A Two-Dimensional Mathematical Model of Liquid-Feed Direct

Energy & Fuels 2002, 16, 1591-1598

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A Two-Dimensional Mathematical Model of Liquid-Feed Direct Methanol Fuel Cells J. R. Fan,* G. L. Hu, J. Yao, and K. F. Cen Institute for Thermal Power Engineering and CE&EE, Zhejiang University, Hangzhou 310027, People’s Republic of China Received November 19, 2001

A two-dimensional across-the-channel mathematical model for the simulation of a direct methanol fuel cell is described. The model accounts simultaneously for electrochemical kinetics, hydrodynamics, and multicomponent transport. The two-dimensional distributions of concentrations of reactants, the two-dimensional distributions of current densities, the crossover flux of methanol from the anode to the cathode, and current-voltage curve for the fuel cell have been calculated. The obtained results indicate that the concentrations of reactants in the catalyst layers in front of the current collectors are very low, which reduce the utility of catalyst; the current density at the edges of the channels is many times greater than the mean current density, which may lead to local overheat.

1. Introduction The direct methanol fuel cell (DMFC), using solid polymer electrolyte (SPE) membranes (e.g., Nafion), is an attractive source of power, especially for vehicular applications. Advantages of this fuel cell include easy and secure storage of the high-energy density liquid fuel, low temperature and pressure operation, possible use of existing fuel distribution infrastructure, and the simple reactor design implies no fuel reforming, which is necessary for the fuel feed of classical hydrogen fuel cells. One of the main drawbacks of the cell to date is methanol permeation through the membrane, with the production of a mixed potential at the cathode and a loss of fuel efficiency. The other is low oxidization activity of methanol at the anode catalyst layer at low operation temperature (typically 80 °C), a highly active, inexpensive, and efficient electrocatalyst is required to reduce the cost of DMFC. In a DMFC, the following catalytically activated reactions of anodic methanol ionization and oxygen reduction at the cathode, which are spatially separated from each other by the polymer membrane, take place:

anode side: CH3OH + H2O f CO2 + 6H+ + 6e- (1) cathode side: 3/2O2 + 6H+ + 6e- f 3H2O

(2)

Reactions 1 and 2 can be combined to give the overall reaction:

CH3OH + 3/2O2 f 2H2O + CO2

(3)

Water management in the membrane has been the focus of most polymer electrolyte fuel cell models because the * Author for correspondence. Fax: +86-0571-88396281. E-mail: [email protected].

PEM must remain hydrated for proper fuel cell operation.1,2 Comparing the liquid feed DMFC with a fuel cell using hydrogen or a vaporized methanol/water solution as the fuel, the liquid feed DMFC has no problem of hydration of the polymer membrane. To better understand, and hence optimize, the DMFC system, a number of groups have attempted modeling and computer simulation of this cell. Verbrugge3 has developed a simple diffusion-only model of methanol through a PEM, assuming dilute solution theory and no significant concentration-dependent methanolmembrane interactions. Scott et al.4 has presented a fairly simple model for a low-temperature PEM fuel cell with a vaporized methanol/oxygen feed. Considering the anode as a porous electrode consisting of an electronically conducting catalyst structure thinly coated with an ion-selective polymer electrolyte, Baxter et al.5 has presented a mathematical model for the fuel cell anode. Sundmacher et al.6 has analyzed the dynamic behavior of a DMFC system, with a simulation model of all relevant physicochemical phenomena: a model using a four-step reaction mechanism as the basis for the anode kinetics. Their research proved that methanol crossover could be reduced by periodic pulsed methanol feeding. Kulikovsky et al.7,8 has developed a two-dimensional, across-the-channel mathematical model for gas feed and liquid feed DMFC, respectively. However, in both these (1) Nguyen, T. V.; White, R. E. J. Electrochem. Soc. 1993, 140, 21782186. (2) Ge, S. H.; Yi, B. L.; Xu, H. F. J. Chem. Ind. Eng. (China) 1999, 50, 39-48. (3) Verbrugge, M. W. J. Electrochem. Soc 1989, 136, 417-423. (4) Scott, K.; Tamma, W.; Cruickshank, J. J. Power Sources 1997, 65, 159-171. (5) Baxter, S. R.; Battaglia, V. S.; White, R. E. J. Electrochem. Soc. 1999, 146 (2), 437-447. (6) Sundmacher, K.; Schultz, T.; Zhou, S., et al. Chem. Eng. Sci. 2000, 56, 333-341. (7) Kulikovsky, A. A.; Divisek, J.; Kornyshev, A. A. J Electrochem. Soc. 2000, 147 (3), 953-959. (8) Kulikovsky, A. A. J. Appl. Electrochem. 2000, 30, 1005-1014.

