the research work at Dalian Institute of

Chemical Physics (DICP) has been financially supported by the Chinese national ‘‘863’’ key project for

Electric Vehicle and a key project of the Chinese Academy of Science.

Address correspondence to Dr. Bengt Sunden, Division of Heat Transfer, Lund Institute of

Technology, Box 118, S-22100, Lund, Sweden. E-mail: [email protected]

Numerical Heat Transfer, Part A, 46: 669–694, 2004

Copyright # Taylor & Francis Inc.

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407780490487731

669

cathode duct, water flooding may appear. In this case, liquid water blocks theoxidant (oxygen) flow to the reaction sites and consequently affects the cellperformance.

It is a widely accepted fact that the PEMFC cathode is the performance-limiting component. This is so because the potential water flooding, as mentionedabove, and occurrence of slower kinetics of the oxygen reduction reaction affectthe cell performance. In addition, mass transfer limitations may occur because ofnitrogen barrier-layer effects in the porous layer; see [1]. Two-phase flow and itseffects on heat transfer, concentration variation, and fuel cell performance arevery important. Because of several limitations and difficulties, experimental in-vestigations of PEMFCs are still limited. For the purpose of supplying guidance fordesign and optimization, comprehensive studies have been undertaken over decadesto simulate and analyze water transport, gas utilization, power produced, and

NOMENCLATURE

a width of porous layer, m

A area, m2

b width of flow duct, m

B microscopic inertial coefficient, 1=m

cp specific heat, J=kg K

D diffusion coefficient, m2=s

Dh hydraulic diameter, m

Dhr diameter ratio

F Faraday constant (96,487 C=mol) or

Forchheimer coefficient

hd height of the duct, m

hp thickness of porous layer, m

hr thickness ratio (¼ hp=hd)

hwl water latent heat, J=kg

H enthalpy, J=kg

I current density, A=m2

J mass flux of species, kg=m2 s

k thermal conductivity, W=m K

kr thermal conductivity ratio (¼ keff=kf)

M molecular weight, kg=kmol

MEA membrane electrolyte assembly

O2,trans transfer coefficient for the oxygen

reaction

P pressure, Pa

q heat flux, W=m2

Re Reynolds number (¼ UDh=n)Rem wall Reynolds number (¼ VmDh=n)s liquid water saturation

S source term; entropy, J=kg K

Sdi source term in momentum equations

T temperature, �CUi velocity components in x, y, and z

directions, respectively, m=s

v velocity vector, m=s

V volume of control volumes at active

site, m3

Vcell cell voltage, V

Vm mass transfer velocity at bottom wall,

m=s

Vo open-circuit potential, V

x; y; z Cartesian coordinates

a net water transport coefficient

bi permeability of diffusion layer, m2

e porosity

Z relative humidity, %

m dynamic viscosity, kg=m s

n kinematic viscosity, m2=s

r density, kg=m3

f mass fraction

Subscripts

a anode or air

active at active site

av average

b bottom wall

bulk bulk fluid condition

c cathode

e electrolyte

eff effective parameter

f fluid or fuel

H2O water vapor

in inlet

m mass transfer

O2 oxygen

out outlet

p porous layer or permeation

s solid layer

sat saturation

w wall

wl water liquid

wp water phase change

wv water vapor

670 J. YUAN ET AL.

electrical current density distribution at both the unit and cell levels of PEMFCs; see[1–15]. It is worthwhile to note that most of the models are one- or two-dimensionalones, and isothermal and=or isobaric assumptions are involved, which may not betrue in PEMFCs, as will be discussed in this study.

Based on a multiple-phase and multiple-component mixture approach, a three-dimensional computational method has been further developed to predict waterphase change, and to analyze its effects on gas flow, heat transfer, and cell perfor-mance (current density) for a cathode duct. Momentum, heat transport, and speciesequations have been solved by coupled source terms and variable thermophysicalproperties of a multicomponent mixture. The duct under consideration consists of aflow duct, porous layer, and solid structure. Advanced boundary conditions areapplied in the analysis, such as combined thermal boundary conditions of heat fluxon the active surface and thermal insulation on the remaining surface. Moreover,mass consumption and generation appearing on the active surface are considered,together with interfacial conditions of velocity, temperature, and species con-centration between the flow duct and the porous layer, etc. It has been found thattwo-phase flow and current density are sensitive to the operating and configurationparameters.

