Improved Tank in Series Model for the Planar Solid Oxide Fuel Cell

1056

Ind. Eng. Chem. Res. 2011, 50, 1056–1069

Improved Tank in Series Model for the Planar Solid Oxide Fuel Cell Shahin Hosseini,† Valery A. Danilov,†,‡ Periasamy Vijay,† and Moses O. Tade´*,† Center For Process Systems Computations, Department of Chemical Engineering, Curtin UniVersity of Technology, GPO Box U1987, Perth, WA 6845, Australia, and Department of Processes and Unit Operations of Chemical Technology, Kazan State Technological UniVersity, Kazan 420015, Russian Federation

Models of different complexity are required in the iterative process of designing a solid oxide fuel cell (SOFC). Models having less complexity and computational dexterity are the ideal ones at the early stages. This work presents the development of an improved tank in series reactor model of the SOFC operating in cocurrent, countercurrent, and cross-current flow directions. The model, which accounts for the charge balances in the electrodes and electrolyte in addition to the component balances and the energy balances, is used for simulating the potentiostatic operation of the cell. The simulation results from the TSR model indicate the influence of flow direction on the steady state and dynamic performances of the cell. Among different flow directions, the coflow case is the most favorable for the planar SOFC, with improved performance. In response to a voltage step increase, the coflow case provides the most uniform transient behavior at different points of the cell. Despite the coflow direction, in which temperature dominates the slow dynamics of the local current density, in the low temperature regions of the counterflow and cross-flow cases, the slow dynamics of the current density tends to be characterized by the initial undershoot followed by a slower transient response that is due to the combined effects of the diffusion resistance within the porous electrode, hydrogen accumulation toward the fuel outlets, and the influence of the PEN temperature. 1. Introduction Solid oxide fuel cells (SOFCs) are potentially attractive electrochemical reactors due to their high efficiency and fuel flexibility.1The planar-type solid oxide fuel cell (SOFC) is desirable for many applications due to its compactness.2 In order to efficiently develop and optimize the planar SOFC stacks, it is convenient to have the capability of modeling different processes taking place in the cells, considering the effects of the operating conditions, the geometry, and the relative orientations of the fuel and air flows on the operation and performance. Modeling efforts for SOFCs are ongoing and can be broadly classified into two types, namely transport approaches and system approaches.3 Transport models range from onedimensional (1D) to three-dimensional (3D), which can be used to model transport processes occurring within the cell components. In 3D models the impact of geometry, flow configuration, and operating conditions on the overall performance of the cell and stack is the most important objective. For instance, Ferguson et al.4 introduced a 3D model for a typical H2/Air planar SOFC, which allows computation of the local distribution of temperature and the concentration of the chemical species. It was found that the counterflow configuration is optimal in the case of electrical efficiency in comparison with coflow and cross-flow cases. 2D models are generally used for unit cell simulations, and more examples of 3D and 2D models can be found in refs 2, and 5-8. Usually, two- and three-dimensional SOFC models are solved using the commercial CFD package. Although CFD techniques are powerful tools in the study of flow and transport phenomena inside fuel cells, such approaches for complete SOFC modeling, including chemical and electrochemical reaction mechanisms, lead to highly complex descriptions, which are time-consuming with respect to set up and simulation. Moreover, such models * To whom correspondence should be addressed. Telephone: 61+89266-7581. Fax: 61+8-9266-2681. E-mail: [email protected]. † Curtin University of Technology. ‡ Kazan State Technological University.

are not applicable for dynamic simulations and for control purposes.9 In this regard, quasi 2D, reduced 1D, and lumped parameter (0D) SOFC models seem to be more promising. Some work which deals with such models can be found in refs 10-14. In 1D SOFC models, generally the PEN structure is considered as a thin layer separating the fuel and the air channels, which are modeled as the plug flow reactors. Not all reactors are perfectly mixed (CSTR), nor do all reactors exhibit plug flow behavior.15 Thus, using ideal reactor networks as a nonideal SOFC model could be a computationally fast method facilitating prediction of the conversions and product distributions for solid oxide fuel cell that takes into account the reactor flow patterns. Ideal reactor networks have been successfully applied by some researchers. Krewer et al.9 developed a tank in series hydrodynamic model to investigate the DMFC (direct methanol fuel cell) anode flow fields of different parallel, spot, and rhomboidal designs. In further studies, the reduced models were integrated into full DMFC models.16 This enabled determination of the influence of flow field design on the steady state and dynamic behavior of DMFC. The purpose of the present work is to demonstrate the steady state and dynamic performances of an SOFC operated with different flow patterns using a tank in series (TIS) dynamic model of a planar SOFC. The model presented in this work accelerates 2D simulation of the planar SOFC under potentiostatic operating mode to get the effects of different coflow, counterflow, and cross-flow directions on the fuel and air utilization, distribution of current, temperature and species concentration and finally dynamic responses to a voltage step change. This model includes species mass balances to the gas channels and diffusion layers and energy balances for the fuel and air channels and also for the PEN structures. Moreover, charge balance equations were defined as the boundary condition at the electrode-electrolyte interfaces and facilitate analyzing the charge transfer process and calculation of the activation overpotential.