10.1021/ef0102723 CCC: $22.00 © 2002 American Chemical Society Published on Web 10/12/2002

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Figure 1. Sketch of the fuel cell. Abbreviations are CC, current collector; CH, fluid channel; DA, anode diffusion layer; RA, anode reaction layer; MM, membrane; RC, cathode reaction layer; and DC, cathode diffusion layer. Notions of positions of the interfaces are shown beneath.

papers the important phenomenon of methanol crossover to form mixed potential at the cathode was ignored. Wang et al.9 has established an advanced two-phase DMFC model. The model fully accounts for the mixed potential effects of methanol oxidation at the cathode as a result of methanol crossover caused by diffusion, convection and electro-osmosis. Simulation of the direct methanol fuel cell has been described in a series of papers.10-12 Thermodynamic framework for a multicomponent membrane was established to provide a means to describe the gradients in electrochemical potential for species in the membrane.10 The driving force for transport and kinetic phenomenon are quantified explicitly based on thermodynamic analysis.11 Using these results, the third paper12 simulated a DMFC operating in the steady state, explicitly considering the problem of methanol concentration and the effect of methanol crossover on cell performance and attempts to optimize the efficiency of the DMFC with the materials that are presently available. The major objective of this work is to develop an twodimensional mathematical with consideration of existence of the channel, which can be used to study the distributions of current densities, distributions of reactants concentrations, methanol crossover through the membrane, and polarization characteristics. A more complex mathematical model including thermal model will be discussed in later paper.

Fan et al.

membrane spatially separates the fuel cell. The catalyst layers between the membrane and the diffusion layer are domain for electrochemical reactions. The catalyst layers provide electron, proton, and reactants channels. In this work, the computational domain consists of five layers: anode diffusion layer, anode catalyst layer, polymer membrane, cathode catalyst layer, and cathode diffusion layer. The assumptions used in this model are 1. conditions are isothermal; 2. porosity is uniform; 3. Tafel kinetics governs the electrochemical reaction; 4. the pressure of the liquid mixture is kept fixed in the channels; 5. the pressure of gas at the cathode side is constant; 6. that there are two different types of pores in the catalyst and diffusion layers: hydrophobic pores and hydrophilic pores. The gas and liquid are transported via hydrophobic and hydrophilic pores, respectively; 7. at the cathode side, in the hydrophobic pores, water saturation conditions exist; 8. the water produced by electrochemical reaction at the cathode side is accounted for in the overall liquid balance in the cell. 2.1. Transport of Gases on the Cathode Side. The diffusion flux of the kth component in a multicomponent mixture of gases in a free space is defined by the Stefan-Maxwell relation:10 n

∑ l)1

The conventional scheme of a DMFC fuel cell is shown in Figure 1. The fuel cell can be divided into seven segments. The two outer segments are graphite-plate current collectors with flow channels for reactants and products to enter and exit the cell, respectively. The adjacent segments are electronically conducting porous diffusion layers that allow for even distribution of reactants to the anode and cathode. The centric polymer (9) Wang, Z. H.; Wang, C. Y. Mathematical Modeling of Liquid-Feed Direct Methanol Fuel Cells. In Proceedings of the 199th Conference of the ECS, 2001. (10) Meyers, J. P.; Newman, J. J Electrochem. Soc. 2002, 149, A710A717. (11) Meyers, J. P.; Newman, J. J Electrochem. Soc. 2002, 149, A718A728. (12) Meyers, J. P.; Newman, J. J. Electrochem. Soc. 2002, 149, A729-A735.