PROBLEM STATEMENT AND ASSUMPTIONS

Figure 1 shows a schematic drawing of a PEMFC cathode duct, which includesthe flow duct, porous layer, and solid current collector. Many essential processes areinvolved, such as transport of reactants (oxygen from air in the cathode) and pro-ducts (water and heat if pure hydrogen is used as the fuel), and electrons as well. Thefocus of this study is to predict the two-phase flow in the composite duct, and thenassess effects of the liquid-water saturation level on the current density distributionand cell performance (in terms of the V–I curve). The following assumptions areemployed in this study.

Figure 1. Schematic drawing of a PEMFC cathode duct.

CELL PERFORMANCE OF PEMFC 671

The analysis is focused only on the cathode duct, and the effects of membrane dryinghave not been included in this model.

Axial velocity and temperature distribution of the species at the inlet of flow duct areuniform; a constant flow rate U¼Uin for the saturated gas mixture at pressurePin¼ 105 Pa is specified as a base condition at the inlet of the flow duct, whileU¼ 0 is specified at the inlet for the solid layer and porous layer.

Species in the duct are perfect and saturated with water vapor; there is negligibleeffect of the dissolved gases on water balance and no interaction between liquidwater and other gas species; liquid water has the same pressure as the gas species.

Species change and heat generation associated with the reaction appear on the activesurface (bottom surface in Figure 1).

The porous layer is homogeneous, and both the gas and liquid phases are in thermalequilibrium with the solid matrix, i.e., sharing the same temperature field; liquidwater appears in the form of small droplets in the species; and a multiple-phasemixture model is employed to describe two-phase flow and heat transfer in thecomposite duct.

Only the right half of the duct is considered, by imposing symmetry conditions at themid-plane.

The following transport processes and parameters are considered in the presentmodel; multicomponent mixture flow based on convection and diffusion in the flowduct and porous layer; species mass fraction for O2, H2O

(v), and H2O(l); mass

generation and consumption and effects on the species composition change by thereaction on the bottom surface in Figure 1; heat generation by the reaction; heatgeneration=absorption caused by water phase change (condensation=vaporization)and heat transfer (convection and=or conduction in all the components of the duct).Moreover, nonuniform gas pressure and temperature distributions in the duct havebeen considered, together with variable thermal-physical properties based on the gasspecies composition and=or temperature.

GOVERNING EQUATIONS

The governing equations to be solved are the continuity, momentum, energy,and species equations. The mass continuity equation is written as

H � ðreffvÞ ¼ Sm ð1Þ

The source term Sm in the above equation accounts for the mass balancecaused by the reaction from=to the active surface Aactive (bottom surface in Figure 1).It corresponds to the consumption of oxygen and generation of water in the cathodeside, respectively. It reads [3, 4]

Sm ¼ Sm;O2þ Sm;H2O ¼ � I

4FMO2

þ ð1þ 2aÞI2F

MH2O

� �Aactive

Vð2Þ

in which V is the volume of control volumes at the active site.

672 J. YUAN ET AL.

The momentum equation reads

H � ðreffvvÞ ¼ �HPþ H � ðmeffHvÞ þ Sdi ð3Þ

Equation (3) is valid for both the porous layer and the flow duct, by including asource term Sdi:

Sdi ¼ � meffvb

� �� reffBVi vj j ð4Þ

The first term on the right-hand side accounts for the linear relationship betweenthe pressure gradient and flow rate by the Darcy law, while the second term is theForchheimer term taking into account the inertial force effects, i.e., the nonlinearrelationship between pressure drop and flow rate. In Eq. (4), b is the porous-layerpermeability, and V represents the volume-averaged velocity vector of the speciesmixture. For example, the volume-averaged velocity component U in the x directionis equal to eUp, in which e is the porosity and Up is the average pore velocity (orinterstitial velocity in the literature). This source term accounts for the linearrelationship between the pressure gradient and flow rate by the Darcy law. It shouldbe noted that Eq. (3) is formulated to be generally valid for both the flow duct andthe porous layer. The source term is zero in the flow duct, because the permeability bis infinite. Equation (3) then reduces to the regular Navier–Stokes equation. For theporous layer, the source term Eq. (4) is not zero, and the momentum Eq. (3) with thenonzero source term in Eq. (4) can be regarded as a generalized Darcy model.

The energy equation reads

H � ðreffvTÞ ¼ H � keffcpeff

HT� �

þ Swp ð5Þ

in which Swp is the heat source associated with the water phase change (con-densation=vaporization) when two-phase flow is considered as in this study; see [4].

Swp ¼ Jwl � hwl ð6Þ

where Jwl is the mass flux of liquid water by phase change and hwl is the water latentheat.

The species conservation equations are formulated into a general one,

H � ðreffvfÞ ¼ H � ðreffDf;effHfÞ þ Sf ð7Þ

where f is mass fraction. The above equation is solved for the mass fraction of O2,H2O

(v), and H2O(l).

