Three-dimensional two-phase mass transport model for direct methanol fuel cells

Transcription

1 Available online at Electrochimica Acta 53 ( Three-dimensional two-phase mass transport model for direct methanol fuel cells W.W. Yang, T.S. Zhao,C.Xu Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, SAR, China Received 18 June 2007; received in revised form 29 July 2007; accepted 30 July 2007 Available online 6 August 2007 Abstract A three-dimensional (3D steady-state model for liquid feed direct methanol fuel cells (DMFC is presented in this paper. This 3D mass transport model is formed by integrating five sub-models, including a modified drift-flux model for the anode flow field, a two-phase mass transport model for the porous anode, a single-phase model for the polymer electrolyte membrane, a two-phase mass transport model for the porous cathode, and a homogeneous mist-flow model for the cathode flow field. The two-phase mass transport models take account the effect of non-equilibrium evaporation/ condensation at the gas-liquid interface. A 3D computer code is then developed based on the integrated model. After being validated against the experimental data reported in the literature, the code was used to investigate numerically transport behaviors at the DMFC anode and their effects on cell performance Elsevier Ltd. All rights reserved. Keywords: Direct methanol fuel cell (DMFC; Two-phase; Mass transport; Model; Methanol crossover 1. Introduction Many efforts have been directed at the development of direct methanol fuel cells (DMFC for powering portable electronic devices because this type of fuel cell offers the promises of high efficiency, high energy density, low emission and simple structure [1]. However, widespread applications of the DMFC technology is still hindered by several technological challenges, such as the low rate of methanol oxidation reaction (MOR on the anode, and the mixed over-potential caused by the methanol crossover to the cathode. In addition, coupled electrochemical reactions and transport of different species including reactants, products, electrons, protons as well as heat, make it difficult to achieve the optimal design and operation of the DMFC. In an effort to resolve these challenging problems, mathematical modeling of the DMFC system plays an import role, as it can provide a powerful and economical tool to analyze the intricate transport processes, which are rather difficult to be studied experimentally. Corresponding author. Tel.: ; fax: address: metzhao@ust.hk (T.S. Zhao. Previous in situ flow visualization studies [2 4] have revealed that the mass transport of methanol virtually takes place in the liquid gas two-phase flow through the porous electrode structure. Hence, to realistically simulate the mass transport process at the DMFC anode, the development of the two-phase mass transport model is needed. To date, several two-phase models for DMFCs have been reported in the literature. Kulikovsky [5] presented a simple model of the flow with gaseous bubbles in the anode channel, which is based on the experimental findings of Yang et al. [2]. In the model, the effect of bubbles on cell performance is virtually equivalent to lowering the stoichiometry of anode flow. Murgia et al. [6] presented a one-dimensional (1D model based on phenomenological transport equations. In order to consider the two-phase flow interaction in the diffusion layer, they introduced a Gaussian function to approximately account for the influence of the capillary pressure on the effective gas porosity. Wang and Wang [7] modeled the DMFC using the two-dimensional (2D two-phase mixture flow model, in which the species in the liquid and gas phase are assumed to be at the thermodynamic equilibrium condition and the catalyst layers are treated to be an interface without thickness. Birgersson et al. [8] studied numerically the anode mass transport in two-phase flow in a 2D domain using the model by Wang and /$ see front matter 2007 Elsevier Ltd. All rights reserved. doi: /j.electacta

2 854 W.W. Yang et al. / Electrochimica Acta 53 ( Wang [7]. Divisek et al. [9] presented a 2D two-phase model for the DMFC with the computational domain restricted to the membrane electrode assembly only. In their work, the presence of hydrophilic and hydrophobic pores was taken into account by introducing an experimental-data-fitted correlation equation between capillary pressure and saturation, which is different from the Leverette function as used elsewhere [7]. Although methanol crossover was considered, the mixed potential effect was not addressed in their work. Rice and Faghri [10] proposed a 2D transient, multiphase model for a passive DMFC. Noteworthy is that the evaporation/condensation of methanol and water was formulated in a manner to capture non-equilibrium effects