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Computational analysis on the electrode geometric parameters for the reversible solid oxide cells
Accepted Manuscript Title: Computational analysis on the electrode geometric parameters for the reversible solid oxide cells Authors: Seoung-Ju Lee, Chi-Young Jung, Sung-Chul Yi PII: DOI: Reference:
S0013-4686(17)30968-4 http://dx.doi.org/doi:10.1016/j.electacta.2017.04.174 EA 29432
To appear in:
Electrochimica Acta
Received date: Revised date: Accepted date:
22-12-2016 10-4-2017 30-4-2017
Please cite this article as: Seoung-Ju Lee, Chi-Young Jung, Sung-Chul Yi, Computational analysis on the electrode geometric parameters for the reversible solid oxide cells, Electrochimica Actahttp://dx.doi.org/10.1016/j.electacta.2017.04.174 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Computational analysis on the electrode geometric parameters for the reversible solid oxide cells
Seoung-Ju Leea, Chi-Young Jungb, Sung-Chul Yia,c*
a
Department of Hydrogen and Fuel Cell Technology, Hanyang University, 222 Wangsimni-ro,
Seongdong-gu, Seoul 133-791, Republic of Korea b
Hydrogen and Fuel Cell Center, Korea Institute of Energy Research (KIER), 20-41,
Sinjaesaengeneogi-ro, Haseo-myeon, Jellabuk-do 56332, Republic of Korea c
Department of Chemical Engineering, Hanyang University, 222 Wangsimni-ro,
Seongdong-gu, Seoul 133-791, Republic of Korea
* Corresponding author. Tel.: +82-2-2220-0481; fax: +82-2-2298-5147. E-mail address: [email protected]
Abstract
Increasing global energy demands have been accelerating the research and development of reversible electrochemical systems that can realize an efficient use of the intermittent renewable energy 1
resources. This paper thus describes a numerical investigation of reversible solid oxide cells (RSOCs), for their high energy efficiency delivered from the high operating temperatures ranging from 600 to 1000 °C. Unlike the previous studies, a model-based strategy is applied for the simultaneous integration of different operating modes (namely, fuel cell and electrolysis cell modes) to enable more realistic predictions on the trade-off behavior of the effects of electrode design parameters on the cell performance. This approach was taken to investigate the effects of various geometric designs and operating parameters (electrode backing layer thickness; interconnector rib size; fuel gas composition) on the current-potential characteristic and the round-trip efficiency. The cell performance was significantly affected by the rib size, particularly when the backing layer was thin, because of the uneven distribution of the reactant species. Overall, this study provides insights into key geometric design parameters that dominate the performance of dual-mode RSOCs.
Keywords: Solid oxide cells; Reversible operation; Model; Optimization
Nomenclature a Av b c C
Exponential constant Active surface area per unit volume (m2 m-3) Exponential constant Exponential constant Molar concentration (mol m-3)
Cp dp D
Heat capacity (J kg -1K-1) Pore diameter (m) Diffusion coefficient (m2 s-1) 2
E F i0 I
Energy (J mol-1) or electric potential (V) Faraday constant (96485 C mol-1) Exchange current density (A m-2) Current density (A m-2)
k K l M ne nel nt p P r
Thermal conductivity (W m-1 K-1) permeability (m2) length (m) Molecular weight (kg mol-1) Number of electrons transferred per reaction Number fraction of electron-conduction particles Total number of particles in unit volume (m-3) Probability of percolation Pressure (atm) Radius (m)
R Rcon S t T
Universal gas constant (8.314 J mol-1 K-1) Contact resistance (Ω cm2) Source term (mol m-3 s-1 or kg m-3 s-1 or J m-3 s-1) Thickness (m) Temperature (K)
u V w x Z
velocity (m s-1) Cell potential (V) Characteristic length (Å) Mole fraction Average coordination number
Greek letters α γ ε η θ κ λ μ
Charge transfer coefficient Pre-exponential factor Porosity Overpotential (V) Contact angle between electron and ion-conducting particles (°) Charge conductivity (S m-1) Round-trip efficiency viscosity (kg m-1 s-1)
ξ ρ τ φ
Stoichiometric coefficient density (kg m-3) Tortuosity Volume fraction of electron-conducting particles 3
Φ ΩD
Charge potential (V) Dimensionless collision integral
Subscripts act ABL AFL b ch el eq f FBL
Activation Air electrode backing layer Air electrode functional layer Backward reaction Gas Channel Electron-conducting particles Equilibrium Forward reaction Fuel electrode backing layer
FFL GDC H2 H2O i
Fuel electrode functional layer GDC layer Hydrogen Steam Species or phase
io j K l m ma N2 O2 rib s SOEC SOFC YSZ
Ion-conducting particles Species Knudsen diffusion electrolyte phase , electrolyte material Species mass Nitrogen Oxygen Interconnector rib Solid phase, electrode material SOEC operation SOFC operation YSZ layer
Superscripts air eff emf fuel
Air electrode Effective Electromotive force Fuel electrode 4
1. Introduction
