# An Improved Tank in Series Model for the Direct Methanol Fuel Cell

Authors: Valery A. Danilov; Joeri Denayer; Il Moon
Journal: International Journal of Electrochemistry
 (2011)

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## Abstract

This study presents an improved TSR model for Direct Methanol Fuel Cell (DMFC) with mass and energy balance equations taking into account gas evolution in the anode channels. The TSR model includes a modified charge balance equation suitable for potentiostatic fuel cell operation mode.  Polarization curves calculated with the improved TSR model agree with experimental data from the literature. The developed TSR model is able to approximate two-dimensional concentration, temperature, and current density profiles in DMFC with parallel flow field.


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## Body

## 1. Introduction

The direct methanol fuel cell is a potential energy source to replace batteries for portable electronics. The main attractive features of DMFC include high energy density and nearly zero recharge time. Numerous 1D and 3D models are available for DMFC in the literature. Kulikovsky presented a 1D isothermal model for DMFC [1]. Murgia et al. developed 1D multicomponent steady-state isothermal model for liquid-feed DMFC [2]. Hyun et al. predicted the DMFC performance for different anode flow field designs using computational simulations [3]. Danilov et al. developed the two-phase model with a submodel for interphase transfer. They used CFD-based DMFC model to visualize and analyze the gas evolution and flow patterns in the anode channels [4–6]. In contrast to the CFD technique with highly complex process description, lumped parameter models with a reactor network are computationally fast, approximating flow patterns and concentration profiles in fuel cells. Sundmacher et al. [7–10] developed CSTR and TSR models for DMFC operating in galvanostatic mode. They used a tank in series reactor model to investigate the influence of the anode flow fields on the steady-state and dynamic behavior of DMFC with parallel, spot, and rhomboidal designs. Our previous studies focused on TSR model development for solid oxide fuel cells [11, 12]. The objective of this research is to modify this TSR model for DMFC operating in potentiostatic mode.