between phases, which differed from the thermodynamic equilibrium assumption employed elsewhere [7,8]. More recently, Yang and Zhao [11] presented a 2D two-phase mass transport model, which was developed based on multiphase flow theory in a porous medium without defining the mixture pressure of liquid and gas phase. Another feature of this model is that the assumption of constant gas-phase pressure is eliminated. Relatively, few papers have been reported on the development of a three-dimensional (3D two-phase DMFC model, which can capture the species distributions and the transport limitations along any direction inside the DMFC. Ge and Liu [12] reported a 3D two-phase model for liquid-feed DMFC and numerically studied various transport phenomena, the cell performance and the effects of methanol crossover. More recently, a similar 3D model based on the multiphase mixture theory for liquid-feed DMFC was also presented by Liu and Wang [13]. The model was used to numerically investigate the distributions of methanol, oxygen, methanol crossover as well as current density. Most recently, the same model was used to study the transport of liquid water in a DMFC [14]. The objective of this work was to extend our 2D two-phase mass transport model for liquid-feed direct methanol fuel cells [11] to a 3D one. The two-phase mass transport in the anode and cathode porous regions is formulated based on multiphase flow theory in porous media without defining the mixture pressure of gas and liquid and assuming a constant gas pressure in the porous regions. The interaction between the phases in this 3D model is captured by taking into account the effect of nonequilibrium evaporation/condensation at the phase interface, as opposed to the assumption of thermodynamic equilibrium condition between the phases in other models [7,8,13,14]. In addition, the model eliminates the assumption of constant gas pressure, as were used previously [15,16]. The rest of the article is organized as follows: We first present the mathematical model, followed by the validation of the model. Finally the model is used to investigate numerically transport behaviors at the DMFC anode and their effects on cell performance. 2. Mathematical model 2.1. Formulation We consider a three-dimensional single DMFC sketched in Fig. 1, which consists of anode and cathode flow channels, dif- Fig. 1. Computational domain. fusion layers, catalyst layers, and a membrane. The 3D model is presented by dividing the entire computational domain into five regions: an anode porous region (ADL and ACL, a cathode porous region (CDL and CCL, a polymer electrolyte membrane (MEM region, an anode flow channel (AFC, and a cathode flow channel (CFC Anode porous region On the DMFC anode, methanol is supplied in the anode flow channel, transfers through the diffusion layer to the catalyst layer, where part of methanol is electro-chemically oxidized to form gas carbon dioxide and current, while the remainder of methanol permeates the membrane to the cathode. The produced carbon dioxide will transfer back through the diffusion layer to the flow channel, and is finally removed by the methanol solution stream. In the anode porous region, we are interested in four major variables, including liquid methanol concentration (C M,l, liquid pressure (p l, gas-void fraction (1 s 1 and methanol vapor concentration (C MV,g. The governing equations corresponding to each of these variables can be given by extending the 2D model [11]: C M,l : Kk rl p l μ l ] C M,l DM,l eff C M,l = Ṙ M,l (1 ( p l : Kk rl p l = ṁ l (2 μ l /ρ l 1 s l : Kk ([ ] ] rg dp c (1 s l + p l μ g /ρ g d(1 s l = ṁ g (3 and C MV,g : Kk ] rg p g C MV,g DMV,g eff μ C MV,g g = Ṙ MV,g (4

3 W.W. Yang et al. / Electrochimica Acta 53 ( Table 1 Constitutive correlations and definitions Parameters Expressions Capillary pressure p c = p g { p 1 = σ cos θ c (ε/k 0.5 J(s 1.417(1 s 2.120(1 s (1 s 3 0 <θ J(s = c s 2.120s s 3 90 <θ c < 180 Relative permeabilities k rl = s 3 Liquid k rg =(1 s 3 Gas Effective diffusion coefficients of species [11] Di,g eff = D i,gε 1.5 (1 s 1.5 i :O 2, WV, MV D M,l ε 1.5 s 1.5 ADL DM eff = ε + ε N ε/(d M,l ε 1.5 s (ε N /(D M,N ε 1.5 N ] ACL D M,N ε 1.5 MEM { MH2 O R W M M R M ADL General generation rate of mass in liquid phase ṁ l,a = (M H2 O + M M j a 6F M H 2 O R W M M R M ACL { MH2 ( O R w CDL ṁ l,c = jc M H2 O 2F I p M H2 O 6Fδ R w CCL { ccl MH2 O R W + M M R M ADL General generation rate of mass in gas phase ṁ g,a = j a M CO2 6F + M H 2 O R W + M M R M ACL { MH2 O R w CDL ṁ g,c = j c M O2 4F + M I para CO 2 + M H2 O 6Fδ R w CCL { { ccl 0 R Mole generation rate of species Ṙ O2 =, Ṙ WV,c = w CDL Ṙ M,l = j c { 4F R M R w j a 6F R M, Ṙ MV,a = CCL { R M, Ṙ R WV,a = M { R w ADL R w ACL The interfacial transfer rate of methanol between the liquid and gas in the anode