Mathematical Modeling of a Porous Media Burner Based Methane Flame Fuel Cell

Adetailedtwo-dimensionalaxisymmetriccomputationalmodelofaﬂamefuelcell(FFC)unitwasdevelopedandpresented.TheFFCunitisbasedontheintegrationofafuel-richmethaneﬂameinaporousmediaburnerandamicro-tubularsolidoxidefuelcell(SOFC). Themodelconsideredthecouplingeffectsofthechemicalreactionsandelectrochemicalreactionsandtheheat-transport,mass-transportandcharge-transportprocessesintheFFC.Thesimulatedtemperaturedistributionandelectrochemicalcharacteristics showedgoodagreementwithexperimentaldata.Thecouplingmechanismofthefuel-richﬂameandtheSOFCanodewereclariﬁed.TheNicatalystintheanodeandtheelectrochemicalreactionspromotedtheconversionofCH 4 in porous media fuel-rich combustion.©TheAuthor

A flame fuel cell (FFC) is a novel kind of solid oxide fuel cell (SOFC) in which a fuel-rich flame is directly integrated with an SOFC. 1 The fuel-rich flame acts as a partial oxidation reformer for the SOFC at the anode to convert C x H y to CO and H 2 . Meanwhile, it also provides heat for the SOFC to start up and operate. [2][3][4] FFCs have emerged as an attractive system owing to its simple setup and rapid startup. Until now, FFCs were investigated using various combustion concepts and different SOFC configurations. [5][6][7][8][9][10][11][12][13][14][15] The effects of operational conditions on FFC performance were studied and discussed in these early experimental studies. In a previous study, an FFC unit was implemented and studied by integrating a porous media burner with a micro-tubular SOFC. 16 In an FFC unit, the anode of the micro-tubular SOFC is directly integrated with the fuel-rich flame. The chemical and electrochemical reactions, as well as the heat transport, are coupled between the anode and the fuel-rich flame. The coupling effects will further influence the combustion characteristics of the porous media burner as well as the electrochemical performance of the SOFC. However, it is a difficult task to clarify the coupling effects via experimental methods due to the chemical and thermal complexity within the FFC unit. 17 Consequently, a modeling technique is necessary to clarify the complex physical and chemical processes.
Compared with the vast number of experimental studies on FFCs, numerical studies have been relatively lacking. Vogler et al. developed a computational model of an FFC based on a flat-flame burner and a planar SOFC. 17 They found that the products of the electrochemical conversion in the SOFC did not influence the flame, and that the flame and the SOFC were chemically decoupled. However, this conclusion was drawn given the fact that a stagnation flame takes place at a certain flamelet that was far away from the SOFC. However, when a porous media burner was applied in the FFC unit, the fuel-rich combustion exhausts flowed along the anode of the micro-tubular SOFC, and chemical reactions still occurred due to the high temperature of the SOFC region.
To that end, a modeling approach was undertaken to investigate the coupling effects between the SOFC anode and the fuel-rich combustion inside a porous media burner. A two-dimensional axisymmetric model was developed that couples the chemical reactions in the fuel-rich flame, the chemical and electrochemical reactions in the anode, the heat-and mass-transport processes in the porous media * Electrochemical Society Student Member. z E-mail: shyx@tsinghua.edu.cn as well as in the anode, and the charge transport in the anode. The numerical results were validated against the experimental results. The coupling effects of the fuel-rich flame and the SOFC anode were studied.

