A comprehensive simulation of gas concentration impedance for solid oxide fuel cell anodes

Energy Conversion and Management 106 (2015) 93–100

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

A comprehensive simulation of gas concentration impedance for solid oxide fuel cell anodes M. Fadaei a,⇑, R. Mohammadi b a b

Department of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, Iran

a r t i c l e

i n f o

Article history: Received 8 May 2015 Accepted 28 August 2015

Keywords: Solid oxide fuel cell Electrochemical impedance spectroscopy Impedance modeling Concentration impedance Anode

a b s t r a c t This paper presents electrochemical impedance simulation of a solid oxide fuel cell anode. The model takes in to account the gas-phase transport processes both in the gas channel and within the porous electrode and couples the gas transport processes with the electrochemical kinetics. The gas phase mass transport is modeled using the transient conservation equations (mass, momentum and species equations) and Butler–Volmer equation is used for the anode electrochemistry. In order to solve the system of non linear equations, an in-house code based on the finite volume method is developed and utilized. Results show a depressed semicircle in the Nyquist plot, which originates from gas transport processes in the gas channel, in addition to a Warburg diffusion impedance originates from gas transport in the thick porous anode. The simulation results are in good agreement with published experimental data. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Today, existence of abundant manufactures and automobiles leads to environmental pollution and global warming. Therefore, tremendous effort has been put into environmentally friendly power sources such as fuel cells over the last several decades. Solid oxide fuel cells (SOFCs) are the safest type of high temperature fuel cells that generate electricity. Tubular, planar, and monolithic structures are primary SOFC structures. The solid electrolyte of SOFCs is made of a ceramic material, such as yttria-stabilized zirconia (YSZ), which requires the operating temperature range of 600–1100 °C. The high operating temperature allows for use of a wide range of fuels. Therefore, in the SOFC, not only hydrogen but also several hydrocarbons can be used as fuels. SOFCs can also be used in a wide range of applications. They can be used in hybrid power generating systems. They can be coupled with a gas turbine or a biomass gasifier, and as such can be integrated with other renewable technologies. The high operating temperature of the SOFC also provides excellent possibilities for combined heat and power generation [1]. Electrochemical impedance spectroscopy (EIS) appears to play an important role in fundamental and applied electrochemistry and materials science in the coming years. In EIS experiments, usually a harmonic excitation is imposed to the electrochemical

⇑ Corresponding author at: P.O.B. 87197-73473, Iran. Tel.: +98 361 5425253. E-mail address: [email protected] (M. Fadaei). http://dx.doi.org/10.1016/j.enconman.2015.08.073 0196-8904/Ó 2015 Elsevier Ltd. All rights reserved.

system and the amplitude and phase shift of the resulting response are measured. Measurements can be conducted over a wide range of frequencies until an impedance spectrum is obtained [2]. This non-destructive technique provides useful information on transport properties, materials characterization, and contributions from each type of losses in SOFC operation [3]. EIS measurements of SOFC anodes show that gas transport in the gas channel and within the porous electrode has a significant effect on the impedance spectra and results in low-frequency arc in the impedance spectra [4–6]. This feature is often referred to as gas diffusion and conversion impedance [4,5] or, more generally, gas concentration impedance [7,8]. The common approach for analyzing the experimental data of EIS is fitting the impedance spectra to an equivalent electrical circuit [6]. This approach is good enough for overall performance comparison, but a valid physical interpretation of the values obtained from this type of fitting is often not possible, especially due to the usually overlapping arcs of the spectrum. Recently, electrochemical impedance simulation using physical models appears as a useful tool for both analyzing the experimental results of EIS and study the fuel cells performance [9–16]. The effect of mass transport on the impedance spectra has been modeled by several authors [4,5,7–9,8,12,15,16]. Most of the mentioned researches on concentration impedance simulation of SOFC anodes mainly emphasize the mass transport in the porous electrode only [9] or in the gas channels only [7,8,12,15], where the electrode itself is treated as reactive channel wall without thickness on its own, thereby neglecting additional gas transport

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Nomenclature a b C Di;j Deff i;j

empirical exponent of the exchange current density empirical exponent of the exchange current density total molar concentration (mole m3) binary diffusion coefficient between species i and j (m2 s1) effective diffusion coefficients (m2 s1)

DBi

Bosanquet diffusion coefficient (m2 s1)

Dki

Knudsen diffusion coefficient (m2 s1)