porous region is given by: R M = A lg h lg s l (1 s l (psat MV p MV RT where p sat MV denotes the saturation pressure of methanol vapor. Note that the interfacial transfer of methanol between the phases is finally embodied in the general source terms on the right-hand sides of Eqs. (1 (4. The detailed expressions of the source terms are listed in Table Cathode porous region On the DMFC cathode, oxygen, traveling with the air or pure oxygen stream in the flow channel, transfers through the diffusion layer to the reaction layer, where the oxygen reduction reaction (ORR takes place to form water. The produced water, along with the water permeated from the anode, transfers back through the diffusion layer to the flow channel, and is swept out by the flow stream. In the cathode porous region, four major variables we are interested in, including oxygen concentration (C O2,g, gas pressure (p g, liquid saturation (s l and water vapor concentration (C WV,g. The governing equations of mass conservation corresponding to the four variables are, respectively, given by [11]: (5 C O2,g : Kk rg p g,c μ g ] C O2,g DO eff 2,g C O 2,g = Ṙ O2,g (6 p g : Kk rg μ g /ρ g s l : Kk rl μ l /ρ l ] p g = ṁ g (7 ( dp c ds s l + p g ] = ṁ l (8 and C WV,g : Kk ] rg p g C WV,g DWV,g eff μ C WV,g g = Ṙ WV,g (9 Note that the interfacial transfer rate of water between the liquid and gas phase is introduced here to account for the effect of water evaporation and condensation [15]: R w = k e ( εsl ρ l M H2 O k c ( ε(1 sl y WV RT (y WV p g p sat WV, (y WV p g p sat WV, y WVp g <p sat WV y WVp g >p sat WV (10 where p sat WV and y WV represent the saturation pressure of water vapor and mole fraction of water vapor in the cathode gas phase,

4 856 W.W. Yang et al. / Electrochimica Acta 53 ( respectively. Finally, this interfacial transfer of water between the phases is considered in the source terms on the right-hand sides of Eqs. (7 (9. The detailed expressions of the source terms are listed in Table Membrane Unlike in the anode and cathode porous regions, in the polymer electrolyte membrane, transport of liquid phases (dissolved methanol and liquid water needs to be considered only, as the membrane is usually regarded as a gas insulator. Transport of methanol through the membrane generally depends on molecular diffusion, electro-osmotic drag and convection. Accordingly, the flux of methanol crossover (N M is given by: N M = D eff M C M + n d,m I F ( K μ l p l,c a δ mem C M (11 With respect to the water transport, the flux due to diffusion mode can be ignored if the membrane is in equilibrium with liquid water on the both sides. Hence, the flux of water crossover (N W is given by: I N W = n d,h2 O F ρ l M H2 O K μ l p l,c a δ mem ( Anode flow field In the anode flow field, the onset of two-phase flow can be observed even at very low current densities due to the release of gas CO 2 under operating conditions [2,4]. Accordingly, the variables in the flow channel that we are interested in include methanol concentration ( C MeOH, gas-void fraction (1 s l, and liquid pressure (p l. To date, only a few investigators have taken account of the effect of two-phase flow on the mass transport of methanol in the anode flow field, in most of them [7,12 14] the homogeneous model was used. Note that the use of the homogeneous model can be justified only when gas bubbles are rather small and homogeneous distributed in the channel. However, the visualization of the flow behavior has revealed that bubbly/slug flow is the most frequently-encountered flow pattern in the anode-flow channel under typical DMFC operating conditions. The large slip velocity between the liquid and gas bubble makes the assumption of the same velocity between liquid and gas in the homogeneous model be unrealistic. In the present work, the modified one-dimensional drift-flux model [11] is used and the governing equations of mass conservation corresponding to each variable are given by: C MeOH : ( s l ū l C MeOH = N M,afc A afc (13 1 s l : (ρ l s l ū l + [ρ g (1 s l ū g ] = m l,afc A afc and p l : + m g,afc A afc ( (ρ l s l ū l ū l = p l 1 2 (ρ l s l f m D h (ū l 2 (15 The gas velocity is related to the liquid velocity according to the drift-flux model: ū g = C 0 1 s l ū g + s l ū l ] + u gj (16 where the distribution factor C 0 and drift velocity u gj, are given, respectively by: C 0 = ( ρ g /ρ l {1.0 exp[ 18.0(1 s l ]} (17 and u gj = ( H s (ρ l ρ g gh w (18 H w ρ l Cathode flow field Under typical DMFC operating conditions, there is predominant gas flow through the cathode channel. For simplicity, the homogeneous model is used to model the two-phase mass transport behavior in the cathode flow field, in which the velocity of both phase are assumed to be the same. Note, again, that the application of the homogeneous model can be justified when liquid