Dkl

) -c∇ξk

(4)

where n is number of components, ξk is the mole fraction of the component k, Nk is its molar flux, c is the total molar concentration of the mixture, and Dkl ) Dlk is the binary diffusion coefficient of molecules k in the gas of molecules l. However, Stefan-Maxwell relations (eq 4) define only n - 1 fluxes. In view of the n ξk ) 1, the nth relation is a sum of the equality ∑k)1 other n - 1.So this equation system is not closed. Dohle11 offered the following model equation with a superposition of Stefan-Maxwell diffusion and Knudsen diffusion to express the mass transport in the diffusion layer and catalyst layer:

Nk 2. Description of Mathematical Model

ξlNk - ξkNl

DK k

n

+

∑ l)1

ξlNk - ξkNl Dkl

) -c∇ξk

(5)

The first term on the left-hand side of eq 5 describe the Knudsen diffusion. The model allows one to interpolate between both mechanisms of gas transport in the diffusion layer (the fluxes is mainly supported by molecular diffusion) and catalyst layers (the fluxes is mainly supported by Knudsen diffusion) by proper diffusion coefficient. In the catalyst layers, the gas reactants dissolve in water before they are subjected to the electrochemical reaction. In this model, we ingore this phenomena and assume that they are transported as components of the gas mixtures. Under steady-state conditions, the continuity equation for species k has the form

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∇Nk ) Rk

(6)

where Rk is the rate of kth species production/consumption in the electrochemical reaction. In the cathode compartment, since nitrogen does not participate in the electrochemical reaction, there is no flux of nitrogen. It is assumed that water saturation conditions exist in the hydrophobic pores on the cathode side; the flux of water vapor is also zero. The model equation of gas flow reduces to a single equation for the oxygen mole fraction, ξO2:

∇(-c∇ξO2) ) NO2∇

(

)

ξN2 ξW 1 + + + K DO2 DN2O2 DO2W

(

)

ξN2 ξW 1 + + ∇NO2 (7) K DO2 DN2O2 DO2W

where the flux NO2 and its derivative ∇NO2 are defined by

(

)

ξN2 ξW 1 + + NO2 ) -c∇ξO2 K DO2 DN2O2 DO2W ∇NO2 ) -

SO2 (i + iM c ) nF c

I)

1 H

v)

where ic is the rate of charged particle loss in the cathodic reaction and iM c is the rate of the parasitic reaction with the products of methanol ionization in the cathode catalyst layer: this reaction consumes oxygen and does not contribute to current generation. 2.2. Potentials. We introduce the electrical potentials of the membrane phase φm and carbon phase φa,φc following the concept of mean parameters. The two types of potential govern the motion of protons and that of electrons, respectively. Taking local electroneutrality into account, the electronic current and ionic current produced or consumed in the catalyst layers lead to a voltage drop via Ohm’s law according to

(10)

∫0Hdy∫0L∇(σm∇φm) dx

(15)

where H is the height of the catalyst layer and L the width. As φc|x)x5 is taken to be zero, then φa|x)x0 - φc|x)x5 ) φa|x)x0 gives total voltage drop across the whole fuel cell. Thus, the current-voltage (I-V) curve is I vs φc|x)x0. 2.3. Velocity and Pressure of Liquid. Taking into account the two forces acting on the fluid: drag caused by ion movement under electric field (electro-osmotic effect) and pressure gradient (Poiseuille flow in hydrophilic pores), we yield the convective flow velocity of the liquid in the polymer membrane related to the gradients of membrane phase potential φm and pressure p according to Schlo¨gl:12

(8)

(9)

j ) (σ∇φ

the anode or cathode catalyst layer volume, and considering that the ion current vanishes at the catalyst/ backing layers interface and that the sum of ion currents through the top and bottom is also zero, we obtain

Kp Kφ cfzfF∇φm ∇p µ µ

(16)

In the gas diffuser, eq 16, with only the pressure gradient term, is used to characterize the liquid flow since the fluid is not charged. In the absence of specific absorption of pore-fluid ions onto the membrane structure, the following electroneutrality expression applies:

zfcf )

∑i zici

(17)

For the membrane of polymer membrane electrolyte fuel cell, the only mobile ions in the membrane pore fluid are hydrogen ions; thus, eq 17 can be written as

- zfcf ) cH

(18)

where CH is the proton concentration in the membrane phase. In accordance with eq 18, eq 16 becomes