10.1021/ie101129k  2011 American Chemical Society Published on Web 12/20/2010

Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011

2. Model Formulation The main components of an SOFC consist of the fuel and air gas channels and a three layer region often referred to as the PEN structure (positive electrode/electrolyte/negative electrode). This model is based on the following assumptions: • Each fuel and air compartment is treated as a CSTR (continuous stirred tank reactor). • Electrochemical reactions occur at the electrode-electrolyte interfaces. • Fuel cell operates with hydrogen/air under the potentiostatic operating mode and constant total pressure. • Ohmic drops in current collectors and electric connections are negligible. • The gas mixtures in fuel and air compartments are treated according to the ideal gas law. • Radiation heat transfer is not considered. 2.1. CSTR Model of the Planar SOFC. Various phenomena occur in the fuel cell such as mass and heat transfer, charge transfer, and the electrochemical reactions. Component balance equations for fuel and air gas channels are written as follows: f FfmolVgas

dyH2 H2 ) yHin2Ffin - yH2Ffout - Ffmolkeff Sij(yH2 - yHcat2 ) dt

CCdL

dyH2O ) yHin2OFfin - yH2OFfout dt H2O Ffmolkeff Sij(yH2O - yHcat2O)

a FamolVgas

f FfmolVgas-i,j

f FfmolVgas-i,j

(2)

O2

dy O2 ) yOin2Fain - yO2Faout - Famolkeff Sij(yO2 - yOcat2 ) dt

(3) In fuel cells, the reacting species are diffusing from the gas channel through the porous electrode to the electrode/electrolyte interface; then the component balance equations for catalyst (diffusion) layers are written as follows: dyHcat2 H2 ) Ffmolkeff Sij(yH2 - yHcat2 ) - rH2 dt

(4)

dyHcat2O H2O ) Ffmolkeff Sij(yH2O - yHcat2O) + rH2O dt

(5)

dyOcat2 O2 ) Famolkeff Sij(yO2 - yOcat2 ) - rO2 dt

(6)

FfmolVfcat FfmolVfcat

FamolVacat

The energy balances for the fuel and air gas channels and the PEN structure (solid volume) are written as follows: f Vgas FfmolCfp

dTf ) (Ffin∆hfin - Ffout∆hfout) + qf dt

(7)

a Vgas FamolCap

dTa ) (Fain∆hain - Faout∆haout) + qa dt

(8)

dTPEN ) qPEN dt

(9)

VPENFCPEN p

CAdL

dηA ) (icell - iA) dt

a FamolVgas-i,j

dyHi,j2 H2 f f ) yi,j-1 - yHi,j2Fi,j Fi,j-1 dt f H2 2 Fmolkeff ) (12) Si,j(yHi,j2 - yHcat,i,j dyHi,j2O H2O f ) yi,j-1 Fi,j-1 - yHi,j2OFfi,j dt H2O 2O Ffmolkeff Si,j(yHi,j2O - yHcat,i,j )

(13)

dyOi,j2 O2 a a ) yi,j-1 Fi,j-1 - yOi,j2Fi,j dt a O2 2 Fmolkeff Si,j(yOi,j2 - yOcat,i,j )

(14)

In these equations, the first and second terms on the righthand side represent the convective mass transfer rates associated with the fuel and air flow streams. The last term on the right side of the above equations presents the species mass diffusion between the gas channels and catalyst layer. Additionally, Vgas-i,jf and Vgas-i,ja are the volume of each fuel and air gas compartment (tank i,j), which are calculated as follows: f f Vgas-i,j ) Vgas /(ninj)

(15)

a a Vgas-i,j ) Vgas /(ninj)

(16)

Here Vgasf and Vgasa are the total volume of the fuel and air gas channels, respectively, and n is the number of tanks in the i and j directions. In the cocurrent flow scheme, the fuel and air exit streams from tank i, j - 1 are the inlet streams into tank i, j. Thus, the exiting flow rates from the gas channels of each tank (Fi,jf and Fi,ja) depend on the flow rates of the previous tank as well (Fi,j-1f and Fi,j-1a). f H2 2 Ffi,j ) Fi,j-1 - Ffmolkeff Si,j(yHi,j2 - yHcat,i,j )f H2O 2O Fmolkeff Si,j(yHi,j2O - yHcat,i,j ) a O2 2 Fai,j ) Fi,j-1 - Famolkeff Si,j(yOi,j2 - yOcat,i,j )

The electrode-electrolyte interface is analogous to that of a capacitor. The next charge balance equations are valid for anode and cathode/electrolyte interfaces.

(10)

(11)

Munder et al.17 used the above charge balance equations for a solid oxide fuel cell under the galvanostatic operating mode (Icell ) const). As shown in our previous papers,18,19 in the case of the potentiostatic mode, the source term on the right side of eqs 10 and 11 is the difference of currents in electrode and electrolyte media in compliance with the electromagnetic theory.20 (Further explanation is given in Appendix-A.) 2.2. TSR Models of the SOFC for Different Flow Directions. 2.2.1. TSR Model for the Cocurrent Flow Direction. Several CSTR fuel cells can be connected as the reactor network to approximate more complicated flow fields. Figure 1a illustrates the application of the tank in series reactor model (TSR) for the cocurrent flow of fuel and air in a planar solid oxide fuel cell. For the TSR model of fuel cells with cocurrent flow direction, the mass balance equations for the components in the fuel and air gas channels are written as follows:

(1) f FfmolVgas

dηC ) (-icell - iC) dt

1057

(17) (18)

In these equations, Fmol is the gas molar density; Si,j is the electrode area of each tank, and yi,j is the species molar fraction in each gas compartment. The effective diffusivity and species molar fraction at the catalyst layer of each compartment are represented as keff and ycat,i,j, respectively.

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Figure 1. Schematic view of different fuel and air flow directions for one of the compartments, CSTR (i, j), in the tank in series reactor (TSR) model of a planar SOFC: (a) CSTR (i, j), with coflow direction; (b) CSTR (i, j), with counterflow direction; (c) CSTR (i, j), with cross-flow direction.