water transferred from the diffusion layer can be quickly removed by the sufficiently high-speed gas stream. In addition, since the ratio of channel length to height is rather large, the two-phase flow and associated mass transport can be treated as a one-dimensional problem along the flow direction. With these assumptions, the two-phase mass transport equations for each species in the cathode flow field are given, respectively, by: C i : s l : and 1 s l ū g C i ] = N i,cfc A cfc i :O 2, Vapor (19 (ρ l s l ū l + [ρ g (1 s l ū g ]= m l,cfc A cfc + m g,cfc A cfc (20 1 p g : 2 (ρ l s l ū l ū l [ρ g(1 s l ū g ū g ] = p g 1 2 ρ f m g ū 2 g (21 D h Up to this point, we have presented all the governing equations that describe the two-phase mass transport in all the regions of a DMFC. To make the above governing equations closed, some constitutive correlations and definitions are needed. These include capillary pressure, relative permeability for both gas and liquid phase, effective diffusion coefficients for each species and the source terms. All these correlations and associated nomenclatures are listed in Table Boundary conditions At the inlet of each flow field, all the variables are given based on the reactant supplying conditions, while the boundary conditions at the outlet of each flow fields are given by assuming that the flow is fully developed and the gradient of concentration of each species is zero. With respect to the external walls, the non-slip flow boundary condition for flow and the impermeablewall condition for species and mass transfer are specified. The

5 W.W. Yang et al. / Electrochimica Acta 53 ( internal interfacial conditions in the sandwiched DMFC are given based on the principle that the continuity and mass/species flux balance are required for each interface to satisfy the general mass and species conservation of the entire cell. The details of interfacial boundary condition can be found elsewhere [11] Electrochemical kinetics On the DMFC anode, the Tafel-like expression is used to model the MOR kinetics: ( γ ( j a = A v,a j0,meoh ref C M αa F CMeOH ref exp RT η a (22 where the reaction order, γ, is related to the methanol concentration and assumed to be the zero-order when methanol concentration is higher than a reference value. Otherwise, the first-order reaction is specified. With respect to the electro-reduction reaction of oxygen on the cathode, the modified first-order Tafel-like kinetics based on a liquid-film covered agglomerate model [11] is used, which gives: ( j c = A v,c j0,o ref C O2 /k H,O2 2 CO ref 2 ( αc F exp RT η c ξ 1 ξ 2 (23 where C O2 represents oxygen concentration in the gas pores. Three factors k H,O2, ξ 1 and ξ 2 denote the Henry factor to capture the effect of dissolving process, the factor in view of the transfer resistance in liquid film and the factor in view of the transport resistance in the agglomerate, respectively Current balance and cell voltage The protons and electrons produced by the MOR on the anode transfer to the cathode through the membrane and the external circuit, respectively. The cell current density can be calculated by: ACL I cell = j a dxdydz (24 A cell To account for methanol crossover, the parasitic current density is used to express the rate of methanol crossover: I para = 6F N M dx dz (25 A cell where the molar flux of methanol crossover, N M, is given by Eq. (11. On the cathode, it is assumed that both the cell current and the parasitic current are completely consumed by the ORR, i.e.: CCL I cell + I para = j c dx dy dz (26 A cell For a given anode overpotential, η a, the cell current density, I cell, and the parasitic current density, I para, can be determined from Eqs. (24 and (25, respectively. Then, the cathode mixed overpotential, η c, with the effect of methanol crossover, can be obtained with the help of Eqs. (23 and (26. Finally, the cell voltage can be determined from: ( V Cell = V 0 η a η c I cell R contact + δ mem (27 κ where V 0, R contact and κ denote the thermodynamic equilibrium voltage of a DMFC, the contact resistance and the proton conductivity of the membrane, respectively. 