Eact

activation energy of the exchange current density (kJ mol1) Faraday’s constant (96,484.56 C mol1) frequency (Hz) channel height (m) current density (A m2) exchange current density (A m2) cell length (m) anode thickness (m) species molar mass (kg kmol1) inlet partial pressure of species (Pa) partial pressure of reactants and products at the reaction sites (Pa) absolute atmospheric pressure (Pa) outlet pressure (Pa) universal gas constant (8.314 kJ kmol1 K1)

F f Hch i i0 L La Mi pin i pra i pref pout R

processes inside the porous structure. Moreover, most of the mentioned researches neglect the coupling of diffusive and convective transport occurring in any practical flow configuration, thereby considering a purely diffusive mass transport [4,5,9,12,16]. This paper improves previously published models by the authors for SOFC gas concentration impedance [15,16]. In order to simulate the gas transport-related impedance, a quasi-2D model was developed by the authors in Ref. [16], where only the diffusive transport was considered. In the present study, the model is extended to a more comprehensive 2-D model in order to include convective flow in gas channel. Moreover, the previous reported model [15] is improved in order to take in to account the effect of gas transport within the porous electrode. In addition, the non-linear kinetic is used to simulate the electrochemical kinetics. This is the first time to our knowledge a comprehensive 2D model for gas concentration impedance of SOFC anodes is reported which includes both the gas transport in the fuel channel and within the porous electrode considering the full coupling of diffusion and convection. In this study, a planar SOFC is modeled and the steady state behavior and electrochemical spectra of the anode are obtained. The developed model couples the electrochemical kinetics with mass transport. The Butler–Volmer equation is used for the anode electrochemistry, and the transient conservation equations (momentum and species equations) are used for mass transport. In order to solve the system of the nonlinear equations, an in-house code based on finite-volume method is developed and utilized. Furthermore, a parametric study is carried out and the influence of electrode thickness, electrode porosity, inlet fuel velocity, inlet fuel composition and temperature on the impedance spectra is also investigated.

Ract Rg rp t T U in W W ch W rib Xi Y Z

activation resistance (ohm m2) gas transport resistance (ohm m2) average pore radius time (s) temperature (K) inlet velocity (m/s) unit cell width (m) channel width (m) rib width (m) species mole fraction complex admittance (ohm1) complex impedance (ohm)

Greek symbols q gas density (kg m3) gconc concentration overvoltage (V) gact activation overvoltage (V) ganode anode overvoltage (V) gsteady steady-state overvoltage (V) gexcitation excitation amplitude (V) s anode tortuosity h time period (s) c pre-exponential factor of the exchange current density (A m2) e anode porosity

In this study the electrolyte and cathode overvoltages are ignored while the SOFC anode overvoltage (ganode) is analyzed. In addition, the anode ohmic overvoltage is neglected due to the high anode electronic conductivity [17–19]. Tables 1 and 2 show the cell geometry, as well as the assumed SOFC operational conditions. 2.1. Electrochemical model The Nernst equation, which is used to determine the electromotive force of an electrochemical reaction, for SOFC is as follows [1]:

V Nernst

PH2 P1=2 Dg 0 RT O2 ln ¼ f þ 2F 2F PH2 O

!

ð1Þ

2. Modeling and simulation approach Fig. 1 depicts the schematic view of a planar anode-supported SOFC and the computational domain.

Fig. 1. Schematic diagram of a planar SOFC and computational domain [16].

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2.2. Mass transfer model

Table 1 Cell geometry [20]. Element

a b

Size

Cell length Unit cell width Channel width Channel height Anode thickness

L W Wch Hch La

Anode Porosity Anode tortuosity Average pore radius Rib width

e s

19 mm 2 mm 1 mm 1 mm 300 lma 700 lmb 0.3 8.5 1.07 lm 0.5 mm

rp W rib

Corresponds to the impedance simulation. Corresponds to the steady state simulation.

where Pi denote the partial pressure of products and reactants in bulk flow. Therefore the cell voltage is obtained by:

V cell ¼ V Nernst  ganode

ð2Þ

The anode overall loss is given by:

ganode ¼ gact þ gconc

ð3Þ

ra Pin RT H P H2 O ln in2 ra ¼ 2F P H2 O P H2

! ð4Þ

where Pra refers to the reactant and product partial pressure at the reaction sites. The activation overvoltage is determined by the Butler–Volmer equation [1]:

 0

iðxÞ ¼ i

 exp

In the present model, the gas phase is modeled as a binary mixture of hydrogen and steam. Furthermore, temperature is assumed to be constant through the entire gas channel and within the porous electrode. 2.2.1. Mass transfer in gas channel Two-dimensional gas-phase convection and diffusion in the gas channel are described using the Navier–Stokes equations. The transient governing equations are given by the continuity equation, the momentum equation, and the species conservation equations. The equation system is closed through the equation of state assuming ideal gas. Further details on the mass transfer modeling in the gas channel can be found in Ref. [15]. 2.2.2. Mass transfer in porous electrode The gas transport within the porous electrode is modeled using the species conservation equation in the porous media, with the effective transport coefficients, as given by:

    @ðC eX i Þ @ @X i @ @X i ¼ CDeff CDeff þ i;j i;j @t @x @y @x @y

ð7Þ

where e is the porosity and Deff i;j is the effective diffusion coefficient in the porous electrode which is given by:

where the anodic concentration overvoltage is [12]:

gconc;a

95

   F F gact  exp  gact RT RT

ð5Þ

2s

1

1

DBi

  a  b P H2 P H2 O Eact 0 i ¼c exp  Pref Pref RT 

ð6Þ

where c is the pre-exponential factor and P ref is the absolute atmospheric pressure (Pref ¼ 1 atm). Also Eact is the activation energy and a and b are empirical exponents values are extracted from experimental results [22] and listed in Table 2.

¼

Dki

where

ðDBi þ DBj Þ

þ

1 Di;j

ð8Þ ð9Þ

s is the tortuosity, DBi is the Bosanquet diffusion coefficient,

Dki

is the Knudsen diffusion coefficient. Knudsen diffusion has and significant effect at small pore diameters and is determined from [23]:

0

where i(x) is current density and i is the anode exchange current density. Considering an Arrhenius-type temperature dependency for exchange current density and a power law ansatz partial pressure dependency, it is normally expressed as [21]:

e

Deff i;j ¼

Dki

2rp ¼ 3

sffiffiffiffiffiffiffiffiffi 8RT pM i

ð10Þ

where r p is the average radius, and Mi is the species molar mass. 2.3. Electrochemical impedance model In order to simulate the impedance spectra, the mentioned transient model is subjected to a harmonically varying overvoltage and the current response is used for impedance calculations. Further details on the impedance model can be found in Ref. [12]. 2.4. Boundary conditions

Table 2 SOFC operational conditions. Description

Value

The general boundary conditions for the computational domain are: At the channel inlet (x = 0), the inlet velocity and the species concentration are set as boundary conditions. The species mass flux in electrolyte-anode interface (due to the local convention and species mass diffusion) is given by [24]:

Gas inlet velocity Outlet pressure Hydrogen inflow mole fraction

uin ðcm=sÞ P out ðPaÞ xin H2

Water inflow mole fraction

xin H2 O

1%

Temperature Activation energy of the exchange current density Pre-exponential factor of the exchange current density Empirical exponent of the exchange current density Empirical exponent of the exchange current density Amplitude in impedance simulation

T ðCÞ

750 105.04

_ i ¼ Deff m ij q

c ðA m2 Þ

1:83  106 T

where

a

-0.1

vf ¼

b

0.3

gvar ðmVÞ

1

Eact ðkJ mol

38 101325 99%

1

Þ

@yi þ qyi v f @y

Xm _i

ð12Þ

q

_ H2 ¼ M H2 m

ð11Þ

i 2F

ð13Þ

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_ H2 O ¼ M H2 O m

i 2F

ð14Þ

At the channel outlet (x = L), the outlet pressure is set as boundary. The mass diffusion fluxes at the inlet and outlet of channel are neglected and at the top of the channel wall the mass diffusion flux is set to zero. At the interface of channel and electrode, outlet diffusion flux from the channel is set equal to the inlet diffusion flux into the electrode.

2.5. Simulation procedure In order to solve the system of the nonlinear equations, an in-house code based on finite volume method is developed. The governing transient mass transport equations (momentum and species equations) are solved by the SIMPLE algorithm. At a specified frequency the governing equations are time integrated to determine the response current. Then, the electrochemical impedance of assumed frequency is obtained. The process for a range of frequencies (103 to 106 Hz) is repeated until an impedance spectrum is obtained. The computational domain in gas channel is discretized by a 50  6 grid in X and Y directions and in porous electrode is discretized by a 50  10 grid in X and Y directions respectively (see Fig. 1). The developed simulation code has the capability of analyzing the steady state and sinusoidal behavior of SOFCs, simulating the electrochemical impedance spectra, as well as investigating the effects of various parameters on the electrochemical impedance spectra of SOFCs.