The concentration of the inert species, nitrogen, is determined from a sum-mation of the mass fractions of the other species. The source term in Eq. (7) includeswater vapor and liquid water caused by phase change. It is written as [4]

Swv ¼ �Swl ¼ MH2OPw;sat � Pwv

P� Pw;sat

� �Xi

massiMi

� �ð8Þ


In Eq. (8), massi is the mass of species i; Pwv is the partial pressure of water vapor,Pw;sat is the saturation pressure at the local temperature, while P is local pressure. Ifthe partial pressure of the water vapor is greater than the saturation pressure, watervapor will condense, and a corresponding amount of liquid water will be formed.Water-vapor partial pressure Pwv in the above equations is calculated based on itsconcentration and local pressure of the gas mixture, while its saturated pressure Pw;sat

at local temperature reads [2]

log10 Pw;sat ¼ �2:179þ 0:029T� 9:183� 10�5T2 þ 1:445� 10�7T3 ð9Þ

BOUNDARY AND INTERFACIAL CONDITIONS

The boundary conditions on the solid active surfaces can be written as:

U ¼ V� Vm ¼ W ¼ 0 � keffqTqy

¼ qb

� reffDf;effqfqy

¼ Jf at the bottomwall ðy ¼ 0Þ ð10Þ

U ¼ V ¼ W ¼ 0 q ¼ 0 ðor T ¼ TwÞ Jf ¼ 0 at the top and sidewalls

ð11Þ

qUqz

¼ qVqz

¼ W ¼ qTqz

¼ qfqz

¼ 0 at themid-plane z ¼ a

2

� �ð12Þ

It should be noted that all the walls for the above boundary conditions are on theexternal surfaces of the solid layer and porous layer. In Eq. (10), Vm is the wallvelocity of mass transfer caused by the electrochemical reaction:

1=2O2 þ 2e� þ 2Hþ ! H2O ð13Þ

The detailed procedure to obtain this value (Vm) was discussed in [16], and the finalform is as follows:

Sm ¼ reff RemnDh

a

Að14Þ

where Rem¼VmDh=n is the wall Reynolds number caused by the electrochemicalreaction. The other variables can be found in the Nomenclature list. qb in Eq. (10) isthe heat source caused by the reaction and is given by [3]

qb ¼ � I

2FDHH2OMH2O � I � Vcell ð15Þ

where DH is the enthalpy change of water formation. The first term in the aboveequation accounts for the quantity of water, and the second one takes care of thecurrent density generated by the electrochemical reaction. When inhomogeneous

674 J. YUAN ET AL.

current density is involved, the local heat generation should be defined for variouszones and locations.

Among various interfacial conditions between the porous layer and gas flowregion, the continuity of velocity, shear stress, temperature, heat flux, mass fraction,and flux of species (for the oxygen, water vapor, and liquid water, respectively) areadopted:

U� ¼ Uþ meffqUqy

� ��¼ mf

qUqy

� �þ

ð16Þ

T� ¼ Tþ keffqTqy

� ��¼ kf

qTqy

� �þ

ð17Þ

f� ¼ fþ reffDf;effqfqy

� ��¼ reffDf;eff

qfqy

� �þ

ð18Þ

Here subscriptþ (plus) is for the fluid side, while7 (minus) is for the porous layerside. Moreover, the thermal interfacial condition Eq. (17) is also applied at aninterface between the porous layer and the solid layer with ks instead of keff.

ADDITIONAL EQUATIONS

It should be noted that the properties in the above equations with subscript effare effective ones. For the flow duct, the effective properties are reduced to regularvalues of the species mixture based on the species composition, or regarded asconstant values in some cases; while in the porous layer, there are many factorsaffecting the effective properties, such as the microstructure of the porous layer,species composition, local temperature, etc. It is not easy to obtain more accuratevalues because the available data of the porous layer structure are still limited. It hasbeen found that setting meff ¼ mf and reff ¼ rf provides good agreement withexperimental data [17]. For the sake of simplicity, this approach is adopted here.

To reveal the porous-layer effects, on the other hand, parameter studies arecarried out for the conductivity keff and species diffusion coefficients Df,eff byemploying the ratios y:

yk ¼keffkf

ð19Þ

yD ¼ Df;eff

Dfð20Þ

In Eq. (19), kf is the species mixture conductivity in the porous layer, and is esti-mated by a typical method [18]:

kf ¼1

2�

Xi

xikfi þP

xikfi

� ��1" #

ð21Þ


where xi is the mole fraction and kfi is the conductivity of the species component.The diffusion coefficients Df in Eq. (20) are the values of the species components inthe species mixture, i.e., DO2

, DH2OðvÞ, and DH2OðlÞ for oxygen, water vapor, andliquid water, respectively. However, the binary diffusion coefficients of the compo-nents in pure air are used as estimation of Df in the calculations [5].