Component balance equations for species in anode and cathode catalyst (diffusion) layers are written as follows: f FfmolVcat-i,j

f FfmolVcat-i,j

2 dyHcat,i,j H2 2 ) Ffmolkeff Si,j(yHij 2 - yHcat,i,j ) + rHi,j2 dt

(19) PEN VPEN i,j FCp

(20) 2 dyOcat,i,j O2 2 ) Famolkeff Si,j(yOij 2 - yOcat,i,j ) + rOi,j2 dt

(21)

Here the variable ri,j represents the mass source term of the reactants in each compartment via electrochemical reaction. Faraday’s law relates the flux of reactants to the electric current arising from the electrochemical reaction as eqs 65-67. Again, Vcat-i,jf and Vcat-i,ja are the volumes of the catalyst layer of each fuel and air compartment (tank i, j), which are calculated as follows: f Vcat-i,j ) Vfcat /(ninj) a Vcat-i,j ) Vacat /(ninj)

CCdL

(27)

dηCi,j C ) (-icell i,j - ii,j) dt

(28)

For the tanks, j ) 1, which represents the inlet sections of the fuel and air gas channels, we obtain the component balances as follows: f FfmolVgas-i,1

(24)

(26)

dηAi,j cell A ) (ii,j - ii,j ) dt

CAdL

(23)

dTfi,j f f ) (Fi,j-1 ∆hi,j-1 - Ffi,j∆hfi,j) + qfi,j dt

dTPEN i,j ) qPEN i,j dt

The governing equations for the fuel and air temperature are formulated with the inlet and outlet energy transfer rates from the gas flow streams (Fi,j-1∆hi,j-1 and Fi,j∆hi,j) and the source terms qi,jf and qi,ja, which represent the amount of heat given from the PEN structure. The variable qi,jPEN represents a source term for the amount of heat accumulation in the PEN structure (solid volume). The energy source terms can be calculated from eqs 68-70. The charge balance equations for the anode/electrolyte and the cathode/electrolyte interfaces are given as follows.

(22)

The energy balances for the fuel and air gas channels and the PEN structure for each compartment (tank i, j) are written as follows: f FfmolVgas-i,j Cfp

dTai,j a a a a ) (Fi,j-1 ∆hi,j-1 - Fi,j ∆hi,j ) + qai,j dt

(25)

2O dyHcat,i,j H2O 2O ) Ffmolkeff ) + rHi,j2O Si,j(yHij 2O - yHcat,i,j dt

a FamolVcat-i,j

a FamolVgas-i,j Cap

dyHi,12 H2 2 ) yHin2Ffi,in - yHi,12Ffi,1 - Ffmolkeff Si,1(yHi,12 - yHcat,i,1 ) dt (29)

Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011 f FfmolVgas-i,1

dyHi,12O dt

) yHin2OFfi,in - yHi,12OFfi,1 H2O 2O Ffmolkeff Si,1(yHi,12O - yHcat,i,1 )

a FamolVgas-i,1

(30)

dyOi,12 O2 2 ) yOin2Fai,in - yOi,12Fai,1 - Famolkeff Si,1(yOi,12 - yOcat,i,1 ) dt (31)

H2 2 Ffi,1 ) Ffi,in - Ffmolkeff Si,1(yHi,12 - yHcat,i,1 )f H2O 2O Fmolkeff Si,1(yHi,12O - yHcat,i,1 ) O2 2 Fai,1 ) Fai,in - Famolkeff Si,1(yOi,12 - yOcat,i,1 )

For the tanks, j ) nj, corresponding to the outlet section of fuel and air gas channels, we define mean outlet variables as ni

ni

Ffout )

∑ i)1

f Fi,n ; yHout2 ) j

∑ i)1

tank (i, j + 1) are also the inlet streams into tank (i, j); therefore, the exiting flow rates from the fuel gas channels depend on the flow rates of the previous tank, and the exiting flow rate from the air gas channels depends on the flow rate of a tank forward. f H2 2 Ffi,j ) Fi,j-1 - Ffmolkeff Si,j(yHi,j2 - yHcat,i,j )f H2O 2O Fmolkeff Si,j(yHi,j2O - yHcat,i,j )

a O2 2 Fai,j ) Fi,j+1 - Famolkeff Si,j(yOi,j2 - yOcat,i,j )

(32) (33)

1059

(40)

The species balance equations for the ij-tank volume in the anode and cathode catalyst layer are written as eqs 17-19. The energy balances for the fuel and air gas channels for each compartment (tank i, j) are written as follows, and those for the PEN structure are the same as those in eq 9.

ni

H2 f yi,n F j i,nj

Ffout

; Tfout )

∑T i)1

f f i,njFi,nj

f FfmolVgas-i,j Cfp

Ffout

dTfi,j f f f ) (Fi,j-1 ∆hi,j-1 - Fi,j ∆hfi,j) + qfi,j dt

(41)

(34) ni

ni

Faout )

∑F i)1

a i,nj ;

yOout2 )

∑ i)1

ni

O2 a yi,n F j i,nj

Faout

; Taout )

∑ i)1

a Ti,n Fa j i,nj

a FamolVgas-i,j Cap

(42)

The developed mathematical model takes into account the following phenomena: • convective mass transport in fuel and air gas channels of each compartment. • transverse transport of gases in the diffusion layers (catalyst layers) of each compartment. • electrochemical oxidation of hydrogen at the anode/ electrolyte interface. • electrochemical reduction of oxygen in the cathode/ electrolyte interface. • charge balances at electrodes/electrolyte interfaces. • energy balances in fuel and air gas channels and for the PEN structure (solid phase) of each compartment. 2.2.2. TSR Model for the Countercurrent Flow Direction. Figure 1b illustrates the application of the tank in series model (TSR) for countercurrent flow of fuel and air in a planar solid oxide fuel cell. An appropriate unsteady-state TSR model with counterflow of fuel and air is obtained from material balances as follows:

f FfmolVgas-i,j

a FamolVgas-i,j

dTai,j a a a a ) (Fi,j+1 ∆hi,j+1 - Fi,j ∆hi,j ) + qai,j dt

Faout

(35)

f FfmolVgas-i,j

(39)

dyHi,j2 H2 f ) yi,j-1 Fi,j-1 - yHi,j2Ffi,j dt H2 2 Ffmolkeff Si,j(yHi,j2 - yHcat,i,j )