The governing equation for each sort of current follows the current conservation

Kp Kφ v ) - cHF∇φm - ∇p µ µ

∇j ) i

Neglecting the change in flux of liquid caused by the anodic reaction for the small fraction of methanol in the mixture, and taking into account the significant amount of water produced in the cathodic reaction, we define the velocity of the liquid mixture by the mass conservation

(11)

Thus, the three potentials obey the following equations:

[

-ia, x1 < x e x2 x2 < x < x3 ∇(σm∇φm) ) 0, ic, x3 e x < x4

[

0, x < x ∇(σa∇φa) ) i , x e x1 < x a 1 2 ∇(σc∇φc) )

[

]

] ]

-ic, x3 < x e x4 0, x4 < x

(12)

F∇v ) RW c (13) (14)

where ia is the rate of charged particles production in the anodic reaction. After obtaining the phase potentials, the current density in the external circuit I can be calculated by applying Gauss’ theorem to eq 12. Integrating eq 12 over

x3 e x e x4

(19)

(20)

where RW c is the rate of water produced in the cathodic electrochemical reaction. Taking the divergence of eq 19 and combining it with eq 20, we can obtain the relation for pressure distribution as follows:

µ ∇(kp∇p) ) - cHF∇(kφ∇φm) - Rw F c

(21)

Equation 21 means that there is no volume source of pressure in the diffusion layers, one volume source,

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Energy & Fuels, Vol. 16, No. 6, 2002 Table 1. Conditions and Parameters quantity

Operating temperature (°C) pressure of anode channel (atm) pressure of cathode channel (atm) channel width (cm) fuel cell width (cm) back layer thickness (cm) catalyst layer thickness (cm) membrane thickness (cm) Ra Rc γa γc methanol concentration in indthe channel (mol cm-3) oxygen number density fraction in the channel water vapor number density fraction in the channel nitrogen number density fraction in the channel transfer current density o at anode, ia,ref (A cm-3) transfer current density at o (A cm-3) cathode, ic,ref methanol diffusion coefficient in the water, DM (cm2 s-1) methanol diffusion coefficient in the water, DM (cm2 s-1) Knudsen diffusion coefficient of oxygen, DkO2 (cm2s-1) binary diffusion coefficient DN2O2 (cm2 s-1) binary diffusion coefficient DO2W (cm2 s-1)

value 80 2.5 1.6 0.15 0.3 0.015 0.002 0.003 0.5 2.0 1.0 0.5 0.002 0.65 0.35 0.0 0.01 1.0× 10-5 6.71× 10-5 2.15 × 10-5 0.07853 (backing layer) 0.00785 (catalyst layer) 0.02669 0.03698

electro-osmotic flux in the anode catalyst and membrane layers, and two volume sources in the cathode catalyst layer: electro-osmotic flux and water production. 2.4. Transport of Methanol. Under a given membrane phase potential, solving eq 21 gives the pressure distribution in the fuel cell. Then we can obtain the velocity distribution of liquid methanol/water solution. The methanol flux consist of the molecular diffusion flux due to the concentration gradient (Fick’s law) and the

Figure 2. Pressure distribution in the electrode.

Fan et al.

convective flux caused by the liquid flow as a whole. Thus, the expression for this flux reads:

NM ) -DM∇CM + vCM

(22)

The mass conservation equation for methanol can be described by the source-sink terms of the material linked via Faraday’s law to the local transfer current density, as follows:

[

SM ia, x1 < x e x2 nF ∇NM ) S M - iM , x3 e x < x4 nF c -

]

(23)

where iM c is the rate of the “parasitic” reaction at the cathode catalyst layer caused by the methanol crossover through membrane from the anode side. 2.5. Transfer Currents. Methanol oxidization on the anode catalyst layer and oxygen reduction on the cathode catalyst layer, both involve several electrons transfer, are complicated multistage process.6 Considering the normal operation voltage, this model employs the Tafel equation by a fit of available experimental data to describe the transfer currents at the anode:7

ia ) i0a,ref

( ) ( CM CM,ref

γa

exp

RaF (φ - φm) RT a

)

(24)

At the cathode catalyst layer, the oxygen reduction current must provide not only the net cell current density (through the external circuit) but also the parasitic current density from methanol crossover. The rate equation at the cathode is as follows: 0 ic + iM c ) ic,ref

( ) ( cO2

cO2,ref

γc

exp

R cF (φ - φc) RT m

)

(25)

Methanol, which penetrates through the membrane to the cathode catalyst layer, form mixed potentials. The “parasitic” reaction will consume the oxygen, however,

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Figure 3. x-direction velocity distribution in the electrode.