The effective diffusivity ratios are corrected by applying the Bruggemanncorrection [1, 5] to account for the effects of porosity in the porous layer:

yD ¼ e1:5 ð22Þ

It should be noted that the thermal-physical properties of the species mixture, suchas the density rf and viscosity nf, are estimated as functions of the local concentrationas well.

According to Dalton’s law, relative humidity of the species mixture is defined as

Z ¼ Pwv

Pw;sat¼ xwv

P

Pw;satð23Þ

where P is the pressure, Pwv is the water-vapor partial pressure, Pw,sat is thesaturation pressure identified in Eq. (9), and xwv is the water vapor molar fraction.The liquid-phase saturation s is employed to describe the liquid water-volumefraction in the species mixture. It reads [2]

s ¼rfw � rgfwv

rwl � rgfwv

ð24Þ

where f is mass fraction, rg is gas-phase density, and rwl is liquid-phase density. Thedensity of the two-phase species mixture is

r ¼ rgð1� sÞ þ rwls ð25Þ

Local current density I is an important parameter of the cell, and essential forsource term calculations related to mass injection=suction and heat generation by theelectrochemical reaction. It should be noted that a constant value is typically pre-scribed for the current density on the active surface in the literature, and is in ourprevious work [19, 20]. In this study, this limitation is released. The local currentdensity is calculated as a function of the oxygen concentration at the electrode=membrane interface and the activation overpotential, while the effects of the liquidwater are considered when liquid water appears. It reads [1, 21],

I ¼ Ioð1� sÞfO2

fO2;ref

" #b

exp �O2;transF

RTVover

� �ð26Þ

where I is the local current density based on the Tafel equation along the activesurface, Io is the exchange current density per real catalyst area, s is the liquid watersaturation, fO2

is the oxygen species mass concentration, O2;trans is the transfer

676 J. YUAN ET AL.

coefficient for the oxygen reaction, Vover is the activation overpotential, and fO2;ref isthe oxygen reference concentration (e.g., P¼ 1 atm) for the oxygen reaction; othersymbols can be found in the Nomenclature. It should be mentioned that, byemploying (17s), the effect of liquid water saturation on the surface availability ofthe reaction site is accounted for in Eq. (26).

It is well known that the polarization curve, which represents the cell voltagebehavior against operating current density (V–I curve), is the standard measure ofperformance for fuel cells, and depends on both the operating conditions and thecomponent design. The operating conditions include the working temperature,partial pressures of fuel and oxidant and their utilization rates, and=or the waterconcentration in the components. On the other hand, the design parameters could bethe porosity, tortuosity, and thickness of the electrodes (affecting concentrationloss of voltage), and the thickness of the electrolyte (ohmic loss), and electrode=electrolyte interface (activation loss). The cell potential Vcell is thus obtained by

Vcell ¼ Vo � Vover � VO ð27Þ

where Vo is the open-circuit potential for a given temperature and pressure, Vover isthe activation overpotential, and VO is the ohmic losses in the porous layer and themembrane plus contact resistances. The open-circuit potential is determined using amodified Nernst law [22]:

Vo ¼ Vo;ref þDSnF

ðT� ToÞ þRT

nFlnðPH2

Þ þ 1

2lnðPO2

Þ� �

ð28Þ

where P denotes the species partial pressure (atm) and DS is the entropy change.Using standard values for the entropy production, this reduces to:

Vo ¼ 1:229� 0:85� 10�3ðT� 298:15Þ þ 4:31� 10�5T lnðPH2Þ þ 1

2lnðPO2

Þ� �

ð29Þ

where T is in kelvin, while the ohmic losses VO are evaluated by [7]

VO ¼ IR ¼ R1

Aactive

Z Aactive

0

I dA ð30Þ

In this study, the liquid water flow in the porous layer is evaluated by the liquid-water mass flux at the active site, and it is defined as

_mmwl;y

� �b¼ refffwlVm þ reffDwl;eff

qfwl

qy

� �b

ð31Þ

where Vm is the velocity component in the vertical direction and f is the massconcentration. The first part on the right-hand side is the mass flux of the liquidwater by convection, while the second part is the one by diffusion. A positive value


represents liquid water flowing from the active site to the porous layer and theflow duct.