(36)

dyHi,j2O H2O f ) yi,j-1 Fi,j-1 - yHi,j2OFfi,j dt H2O 2O Ffmolkeff Si,j(yHi,j2O - yHcat,i,j )

(37)

dyOi,j2 O2 a ) yi,j+1 Fi,j+1 - yOi,j2Fai,j dt O2 2 Famolkeff Si,j(yOi,j2 - yOcat,i,j )

(38)

The charge balance equations for the anode/electrolyte and the cathode/electrolyte interfaces are the same as those in eqs 23 and 24. For the tanks, j ) 1 and j ) nj, corresponding to the inlet section for counterflow of fuel and air, respectively, we obtain dyHi,12 H2 2 ) yHin2Ffi,in - yHi,12Ffi,1 - Ffmolkeff Si,1(yHi,12 - yHcat,i,1 ) dt (43)

f FfmolVgas-i,1

f FfmolVgas-i,1

dyHi,12O ) yHin2OFfi,in - yHi,12OFfi,1 dt H2O 2O Ffmolkeff Si,1(yHi,12O - yHcat,i,1 )

(44)

dyOi,12 O2 a ) yOin2Fai,in - yi,n F j i,nj dt O2 O2 O2 Famolkeff Si,nj(yi,n - ycat,i,n ) j j

(45)

a FamolVgas-i,n j

H2 2 Ffi,1 ) Ffi,in - Ffmolkeff Si,1(yHi,12 - yHcat,i,1 )f H2O 2O Fmolkeff Si,1(yHi,12O - yHcat,i,1 )

O2 a O2 O2 Fi,n ) Fai,in - Famolkeff Si,nj(yi,n - ycat,i,n ) j j j

(47)

For the tanks, j ) 1 and j ) nj, corresponding to the outlet section for coflow of air and fuel, respectively, we define mean outlet variables as follows: ni

ni

In the counterflow scheme, the fuel side exit streams from tank (i, j - 1) are also the inlet streams into tank (i, j). In the air gas channels of each compartment, the exit streams from

(46)

Ffout )

∑F i)1

f i,nj ;

yHout2 )

∑ i)1

ni

H2 f yi,n F j i,nj

Ffout

; Tfout )

∑T i)1

f f i,njFi,nj

Ffout

(48)

1060

Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011 ni

ni

Faout

)

∑

Fai,1 ;

yOout2

)

∑

ni

yOi,12Fai,1

i)1

Faout

i)1

;

Taout

)

∑

nj

Tai,1Fai,1

nj

i)1

Faout

Faout

)

∑ j)1

Fna i,j ;

yOout2

)

∑ j)1

nj

ynOi,j2 Fna i,j ;

Faout

Taout

)

∑T j)1

a a ni,jFni,j

Faout

(49)

(58)

2.2.3. TSR Model for the Cross-flow Direction. The coupling of reactor elements shown in Figure 1c is the basis for our analysis of planar cross-flow SOFC as a chemical reactor. The detailed species balance equations for cross-flow of fuel and air gas channels at each compartment are written as follows:

2.3. Electrochemical Submodel. 2.3.1. Electrode Current. Assuming the electrochemical reactions at the anode/electrolyte and cathode/electrolyte are given by [H2 + O2- f H2O + 2e-] and [1/2O2 + 2e- f O2-], respectively, the full expression of the Butler-Volmer equation was used for the local current density as follows:

f FfmolVgas-i,j

f FfmolVgas-i,j

dyHi,j2 H2 f ) yi,j-1 Fi,j-1 - yHi,j2Ffi,j dt H2 2 Si,j(yHi,j2 - yHcat,i,j ) Ffmolkeff dyHi,j2O dt

a FamolVgas-i,j

dt

[ (

H2O f ) yi,j-1 Fi,j-1 - yHi,j2OFfi,j H2O 2O Ffmolkeff Si,j(yHi,j2O - yHcat,i,j )

dyOi,j2

(50)

iAi,j ) iA0 exp

(51)

O2 a ) yi-1,j Fi-1,j - yOi,j2Fai,j O2 2 Famolkeff ) Si,j(yOi,j2 - yOcat,i,j

(52)

In the cross-flow scheme, the fuel side exit streams from tank (i, j - 1) are also the inlet streams into tank (i, j). In the air gas channels of each compartment, the exit streams from tank (i 1, j) are also the inlet streams into tank (i, j); therefore, the exiting flow rates from the fuel and air gas channels are defined as follows: f H2 H2 Ffi,j ) Fi,j-1 - Ffmolkeff Si,j(yHi,j2 - ys,i,j )f H2O 2O Fmolkeff Si,j(yHi,j2O - yHcat,i,j ) a O2 2 Fai,j ) Fi-1,j - Famolkeff Si,j(yOi,j2 - yOcat,i,j )

(53) (54)

The species balance equations for the ij-tank volume in the anode and cathode catalyst layers and the charge balance equations for the anode/electrolyte and the cathode/electrolyte interfaces are written the same for co- and counterflow directions. The energy balances for the fuel and air gas channels for each compartment (tanki,j) are written as follows, and again, that for the PEN structure is the same as eq 9. f FfmolVgas-i,j Cfp

dTfi,j f f ) (Fi,j-1 ∆hi,j-1 - Ffi,j∆hfi,j) + qfi,j dt

iCi,j

)

iC0

[ (

) (

)]

) (

)]

RAC F(ηAi,j - ηAeq) RAAF(ηAi,j - ηAeq) - exp RTi,j RTi,j

C RCCF(ηi,j - ηCeq) RACF(ηCi,j - ηCeq) exp - exp RTi,j RTi,j

dTai,j a a ) (Fi-1,j ∆hi-1,j - Fai,j∆hai,j) + qai,j dt

( )