## 2. Model Formulation

### 2.1. TSR Model for Cocurrent Flow DMFC

Fuel and oxidant flow inside the fuel cell with parallel channels can be approximated as a reactor network complying with a tank in series model. Figure1 illustrates the application of a tank in series reactor (TSR) model for cocurrent flow mode in a planar fuel cell with parallel flow field design.Figure 1
TSR model for cocurrent flow DMFC.The developed TSR model for DMFC is based on the following assumptions.(i)
Each anode and cathode compartment is treated as a TSR.(ii)
Electrochemical reactions occur at the electrode/membrane interfaces.(iii)
Fuel cell operates under the potentiostatic operating mode and constant pressure.(iv)
Negligible contact resistances between components.(v)
Negligible evaporation and condensation in cathode channels.(vi)
Uniform distribution of fuel and oxidant in channels.Balance equations are composed for channel compartment and catalyst layer in each tank. For fuel cells with cocurrent flow mode, the mass balance equations for the anode and cathode channels are written as follows:(1)Vi,jAdCA,i,j(k)dt=CA,i,j-1(k)Fi,j-1A-Ci,j(k)FL,i,jA-NA,i,j(k),k=MeOH,CO2,H2O,Vi,jCdCC,i,j(k)dt=CC,i,j-1(k)Fi,j-1C-CC,i,j(k)Fi,jC-NC,i,j(k),k=O2,H2O,N2.
Gas evolution in the anode channels is defined by the following balance equation for the gas phase:
(2)Vi,jAdφG,i,jAdt=φG,i,j-1AFi,j-1A-φG,i,jAFi,jA-rdesorp,i,jAρmol,LA,
In cocurrent flow mode, the fuel and oxidant outgoing streams from tank i,j-1, are the inlet streams into tank i,j. Therefore, the inlet flow rate for each tank depends on the outgoing flow rate of the previous tank:
(3)Fi,jA=Fi,j-1A-(NA,i,jMeOH+NA,i,jCO2+NA,i,jH2O)(CA,i,jMeOH+CA,i,jCO2+CA,i,jH2O),Fi,jC=Fi,j-1C-(NC,i,jO2+NC,i,jH2O)(CC,i,jO2+CC,i,jH2O+CC,i,jN2),
Here Vi,jA, Vi,jC represent the volume of anode and cathode channels in tank (i,j), Vi,jA=VA/(ninj), Vi,jC=VC/(ninj); Si,j electrode area in tank (i,j), Si,j=Scell/(ninj).Component balance equations for the anode and cathode catalyst layers reflect changes due to consumption and production of species via reaction and mass transfer:(4)Vcat,i,jAdCA,cat,i,j(k)dt=NA,i,j(k)-rA,i,j(k),k=MeOH,CO2,H2O,(5)Vcat,i,jCdCC,cat,i,j(k)dt=NC,i,j(k)-rC,i,j(k),k=O2,H2O.
Here Vcat,i,jC, Vcat,i,jA correspond to  the volume of anode and cathode catalyst layer in tank (i,j), Vcat,i,jC=VcatC/(ninj), Vcat,i,jA=VcatA/(ninj).The energy balances for the channels and the MEA structure in tanki,j are written as follows:
(6)Vi,jAρmol,LACp,LAdTL,i,jAdt=(ρmol,LAFi,j-1AΔhL,i,j-1A-ρmol,LAFi,jAΔhL,i,jA)+qL,i,jA+qcoll,i,jA,(7)VG,i,jCρmol,GCCp,GCdTG,i,jCdt=(ρmol,GCFi,j-1CΔhG,i,j-1C-ρmol,GCFi,jCΔhG,i,jC)+qG,i,jC+qcoll,i,jC,(8)Vi,jSρCpSdTi,jSdt=qi,jS-qL,i,jA-qG,i,jC.
Here qG,i,jC,qL,i,jA represent the convective heat transferred from the MEA to the channel in tank i,j; qcoll,i,jC,qcoll,i,jA represent heat source due to heat transfer from channel to the collector; qi,jS represents the heat generation in MEA in tank i,j.The charge balance equations for anode and cathode electrode/membrane interfaces are given by(9)CdlAdηi,jAdt=Ii,jcell-Ii,jA,CdlCdηi,jCdt=-Ii,jcell-(Ii,jC+Ii,jpar).Forj = 1 tanks, representing the inlet sections of the anode and cathode channels, we obtain the following balance equations:
(10)Vi,1AdCA,i,1(k)dt=CA,i,in(k)Fi,inA-CA,i,1(k)Fi,1A-NA,i,1(k),k=MeOH,CO2,H2O,(11)Vi,1CdCC,i,1(k)dt=CC,i,in(k)Fi,inC-CC,i,1(k)Fi,1C-NC,i,1(k),k=O2,H2O,N2,(12)Vi,1AdφG,i,1Adt=φG,i,inAFi,inA-φG,i,1AFi,1A-rdesorp,i,1Aρmol,LA,(13)Fi,1A=Fi,inA-(NA,i,1MeOH+NA,i,1CO2+NA,i,1H2O)(CA,i,1MeOH+CA,i,1CO2+CA,i,1H2O),(14)Fi,1C=Fi,inC-(NC,i,1O2+NC,i,1H2O)(CC,i,1O2+CC,i,1H2O+CC,i,1N2),
where   Fi,inA=FinA/ni is the inlet anode volumetric flow rate for (i,j) tank with j = 1; Fi,inC=FinC/ni is the inlet cathode volumetric flow rate for (i,j) tank with j = 1.Forj=nj tanks, corresponding to the outlet section of fuel and oxidant gas channels, we define mean outlet variables as follows:
(15)FoutA=∑i=1niFi,njA,CoutMeOH=∑i=1niCA,i,njMeOHFi,njAFoutA,TL,outA=∑i=1niTL,i,njAFi,njAFoutA,FoutC=∑i=1niFi,njC,CoutO2=∑i=1niCC,i,njO2Fi,njCFoutC,TG,outC=∑i=1niTG,i,njCFi,njCFoutC.
The developed mathematical model includes the following phenomena:(i)
electrochemical oxidation of methanol at the anode electrode/membrane interface,(ii)
electrochemical reduction of oxygen at the cathode electrode/membrane interface,(iii)
charge balances at anode and cathode electrode/membrane interfaces,(iv)
energy balances in gas channels and in MEA.


### 2.2. The Electrochemistry Submodel

#### 2.2.1. Electrode Current

The rate of electrochemical reactions is defined by Butler-Volmer equation. The anode reaction rate is(16)Ii,jA=I0A(Ccat,i,jMeOHCrefMeOH)0.5×[exp⁡(αAAFRTi,j(ηi,jA-ηeqA))-exp⁡(-αCAFRTi,j(ηi,jA-ηeqA))],
where I0A is the anode exchange current density.The cathode reaction rate is(17)Ii,jC=I0CCcat,i,jO2CrefO2[exp⁡(αACFRTi,j(ηi,jC-ηeqC))-exp⁡(-αCCFRTi,j(ηi,jC-ηeqC))],
where I0C is the cathode exchange current density.Parasitic current is calculated as methanol molar flux from anode catalyst layer to cathode catalyst layer with a linear profile in the membrane:(18)Ii,jpar=neAFDmMeOHδm(CA,cat,i,jMeOH-0).Source terms in the component balance equations reflect changes due to the consumption or production of species via reaction or mass transfer. For component balance equations in anode catalyst layer (4), the rate of electrochemical reaction is given by
(19)rA,i,jMeOH=Si,jνMeOHneAFIi,jA,rA,i,jCO2=Si,jνCO2neAFIi,jA,rA,i,jH2O=Si,jνH2OneAF(Ii,jA)-Si,jnd,i,jIi,jAF,
where nd,i,j represents the electro-osmotic drag coefficient.For balance equations in cathode catalyst layer (5), the rate of electrochemical reaction is
(20)rC,i,jO2=Si,jνO2neCF|Ii,jC|,
The source term for water on the cathode side includes electrochemical reaction and transfer from anode side
(21)rC,i,jH2O=Si,jνH2OneCF|Ii,jC|+Si,jnd,i,j|Ii,jC|F,
An empirical equation for calculating electro-osmotic drag coefficient nd is taken from Ren et al. [13]. For the inert nitrogen component, the corresponding source term is equal to zero (rC,i,jN2=0).