3. Results and discussion The above-described governing equations for the cell geometric dimensions and operating parameters listed in Table 2 subjected to the boundary conditions, along with the constitutive relations shown in Table 1 and electrochemical properties listed in Table 3, are solved numerically using a self-written code, which was developed based on the SIMPLE algorithm with the Finite-Volume-Method Model validation The predicted cell performance of the DMFC fed with 0.5 and 2.0 M methanol solution and experimental data are compared in Fig. 2 under the same anode flow rate of 1.0 ml/min and the same temperature of 75 C. The more details about the experimental data and test conditions can be found elsewhere [17]. It is worth mentioning that all the numerical curves in Fig. 2 are predicted with using the same set of parameters listed in Tables 2 and 3. Generally speaking, the predicted polarization curves are in reasonable agreement with the experimental data, as can be seen in Fig. 2. However, there still exist relatively large deviations between the prediction and experimental data for the case of higher methanol concentration (2.0 M. For instance, at the high current density (4000 A/m 2, the standard maximum deviation in voltage between the numerical and experimental results is about 35%. This large deviation is probably caused by the simplifications and assumptions adapted in the model formulation Fig. 2. Comparison between numerical predictions (solid lines and experimental data (symbols [17] for various methanol concentrations.

6 858 W.W. Yang et al. / Electrochimica Acta 53 ( Table 2 Baseline conditions Parameters Symbols Value Thicknesses of functional layers δ adl, δ acl, δ mem, δ ccl, δ cdl (mm 0.27, 0.02, 0.13, 0.02, 0.27 Width, height and length of channel w c, H c, L c (mm 1.0, 1.0, 30 Width of rib w r (mm 1.0 Operation temperature T (K Inlet pressure p in l,anode, pin g,cathode (Pa , Inlet concentrations of methanol and methanol vapor CM, in anode, Cin MV, anode (mol/m , CMV sat Inlet oxygen concentration CO in 2 (mol/m Inlet water vapor concentration CWV,a in, Cin WV,c (mol/m3 CWV sat Inlet liquid saturation sa in, c 1.0, 0.0 and the uncertainties in the parameters used in calculation. It also can be found from the figure that, for the lower methanol concentration (0.5 M, increasing the current density resulted in a sharp drop in cell voltage, meaning that the methanol is entirely consumed in the catalyst layer and the limiting current density was reached as the result of mass transport limitation. However, for the higher methanol concentration (2.0 M, the cell voltage decreases almost linearly with current density toward zero with no mass transport limitation. This polarization behavior means that for high methanol concentrations, the voltage loss is predominately caused by the sluggish reaction and the mixed overpotential as the result of severe methanol crossover. All these polarization behaviors are consistent with those revealed by experiments. Table 3 Physicochemical properties Parameters Symbols Value Ref. Porosity, permeability ADL, CDL ε, K (m 2 0.7, [18] ACL, CCL ε, K (m 2 0.3, MEM ε, K (m 2 0.3, Contact angle DL, CL θ c ( 120, 100 Nafion volume ACL, CCL ε N 0.4 Diffusivities MeOH in water D M,l (m 2 /s [8] MeOH in Nafion D M,N (m 2 /s e [2436(1/333 1/T] [19] MeOH in vapor D MV,g (m 2 /s T T 2 [7] O 2 in gas D O2,g (m 2 /s ( T [7] O 2 in water D O2,l (m 2 /s [20] O 2 in Nafion D O2,N (m 2 /s [20] Water vapor D WV,g (m 2 /s ( T [7] Viscosity of gas, liquid phase μ g, μ l (kg/m s , [23] Electro-osmotic drag coefficients n d,h2 O, n d,m 2.5, n d,h2 Ox M [22] Evaporation rate constant for water k e (1/(atm s [6] Condensation rate constant for water k c (mol/(atm s cm [6] Interfacial transfer rate constant for methanol h 1g (m 2 /s [9] Specific interfacial area between liquid and gas A 1g (m [9] Proton conductivity in membrane κ ( 1 m 1 7.3e [1268(1/298 1/T] [19] Henry law constant for oxygen k H,O2 e ( 666/T+14.1 /R/T [20] Henry law constant for methanol k H,M (atm 0.096e (T 273 [7] The saturation pressure of water vapor log 10 p sat WV (atm (T (T (T [21] The saturation pressure of methanol vapor p sat MV (atm k Hx M,l [7] Thermodynamic voltage V 0 (V 1.21 [7] Transfer coefficient of anode α a 0.5 Transfer coefficient of cathode α c 1.0 Anode exchange current density A v,a j0,m ref Cathode exchange current density A v,c j0,o ref 2 (A/m Anode reference concentration CM ref 100 [7] Cathode reference concentration CO ref 2 (mol/m [10]

7 W.W. Yang et al. / Electrochimica Acta 53 ( Fig. 3. Distribution of methanol concentration in the anode porous region and membrane (unit: m 3 /s Variations in methanol concentration and gas-void fraction The distributions of liquid methanol concentration in the DMFC anode are shown in Fig. 3. As can be seen from this figure, methanol concentration in the diffusion layer is higher in the open-channel region than under the ribs, demonstrating that the flow-field ribs causes methanol to be distributed unevenly over the reaction area. Also, methanol concentration is the highest at the channel-diffusion layer interface, but becomes lower towards the catalyst layer, where methanol is consumed by the MOR. It can also be observed that, along the channel direction, methanol concentration at the channel-diffusion layer interface decreases from the inlet to the outlet as the result of the MOR. Note that the variation in methanol concentration along any directions in the cell and the mass-transport