H2 γH2 H2O γH2O iA0 ) i*(y exp A cat,i,j) (ycat,i,j)

For the tanks j ) njand i ) ni, corresponding to the outlet section for coflow of fuel and air, we define mean outlet variables as follows: ni

ni

Ffout )

∑F i)1

f i,nj ;

yHout2 )

∑ i)1

ni

H2 f yi,n F j i,nj

Ffout

; Tfout )

∑T i)1

f f i,njFi,nj

Ffout

ECact RT

(61)

(62)

2.3.2. Electrolyte Current. For SOFC fuel cells operated under potentiostatic mode (constant voltage), the current density in electrolyte media is found from the voltage equation as follows:18 OCV icell - Ecell - ηAact,ij + ηCact,ij)/ROhmic ij ) (Eij ij

(63)

A C Here, ηact,ij and ηact,ij (the anode and cathode activation overA potential of each compartment, respectively) are given as ηact,ij A A C C C A C ) ηij - ηeq and ηact,ij ) ηij - ηeq, where ηeq and ηeq are the anodic and cathodic equilibrium potential differences at the electrode/electrolyte interfaces of each compartment and ηAij and is the ηCij are the anode and cathode overpotentials and ROhmic ij ) δe/σe. ohmic resistance, given by ROhmic ij

(

-10300 Tij

)

(64)

2.3.3. Mass Transfer Due to Reaction at the Electrode/ Electrolyte Interface. The component concentration at the electrode/electrolyte interface is calculated from the flux continuity equation. In fuel cells, the reacting species are transferred from the channel through the porous electrode to the electrode/electrolyte interface. The effect of electrochemical reaction on the species concentrations is included in the component flux from the channel to catalyst layer. (Si,j ) Scell/(ninj)) rHi,j2 ) Si,j

(57)

EAact RT

( )

O2 γO2 iC0 ) i*(y exp C cat,i,j)

σe ) 3.34 × 104 exp

(56)

(60)

Here i0A and i0C are the anode and cathode exchange current densities, which are defined as follows:

(55) FamolVai,jCap

(59)

A νH2ii,j

nAe F

(65)


rHi,j2O )

νH2OiAi,j Si,j A ne F

rOi,j2 )

νO2iCi,j Si,j C ne F

Table 2. Operating Conditions for the Planar SOFC

(66)

parameter

(67)

fuel flow rate, mol/s oxidant flow rate, mol/s H2 mole fraction in fuel O2 mole fraction in oxidant temp, K pressure, atm

2.3.4. Heat Transfer Submodel. The heat transfer equation defines the continuity of the heat flux condition in fuel and air gas channels of each compartment, and for the PEN structure they are defined as follows:21

(

qPEN ) i,j

qfi,j ) RfSi,j(TPEN - Tfi,j) i,j

(68)

qai,j ) RaSi,j(TPEN - Tai,j) i,j

(69)

) (

∆HR - Ecell Ii,j + Si,jRf + 2F (CHp 2 - CHp 2O) Ii,j (Tfi,j - TPEN i,j ) + 2F COp 2 I (Ta - TPEN Si,jRa + i,j ) 4F i,j i,j

(

)

1061

)

(70)

Constitutive equations for physical and thermophysical properties and the convective heat transfer coefficient are taken from the literature.21-23 3. Simulations and Discussion of the Results For modeling SOFCs with a CSTR network as a tank in series (TSR) model, we used operating conditions and geometrical parameters from the work of Ravussin et al.24 The fuel gas channels are fed with a gas mixture of 97% H2 and 3% H2O, and the inlet air composition at the air gas channels is 21% oxygen and 79% nitrogen. The number of tanks (CSTR reactor) in the TSR model can be estimated from bench or numerical experiments. For planar SOFC, we set the number of tanks as ni ) 8 and nj ) 8, corresponding to the TSR model with 704 nonlinear coupled first order ordinary differential equations. This model was implemented in MATLAB, and it was initialized with the feed composition and temperature. The geometry and electrochemical parameters of the anode and cathode catalyst layers and the operating conditions are listed in Tables 1 and 2, respectively. A comparison of the simulated and experimental V-I curves is shown in Figure 2. It was observed that the predicted V-I curves had a similar shape to that of the experimental data24 reported for the countercurrent flow of fuel and air. Although the limiting current deviates from the experiments, the simulation results showed similar tailing effects at high current density. The deviation of the TSR model with ni ) 8, nj ) 8 indicates that the flow patterns in the channels are changed with flow

5.07 × 10-5 6.17 × 10-4 0.97 0.2 1073 1

rate. The voltage drop at high current density was largely due to the concentration loss, especially at the anode side; therefore, the main reason for this discrepancy was due to nonuniformity of the fuel distribution, and at high current density this effect is more significant. The high volumetric flow rate of the fuel flow leads to increased uniformity of the fuel distribution and also the reaction rates across the active area. Low volumetric flow rates of the fuel flow, on the other hand, lead to inhomogeneous distribution of fuel within the anode compartments; therefore, assuming uniform flow distribution for this study, the predicted V-I curve for the SOFC operating with 400 mL/min of hydrogen fuel, agreed best with the experimental values. The normalized least-squares error (Ei) in the cell current was calculated for the three cases with different fuel inlet flow rates, so as to quantify the deviation of the model from the experimental results. Ei )