NUMERICAL APPROACHES AND MODEL VALIDATION

A finite-volume method is employed to solve the governing equations with theboundary conditions and the interfacial conditions as presented in [17]. The solutiondomain is discretized with a uniform grid size in the cross section because similartransport processes are involved and coupled in various subdomains. A nonuniformdistribution of grid points is applied in the main flow direction, with an expansionfactor to get finer meshes in the entrance region of the duct. The Cartesiancoordinate system in the physical space is replaced by a general nonorthogonalcoordinate system. The momentum equations are solved for the velocity componentson a nonstaggered grid arrangement. The Rhie-Chow interpolation method is usedto compute the velocity components at the control-volume faces. Algorithms basedon the tridiagonal matrix algorithm (TDMA) and a modified strongly implicitprocedure (SIP) (used in this study) are employed for solving the algebraic equations.The convective terms are treated by QUICK, while the diffusive terms are treated bya central difference scheme. The SIMPLEC algorithm handles the linkage betweenvelocities and pressure. An in-house computational fluid dynamics (CFD) code isfurther developed to include variable thermophysical properties and the liquid=vapor water two-phase flow.

The important feature of this model is based on the approach of the multi-component two phase mixture. The phase change and its effects on the gas flow andheat transfer are considered. The amount of water undergoing phase change iscalculated based on the partial pressure of water vapor and the saturation pressure. Itis worthwhile to note that two-phase flow and heat transfer have been included andimplemented to get local pressure, temperature, and species component composition;this model is therefore considered a nonisothermal and nonisobaric. It is alsoworthwhile to note that the source term is added to the species conservation equationsfor water phase change. When the partial pressure of water vapor is greater than thesaturated pressure, water vapor will condense to liquid water. Consequently, its massfraction will be reduced, together with a release of water latent heat until the partialpressure equals the local saturation pressure. On the other hand, if the partial pressureis lower than the saturation pressure, the liquid water will evaporate if liquid water isavailable. It should be mentioned that the source term concerning the water phasechange and the associated heat source term correspond to the control volumes wheretwo-phase water appears, and these are not treated as the boundary conditions.

The in-house CFD code has been further developed to handle variable prop-erties of gas species. These properties are calculated as functions of pure componentproperties and the species concentration, and are updated for every new iteration [16,19]. To check the dependence of the results on the number of grid points, numericalcalculations were carried out using various numbers of grid points and expansionfactors. The final number of grid points 90650630 (x6y6z) is then chosen for allcases investigated.

Although no experimental data are available for verification of the two-phaseflow simulations, comparisons with other models [1, 2] show qualitative agreement

678 J. YUAN ET AL.

regarding the 10% liquid water saturation in the porous layer close to the exit.Furthermore, the polarization curves obtained from this study are compared withexperimental data in Figure 2. From the figure it is found that the agreementbetween calculations and experimental results is good for both single- and two-phasemodels at low current densities, but not for the case of single-phase modeling at highcurrent densities. It is clear that the single-phase simulation produces a higher cur-rent density, while the two-phase model fits fairly well the experimental data at highcurrent densities. As confirmed in [7], the low current density of the experimentalresults may be caused by the presence of liquid water in the catalyst layers and thegas diffusion layers, but the single-phase model neglects the volume occupied byliquid water in the cathodes and effects on the cell performance.

RESULTS AND DISCUSSION

Parameters of a conventional PEMFC cathode duct, similar to those in [3, 7],are applied as a base case in this study. Table 1 shows the geometry parameters,while Table 2 lists the operation and transport ones. For the porous layer, the

Figure 2. Polarization curves comparison of the modeling results with the experimental data [7].

Table 1. Geometries of the PEMFC cathode duct (cm)

Length (x) Depth (y) Width (z)

Overall duct 10 0.20 0.16

Gas flow duct 10 0.12 0.08

Diffusion layer 10 0.04 0.16


parameters are chosen as: thermal conductivity ratio yk ¼ 1:0, porosity e ¼ 0:5; andpermeability b ¼ 2�10�10 m2. It should be noted that all the results presentedhereafter are for the base-case condition unless otherwise stated.

Figure 3a shows the velocity profile along main flow direction, in which thescale of the vector plots (i.e., 2 m=s) is a reference value of the maximum velocity.As shown in the figure, a parabolic profile is clearly observed in the flow duct. Onthe other hand, because the species have difficulty penetrating into the porouslayer, the velocity in the porous layer is very small except in the region close to theflow duct. It is clear that the temperature increases along the main flow direction,see Figure 3b. Variation in temperature can also be observed in the verticaldirection with slightly larger temperature and gradients close to the bottom sur-face. These are created by the heat generation due to both the reaction close to theactive surface and the latent heat release by water condensation in the two-phaseregion, caused by the increase in the water-vapor concentration in this area. It isworthwhile to note that the temperature is nonuniformly distributed. By con-sidering the local temperature distribution, the effects on the saturation pressurecan be found, which is not the case when isothermal assumption is employed, asdone by some authors in the literature.