limitation can only be captured by the present 3D model. The distribution of gas-void fraction in the anode porous region is shown in Fig. 4. Note that the gas-void fraction in the anode porous region (α was obtained by subtracting anode liquid saturation from unity. Apparently, the gas-void fraction is higher in the catalyst layer than that in the regions close to the flow channel due to the fact that carbon dioxide, generated by MOR in the catalyst layer, is transferred back from the catalyst layer to the flow channel and then removed by the bulk stream in the flow channel. Moreover, the gas tends to accumulate in the region under the flow-field ribs, resulting in a higher gas coverage level there. It also can be observed that, along the channel direction, the gas-void fraction in the downstream region is much higher than that in the upstream region as the result of the accumulation of gas bubbles in the channel, which in turn hinders the gas bubble transferring out of the porous region. The gas in the anode channel along with the trapped gas bubbles in the anode electrode reduces the available space for liquid methanol transport, thus limiting the access of methanol from the channel to the catalyst layer and lowering the cell performance. The variations in cross-sectional averaged methanol concentration and gas-void fraction in the anode flow field as well as the current density distribution along the channel direction are shown in Fig. 5. It can be seen from Fig. 5a that methanol concentration lowers about 6% from the inlet to the outlet as the result of the MOR, while the local current density, shown in Fig. 5c, lowers about 20% along the flow direction. It is understood that the local current density is almost proportional to the local molar flux of methanol transferred to the catalyst layer at the limiting current condition, i.e.: I = 6Fk(C AFC M CACL M 6FkCAFC M (28 where k and CM AFC, respectively, denote the local mass transfer coefficient of methanol and local methanol concentration in the anode channel. Eq. (28 implies that the variation in current density along the channel is attributed not only to the variation Fig. 4. Distribution of gas-void fraction in the anode porous region. Fig. 5. Variations in (a cross-sectional averaged methanol concentration, (b cross-sectional averaged gas-void fraction, (c local current density along the channel length.

8 860 W.W. Yang et al. / Electrochimica Acta 53 ( in methanol concentration but also to the variation in the mass transfer coefficient. Comparing Fig. 5a and c, indicates a significant decrease in the mass transfer coefficient along the channel direction. The decrease in the mass transfer coefficient is caused by the gas blockages at the channel-diffusion layer interface, as evident from the variation in gas-void fraction along the channel direction shown in Fig. 5b, which shows that the gas-void fraction in the channel significantly increases along the flow direction Effect of methanol flow rate Fig. 6 shows the variations in methanol concentration and gas-void fraction along the channel at various methanol flow rates. As can be seen from this figure, the distribution of methanol becomes more uniform with increasing the methanol flow rate. Correspondingly, as shown in Fig. 6b, the gas-void fraction in the channel lowers with increasing the flow rate. A more uniform distribution of methanol and lower gas coverage level tends to result in better cell performance. Fig. 7 shows the variations in the limiting current density for various methanol concentrations and at different flow rates. Fig. 7. Limiting current densities for various methanol concentration and anode flow rate. Note that the limiting current density represents the maximum flux of methanol transferring from the channel to the catalyst layer. For a given methanol concentration, it is clear that the limiting current density increases with the flow rate. Also, it can be found that for each given flow rate the limiting current density increases almost linearly with methanol concentration. This finding is consistent with the experimental results reported elsewhere [17]. It must be pointed out that although a higher flow rate in general will lead to better cell performance, an increase in the flow rate results in a larger pressure drop through the flow field, which, in turn, will cause a larger pumping power Overall mass transport coefficient The overall mass transfer coefficient of methanol from the flow field to the catalyst layer can be represented by [17]: k tot = I limitng 6F C Channel Methanol (29 Fig. 6. Variations in (a cross-sectional averaged methanol concentration and (b cross-sectional averaged gas-void fraction along the channel length for various anode flow rate. Fig. 8. Overall mass-transfer coefficient for various methanol flow rate.