#### 2.2.2. Electrolyte Current

For fuel cells operating under potentiostatic mode (Ecell=const), the current density in PEM electrolyte media is estimated from the voltage equation:
(22)Ii,jcell=(Ei,jOCV-Ecell-ηact,i,jA+ηact,i,jC)Ri,jOhmic,
where ηact,i,jA is the anode activation overpotential, ηact,i,jA=ηi,jA-ηeqA; ηact,i,jC is the cathode activation overpotential, ηact,i,jC=ηi,jC-ηeqC; ηeqA, ηeqC is the anodic and cathodic equilibrium potential differences;  ηi,jA, ηi,jC is the anode and cathode potential differences at the electrode/membrane interfaces; Ri,jOhmic is the ohmic resistance.As shown in  AppendixB, voltage (22) is applicable for calculating the current in PEM under a linear membrane phase potential profile. Membrane conductivity is given by the following empirical relation [14]:
(23)σm=(0.0005139Cm-0.000326)exp⁡(1268(1303-1T)).


### 2.3. Heat Transfer Submodel

The source terms in energy balance equations for anode (6) and cathode channels (7) are defined as follows:
(24)qi,jA=αASi,j(Ti,jS-Ti,jA),(25)qi,jC=αCSi,j(Ti,jS-Ti,jC).
The source term in energy balance equation (8) for MEA is
(26)qi,jS=δmSi,jRi,jOhmic(Ii,jcell)2-ΔHRNC,i,jH2O+Si,jEcellIi,jcell+(Si,jαC+∑kCp(k)NC,i,j(k))(Ti,jC-Ti,jS)+(Si,jαA+∑kCp(k)NA,i,j(k))(Ti,jA-Ti,jS).
Additional source terms in energy equations (6) and (7) take into account heat transfer from channels to the anode and cathode collectors
(27)qcoll,i,jA=αcollASi,j(Tcoll-Ti,jA),(28)qcoll,i,jC=αcollCSi,j(Tcoll-Ti,jC).
Convective heat transfer coefficients αC and αA are calculated from empirical correlations for laminar heat transfer in channels. Heat transfer coefficients αcoll are found by a technique proposed by Siegel et al. [15].


### 2.4. Mass Transfer Submodel

Component molar fluxes for species on the cathode side are defined by mass transfer in gas phase:(29)Ni,j(k)=Si,jρmol,GβG,eff(k)(yi,j(k)-ycat,i,j(k)),k=O2,H2O,
where βG,eff(k) is the effective mass transfer coefficient in gas phase for k component; yi,j(k),ycat,i,j(k) are mole fraction of k component in (i,j) tank in channel and catalyst surface, respectively.The multicomponent mixture in gas phase includes carbon dioxide, methanol and water in equilibrium with the liquid phase. Component molar fluxes for species on the anode side are defined by mass transfer in liquid phase:(30)Ni,j(k)=Si,jρmol,LβL,eff(k)(xi,j(k)-xcat,i,j(k)),k=MeOH,CO2,H2O.


### 2.5. Gas Evolution

Gas evolution results from interphase mass transfer of carbon dioxide in the anode channels. The source term in balance equation for gas phase (2) describes the rate of desorption and absorption processes in the anode channels. The conventional submodel for estimating the source term is based on mass transfer equation
(31)rdesorp,i,j=Si,jkVρmol,G(yi,jCO2-ysat⁡,i,jCO2),
where kV is an empirical volumetric mass transfer coefficient.It is generally assumed that the multicomponent gas phase is in equilibrium with the liquid phase. Using the equilibrium condition, Danilov et al. [4–6] proposed the following equation for estimating the source term in anode channels:
(32)rdesorp,i,j=γ(ψ+rA,i,jCO2),
where γ is the splitting factor; ψ is the coefficient. The derivation of the auxiliary equation for coefficient ψ is given in [4–6]. The local splitting factor γ is found from solving the equilibrium flash equation [4–6]. It should be noted that new submodel (32) corresponds to an equilibrium model of multicomponent mass transfer between liquid and gas in the anode channels [4–6].According to the physical meaning of the mass balance equations, molar component concentration is defined as mixture concentrations for gas-liquid flow in channels. Component mole fraction in liquid phase(x(k)) can be expressed through mixture concentration (C(k)) with known gas volume fraction (φG). As shown in Appendix C, the condition of thermodynamic equilibrium in the multicomponent gas-liquid mixture gives the next relationship for the specie mole fraction in liquid phase
(33)x(k)=C(k)K(k)φGρmol,G+(1-φG)ρmol,L,
where K(k) is the distribution coefficient for k-component, K(k)=y(k)/x(k).