It has been revealed from the calculated results that oxygen mass concentrationdecreases along the main flow direction in both the flow duct and the porous layer(not shown here). Water activities in the duct are shown in Figure 4. The massconcentration of the water vapor at the entrance is 23%, which corresponds to thesaturated one at the base-case condition (Tin¼ 70�C). It can be observed that watervapor is generated at the bottom surface, and is transported back to the flow ductthrough the porous layer. For this reason, larger mass fractions of water vapor canbe found in the porous layer close to the bottom surface. Therefore, the partialpressure of water vapor is larger in the regions mentioned above, and smaller in theinterfacial region and the flow duct. Based on the calculated partial pressure and thesaturation one, the liquid water mass concentration was predicted and shown inFigure 4b. It is found that the liquid water appears in both the flow duct and theporous layer, with the largest mass fraction (around 10%) in the porous layer closeto the exit. Because the saturation pressure is proportional to the local temperature,smaller saturated pressures can be expected for the flow duct compared to the porouslayer. This is why the liquid water can appear in the flow duct as well, but withsmaller mass fractions (less than 5%). A proper gradient of the liquid water con-centration should be kept for the liquid water to be driven out of the porous layer.

Table 2. Parameters employed as the base case in the study

Oxidant inlet temperature, Tin 70�CInlet relative humidity, Z 100%

Oxygen binary diffusion coefficient, DO2,f2.8461075 m2=s

Water vapor diffusion coefficient, DH2OðvÞ ;f 1.2561075 m2=s

Liquid water diffusion coefficient, DH2OðlÞ ;f 1.061075 m2=s

Inlet Reynolds number, Rein 50

Exchange current density, Io 0.01 A=m2

Transfer coefficient for the oxygen reaction, O2,trans 0.5

Activation overpotential, Vover 0.3 V

680 J. YUAN ET AL.

Figure 5 shows the corresponding liquid-water saturation level along the mainflow direction of the composite duct. As shown in Eq. (24), liquid saturationrepresents the volume occupied by the liquid phase divided by the total volume of theduct. It is clear that the liquid saturation s is zero in the single-phase gas flow region.This is so because the species density reduces to the gas-phase density, and the water

Figure 3. (a) Velocity vectors. (b) Contours of temperature T along the main flow direction of a PEMFC

cathode duct at the base case.


mass fraction equals the mass fraction of the water vapor as well. The liquidsaturation s is 1.0 for the case of single-phase liquid flow. As shown in Figure 5, theliquid saturation s is zero in the inlet region and increases along the flow direction.It is also true that the liquid saturation s decreases from the active site to the flowduct at the same x due to the liquid water transport, as discussed later in this article.

Figure 4. Mass concentration profiles for (a) water vapor and (b) liquid water along main flow direction of

PEMFC cathode duct for the base case.

682 J. YUAN ET AL.

The liquid water occupies more volume in the porous layer, with the largest value ofs appearing in the corner close to the active surface at the exit of the duct, while thesingle-phase gas species occupy most of the flow duct except that small values of scan be observed in the interface region after a certain distance from the inlet.

Figure 6 presents liquid water saturation in two cross sections, one located at50% duct length downstream the inlet and the other at the exit. It is found that theliquid water saturation is almost uniformly distributed in the two-phase region inthe flow duct; however, a large saturation can be observed in the porous layer belowthe solid layer, with the largest value in the active site close to the vertical side wall.This is because the water vapor generated under the flow duct is removed relativelyquickly due to the shorter transport distance along the porous-layer thickness, whilethe water vapor generated below the solid layer accumulates in the active regionclose to the side wall. A large partial pressure of water vapor is expected to appear;consequently more liquid water is condensed in this region. By comparing Figures 6band 6a, it is revealed that the characteristics of the liquid water saturation are similarto each other, except that it appears in most of the flow duct for the cross section ofthe duct for the position at the exit (Figure 6b). Moreover, it can be observed that theliquid water saturation s in the porous layer is higher than that in the flow duct,which indicates that the liquid water flows toward the flow duct from the active site.

As discussed above, the liquid-water mass composition and saturation level atthe active surface are the highest, consequently greater contribution to the speciesflow and heat transfer can be expected.