Renewable energy resources, including solar, wind, tidal, and biomass, are subjects of on-going research in the field of sustainable development. They can replace conventional fossil fueled engines with more eco-friendly ones. However, their intermittency produces undesirable time lags between demand and supply. It requires an energy storage device in which excess energy is stored in the form of an energy carrier and released as needed. Among various candidates for energy storage, solid oxide electrolysis cells (SOECs) and solid oxide fuel cells (SOFCs) have gained popularity for fuel flexibility, contaminant tolerance and fast electrochemical reaction kinetics due to operating temperatures ranging from 600 to 1000 °C [1,2,3,4]. Typically, solid oxide cells consist of oxide ion-conducting electrolytes sandwiched between porous fuel and air electrodes, flow fields, and bipolar plates. The electrodes play an important role in promoting the kinetics of the electrochemical reactions, and they are used as the mechanical supports for the cell assembly. Thus, the structural design parameters of the electrodes – such as the electrode thickness and the rib size (interconnector–electrode interface) – has to be optimized in order to achieve superior cell performance by ensuring sufficient mass transport of reactant species and reaction sites. In order to reduce the trial-and-error efforts in finding the optimal electrode design, a number of researchers have investigated the effects of various design parameters on the performance of SOECs and SOFCs, individually, using the model-based approaches. Kenney and 5
Karan emphasized the need for optimization of electrode microstructure and geometry by estimating their impact on mass transport and reaction characteristics in SOFC cathode [5]. Subsequently, Ni et al. used a mathematical model to evaluate the effect of pore structural parameters of electrode on the overpotentials of SOEC [6], and Shi and Xue developed a two-dimensional (2-D) numerical model to predict the performance of bi-electrode supported SOFC with various pore distributions of electrodes [7]. In terms of stack operation, Laurencin et al. assembled an in-house multiphysics model to study the effects of cell structural design parameters and operating conditions on irreversible losses in SOECs [8]. Hosseini et al. developed a model to optimize the macro- and micro-structural parameters of SOFCs, including the thickness and porosity of the diffusion layer [9]. The optimization of rib size in SOFCs was performed using numerical approach [10,11,12]. In addition, the characteristics of the SOFC electrodes were been explored in detail, estimating the electrochemical active thickness, which means electrochemically active region in the electrodes [13,14,15,16]. For SOECs, the competitive effects of electrochemical reactions and chemical reactions were investigated by varying the thickness, porosity, particle size and ionic conductivity of electrodes [17, 18]. On the other hand, reversible solid oxide cells (RSOCs) are still at an early stage of development [19,20,21,22,23]. The ability to switch between electrolysis cell and fuel cell modes is one of the foremost features that facilitate storing/generating energy in a cost-effective manner. However, the optimized parameters on the designs of SOECs or SOFCs, solely, would not 6
necessarily result in high performance of RSOCs because these devices differ in electrode kinetics, gas environment, heat generation, and chemical stability. In order to address this issue, Njodzefon et al. developed a zero-dimensional RSOC model to predict the current–potential characteristics at various operating temperatures [24]. Inspired by their results, Li et al. developed a theoretical model coupled with heterogeneous elementary reaction paths to enable realistic prediction on the RSOC performance with different air electrode thicknesses and microstructural features [25]. Energy storage system was designed later using RSOC to examine the influence of operating temperature, pressure and fuel utilization on the system efficiency [26]. Though some of the geometric parameters of the electrodes were addressed in [25, 27], most of the existing studies have been focused on the influence of the operating conditions on the RSOC performance. In particular, there is lack of information on the optimization of rib size considering simultaneously both the SOEC and SOFC operations. Therefore, it is still unknown and now under active debate whether all the electrode design parameters are significant in improving the overall performance of RSOC. In this work, we report a 2-D model to evaluate the trade-off between various design parameters for the individual SOEC and SOFC operations, thus the RSOC performance. The proposed model takes into account both the interconnector rib and the gas flow channel at external boundaries. The effects of the electrode backing layer thickness, the rib size, and the fuel gas composition on the performance of RSOCs were investigated with consideration of suitable conditions for their dual operations. In addition, the distribution of reactant is approximated as an indicator for the cell 7
polarization during the RSOC operations.