∑

κj)1,..., m

2 ei,κ j

(71)

where ei,κj ) (iexp,κj - isim,κj)/(iexp,κj - isim,κj)max is the normalized error between the experimental and the simulated values of the current density (i). κj represents the “m” number of voltage points for which the errors in current density are calculated; iexp and isim represent the experimental and the simulated values of the cell current density, respectively. The normalized leastsquares error for the case of 400 mL/min inlet fuel flow rate was 1.9, and those for the cases of 260 mL/min and 180 mL/ min fuel inlet flow rates are 2.26 and 2.32, respectively. It can be seen that the model accuracy is better at high fuel flow rates. Additionally, average temperature, current density, and fuel and air utilization with different fuel and air flow directions are itemized in Table 3. The average anode current density at the electrode/electrolyte interface was calculated as the mean surface. Under steady state conditions, differences among the fuel and air utilization, average current density, and temperatures for the fuel and air

Table 1. Geometry and Electrochemical Parameters for the Planar SOFC parameter

anode

cathode

catalyst width δ, m porosity ε pre-exponential kinetic factor i*, A m-2 activation energy, J · mol-1 charge transfer coefficient RA charge transfer coefficient RC channel height, m section area, m2 electrolyte width, m

0.25 × 10-3 0.4 2.9 × 108 120000 2 1 0.75 × 10-3 5 × 10-3 1 × 10-5

0.03 × 10-3 0.4 7.0 × 108 120000 1.4 0.6 Figure 2. Comparison of simulated and experimental V-I data24 for the countercurrent direction (ni ) 8, nj ) 8) of fuel and air flows.

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Table 3. Single Cell Stack Simulation Results (Cell Voltage ) 0.8 V) parameter

air Tavg

air Tout

fuel Tavg

fuel Tout

PEN Tavg

A iavg

cross-flow 1148 1223 1177 1134 1178 1688 coflow 1150 1216 1174 1237 1177 1708 counterflow 1136 1235 1163 1094 1168 1741

FU, % AU, % 57 58.5 52.5

12.2 12.1 12.3

channels and for the PEN structure of coflow, counterflow, and cross-flow directions provide support for the influence of the flow direction on the SOFC performance. Fuel utilization and air utilization coefficients are defined as follows: FU ) 1 -

AU ) 1 -

yHout2 FAout yHin2FAin yOout2 FCout yOin2FCin

(72)

(73)

The TSR model is able to predict a two-dimensional distribution of concentration, temperature, and current density in planar SOFC. Using arrays of several tanks in series, we were able to detect current heterogeneities along the cell array and regions of higher-current densities in the fuel cell. The steady state simulation results for SOFC with different flow directions are given in Figures 3-5. It is shown that decreasing of the oxygen concentration through the air flow direction results from the electrochemical reaction at the cathode catalyst layer. For the species concentration among different flow configurations, the primary difference was the twodimensional nature of the cross-flow case in comparison with coflow and counterflow cases. For the coflow and counterflow directions, the H2 concentration profile across the full width of the cell decreases uniformly, but for the cross-flow case, we have a minimum point at one side and a maximum point at the other side of the cell width and along the fuel flow direction. When trying to maximize fuel utilization, the minimum point increases the risk of localized fuel depletion. Fuel depletion may contribute to creation of cold spots and increases the severity of the thermal stresses. The concentration of water in fuel gas channels is increased due to the component flux from the anode catalyst to the gas channel. In each flow configuration, the PEN temperatures increased along the air flow direction and reached a maximum near the air outlet, that is, due to the air flow that was most effective in cooling near the air inlet and carried heat generated in the PEN structure toward the air outlet. Of the three flow configurations, the coflow direction had the most uniform temperature distribution and smallest temperature difference from the air and fuel inlets to outlets, and this is due to the offsetting effects of the air inlet that is aligned with the fuel inlet. For the counterflow and cross-flow cases, the PEN temperature was highest near the fuel inlet and air outlet. Despite the coflow and counterflow directions, again, the PEN temperature of the cross-flow case had a two-dimensional nature and the most nonuniform distribution. The temperature variation is important for evaluation of thermally induced stresses due to the temperature gradient. Consistent with the temperature distribution, the current density was most uniform in the coflow case. In the cross-flow and counterflow directions, current density was high near the fuel inlet and air outlet. The twodimensional nature of the current density for the cross-flow case near the fuel inlet was because of the cooling effect at the air inlet and the lower concentration of oxygen at the air outlet. Generally, the cell performance decreases considerably with temperature decrease. This is mainly caused by an increase in both the Ohmic and activation overpotentials.26 On the other

hand, overpotential is strongly dependent on temperature and current density.27 Increasing the current density causes the activation overpotential to be increased;28 meanwhile, current density also takes into account the effects of the reactants’ concentration. As a result, histograms for the coflow and counterflow cases showed opposite trends in the spatial domain, except for overpotential and current density. For the coflow SOFC, activation overpotential decreased along the gas flows while the maximum current density was roughly midstream from the gas inlet to outlet due to combined effects of the PEN temperature increase and hydrogen concentration decrease. For the counterflow case, the high temperature region is interchanged and then the maximum current density is placed at the fuel inlet, which causes the activation overpotential to become a bit higher in this region. Afterward, under the effects of temperature and current density, the activation overpotential decreased, followed by a gradual increase. The nonuniform current density reveals the influence of reactants’ concentration and temperature distribution. Analysis of the steady-state current and temperature profiles shows that the cocurrent direction of fuel and air provides the most uniform current and temperature distribution and the highest fuel utilization. Additionally, steady state results indicate that the TSR model presented in this work is able to predict the spatial distribution of the variables and can be used for preliminary studies instead of the more complex CFD models. Finally, an investigation of transient behavior is studied. Such transient behaviors are critical to SOFC control development. SOFC operation is often subjected to transient conditions in external load, which can be either load current or load voltage. Since this model was used for simulating the potentiostatic operating mode, in this section we analyzed transient behaviors for the coflow, counterflow, and cross-flow SOFC induced by the voltage step change from 0.8 to 0.85 V after 20 s, with the rest of the boundary conditions kept constant. Figures 6-8 show the cell dynamic responses for different fuel and air flow directions at three different representative points of the cell: CSTR (3, 3), CSTR (4, 4), and CSTR (5, 5). As one can see from Figures 6-8, numerical dynamic results for the coflow, counterflow, and cross-flow direction show that the cell can obtain very quick current decrease when subjected to a voltage step increase, followed by a slow transient period, leading to the new steady state current. However, the transient response of the cell current density is very complex and related to all parametric changes, which are associated with the entire fuel cell activity. The sudden step increase of the cell voltage induces the quick response of the charge transport within the PEN assembly, which leads to an immediate decrease of the current density. At the instant of this momentary decrease of the current density, a sudden change in the mass source term of the component balance equation occurs, and consequently, the hydrogen concentration instantaneously increases, followed by a slow increase that is due to the diffusion effect, until the overall system response becomes steady. Simultaneously, the fraction of hydrogen which has not been consumed in the electrochemical reaction is accumulated toward the fuel outlets, which can influence the slow dynamic of the current density along the fuel flow direction. For better comparison with the numerical steady state results, the hydrogen concentration is normalized with the value of the inlet boundary condition. Reduction of the current density also decreases the heat production, which results in a temperature decrease. The PEN temperature has the gentlest response and a substantial effect on the electrolyte conductiv-