Figure 7a shows a comparison of the liquid water mass fluxes by convection,diffusion, and total value for the base case. It is found that both convection anddiffusion mass fluxes have very small values at the inlet region. Due to the phase

Figure 5. Liquid saturation profile along the main flow direction in a PEMFC cathode duct for the

base case.


change when the water vapor is generated and the saturation condition is reached,the liquid water can be accumulated at this region. Its mass concentration andgradient along the duct become larger, so larger values can be observed for both theconvection and diffusion after a certain distance downstream of the inlet. As shownin Figure 7a, the liquid water diffusion dominates the liquid water flow in the porouslayer to the flow duct. By comparing the value of the convection and diffusionfluxes, it is found that the convection is weak (15% or less contribution to the totalflux). Consequently, the total flux from Eq. (31) is controlled by the diffusion forthis specific case. Figure 7b shows the predicted total values of the cross-sectional

Figure 6. Cross-sectional liquid saturation distribution at (a) the half-length from the inlet and (b) the exit

of a PEMFC cathode duct for the base case.

684 J. YUAN ET AL.

liquid-water mass flux at various stations along the main flow stream. The mass fluxis zero at the inlet region, referring to Figure 7b. It is found that the total mass fluxhas almost uniform values in the cross sections, except for a weak liquid water flowfor the site below the solid layer, which has a long transport distance to the flow duct.

As mentioned earlier, the local current density I is one of the most criticalparameters for fuel cell performance, and for modeling as well. It is possible to

Figure 7. Liquid water flux at the active surface for (a) the main flow direction and (b) the cross-sections in

a composite duct for the base case.


calculate the distribution of the local current density once the detailed distribution ofthe oxygen at the active site is available. It is assumed that the catalyst load is evenlydistributed at the active site, and the activation overpotential therefore is evaluatedby a constant value.

Figure 8 shows a local current density distribution on the active surface (in thex–z plane). It is found that the current density is high near the entrance, and thendecreases along the main flow direction. This is so because the oxygen transfer to the

Figure 8. Local current density distribution in a cathode duct of a PEMFC for the base case.

Figure 9. Current density distribution for the cross sections along a PEMFC cathode duct for the

base case.

686 J. YUAN ET AL.

reaction site is larger near the entrance region, which is dominated by the oxygenconvection. The reduced current density downstream is due to the small oxygentransport rate controlled by the diffusion [20]. It is also true that the current densityis nonuniformly distributed in the cross section with a smaller value in the cornerregion beneath the solid current collector. This is due to a longer transport distancefrom the flow duct to the active site. From Figure 9, a similar conclusion can bedrawn, i.e., the current density is nonuniform along the main flow direction and inthe cross section as well. It should be noted that the liquid water saturation affectsthis nonuniform distribution as indicated by Eq. (26).

Figure 10 shows the current density distribution along the main flow directionand reveals the effects of the liquid water saturation keeping the same overpotential.As discussed earlier, the liquid-water saturation level is low in the entrance region,and consequently almost identical current density can be observed for both cases inFigure 10. After a certain distance downstream of the inlet, the effects of the liquidwater saturation become more clear; about 20% lower current densities are predictedat the exit if two-phase effects are considered. As liquid water appears, the oxygensupply to the active surface is impeded and subsequently the local current density,which depends on the liquid-water saturation level, is reduced.

Sensitivity studies for the effects of the liquid water saturation on the currentdensity by varying some of the parameters from the base condition are conductedand presented hereafter. The effect of the inlet temperatures (i.e., the cathodehumidification temperatures with 100% relative humidity) on the liquid watersaturation is shown in Figure 11. At a high temperature (80�C), in Figure 11a, liquidwater saturation is smaller compared to the case at a low temperature (60�C inFigure 11b) in both the porous layer and the flow duct. This indicates that a low inlettemperature contributes to a large liquid-water saturation level (with the highestvalue of 0.145 versus 0.100 in the active region close to the vertical side wall).

Figure 10. Liquid water effects on the current density distribution along a PEMFC cathode flow direction.


The effect of the inlet humidification temperature on the current density isshown in Figure 12a. At a high temperature (80�C), it is revealed that liquid watersaturation is smaller compared to the case at a low temperature (60�C) in both theporous layer and the flow duct. This indicates that a high inlet temperature con-tributes to a smaller liquid-water saturation level, and less effect on the decrease in

Figure 11. Cross-sectional liquid saturation distribution for (a) Tin¼ 80�C and (b) Tin¼ 60�C at the exit of

a composite duct for the base case.