2. Model Formulation 2.1. System description and model assumptions A cross-sectional 2-D domain of RSOC was considered in order to elucidate the influence of the interconnector design by computational fluid dynamics (CFD) analysis. As illustrated in Fig. 1, the symmetric domain was based on a planar RSOC design comprising six layers: a Ni-yttria-stabilized zirconia (YSZ) fuel electrode backing layer (FBL); a Ni-YSZ fuel electrode functional layer (FFL); a YSZ electrolyte; a gadolinia-doped ceria (GDC) interdiffusion barrier layer; a La0.8Sr0.2CoO3 (LSC)GDC air electrode functional layer (AFL); and an LSC air electrode backing layer (ABL).
The
electrochemical reactions in SOEC and SOFC operations will most likely occur at the functional layers near the electrode–electrolyte interfaces, as they mainly depend on the ionic charge transport [18]. Considering extension of the reaction zones, the FBL, FFL, AFL, and ABL were defined as the electrochemical reaction zone in the model domain. The major assumptions used to develop the RSOC model are listed below: 1) Steady-state operation; 2) Isothermal condition at the interconnector–electrode and gas channel–electrode interfaces; 3) Ideal gas mixture; 4) Isotropic microstructural and material properties; 8
5) Cell potential ranging from 0.4 to 1.6 V for SOFC and SOEC operation. . 2.2. Mass transport In the porous electrode layers, the fuel and air are transferred as they are produced or consumed by the electrochemical reactions. To describe the mass transport, the governing equation was adopted as: eff ∇ ∙ (𝑢𝐶𝑖 ) = ∇ ∙ (𝐷𝑖,𝑚 ∇𝐶𝑖 ) + 𝑆𝑖
(1)
where Deff i,m , Ci, and Si are the effective diffusion coefficient, molar concentration, and generation/consumption rate of species i, respectively. The fluid velocity in porous media (u) can be written in accordance with Darcy’s law: 𝐾
𝑢 = − ∇𝑃
(2)
𝜇
where K is the permeability of porous media and μ is the viscosity of fluid. In order to calculate the pressure (P) field, the conservation equation was employed as follow [28]: 𝐾
0 = ∇ ∙ (𝜌 ∇𝑃) + 𝑆ma
(3)
𝜇
where ρ is the density of fluid and Sma is the rate of mass generation/consumption due to the electrochemical reactions. Species diffusion in the porous electrodes can proceed through molecular diffusion or Knudsen diffusion, depending on the mean free path of the molecules and the pore size. The mean free path of the gas species was approximately 0.2–0.4 μm in this study; the comparable pore radii (0.25 and 2.4 μm) were considered for the computational analysis. Therefore, the effective diffusion coefficient 9
was defined, involving the two parallel diffusion mechanisms, as: 1 eff 𝐷𝑖,𝑚
𝜏
1
= ( 𝜀 𝐷
𝐾,𝑖
+
1
8𝑅𝑇
3
𝜋𝑀𝑖
𝐷𝐾,𝑖 = 𝑑p √ 𝐷𝑖𝑗 =
1 𝐷𝑖𝑗
)
(4) (5)
2.6×10−7 𝑇 1.5
(6)
𝑤𝑖 +𝑤𝑗 2 2 ) 𝛺𝐷,𝑖𝑗 1 1 2 √ + 𝑀𝑖 𝑀𝑗
𝑃(
where Dij, DK,i, and xi are the binary diffusion coefficient, Knudsen diffusion coefficient, mole fraction of species i , respectively; τ is the tortuosity and ε is the porosity; dp, Mi, R, and T are the pore diameter, the molecular weight of species i, the universal gas constant, and the operating temperature, respectively; wi is the characteristic length of species i [29, 30] and ΩD,ij is the dimensionless collision integral [30]. The source term arising from the electrochemical reactions (Si) can be written as: 𝑆𝑖 =
𝜉𝑖 ∇∙𝐼
(7)
𝑛𝑒,𝑖 𝐹
where ξi, ∇ ∙ 𝐼, ne,i, and F are the stoichiometric coefficient of species i, the rate of charge generation/depletion, the number of electrons transferred per electrochemical reaction, and the Faraday constant, respectively. The mass transport was coupled with the electrochemical reaction kinetics via ∇ ∙ 𝐼, which is described in detail in Section 2.3.