## 2.1. TSR Model for Cocurrent Flow DMFC

Fuel and oxidant flow inside the fuel cell with parallel channels can be approximated as a reactor network complying with a tank in series model. Figure1 illustrates the application of a tank in series reactor (TSR) model for cocurrent flow mode in a planar fuel cell with parallel flow field design.Figure 1
TSR model for cocurrent flow DMFC.The developed TSR model for DMFC is based on the following assumptions.(i)
Each anode and cathode compartment is treated as a TSR.(ii)
Electrochemical reactions occur at the electrode/membrane interfaces.(iii)
Fuel cell operates under the potentiostatic operating mode and constant pressure.(iv)
Negligible contact resistances between components.(v)
Negligible evaporation and condensation in cathode channels.(vi)
Uniform distribution of fuel and oxidant in channels.Balance equations are composed for channel compartment and catalyst layer in each tank. For fuel cells with cocurrent flow mode, the mass balance equations for the anode and cathode channels are written as follows:(1)Vi,jAdCA,i,j(k)dt=CA,i,j-1(k)Fi,j-1A-Ci,j(k)FL,i,jA-NA,i,j(k),k=MeOH,CO2,H2O,Vi,jCdCC,i,j(k)dt=CC,i,j-1(k)Fi,j-1C-CC,i,j(k)Fi,jC-NC,i,j(k),k=O2,H2O,N2.
Gas evolution in the anode channels is defined by the following balance equation for the gas phase:
(2)Vi,jAdφG,i,jAdt=φG,i,j-1AFi,j-1A-φG,i,jAFi,jA-rdesorp,i,jAρmol,LA,
In cocurrent flow mode, the fuel and oxidant outgoing streams from tank i,j-1, are the inlet streams into tank i,j. Therefore, the inlet flow rate for each tank depends on the outgoing flow rate of the previous tank:
(3)Fi,jA=Fi,j-1A-(NA,i,jMeOH+NA,i,jCO2+NA,i,jH2O)(CA,i,jMeOH+CA,i,jCO2+CA,i,jH2O),Fi,jC=Fi,j-1C-(NC,i,jO2+NC,i,jH2O)(CC,i,jO2+CC,i,jH2O+CC,i,jN2),
Here Vi,jA, Vi,jC represent the volume of anode and cathode channels in tank (i,j), Vi,jA=VA/(ninj), Vi,jC=VC/(ninj); Si,j electrode area in tank (i,j), Si,j=Scell/(ninj).Component balance equations for the anode and cathode catalyst layers reflect changes due to consumption and production of species via reaction and mass transfer:(4)Vcat,i,jAdCA,cat,i,j(k)dt=NA,i,j(k)-rA,i,j(k),k=MeOH,CO2,H2O,(5)Vcat,i,jCdCC,cat,i,j(k)dt=NC,i,j(k)-rC,i,j(k),k=O2,H2O.
Here Vcat,i,jC, Vcat,i,jA correspond to  the volume of anode and cathode catalyst layer in tank (i,j), Vcat,i,jC=VcatC/(ninj), Vcat,i,jA=VcatA/(ninj).The energy balances for the channels and the MEA structure in tanki,j are written as follows:
(6)Vi,jAρmol,LACp,LAdTL,i,jAdt=(ρmol,LAFi,j-1AΔhL,i,j-1A-ρmol,LAFi,jAΔhL,i,jA)+qL,i,jA+qcoll,i,jA,(7)VG,i,jCρmol,GCCp,GCdTG,i,jCdt=(ρmol,GCFi,j-1CΔhG,i,j-1C-ρmol,GCFi,jCΔhG,i,jC)+qG,i,jC+qcoll,i,jC,(8)Vi,jSρCpSdTi,jSdt=qi,jS-qL,i,jA-qG,i,jC.
Here qG,i,jC,qL,i,jA represent the convective heat transferred from the MEA to the channel in tank i,j; qcoll,i,jC,qcoll,i,jA represent heat source due to heat transfer from channel to the collector; qi,jS represents the heat generation in MEA in tank i,j.The charge balance equations for anode and cathode electrode/membrane interfaces are given by(9)CdlAdηi,jAdt=Ii,jcell-Ii,jA,CdlCdηi,jCdt=-Ii,jcell-(Ii,jC+Ii,jpar).Forj = 1 tanks, representing the inlet sections of the anode and cathode channels, we obtain the following balance equations:
(10)Vi,1AdCA,i,1(k)dt=CA,i,in(k)Fi,inA-CA,i,1(k)Fi,1A-NA,i,1(k),k=MeOH,CO2,H2O,(11)Vi,1CdCC,i,1(k)dt=CC,i,in(k)Fi,inC-CC,i,1(k)Fi,1C-NC,i,1(k),k=O2,H2O,N2,(12)Vi,1AdφG,i,1Adt=φG,i,inAFi,inA-φG,i,1AFi,1A-rdesorp,i,1Aρmol,LA,(13)Fi,1A=Fi,inA-(NA,i,1MeOH+NA,i,1CO2+NA,i,1H2O)(CA,i,1MeOH+CA,i,1CO2+CA,i,1H2O),(14)Fi,1C=Fi,inC-(NC,i,1O2+NC,i,1H2O)(CC,i,1O2+CC,i,1H2O+CC,i,1N2),
where   Fi,inA=FinA/ni is the inlet anode volumetric flow rate for (i,j) tank with j = 1; Fi,inC=FinC/ni is the inlet cathode volumetric flow rate for (i,j) tank with j = 1.Forj=nj tanks, corresponding to the outlet section of fuel and oxidant gas channels, we define mean outlet variables as follows:
(15)FoutA=∑i=1niFi,njA,CoutMeOH=∑i=1niCA,i,njMeOHFi,njAFoutA,TL,outA=∑i=1niTL,i,njAFi,njAFoutA,FoutC=∑i=1niFi,njC,CoutO2=∑i=1niCC,i,njO2Fi,njCFoutC,TG,outC=∑i=1niTG,i,njCFi,njCFoutC.
The developed mathematical model includes the following phenomena:(i)
electrochemical oxidation of methanol at the anode electrode/membrane interface,(ii)
electrochemical reduction of oxygen at the cathode electrode/membrane interface,(iii)
charge balances at anode and cathode electrode/membrane interfaces,(iv)
energy balances in gas channels and in MEA.