688 J. YUAN ET AL.

the current density, which can be observed in Figure 12a. It is known that humi-difying the inlet species can increase the water vapor concentration (not shown here).Consequently, it is easier to form two-phase flow in the duct, particularly in theporous layer, where water vapor is generated as well. This is because water con-densation will occur only when the water vapor pressure exceeds the saturatedpressure. As discussed in [19], a decrease in the inlet humidity alone would lead to asmall liquid water saturation in the porous layer and the flow duct, and consequentlyless effect on the decrease in the current density. It is found in Figure 12b that the dryinlet species (relative humidity Z¼ 0) will contribute to a smaller current densitydecrease compared with the base case (relative humidity Z¼ 100%).

Figure 12. Effects of (a) the inlet temperature and (b) the inlet relative humidity of the gas species on the

current density distribution along a PEMFC cathode flow direction.


Figures 13 and 14 show the overpotential effects on the liquid water saturationand local current density distribution, respectively, along the main flow direction atthe base condition. As confirmed in [7, 12], the overpotential has important effects onthe electrochemical reaction rate. When the overpotential is small, the oxygen con-sumption is small and thus the reduction reaction rate is low (not shown in this

Figure 13. Liquid-water saturation profile for the effects of the overpotential: (a) Vover¼ 0.25 V and (b)

Vover¼ 0.35 V along the main flow direction of PEMFCs.

690 J. YUAN ET AL.

study); consequently, the local current density and the liquid water saturation aresmall. Figure 13a shows the liquid-water saturation profile for the case of a smalloverpotential (Vover¼ 0.25 V). By comparing with Figure 13a, it is found fromFigure 13b that the liquid water saturation increases (s¼ 0.11 versus s¼ 0.04 at theexit of the duct) when the overpotential increases (Vover¼ 0.35 V). More effects oftwo-phase flow in decreasing the local current density are thus expected for the caseof a large overpotential, which is found in Figure 14.

Similar to the inlet temperature and relative humidity, the cathode species flowrate and pressure at the inlet affect various transport properties in the PEMFCs. Thepredicted local current densities are shown for various gas flow rates in Figure 15aand for pressures in Figure 15b. The higher flow rate (Rein¼ 100 or 200 versus 50)and pressure (Pin¼ 3 or 6 atm versus 1 atm) at the cathode inlet imply a higheroxygen mass concentration at the active surface, and then higher local currentdensities along the main flow direction, which is found in Figure 15. The reason forthe improved performance is the increase of the partial pressure and oxygen content,as well as a higher open-circuit voltage based on the Nernst equation, when inletpressure and flow rate increase. From Figures 15a and 15b, it is also true that theeffects of the two-phase flow on the local current densities are less significant atcathode inlet areas than farther downstream. By referring to Figure 15b, it is foundthat the local current density and two-phase flow effects are less sensitive whenpressure Pin is larger than 3 atm. It is a fact that the change of the inlet pressurecontributes to the change of the inlet gas compositions (assuming the inlet gases arefully humidified), the exchange current density, and the gas diffusion coefficient.However, the change in the inlet oxygen=water vapor composition is stronger inthe range from 1 to 3 atm, if compared with that in the range from 3 to 6 atm.

Figure 14. Effects of the overpotential Vover on the current density distribution along a PEMFC cathode

flow direction.


Overall, the cell configuration and operation parameters should be carefully designedto reach an optimum, by considering the gain in the cell performance and additionalcosts of a more powerful blower or=and compressor.

CONCLUSIONS

This study deals with three-dimensional predictions of simultaneous speciescomponent distribution and heat transfer, with main focus on the two-phase flowand its effects on the current density distribution appearing in the cathode duct of

Figure 15. Effects of (a) inlet velocity Rein and (b) inlet pressure Pin on the local current density along the

main flow direction for the base case.

692 J. YUAN ET AL.

PEMFCs. The important characteristic of this study is based on a multiple-phaseand multiple-component mixture model by in incorporating the Tafel formula and aliquid-phase saturation function. It has been revealed that the nonuniformly dis-tributed current density is due to the nonuniformity of the oxygen distribution andtransport rate to the active surface, as well as the liquid-water saturation effects. It isfound that a high saturation level reduces the current density, as in the cases of lowinlet temperature=overpotential and high inlet humidity of gas species. High inletflow rate and pressure can improve the two-phase flow and its effects on the localcurrent density. Good agreement has been demonstrated between the presentnumerical predictions and measured results in terms of the polarization curve.

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THREE-DIMENSIONAL ANALYSIS OF TWO-PHASE FLOW AND ITS ... · research project at Lund Institute of Technology (LTH), while the research work at Dalian Institute of Chemical Physics