2.3. Electrochemistry To calculate the electrode and electrolyte potentials in the two operating modes, the governing equations for charge transport were defined as: 10
eff 0 = ∇ ∙ (𝜅𝑠,𝑖 ∇𝛷𝑠 ) + ∇ ∙ 𝐼𝑠
(8)
eff 0 = ∇ ∙ (𝜅𝑙,𝑖 ∇𝛷𝑙 ) + ∇ ∙ 𝐼𝑙
(9)
where the subscripts s and l indicate the solid phase (electrode material) and electrolyte phase (electrolyte material), respectively; Φ is the potential of the electrode or electrolyte; and I is the current density of the fuel or air electrode. The effective electronic or ionic conductivity ( κeff ) was estimated as [31]: 𝜅𝑠eff = 𝜅𝑠 ( 𝜅𝑙eff = 𝜅𝑙 (
𝜑𝑒𝑙 −0.294
)
𝜑𝑖𝑜 −0.294
)
2
(10)
𝜀 1+1+𝜀−0.294
2
(11)
𝜀 1+ −0.294 1+𝜀
where φ is the volume fraction of electron or ion-conducting particles. The amount of charge generation in the solid and electrolyte phases (∇ ∙ 𝐼𝑖 ) was evaluated using the Butler–Volmer equation: ∇ ∙ 𝐼𝑖fuel = 𝐴v 𝑖0fuel {exp ( ∇ ∙ 𝐼𝑖air = 𝐴v 𝑖0air {exp (
fuel fuel 𝐹𝛼b 𝜂
𝑅𝑇
air air 𝐹𝛼b 𝜂
𝑅𝑇
) − exp (
) − exp (
𝐹𝛼ffuel 𝜂fuel 𝑅𝑇
𝐹𝛼fair 𝜂air 𝑅𝑇
)}
(12)
)}
(13)
where the superscripts “fuel” and “air” indicate the fuel and air electrodes, respectively; i0, αb, and αf are the exchange current density and the backward and forward charge transfer coefficients, respectively. The electrode overpotential (ηi) and i0 were defined as follows [29, 32]: 𝑖 𝜂 𝑖 = 𝛷𝑠 − 𝛷𝑙 − 𝐸eq
(14)
air fuel 𝐸 emf = 𝐸eq − 𝐸eq = 1.253 [V] − 2.452 × 10−4 [V K −1 ]𝑇 −
𝑖0fuel = 𝛾 fuel 𝑥Ha 2 𝑥Hb 2 O exp (−
fuel 𝐸act
𝑅𝑇
)
11
(16)
𝑅𝑇 2𝐹
ln (
𝑃𝐻 2 𝑂
0.5 𝑃𝐻2 𝑃𝑂 2
) (15)
𝑖0air = 𝛾 air 𝑥O𝑐 2 exp (−
air 𝐸act
𝑅𝑇
)
(17)
where Eeq and Eemf represent the equilibrium and electromotive force potentials, respectively; Eact is the activation energy; γ is the pre-exponential factor; and a, b, and c are exponential constants used to consider the concentration dependence. The active surface area per unit volume (Av) was calculated by taking into account the microstructures of the electrodes, and expressed as follows [33]: 𝐴v = πsin2 𝜃𝑟el2 𝑛t 𝑛el (1 − 𝑛el )
𝑍el 𝑍io 𝑍
𝑝el 𝑝io
(18)
where θ, rel, nt, and nel are the contact angle between the electron- and ion-conducting particles, the radius of the electron-conducting particles, the number of particles per volume, and the numberfraction of electron-conducting particles, respectively; Z, Zel, and Zio are the average coordination number and coordination numbers of the electron- and ion-conducting particles, respectively; and pel and pio are the probabilities of the electron- and ion-conducting particles, respectively. The parameters in Eq. (18) were calculated as the following equations: 𝑛t = 4
1−𝜀
𝑛el =
𝜑el
(19)
3 𝑟 π𝑟 3 [𝑛 +(1−𝑛el )(𝑟io ) ] 3 el el el
(20)
3
𝑟
[𝜑el +(1−𝜑el )/(𝑟io ) ] el
𝑍el = 3 +
𝑍−3 𝑟
2
(21)
2
(22)
[𝑛el +(1−𝑛el )(𝑟io ) ] el
𝑟
𝑍io = 3 +
(𝑍−3)(𝑟io )
2
el
𝑟
[𝑛el +(1−𝑛el )(𝑟io ) ] el
0.4 4.236−𝑍el−el 2.5
𝑝el = [1 − (
2.472
)
]
(23)
12
0.4 4.236−𝑍io−io 2.5
𝑝io = [1 − ( 𝑍el−el =
)
2.472 𝑛el 𝑍
]
(24) (25)
2
𝑟
[𝑛el +(1−𝑛el )(𝑟io ) ] el
𝑍io−io =
(1−𝑛el )𝑍
(26)
2
𝑟 [𝑛el ( io ) +(1−𝑛el )] 𝑟 el
In this study, the electron- and ion-conducting particles were assumed to have the same radius and volume fraction.