## 2.2. The Electrochemistry Submodel

### 2.2.1. Electrode Current

The rate of electrochemical reactions is defined by Butler-Volmer equation. The anode reaction rate is(16)Ii,jA=I0A(Ccat,i,jMeOHCrefMeOH)0.5×[exp⁡(αAAFRTi,j(ηi,jA-ηeqA))-exp⁡(-αCAFRTi,j(ηi,jA-ηeqA))],
where I0A is the anode exchange current density.The cathode reaction rate is(17)Ii,jC=I0CCcat,i,jO2CrefO2[exp⁡(αACFRTi,j(ηi,jC-ηeqC))-exp⁡(-αCCFRTi,j(ηi,jC-ηeqC))],
where I0C is the cathode exchange current density.Parasitic current is calculated as methanol molar flux from anode catalyst layer to cathode catalyst layer with a linear profile in the membrane:(18)Ii,jpar=neAFDmMeOHδm(CA,cat,i,jMeOH-0).Source terms in the component balance equations reflect changes due to the consumption or production of species via reaction or mass transfer. For component balance equations in anode catalyst layer (4), the rate of electrochemical reaction is given by
(19)rA,i,jMeOH=Si,jνMeOHneAFIi,jA,rA,i,jCO2=Si,jνCO2neAFIi,jA,rA,i,jH2O=Si,jνH2OneAF(Ii,jA)-Si,jnd,i,jIi,jAF,
where nd,i,j represents the electro-osmotic drag coefficient.For balance equations in cathode catalyst layer (5), the rate of electrochemical reaction is
(20)rC,i,jO2=Si,jνO2neCF|Ii,jC|,
The source term for water on the cathode side includes electrochemical reaction and transfer from anode side
(21)rC,i,jH2O=Si,jνH2OneCF|Ii,jC|+Si,jnd,i,j|Ii,jC|F,
An empirical equation for calculating electro-osmotic drag coefficient nd is taken from Ren et al. [13]. For the inert nitrogen component, the corresponding source term is equal to zero (rC,i,jN2=0).