1063

Figure 3. Histograms with steady state simulation results predicted by the TSR model for a planar SOFC with a coflow direction of fuel and air (Vcell ) 0.8 V).

ity. Thus, the PEN temperature decrease, on the other hand, causes further reduction of the local current density. The same trends of the transient responses obtained from the simulations of this model were also reported by other authors.28-33 From the figures, it is evident that both the transient behavior and the magnitude of the changes in the responses vary at different representative points, as well as with the flow directions. As was shown in Figure 6, for the coflow direction, the magnitude of the change for current, temperature, hydrogen concentration, and activation overpotential increases toward the fuel and air outlets. This is due to the convective cooling effect of the air flow, which causes the low and high temperature

regions to be placed at the air inlet and outlet, respectively. At the low temperature region, CSTR (3, 3), overpotential is higher, which in turn diminishes the dynamic response of the current density to the voltage step increase.34 The simulation results of the coflow direction indicate that, despite the hydrogen concentration increase, the slow dynamics of the local current density is mainly influenced by the intensity and magnitude of the temperature change, which causes a further decrease in current density. The response of the activation overpotential in Figure 6d is consistent with temperature and current. After the first sudden decrease, it shows a moderate increase toward the new steady state value.

1064


Figure 4. Histograms with steady state simulation results predicted by the TSR model for a planar SOFC with a counterflow direction of fuel and air (Vcell ) 0.8 V).

Contrary to the coflow case, the counterflow direction shows almost opposite transient behavior that is due to interchange of the air inlet and high temperature regions. For the points near the air outlet, i.e. CSTR (3, 3), the magnitude of change for the local current density is considerably higher, which leads to a steep and large decrease of temperature. Thus, for the points near the air outlet, the slow dynamic response of the local current density is mainly influenced by temperature. On the other hand, for the points near the fuel outlet, the temperature change is significantly smaller; meanwhile, as we move toward the fuel outlet, the dynamic response of the current density tends to be characterized by the initial undershoot, followed by a slower

transient that is due to the pronounced effect of the diffusion resistance within the porous electrode, hydrogen accumulation toward the fuel outlets, and less influence of the PEN temperature. As one can see from Figure 8, the dynamic responses of the cross-flow direction are absolutely consistent with the above justifications. Here under the effects of the cross-flow direction of fuel and air, the transient behavior is a little bit different. However, the same as the counter-flow direction, again for the points near the air and fuel inlets, i.e. CSTR (3, 3), a small amount of hydrogen accumulates and the intensive change of temperature mainly affects the slow dynamic of the local current density. At the same time for the points near the outlets,


1065

Figure 5. Histograms with steady state simulation results predicted by the TSR model for a planar SOFC with a cross-flow of fuel and air (Vcell ) 0.8 V).

accumulation of hydrogen increased; therefore, the slow dynamic of the current density is influenced by the combined effects of diffusion, hydrogen accumulation, and temperature. Analysis of dynamic responses to a voltage step change shows that the coflow direction of fuel and air provides the most uniform transient responses at different points of the cell; meanwhile, comparing with the counterflow and cross-flow directions, the PEN temperature dominates the slow dynamic of the current density and almost all dynamic behaviors. The developed unsteady-state tank in the series SOFC model with charge balance will provide a better understanding of the main phenomena governing electrochemical reactions in fuel cells. Additionally, the results are useful in understanding the

sensitivity of the SOFC performance to the fuel and air flow directions. Intensification of electrochemical processes can be achieved by changing the direction of fuel and air flow. 4. Conclusions The tank in series model presented in this work takes into account the charge balances at the electrodes and electrolyte interfaces, species mass balances to the gas channels and diffusion layers, and energy balances for the fuel and air channels and also for the PEN structures. All simulations were done under the potentiostatic operating mode, and using the charge balance equations for the interface between the electron

1066


Figure 6. Dynamic simulation results predicted by the TSR model for a planar SOFC with a coflow direction of fuel and air (∆V ) +0.5 V).

Figure 7. Dynamic simulation results predicted by the TSR model for a planar SOFC with a counterflow direction of fuel and air (∆V ) +0.5 V).

and ion conducting media facilitates calculation of the activation overpotential with no restriction concerning the number of the charge transfer steps. The simulation results from the TSR model indicate the influence of flow direction on the steady state and

dynamic performances of a planar SOFC with coflow, counterflow, and cross-flow directions of fuel and air. Examination of simulation results indicates the possibility of enhancing the fuel cell performance by decreasing the misdistribution of fields.


1067

Figure 8. Dynamic simulation results predicted by TSR model for a planar SOFC with a cross-flow direction of fuel and air (∆V ) +0.5 V).