3. Results and discussions 3.1. Steady state results Fig. 2 compares the polarization curve when the simulation is carried out with both mass transfer models, with the polarization curve when the simulation is carried out without the porous electrode mass transfer model. The latter curve is therefore similar to the previous reported results [15]. In addition, this figure shows the actual experimental data obtained on a planar SOFC [25], for inflow fuel composition of 97% hydrogen and 3% vapor. This figure clearly shows that when the simulation is carried out without the porous electrode mass transfer model, the obtained results are very far from the experimental data. However, there is still a deviation between the simulation results of the present study and

Fig. 2. Polarization curve at 750 °C in gas composition of 97% H2 + 3% H2O.

Fig. 3. Hydrogen (a) and vapor (b) mole fraction contours in the gas channel and within the porous electrode for ganode = 500 mV at 750 °C in gas composition of 99% H2 + 1% H2O.

Fig. 4. Flow velocity contours in the gas channel and within the porous electrode for ganode = 500 mV at 750 °C in gas composition of 99% H2 + 1% H2O.

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97

experimental data, due to the fact that the cathode and electrolyte overvoltages are ignored in this study. Fig. 3 depicts the hydrogen and vapor mole fraction contours in the gas channel and porous electrode for anode overvoltage of 500 mV. In the longitudinal direction, the mole fraction of hydrogen decreases along the fuel channel from 0.99 to 0.1 which is due to strong consumption rate of hydrogen through the anode wall. In the Y direction, hydrogen and water vapor concentrations do not show significant gradients in the gas channel. This is due to the laminar flow regime. However, the results show a linear concentration reduction along the Y direction within the porous electrode. Fig. 4 depicts velocity contours in the gas channel and porous electrode for anode overvoltage of 500 mV. As expected, because of boundary layer development, velocity is increased along the channel, whereas the velocity value is decreased along the porous electrode. This is because of the considerable change in hydrogen molar fraction, which changes the transport properties of the gas mixture. Fuel concentration is decreased until gas mixture flows through anode to reaction interface that it leads to gas mixture density increasing and consequently decreases the gas flow velocity within anode. 3.2. Impedance spectra Fig. 5 depicts the impedance spectra for ganode = 0 V when the impedance simulation is carried out with both mass transfer models. The gas concentration resistance (Rg) is considered as the resistance corresponding to the appeared full impedance arc. Also, the high-frequency intercept with the real axis corresponds to the charge-transfer resistance (Ract). This figure shows that full impedance spectra generally consist of two main features. The first feature appears at lower frequencies, and the second feature appears at relatively higher frequencies.

Fig. 6. Electrochemical impedance spectra rising from both mass transfer models and impedance spectra when eliminating the gas transfer model in the porous electrode.

Fig. 6 compares the impedance spectra when the impedance simulation is carried out with both mass transfer models, with the impedance spectrum when the impedance simulation is carried out without the gas channel mass transfer model. The latter spectrum is therefore originating from mass transport processes within the porous electrode. This spectrum shows the typical shape of a finite length Warburg element (45° slope at high frequencies) [2], with a relaxation frequency of 400 Hz. Therefore, the lower frequency feature in Fig. 5 originates from mass transfer processes in the gas channel with a relaxation frequency of 7.2 Hz. For the verification purposes, the impedance spectrum originated from mass transfer in the gas channel is obtained and compared to Bessler and Gewies’ results [8]. The impedance spectra appear as a semicircle for both studies. Using the same input parameters as Bessler and Gewies’ study for channel flow model (Table 1 of [8]), the present study yields the relaxation frequency of 4.5 Hz and the resistance of 2.2 X cm2. These values are almost equal to Bessler and Gewies’ results of 1.6 Hz and 4.8 X cm2 (Fig. 6 of [8]). 4. Parametric study 4.1. Effect of overvoltage Fig. 7 depicts impedance spectra for different overvoltages. As shown, the gas concentration resistance decreases when anode overvoltage rises up to 300 mV. In addition, this figure shows that anode overvoltage variations impose almost no changes to the relaxation frequencies. 4.2. Effect of inlet fuel concentration

Fig. 5. Electrochemical impedance spectra for ganode = 0 V at 750 °C in gas composition of 99% H2 + 1% H2O. (a) Nyquist plot, (b) Bode plot.