Models Development
Model geometry.-A two-dimensional axisymmetric model was developed to represent the experimental setup in the previous study. 16 A micro-tubular SOFC is placed on top of a two-layer porous media burner. Fig. 1 shows the schematics of the experimental setup and the model domain. The model considers the heat-, mass-, and chargetransport processes and the chemical and electrochemical reactions. Since the focus of the numerical simulation is the coupling effects of the anode and the fuel-rich combustion, the cathode channel was neglected, and the gas at the interface of the cathode and the cathode channel are considered to be air for simplification. The following assumptions were made in the model: (1) Gases were assumed to be incompressible ideal gases, (2) The porous media were considered to be isotropic and homogeneous, (3) The porous media were optically thick and solid-phase radiation was taken into account using the Rosseland approximation, (4) Only heat conduction was considered in the electrodes, and the gas temperature was considered to be the same as the solid temperature in the electrodes, (5) The reaction active sites were uniformly distributed in the electrodes, and the two conducting phases were continuous and homogeneous in each layer, (6) Darcy flow was considered in the electrodes.
Governing equations.-The model was developed based on the model of fuel-rich combustion in a porous media burner, which was described in detail in the previous study. 18 The governing equations of the combustion region and the porous anode channel were the same as those presented in the previous paper. 18 Consequently, only governing equations for the micro-tubular SOFC region are presented in this paper. 19 Charge conservation.-The electronic and ionic charge balance at the electrodes and the ionic charge balance at the electrolyte are shown as follows: Electronic charge balance at the anode: C dl,an S act,an ∂ V el,an − V ion,an ∂t + ∇ · (−σ el,an ∇V el,an ) = Q el,an = −Q ion,an [2] Ionic charge balance at the cathode: Electronic charge balance at the cathode: [4] Ionic charge balance at the electrolyte: In Eqs. 1-5, t is time, C dl is the specific interface double-layer capacitance between electronic and ionic conductor phases, Q is the transfer current source, and V el and V ion are the electric potentials of the two conductor phases.
Mass conservation.-The mass conservation equation of species i in the anode can be expressed as: where ρ fuel is the gas density at the anode, ε is the porosity, u is the velocity, R i is the mass source of species i. The mass sources from the chemical reactions and the electrochemical reactions in the anode were considered. The water-gas shift reaction (WGS) and the internal reforming reaction (DIR) are taken into consideration: The global expressions of the reaction rates are 20 The electrochemical oxidation of H 2 and CO is considered to be at the anode and the electrochemical reduction of O 2 are considered at the cathode: The mass source from electrochemical reactions can be expressed as follows: The mass source term of species i can be calculated as follows: Momentum conservation.-The momentum conservation in the electrodes were described using Darcy's law: [28] where μ g is the dynamic viscosity, and α is the permeability.
Energy conservation.-Only heat conduction was considered in the electrodes and the electrolyte: where Q heat is the heat source term, which can be calculated as follows: where, the Q ohm is the ohmic hea, which can be calculated as, where, i is local electronic or ionic current density and σ is the electric conductivity. Q rev is the reversible heat effect of entropy change in electrochemical reactions, which can be calculated as, , anode [32] where S denotes the molar entropies. Q irr is the irreversible heat generation due to activation polarizations, which can be formulated as, Boundary conditions.-The boundaries are specified in Fig. 1 and the detailed settings of the boundary conditions are listed as below.
Momentum balance and mass balance: The velocity and mass fraction of species i were specified at the inlet ∂ inlet . A free flow condition was set at the outlet ∂ outlet . The gas insulation condition was set at the electrode/electrolyte interfaces ∂ a|e and ∂ c|e . The wall boundaries ∂ wall were set as walls with no slip. Other boundaries were set as internal boundaries.
Energy balance: At the inlet, the gas temperature was set as a constant temperature T g = 300K and the solid temperature was set as the radiant boundary λ s,eff Ionic charge balance: At the electrode/electrolyte interfaces ∂ a|e and ∂ c|e , the continuity boundary condition was applied. Other boundaries were set as electric-insulation condition.
Electronic charge balance: At the electrode/channel interfaces ∂ a|c and ∂ c|c , the electric potentials V cell,an and V cell,ca were specified, respectively. Other boundaries were set as the electric-insulation condition.
Model parameters.-The parameters related to the combustion models were presented in the previous paper. 18 The parameters related to the other regions are shown in Tables I and II. Model solution.-The model was solved by setting a given cell voltage. The partial differential equations were solved using the commercial software ANSYS Fluent (ANSYS, Inc., Canonsburg, PA, USA).