The results demonstrate a strong coupling between the temperature, concentration, activation overpotential, and current density distributions. Among different flow directions, the coflow case is the most favorable for a planar SOFC with improved performance, and in response to a voltage step increase, it provides the most uniform transient behavior at different points of the cell. For the coflow direction, temperature dominates the slow dynamics of the local current density, while, in the low temperature regions of the counterflow and cross-flow cases, the slow dynamics of the current density tends to be characterized by the initial undershoot, followed by a slower transient response. This is due to the combined effects of the diffusion resistance within the porous electrode, hydrogen accumulation toward the fuel outlets, and the influence of the PEN temperature. Design is an iterative process where more and more refined models are required in each phase. The model presented in this work can serve as an early step in the iteration by means of accelerating 2D simulation to get the trend in the distribution of important variables. Based on the results from this model, decisions on the need for more refined models may be taken. Using this model, it is possible to take into account the hydrodynamic behavior of different flow configurations. Moreover, the model computational time is much less compared to that for CFD models (less than 3 min to solve 704 ordinary differential equations). Future work will concentrate on the optimization of modeling and model based controller design to improve the performance of SOFC stack. Acknowledgment S.H. gratefully acknowledges Curtin University of Technology for the CSIRS Scholarship. The authors are grateful to the Australian Research Council (ARC) for Grant DP0880483 and to Ceramic Fuel Cells Company for helpful discussions.

Appendix I: Derivation of Charge Balance Equation The electric potential fields are governed by the charge conservation equations. The charge balance at the interface between electron-conducting and ion-conducting phases can be expressed as follows:20 ∂Q + ∇ · is ) an(i1 - i2) ∂t

(A.1.1)

where i1 is the component of current in electron-conducting media normal to the boundary, i2 is the component of current in ion-conducting media normal to the boundary, is is the superficial current density, Q is charge, and an is the normal to the boundary. For the CSTR and TSR models, the net charge flux is zero. ∇ · is ) 0

(A.1.2)

The ionic and electronic phase interfaces behave like a capacitor in which the charge density of electrons and ions depends on the potential difference across this double layer. The charge or discharge rate at the electrode-electrolyte double layer can be represented as follows:25 ∂(Φs - Φe) ∂Q ) CdL ∂t ∂t

(A.1.3)

The electric potential for the electron conducting phase is Φs, the potential for the electrolyte phase is Φe, and they obey the following charge conservation equation: ∂η 1 ) (i - i2) ∂t CdL 1 where η is the overpotential and η ) Φs - Φe.

(A.1.4)

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The improved SOFC models with the new charge balance equation provide a better understanding of the main phenomena governing electrochemical reactions in fuel cells.

T ) temperature (K) V ) volume (m3) y ) species mole fraction

Appendix II: Electrolyte Current

Greek Letters

By definition, the electrolyte current is defined as follows.

RAA ) anodic charge transfer coefficients for anode RCA ) cathodic charge transfer coefficients for anode RAC ) anodic charge transfer coefficients for cathode RCC ) cathodic charge transfer coefficients for cathode ε ) porosity η ) overpotential (V) R ) thermal conductivity (W m-2 K-1) Fmol ) molar density (mol m-3) FCpPEN ) 106 (J m-3 K-1) σ ) conductivity (Ω-1 m-1)

ie ) -σe

∂Φ ∂n

(A.2.1)

Here, n is the normal to the interface, and σe is the electrolyte conductivity. Using a linear approximation of the potential profile in the ionic conducting media, the electrolyte current is defined as follows (δeis the electrolyte thickness):

(

ΦCe - ΦAe ie ≈ -σe δe

)

(A.2.2)

The values of electric potential in ionic conducting media at electrode/electrolyte interfaces can be expressed from the definition of the activation overpotential as follows: ηAact ) ΦsA - ΦAe - ηAeq

(A.2.3)

ηCact ) ΦsC - ΦCe - ηCeq

(A.2.4)

We note that the reversible Nernst potential at the electrode/ electrolyte interface is linked with anodic and cathodic equilibrium potential differences: ηCeq - ηAeq ) EOCV

(A.2.5)

It is known that the difference of the anodic and cathodic electric potentials is equal to the cell voltage Ecell )

ΦsC

-

ΦsA

(A.2.6)

Taking into account eqs A.2.2-A.2.6, we obtain the next expression for the electrolyte current density: ie ) (EOCV - Ecell - ηAact + ηCact)/ROhmic

(A.2.7)

Application of new expression for electrolyte current is shown in our recent papers.18,19 Nomenclature CdL ) double layer capacitance (A s V-1 m-2) Cp ) specific heat (J mol-1 K-1) Ecell ) cell voltage (V) EOCV ) open circuit voltage (V) Ei ) normalized least-squares error Fin, Fout ) inlet and outlet mass flow rate (mol s-1) F ) Faraday’s constant (96485 C mol-1) h ) enthalpy (J mol-1) ∆HR ) reaction enthalpy (J mol-1) I ) current (A) i ) current density (A m-2) i0 ) exchange current density (A m-2) keff ) effective diffusivity (m s-1) n ) number of compartments (CSTR) r ) mass source term (mol s-1) R ) ideal gas constant (J mol-1 K-1) ROhmic ) Ohmic resistance (Ω m2) q ) energy source term (J s-1) Sij ) electrode area in tank ij (m2)

Sub-/Superscripts in ) inlet out ) outlet eff ) effective i, j ) ij cell A ) anode C ) cathode eq ) equilibrium act ) activation cat ) catalyst layer f ) fuel a ) air ν ) stoichiometric factor exp ) experimental sim ) simulation

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ReceiVed for reView May 20, 2010 ReVised manuscript receiVed November 17, 2010 Accepted November 30, 2010 IE101129K

Improved Tank in Series Model for the Planar Solid Oxide Fuel Cell

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