2.4. Energy transport In the RSOC, the electrochemical kinetic and the transport properties of the materials are dependent on the temperature [34], and the temperature gradient in the cell can result in the cell fracture by thermal stresses. Thus, the thermal management is important to ensure reliable operations with high cell performance. In particular, the system efficiency of the SOEC operation is affected by the cell potential for thermally self-sustaining operation, called as thermoneutral potential [35]. However, the thermal issue, which strongly depends on the gas flow distribution and the operating conditions [36, 37], is beyond the scope of this paper. Hence, the energy balance for maintaining the cell temperature was excluded, and the temperature was assumed constant at the boundaries of the model domain. The energy conservation equation for the system was implemented as: ∇ ∙ (𝜌𝐶p 𝑢𝑇) = ∇ ∙ (𝑘 eff ∇𝑇) + 𝑆𝑇 𝑘 eff = 𝑘f 𝜀 + 𝑘s (1 − 𝜀)
(27) (28)
where Cp is the heat capacity of fluid; kf and ks are the thermal conductivity of the fluid and solid, 13
respectively. The heat generation/depletion term (ST) consists of three sources, that is ohmic resistance, thermodynamic irreversibility, and reversible heat by electrochemical reaction. 𝑆𝑇 = (
𝐼𝑙2
eff +
𝜅𝑙
𝐼𝑠2
𝜅𝑠eff
) + |𝜂𝛻 ∙ 𝐼| +
∇∙𝐼 𝑛𝑒 𝐹
𝑇∆𝑆
(29)
where ∆S is the entropy change for the reaction.
2.5. Model implementation The governing equations for mass, charge and energy conservation were coupled and solved by a CFD software package (FLUENT 6.3.26) with a user-defined function written in the computer language of C++. The numerical calculations were carried out according to the procedure which mitigates the solution convergence problem due to the severe coupling of transport phenomena, by using previous case solutions as initial values. In addition, the grid independence of numerical solution was tested to ensure the accuracy of the simulation results for various electrode geometric parameters. Four different grid sizes, 160×50, 320×100, 480×150 and 640×200 were examined during the test. Fig. 2 shows the comparison of the amount of charge generation at 0.4 V computed based on the four grids. The grid independent solutions were obtained with the finer grids, 480×150 and 640×200. Consequently, 480×150 was selected as an optimum grid density. As for boundary conditions, the YSZ and GDC layers were assumed to separate the fuel and air electrodes completely, conducting oxide ions but not electrons. Although the GDC layer has significant electron conductivity, in particular at very low oxygen pressure, the YSZ layer acts as a 14
barrier to prevent the electron transport between the two electrodes. In addition, the electron transport in the thin GDC layer had nearly no effect on the electrochemical reactions in the model predictions. Thus, for the simplicity of the calculation, non-flux condition was applied for the electrolyte potential at the interfaces between the backing layers and gas channel/interconnector, while Dirichlet boundary condition was chosen for the species, electrode potential and energy, as specified in Table 1. In both the SOEC and SOFC operations, the molar composition was set to 50 % H2O and 50 % H2 at the FBL–fuel gas channel interface, and 21 % O2 and 79 % N2 at the ABL–fuel gas channel interface. The remaining external boundaries were treated as symmetric. The contact resistances on the backing layer–interconnector interfaces were set at 0.005 Ω cm2 for both the fuel and air electrode sides of the cell, to consider their effect on optimizing the rib size. Accordingly, the current density at the rib (irib) can be determined as [31]: 𝑖rib =
∆𝑉
(30)
𝑅𝑐𝑜𝑛
where ∆V is the potential drop at the rib and Rcon is the area specific contact resistance at the rib.