Results and Discussion
Model validation.-The combustion model was validated by the measured temperature distribution and gas compositions reported in the previous paper. 18 In this paper, the modeled temperature distribution of the entire FFC reactor was compared with the experimental results obtained at an equivalence ratio of 1.6 and a gas velocity of 0.15 m/s, as shown in Fig. 2. It can be seen that the simulated profile matched the experimental results well. Then, the experimental data of the electrochemical performance published by our group was used to further validate the model. 16 Fig. 3 shows the modeling and experimental IV curves of the FFC unit at various equivalence ratios. The modeled curves agree well with the experimental data, which means that the model developed can be used to represent actual conditions. Effects of Ni catalyst.-In the FFC unit, the anode and the porous media burner are directly integrated. The temperatures and gas compositions were calculated with and without the chemical reactions at the anode to analyze the effects of the Ni catalyst on the fuel-rich reformation. Although not shown here, the modeled temperature distribution varied little after considering the chemical reactions at the Ni catalyst. Simulation results with and without the chemical reactions at the Ni catalyst for the equivalence ratio of 1.6 and the gas velocity of 0.15 m/s are compared in Fig. 4. It should be noted that the species mole fraction shown in Fig. 4 is the calculated average mole fraction for each species within the anode. It can be seen that the mole fractions of H 2 and CO 2 increased, while the mole fractions of CO and CH 4 decreased due to the water-gas shift reaction and the direct internal reformation reaction that occurred at Ni surface. Fig. 5 further shows the composition distribution in the anode and anode channel. When the fuel-rich combustion exhausts entered the SOFC region, the mole fractions of H 2 and CO 2 increased, and a concentration gradient from the anode to the anode channel emerged. In constrast, CO and CH 4 were consumed at the anode. A concentration gradient from the anode channel to the anode emerged, which led to the decrease of CO and CH 4 in the anode channel. Consequently, the Ni catalyst in the SOFC anode promoted the conversion of CO and CH 4 to H 2 , which has a catalytic enhancement effect on the fuel-rich combustion of methane in the porous media burner.
Effects of electrochemical reactions.-Since the fuel utilization efficiency of the FFC unit is only 1%, 16 the average temperature of the SOFC only increased by 5-6 K after considering the heat effects of the electrochemical reactions. Meanwhile, the relatively low fuel utilization efficiency makes it difficult to investigate the effects of the electrochemical reactions on the composition distributions. However, the integration of a micro-tubular SOFC stack with the burner is necessary for the practical application of the FFC. Consequently, in this section, the effects of the electrochemical reactions on the fuelrich combustion were analyzed given a fuel utilization efficiency of 36%. Fig. 6 shows the calculated temperature distribution in the SOFC region with/without electrochemical reactions (at a voltage of 0.6 V). It can be seen that the heat release of electrochemical reactions led to a temperature increase at the anode/electrolyte interface ∂ a|e . The temperature gradient from the anode/electrolyte interface ∂ a|e to the anode/anode channel interface ∂ a|c further led to the temperature increase of the anode and the anode channel. The average temperature of the anode increased by 20-30 K after introduction of the electrochemical reactions. Since there exists a large gradient from the upstream to the downstream of the SOFC, the current density decreased rapidly along the axial direction of the SOFC, as shown in Fig. 7. Consequently, the heat released by the electrochemical reactions decreased rapidly. Upstream of the SOFC region, where the temperature is high, the temperature increased by 50 K due to the heat release of the electrochemical reactions.
The composition distributions along the anode-channel interface ∂ a|c are shown in Fig. 8. The mole fraction of H 2 showed a trend from decrease to increase along the axial direction. H 2 was consumed by the electrochemical reactions at the high-temperature region of the SOFC. However, the consumption rate of H 2 decreased due to the   mole fractions of CO and H 2 O decreased at the low-temperature region, which indicates that the production of H 2 was mainly due to the water-gas shift reaction. In addition, the direct internal reformation of CH 4 reached equilibrium with/without electrochemical reactions since the mole fraction of CH 4 changed little at the low-temperature region. The temperature increase caused by the electrochemical reactions promoted the forward movement of chemical equilibrium of the direct internal reformation reaction of CH 4 , which promoted the conversion of CH 4 to H 2 and CO, and led to a decrease of the mole fraction of CH 4 . The mole fraction of H 2 O increased at the high-temperature region due to the electrochemical reaction of H 2 . The production of H 2 O provided reactant for the DIR and WGS reactions and promoted the conversion of CO and CH 4 to H 2 .

Conclusions
A comprehensive 2D axisymmetric model of a flame fuel cell unit was developed in this paper. The model considers the coupling effects of chemical and electrochemical reactions, momentum transfer, heat transport, and mass transport in both the SOFC and the porous media burner. Numerical predictions of the temperature distribution and IV curves showed good agreement with experimental results for various equivalence ratios. The effects of the Ni catalyst and the electrochemical reactions on methane fuel-rich combustion were analyzed. Ni catalyst in the SOFC anode promoted the water-gas shift reaction and the direct internal reformation of methane in the fuel-rich reformation region. The heat released and the H 2 O produced by the electrochemical reactions promoted the conversion of CH 4 and CO to H 2 , which has a catalytic enhancement effect on the fuel rich combustion of methane in porous media.  an  anode  DIR  direct internal reforming  eff  effective  el  electronic  ion  ionic  irr  irreversible  ref  referenced  rev  reversible  s  solid  trans  transfer  TPB  triple phase boundary  WGS  water gas shift