### 2.2.2. Electrolyte Current

For fuel cells operating under potentiostatic mode (Ecell=const), the current density in PEM electrolyte media is estimated from the voltage equation:
(22)Ii,jcell=(Ei,jOCV-Ecell-ηact,i,jA+ηact,i,jC)Ri,jOhmic,
where ηact,i,jA is the anode activation overpotential, ηact,i,jA=ηi,jA-ηeqA; ηact,i,jC is the cathode activation overpotential, ηact,i,jC=ηi,jC-ηeqC; ηeqA, ηeqC is the anodic and cathodic equilibrium potential differences;  ηi,jA, ηi,jC is the anode and cathode potential differences at the electrode/membrane interfaces; Ri,jOhmic is the ohmic resistance.As shown in  AppendixB, voltage (22) is applicable for calculating the current in PEM under a linear membrane phase potential profile. Membrane conductivity is given by the following empirical relation [14]:
(23)σm=(0.0005139Cm-0.000326)exp⁡(1268(1303-1T)).


## 2.2.1. Electrode Current

The rate of electrochemical reactions is defined by Butler-Volmer equation. The anode reaction rate is(16)Ii,jA=I0A(Ccat,i,jMeOHCrefMeOH)0.5×[exp⁡(αAAFRTi,j(ηi,jA-ηeqA))-exp⁡(-αCAFRTi,j(ηi,jA-ηeqA))],
where I0A is the anode exchange current density.The cathode reaction rate is(17)Ii,jC=I0CCcat,i,jO2CrefO2[exp⁡(αACFRTi,j(ηi,jC-ηeqC))-exp⁡(-αCCFRTi,j(ηi,jC-ηeqC))],
where I0C is the cathode exchange current density.Parasitic current is calculated as methanol molar flux from anode catalyst layer to cathode catalyst layer with a linear profile in the membrane:(18)Ii,jpar=neAFDmMeOHδm(CA,cat,i,jMeOH-0).Source terms in the component balance equations reflect changes due to the consumption or production of species via reaction or mass transfer. For component balance equations in anode catalyst layer (4), the rate of electrochemical reaction is given by
(19)rA,i,jMeOH=Si,jνMeOHneAFIi,jA,rA,i,jCO2=Si,jνCO2neAFIi,jA,rA,i,jH2O=Si,jνH2OneAF(Ii,jA)-Si,jnd,i,jIi,jAF,
where nd,i,j represents the electro-osmotic drag coefficient.For balance equations in cathode catalyst layer (5), the rate of electrochemical reaction is
(20)rC,i,jO2=Si,jνO2neCF|Ii,jC|,
The source term for water on the cathode side includes electrochemical reaction and transfer from anode side
(21)rC,i,jH2O=Si,jνH2OneCF|Ii,jC|+Si,jnd,i,j|Ii,jC|F,
An empirical equation for calculating electro-osmotic drag coefficient nd is taken from Ren et al. [13]. For the inert nitrogen component, the corresponding source term is equal to zero (rC,i,jN2=0).


## 2.2.2. Electrolyte Current

For fuel cells operating under potentiostatic mode (Ecell=const), the current density in PEM electrolyte media is estimated from the voltage equation:
(22)Ii,jcell=(Ei,jOCV-Ecell-ηact,i,jA+ηact,i,jC)Ri,jOhmic,
where ηact,i,jA is the anode activation overpotential, ηact,i,jA=ηi,jA-ηeqA; ηact,i,jC is the cathode activation overpotential, ηact,i,jC=ηi,jC-ηeqC; ηeqA, ηeqC is the anodic and cathodic equilibrium potential differences;  ηi,jA, ηi,jC is the anode and cathode potential differences at the electrode/membrane interfaces; Ri,jOhmic is the ohmic resistance.As shown in  AppendixB, voltage (22) is applicable for calculating the current in PEM under a linear membrane phase potential profile. Membrane conductivity is given by the following empirical relation [14]:
(23)σm=(0.0005139Cm-0.000326)exp⁡(1268(1303-1T)).


## 2.3. Heat Transfer Submodel

The source terms in energy balance equations for anode (6) and cathode channels (7) are defined as follows:
(24)qi,jA=αASi,j(Ti,jS-Ti,jA),(25)qi,jC=αCSi,j(Ti,jS-Ti,jC).
The source term in energy balance equation (8) for MEA is
(26)qi,jS=δmSi,jRi,jOhmic(Ii,jcell)2-ΔHRNC,i,jH2O+Si,jEcellIi,jcell+(Si,jαC+∑kCp(k)NC,i,j(k))(Ti,jC-Ti,jS)+(Si,jαA+∑kCp(k)NA,i,j(k))(Ti,jA-Ti,jS).
Additional source terms in energy equations (6) and (7) take into account heat transfer from channels to the anode and cathode collectors
(27)qcoll,i,jA=αcollASi,j(Tcoll-Ti,jA),(28)qcoll,i,jC=αcollCSi,j(Tcoll-Ti,jC).
Convective heat transfer coefficients αC and αA are calculated from empirical correlations for laminar heat transfer in channels. Heat transfer coefficients αcoll are found by a technique proposed by Siegel et al. [15].