3. Results and Discussion 3.1. Model validation The flow and the distribution of species along the cell length, which are excluded in this study, can cause significant concentration loss, in particular at high current densities. Thus, the proposed model was validated by comparing its results with the experimental current-potential curves previously 15
reported [19],which were based on button cells under low fuel utilization conditions. Specifically, several electrochemical parameters were fitted to the experimental data for approximation, without making the uncertain assumptions (e.g. charge transfer coefficient being same in both the SOEC and SOFC operations) [32]. To reflect the characteristics of electrochemical reactions in the two operating modes, the charge transfer coefficients (αb, αf) were varied independently, obtaining bestfit curves near the zero-current point. In addition, the exponential constants for concentration dependence (a,b,c) were set to values that minimize the error between the predicted and experimental results at high current region that exhibits different mass transport resistances due to the microstructure of electrodes. Since the pre-exponential factors for exchange current density (γ) had less influence on the slope of fitting curves than the previous parameters, they were estimated by considering the average of absolute errors in overall current range. Table 2 lists the operating conditions, material properties, and geometric information; Table 3 presents the electrochemical parameters in the model validation. As presented in Fig. 3, the simulated results were in good agreement with the experimental data for two different microstructures in the FBL (labelled as highporosity 0.38, low-porosity 0.24), having errors within 3 % in the target range of cell potentials. Thus, the model could be used to capture the effect of mass transport on the cell performance in both the operations. In the following studies, the microstructural parameters were fixed as in the case of the high-porosity 0.38, unless mentioned otherwise.
16
3.2. Cell performance with respect to thickness of electrodes In the intermediate-temperature solid oxide cells, the fuel and air electrode-supported designs have been widely accepted, that the thickness of either fuel or air electrode layer is in between 500 and 1000 μm, whereas the other layers are on the order of tens of micrometers. Because the directions of species transport in SOEC and SOFC operations are opposite to each other as shown in Fig. 1b, the variation of electrode thicknesses leads to different degree of mass transport resistance for each operation. Thus in order to estimate its impact on the RSOC performance, either FBL or ABL thicknesses was varied from 50 to 800 μm, while maintaining the opposite electrode backing layer at 100 μm. Fig. 4 presents the simulated current–potential (I–V) characteristic curves for the various FBL and ABL thicknesses. The species diffusion within the fuel electrode could influence the electrochemical reactions in both the two operating modes because H2O and H2 were the reactants for SOECs and SOFCs, respectively. Nevertheless, similar to the experimental results obtained with different microstructures [19], the performance of the SOEC was more sensitive to its FBL thickness at higher current density, relative to that of the SOFC (Fig. 4a). This was mainly attributed to the concentration loss during SOEC operation, related to the consumption of reactants and the concentration dependence factors in Eqs. (15,16,17). In contrast, the cell performance during SOFC operation was severely degraded, with greater mass transport resistance, upon increasing the ABL thickness, while that during SOEC operation remained constant (Fig. 4b). This was due to O2 being the only reactant in the air electrode for the SOFC, with the diffusion of air being slower than that of 17
H2O/H2. In addition, Fig. 4a reveals that the thinnest FBL (50 μm) did not result in the best cell performance for SOEC mode, despite having the shortest species diffusion path. In fact, the electrodes having thickness smaller than the electrochemical active thickness would reduce the number of active sites involved in the electrochemical reactions [15,16]. Under the conditions examined in this study, most of the electrochemical reactions occurred in the FFL and AFL (>98 % based on simulated result at the most severe condition). Thus, it could be deduced that the uneven distribution of reactant species (H2O) rather than the insufficient active sites was likely the cause of the decrease in the SOEC performance when the thinnest FBL was used. On the other hand, the thinnest FBL exhibited the best SOFC performance due to the higher Knudsen diffusivity of H2 preventing the uneven distribution of reactant. Furthermore, the effect of the thinnest ABL on the SOFC performance degradation at high current density was insignificant compared with that of the thinnest FBL for SOEC mode. This was because the influence of the thinnest ABL on the reactant distribution was reduced, under the operation close to the limiting current density. The overall RSOC performance can be represented by the round-trip efficiency, which is defined as the ratio of the input energy in SOEC operation to the output energy in SOFC operation. Assuming the operating time and the magnitude of the current density are same for the two operations, the round-trip efficiency of the cell (λ) can be calculated as [26]: 𝜆 = 𝑉SOFC /𝑉SOEC