9 W.W. Yang et al. / Electrochimica Acta 53 ( Based on Eq. (29, we calculated the overall mass transfer coefficients of methanol for various flow rates. The results are presented in Fig. 8, along with the equation fitted with the experimental data [17]. It is seen that the predicted results in the present work are in good agreement with the experimental data. 4. Conclusion A three-dimensional steady-state model for liquid feed direct methanol fuel cells is presented in this work. This integrated 3D model consists of five sub-models, each simulating a specific component of the DMFC. A 3D computer code is then developed based on the integrated model. After being validated against the experimental data reported in the literature, the code was used to investigate numerically transport behaviors at the DMFC anode and their effects on cell performance. The numerical results indicate that: (1 the non-uniform distributions of methanol and the gas-void fraction over the reaction layer strongly influence the distributions of local current density. These in-plane distributions of species and their effects on cell performance can only be captured by the 3D model; (2 the higher methanol flow rate results in more uniform distribution of methanol along the flow direction and relatively lower gas-void fraction in the channel, thereby increasing the mass transport limitation of methanol from the channel region to the catalyst layer; (3 the overall mass transfer coefficient of methanol predicted by the present 3D model is in good agreement with the experimental results reported in the literature. Acknowledgements The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no Appendix A. Nomenclature A afc cross-section area of anode flow channel (m 2 A cfc cross-section area of cathode flow channel (m 2 A cell cell area (m 2 A lg interfacial specific area between liquid and gas phase (m 2 /m 3 A v specific area (m 2 /m 3 C molar concentration (mol/m 3 C cross-sectional averaged concentration in the flow channel (mol/m 3 D diffusivity (m 2 /s F Faraday constant, 96,478 (C/mol I current density (A/m 2 I cell cell current density (A/m 2 I current vector (A/m 2 I para parasitic current resulting from methanol crossover (A/m 2 j 0 exchange current density (A/m 2 j a anode current density (A/m 3 j c cathode current density (A/m 3 K permeability of porous material (m 2 k c condensation rate (mol/(atm s m 3 k e evaporation rate (1/(atm s k H Henry s law constant (Pa k r relative permeability ṁ source term in mass conservation equation (kg/m 3 s M molecular weight (kg/mol N mol flux (mol/m 2 s n d electro-osmotic drag coefficient p c capillary pressure (Pa p g gas phase pressure (Pa p l liquid phase pressure (Pa R gas constant (J/(mol K Ṙ source term in species conservation equation (mol/(m 3 s R interfacial species transfer rate (mol/(m 3 s R contact Ohmic contact resistance ( m 2 s liquid saturation s cross-sectional average liquid saturation in the channel T temperature (K V 0 thermodynamic equilibrium voltage (V V cell cell voltage (V x coordinate, m, or mole fraction in liquid solution (mol/mol y coordinate, m, or mole fraction in gas mixture (mol/mol Greek letters α gas-void fraction (1-s α a anode transfer coefficient at anode α c cathode transfer coefficient at cathode δ thickness of porous layer (m ε porosity of porous medium γ reaction order of ORR η overpotential (V κ ionic conductivity of membrane ( 1 m 1 μ viscosity (kg/(m s θ c contact angle ( ρ density (kg/m 3 σ interfacial tension (N/m Superscripts eff effective value in inlet condition ref reference value sat saturated value Subscripts a anode acl anode catalyst layer adl anode diffusion layer afc anode flow channel c cathode ccl cathode catalyst layer cdl cathode diffusion layer cfc cathode flow channel g gas phase

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