## 2.4. Mass Transfer Submodel

Component molar fluxes for species on the cathode side are defined by mass transfer in gas phase:(29)Ni,j(k)=Si,jρmol,GβG,eff(k)(yi,j(k)-ycat,i,j(k)),k=O2,H2O,
where βG,eff(k) is the effective mass transfer coefficient in gas phase for k component; yi,j(k),ycat,i,j(k) are mole fraction of k component in (i,j) tank in channel and catalyst surface, respectively.The multicomponent mixture in gas phase includes carbon dioxide, methanol and water in equilibrium with the liquid phase. Component molar fluxes for species on the anode side are defined by mass transfer in liquid phase:(30)Ni,j(k)=Si,jρmol,LβL,eff(k)(xi,j(k)-xcat,i,j(k)),k=MeOH,CO2,H2O.


## 2.5. Gas Evolution

Gas evolution results from interphase mass transfer of carbon dioxide in the anode channels. The source term in balance equation for gas phase (2) describes the rate of desorption and absorption processes in the anode channels. The conventional submodel for estimating the source term is based on mass transfer equation
(31)rdesorp,i,j=Si,jkVρmol,G(yi,jCO2-ysat⁡,i,jCO2),
where kV is an empirical volumetric mass transfer coefficient.It is generally assumed that the multicomponent gas phase is in equilibrium with the liquid phase. Using the equilibrium condition, Danilov et al. [4–6] proposed the following equation for estimating the source term in anode channels:
(32)rdesorp,i,j=γ(ψ+rA,i,jCO2),
where γ is the splitting factor; ψ is the coefficient. The derivation of the auxiliary equation for coefficient ψ is given in [4–6]. The local splitting factor γ is found from solving the equilibrium flash equation [4–6]. It should be noted that new submodel (32) corresponds to an equilibrium model of multicomponent mass transfer between liquid and gas in the anode channels [4–6].According to the physical meaning of the mass balance equations, molar component concentration is defined as mixture concentrations for gas-liquid flow in channels. Component mole fraction in liquid phase(x(k)) can be expressed through mixture concentration (C(k)) with known gas volume fraction (φG). As shown in Appendix C, the condition of thermodynamic equilibrium in the multicomponent gas-liquid mixture gives the next relationship for the specie mole fraction in liquid phase
(33)x(k)=C(k)K(k)φGρmol,G+(1-φG)ρmol,L,
where K(k) is the distribution coefficient for k-component, K(k)=y(k)/x(k).