Figure 4. y-direction velocity distribution in the electrode.

the electrons and ions produced in this reaction do not contribute to current generation. The transfer current 8 iM c of reaction is as follows: 0 iM c ) ia,ref

( ) ( CM CM,ref

γa

exp

RaF (φ - φc) RT m

)

(26)

This rate is similar to the anodic reaction rate given by eq 24 with the cathodic potential difference. 2.6. Boundary Conditions. The computational domain covers a part of the cell assembly, which is repeated periodically along y-direction. Therefore, on the bottom and top ends (y ) 0 and y ) H) periodic boundary conditions for all variables are imposed: f|y)0 ) f|y)H, where f stands for all potentials, concentrations, and pressure.

The x component of proton current density at the catalyst/backing layers interface is equal to zero, giving Von Neumann boundary condition ∂φm/∂x ) 0. Between the channels the carbon phase potentials are fixed (φa0,φc0 ) 0). The solution of the problem for a given φa0 provides the mean current density in the fuel cell, i.e., a point on the I-V curve. The electron current does not flow to the gas channel and the membrane, the normal component of electron current at the channel surfaces is zero ∂φa,c/∂x ) 0, and the same condition is imposed on both sides of the membrane. At the surfaces of the current collectors the normal component of fluid velocity is zero, which means ∂p/∂x ) 0. As the membrane/catalyst interface is assumed to be impermeable to gases, the flux of gases is zero at this

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interface ∂ξO2/∂x ) 0. The same condition is imposed on surfaces of metal electrodes. In the fluid channels, all concentrations are assumed to be fixed. It is assumed that the crossover methanol has been consumed completely at the catalyst/diffusion layers interface on the cathode side; that is, cM ) 0 at x4. 3. Results and Discussion The computational domains of species concentration equations, pressure equation of liquid mixture and potentials equations cover different layer. The problem in the coupled domain is solved without the use of boundary conditions at the interfaces, all the domains are considered as entirety. The method adopted in this work is to use proper boundary value for the parameters, such as diffusion coefficients, rather than boundary conditions for the dependent variables at the interface between layers of different properties. Model equations were converted to a finite-difference form by the control volume method. An orthogonal nonuniform grid for computational discretion was introduced in this work. Equation 23 for methanol concentration is of convection-diffusion type and is formally analogous to eq 7; the hybrid scheme was adopted to solve these equations. The results presented below were obtained for the parameters and conditions listed in Table 1. Figure 2 shows the liquid pressure distribution within the electrode. Pressure drops along the electrode ydirection are small except near the channel/collector corners where both the x-direction and y-direction velocities are highest. Most of the pressure drops are located in the x-direction between the anode catalyst layer and the cathode catalyst layer. The x-direction and y-direction velocities are given in Figure 3 and 4. As supported by the pressure-drop profiles, these velocities are highest near the edges. So, the methanol flux supported by convection process will dominate near the edges of the current collectors. The y-direction velocities are smaller than the x-direction velocities in the most regions. Thus, the fluid mainly flows across the membrane, and the x-direction flux is larger than that of the y-direction, which may cause low concentration in the catalyst layer in front of the collectors. Figure 5 displays the distribution of current densities (mean current density in the cell is 200 mA/cm2): (a) and (c) are the contour lines of electron current densities, and (b) is the contour lines of proton current density. It can be known from this figure that the contour lines of electron current densities in both anode and cathode reveal strong peaks near the edges of current collectors (at y ) 0.075 and y ) 0.225 cm). Physically, the electrons produced opposite the fuel channel flow to the nearest point of the current collector, i.e., the edges collect all the current that is produced in front of the channels. The peak current density is many times higher than the mean current density through the fuel cell, and it may produce local Joule overheating, which will be studied in later work. Correspondingly, the proton density near the edges is higher than other section for the effect of the motion of electrons; however, the collection is not so strong as the current density. The distribution of methanol concentration and oxygen mole fraction is shown as Figure 6. We can find that the reactants concentrations are very low in the catalyst