Fig. 8 depicts impedance spectra at different inlet fuel concentrations. As shown, the gas concentration resistance significantly

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Fig. 7. Electrochemical impedance spectra for different anode overvoltages (a) Nyquist plot, (b) Bode plot. Fig. 8. Electrochemical impedance spectra for different inflow hydrogen concentrations (a) Nyquist plot, (b) Bode plot.

decreases with decreasing the inlet hydrogen concentration from 99% to 90%, so that the resistances of both gas channel and porous electrode-related arcs decrease with decreasing the inlet hydrogen concentration. This is due to the variation of the Nernst potential with hydrogen and water concentration, as described by Eq. (1). However, inflow fuel concentration does not affect the relaxation frequency and it remains constant.

changes to the porous electrode-related arcs, whereas the gas channel-related arcs are thermally activated. In addition, charge-transfer resistance decreases with increasing temperature.

4.4. Effect of anode thickness 4.3. Effect of temperature Fig. 9 depicts impedance spectra for various operating temperatures. It is seen that temperature variations impose almost no

Fig. 10 illustrates the impedance spectra for various anode thicknesses (total thickness of the anode electrode and the anode support) at 750 °C. This figure shows that anode thickness

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99

variations impose almost no changes to the resistance and relaxation frequency of the gas channel-related arcs, whereas variations of the porous electrode-related arcs are significant. It is clear that increasing the anode thickness will increase the diffusion resistance in the porous electrode. As can be seen in Fig. 10, decreasing the anode thickness decreases the resistance and increases the relaxation frequency of the porous electroderelated arcs. When the anode thickness decreases to 30 lm, the porous electrode-related arc almost vanishes and gas transport within the porous electrode does not appear as an additional impedance arc. Therefore, gas transport within the porous electrode with porosity of 0.3 and average pore radius around 1 lm is negligible for the anode thickness less than 30 lm.

4.5. Effect of inflow velocity Fig. 11 shows impedance spectra for various inflow velocities. As can be seen the gas channel-related arc shows strong dependency with inlet velocity as its resistance decreases and its frequency increases with increased velocity, whereas porous electrode-related arcs are independent of inlet velocity.

4.6. Effect of porosity

Fig. 9. Electrochemical impedance spectra for various operating temperatures.

Fig. 10. Electrochemical impedance spectra for various anode.

Fig. 12 shows impedance spectra for various anode porosities. It is clear that increasing the porosity will reduce the diffusion resistance due to a more open structure and subsequently larger effective diffusion coefficient. However porosity variations impose no change to the relaxation frequencies.

Fig. 11. Electrochemical impedance spectra for various inflow velocities.

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produces a tool for analyzing the experimental data of EIS, in order to determine the origin of each appeared arc in the impedance spectra. In addition, the effect of different operating and micro structural parameters on the impedance spectra can be investigated which reduces both the cost and time usually associated with EIS measurements. References

Fig. 12. Electrochemical impedance spectra for various anode porosities.

5. Conclusions This work reports on development of a comprehensive simulation code of concentration impedance using physical models. The emphasis of this study is on the mass transfer phenomena, including diffusive and convective mass transport, in the gas channel and within the porous electrode. The simulated concentration impedance spectra show two dominant features; a high-frequency arc (in the range of 20 Hz to 1 kHz) and a low-frequency arc (less than 20 Hz). The first feature appears as Warburg diffusion impedance originates from gas transport in the porous electrode, whereas the second feature appears as a depressed semicircle in the Nyquist plot, which originates from gas transport processes in the gas channel. However, the presence of the first feature and its relaxation frequency depend on the porous anode thickness. The performed parametric study demonstrates the potential of the developed simulation code to predict the effects of various parameters on the anode concentration impedance. For instance, simulation results show that simulation of gas transport within the porous electrode with porosity of 0.3 is not essential when the anode thickness is less than 30 lm. Moreover, it is found that convective transport plays an important role on the concentration impedance, as gas concentration resistance decreases with increased inflow velocity. The simulation results are in reasonable agreement with reported experimental data. It is found that the EIS simulation

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