## 3. Results and Discussion

Modeling and simulations are valuable tools for improving our understanding of the heat and mass transfer processes in fuel cells. To validate the improved TSR model, we used experimental data reported by Murgia et al. [2] for a 25 cm2 DMFC operating with 1.5 M and 1 M aqueous methanol feeds at 90°C. Operating conditions for this DMFC are listed in Table 1.Table 1
Operating conditions for the DMFC.
ParameterFuel flow rate, cm3/min15Oxidant flow rate, cm3/min1000MeOH  M in fuel1.5 M; 1 MCO2, m.fr. in fuel1.0 × 10−4O2 mol fraction in oxidant0.99Temperature,C90Pressure, atm1The number of tanks is set asni=4 and nj=4, corresponding to a TSR model with 272 nonlinear coupled first-order ordinary differential equations. The accepted number of tanks corresponds to the laminar flow regime in the channels with parallel flow field design. The developed TSR model was implemented in MATLAB, and it was initialized with the feed composition and temperature. The geometry and electrochemical parameters of DMFC are listed in Table 2.Table 2
Geometry and electrochemical parameters for DMFC.
ParameterAnodeCathodeExchange current densityI0, A m−247.10.042Reference concentrationCref, M20.034Charge transfer coefficientαA0.50.8Charge transfer coefficientαC0.51.2Channel height, m1×10−3Electrode areaS, m225×10−4Membrane widthδm, m2×10−4Catalyst widthδcat, m1×10−5Figure2 compares the experimental and calculated polarization curves for DMFC with operating conditions in Table 1. The solid line represents a DMFC model prediction with the improved TSR model.Figure 2
Comparison of the polarization curves predicted by TSR model with experimental data [2].Figure3 displays histograms with steady-state simulation results predicted by the TSR model for DMFC with 1.0 M feed of fuel at the anode and pure oxygen feed at the cathode. It should be noted that each tank is characterized by complete mixing, and distribution of concentration, temperature, and current density is presented by step change for tanks in series. Concentration of methanol in channel is decreased due to the anode electrochemical reaction. For the anode channels, carbon dioxide is the product of the anode electrochemical reaction. Gas content in the anode channels is increased from inlet to the outlet following the trend in molar CO2 concentration. The two-dimensional histograms indicate a strong coupling between the reactants and the current density distribution. The predicted current density profile is following the trend in methanol concentration in anode channels. Murgia et al. [2] obtained experimental polarization data by circulating aqueous methanol from a reservoir where CO2 was released to the atmosphere. The inlet CO2concentration in the liquid fuel in Table 1 corresponds to the gas-liquid equilibrium condition in the fuel reservoir. The predicted temperature profile in MEA is following the main trend in the current density distribution.Histograms with steady-state simulation results predicted by TSR model for DMFC with coflow mode of fuel and oxidant (Ecell=0.2V). Numerical modelling of fuel cell with anode 1.0 M feed of fuel and pure oxygen feed at the cathode. Temperature variation δTi,jS=Ti,jS-TinC.  Methanol molar concentration variation δCi,jMeOH=Ci,jMeOH-CinMeOH.  Carbon dioxide molar concentration variation δCi,jCO2=Ci,jCO2-CinCO2.
(a)(b)(c)(d)(e)(f)Figure4 presents histograms with steady-state simulation results predicted by the TSR model for DMFC with 1.5 M feed of fuel at the anode and pure oxygen feed at the cathode.Histograms with steady-state simulation results predicted by TSR model for DMFC with coflow mode of fuel and oxidant (Ecell=0.2V). Numerical modelling of fuel cell with anode 1.5 M feed of fuel and pure oxygen feed at the cathode. Temperature variation δTi,jS=Ti,jS-TinC. Methanol molar concentration variation δCi,jMeOH=Ci,jMeOH-CinMeOH. Carbon dioxide molar concentration variation δCi,jCO2=Ci,jCO2-CinCO2.
(a)(b)(c)(d)(e)(f)Gas management greatly influences the performance of the fuel cell. On the anode side, carbon dioxide is produced by electrochemical oxidation of the methanol. Inefficient removal of CO2 bubbles may block anode channels and decrease efficiency of the fuel cells due to limited mass transport and maldistribution of reactants. According to the conventional submodel, the interface flux is completely dependent on empirical mass transfer coefficient (kV). The predicted gas content in the anode channels can vary from 0 to 90% depending on the value of the mass transfer coefficient. Simulation results reveal that high values of the mass transfer coefficient (kV>6000s-1) predict gas evolution in the anode channels with high gas content (φ>40%). For low mass transfer coefficients (kV<6000s-1), on the other hand, φ<40%. The volumetric mass transfer coefficient kV=6000s-1 used in gas evolution submodel (31) provides the condition φ≈40% in the anode channels, which is close to the equilibrium condition given by the flash equation.As shown in Figure5, empirical model (31) and new submodel (32) for gas evolution predict similar trends in the gas content distribution. In contrast to nonequilibrium submodel (31) with empirical mass transfer, new rate expression (32) determines the gas content under equilibrium conditions without empirical coefficients.Histograms with gas content in anode channels predicted by TSR model for  DMFC with (a) empirical mass transfer (31) and (b) new gas evolution submodel (32) (Ecell=0.2V).
(a)(b)Comparison of DMFC performance with different fuel composition is given in Table3. Changing the fuel composition from 1 M to 1.5 M gives a small improvement in fuel cell performance. The mass transfer coefficient is presented in the form of limiting current density. Average current density approaches the value of limiting current densities under cell voltage Ecell=0.2V. The possible way of DMFC performance improvement is to intensify mass transfer from the channels to the catalyst layer.Table 3
Single DMFC cell simulation results(Ecell=0.2V).
ParameterOutlet gasMean parasiticMean limitingMean anodeFuelOxidantcontentcurrentcurrentcurrentutilizationutilizationφoutAI̅parI̅limAI̅AFUAUm3/m3A/m2A/m2A/m2%%1.0 M0.2739.170096905.20.018.11.5 M0.36329.8105149819.90.0111.8


## 4. Conclusions

This study presents a TSR model for DMFC operating in potentiostatic mode. The developed TSR model includes mass and energy balance equations in anode and cathode compartments together with gas evolution in anode channels. Modified charge balance equations are defined as interface boundary conditions. The possibility of evaluating two-dimensional profiles in the DMFC is one of the main advantages of the developed TSR model. The simulation results indicate the strong coupling between concentration, temperature and current density distribution in coflow DMFC. Taking into account the assumption of uniform distribution of fuel and oxidant in channels, the TSR model predicts the limiting performance of the DMFC under the given flow regime (number of tanks), mass transfer and catalyst activity. The improved TSR model allows studying the influence of different parameters on the DMFC performance. Results can be used to better understand and investigate the effects of various parameters and operating conditions on DMFC performance.



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*Source: 204937-2011-07-07.xml*