Figure 5. Contour maps of current density (200 mA cm-2): (a) electron current density in the anode compartment, (b) proton current density in the membrane, (c) electron current density in the cathode compartment.

layer in front of the current collectors for the limitation for transport of species. The utility of catalyst is reduced; one can remove catalyst from the shaded regions, but has no effect on the performance of the cell. To verify our model, we compared results with experimental data obtained by Scott.16 Figure 7 shows the simulated and experimental current-voltage characteristics for the LFDMFC. The numerical model has been run with the conditions similar to experimental case and other parameters reported in Table 1. It (13) Bernardi, D. M.; Verbrugge, M. W. AICHE J. 1991, 37, 11511163. (14) Dohle, H.; Divisek, J.; Jung, R. J. Power Sources 2000, 86, 469477. (15) SchlO ¨ gl, R. Ber. Bunsen-Ges. Phys. Chem. 1966, 70, 400-414. (16) Scott, K.; Argyropoulos, P.; Sundmacher, K. J Electroanal. Chem. 1999, 477, 97-110.

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Figure 8. Methanol flux, which permeates through membrane.

increasing electro-osmotic flux, even leads to a growth of inverse convective flux of methanol. 4. Conclusions

Figure 6. Contour maps of reactants concentration: (a) methanol concentration, (b) oxygen relative number density.

A comprehensive two-dimension mathematical model is adopted to study the characteristics of liquid-feed DMFC. The distribution of reactants concentrations, the distributions of current densities and current-voltage curve has been obtained by numerical computation, and then the crossover of methanol is analyzed qualitatively. The computational results indicate the presence of the fluid channels on both sides of the fuel cell leads to the formation of complex two-dimensional fields of reactants concentration and current density. A shaded region forms behind the contact of the current collector plates and the diffusion layer, where there is a lack of methanol and oxygen. The current densities near the edges of the channels are many times higher than those of the other section: theses current densities may produce high temperature in this regions and then cause degradation of membrane/electrode assembly. At the high current density, the inverse pressure gradient can be formed in the cathode catalyst layer. This pressure gradient drives the flux of liquid toward the anode and reduces methanol crossover. Acknowledgment. The authors express their thanks to Natural Science Foundation of Zhejiang Province (People’s Republic of China) and National Nature Foundation of China for support. Nomenclature

Figure 7. Comparison between model and experimental cell polarization data. Experimental data: (]) 2.0M, (O) 0.5M methanol concentration. 90 °C, 2 bar air pressure. Solid lines model data.

indicates that the predicted results agree well with experimental data. Figure 8 shows the crossover of methanol at the different current density. At low current density, the relative value of methanol crossover is very high. The methanol crossover decrease when the mean current density in the cell increases. We know that the electroosmotic flux increases with higher current density; that is, more protons moving from the anode side to the cathode side will cause more methanol flow to the cathode side. However, the pressure of liquid in the cathode catalyst layer increase, this compensates the

C: molar concentration, mol cm-3 CH: proton molar concentration in the membrane, mol cm-3 D: diffusion coefficient, cm2 s-1 F: Faraday constant ) 96487 C/mol H: cell height, cm i: transfer current density, A cm-3 iM c : rate of the parasitic reaction in the cathode catalyst layer, A cm-3 I: mean current density, A cm-2 Kp: hydraulic permeability, cm2 Kφ: electrokinetic permeability, cm2 L: catalyst layer width, cm n: component number and electron number participating in the electrochemical reaction N: molar flux, mol cm-2 s-1 p: pressure of the liquid, atm R: gas constant and production/consumption rate of material

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S: stoichiometric coefficient T: cell temperature, K v: liquid velocity, cm s-1 Zi: charge number (i * f) Greek Symbols R: φ: γ: µ: F: σ: ξ:

transfer coefficient potential, V concentration parameter fluid viscosity, kg cm-1 s-1 density of the liquid, mol cm-3 conductivity, Ω1-cm-1 mole fraction

Fan et al. W: liquid water Subscripts a: anode c: cathode f: membrane species (the fixed charge site species) kl: component kl m: membrane M: methanol N2: nitrogen O2: oxygen ref: reference w: water vapor

Superscripts K: Knudsen diffusion

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