(31)
where V is the operating cell potential. Fig. 5a shows the effect of varying the FBL and ABL 18
thicknesses on the round-trip efficiency at 0.2 to 0.8 A cm -2. The maximum variation of the efficiency with different electrode thicknesses was 0.5% at 0.2 A cm -2, and it reached 2.8 % at 0.8 A cm -2. In the current range studied, the 200-μm-thick FBL or ABL yielded the highest efficiency for each current density. For the thickest backing layers (800 μm), the FBL exhibited higher efficiency (1.1 % higher at 0.8 A cm -2) than the ABL as the current increased. This resulted from the severe performance degradation in the SOFC operation with the thickest ABL as shown in Fig. 4b. In order to evaluate the optimal cell configuration, the round-trip efficiencies were calculated with constant total backing layer thickness of 900 μm, as shown in Fig. 5b. The thin FBL and ABL (100-100 μm case), that can be implemented to metal-supported cells, was also considered as a possible cell configuration in the comparison. Overall, the results in Fig. 5b were consistent with that in Fig. 5a. The fuel electrode-supported cells (thick FBL and thin ABL), which provide sufficient mechanical strength, showed higher efficiency than the air electrode-supported cells (thin FBL and thick ABL) over the whole range of current density. The difference of the efficiency between the 100-100 μm case and 700-200 μm case was only 0.6% at 0.8 A cm -2. Moreover, it is notable that the RSOC performance was most affected by the thinnest electrodes. Therefore, the fuel electrode-supported design is expected to be more suitable for the RSOC operation in practical point of view, and the thickness of the thin electrode should be optimized to enhance the efficiency.
19
3.3. Effect of rib size on cell performance In the flow field design, the interconnector rib size is crucial for balancing the concentration and ohmic losses as it determines the length and area of transport paths for species and charge. Thus, three different rib widths of 0.5, 1.0, and 1.5 mm were examined to identify their influence on mass and charge transport. The simulations were carried out for the three cases in the SOFC and SOEC operations at 0.6 V and 1.4 V, respectively. As shown in Fig. 6, under both of the operating conditions, the magnitude of the current density increased upon decreasing the rib width, regardless of the thickness of the backing layer, due to the easier diffusion of reactants. In addition, the rib size resulted in more drastic changes in the current density of the SOFC upon varying the FBL thickness and as well as the current density of the SOEC upon varying the ABL thickness. Hence, the thickness of the backing layer at the maximum current density had a tendency to increase upon enlarging the rib width. For the SOEC operation, the optimal FBL thickness for the maximum current density was found to be only 100 μm for a rib width of 0.5 mm, while it increased to approximately 200 μm for a rib width of 1.0 mm. To further discuss the relationship between the optimal electrode thickness and the rib size, the distributions of the H2O concentration and fuel electrode overpotential were analyzed by selecting cases at 1.4 V for the SOEC operation (Fig. 7). As expected, the reactant depletion became more prominent upon enlarging the rib width, due to the increased diffusion distance. Thus, in Fig. 7c, the electrode overpotential was increased specifically at the area under the rib. On the contrary, as shown 20
in Figs. 7a and 7b, it was clear that the thicker FBL alleviated the reactant depletion under the rib. In particular, in the 200-μm-thick FBL having a 0.5-mm rib width (lrib), the H2O distribution was the closest to a one-dimensional profile, meaning that the performance improved as a result of uniform distribution of reactant with higher utilization of the electrode materials [44]. Consequently, for the 200-μm-thick FBL in Fig. 7d, the rib size had little effect on the distribution of the electrode overpotential. However, the benefit from mitigating the reactant depletion under the rib was not observed upon further increasing the thickness of the backing layers as the concentration loss was more dominant. In other words, the influences of the thickness and the rib size on the overall RSOC performance should be simultaneously considered to determine the optimal electrode design. Fig. 8 illustrates the comparison of the round-trip efficiencies of the RSOC at 0.8 A cm -2, according to the variation of the thickness of the backing layers and the rib size. The average of efficiency with respect to the three rib widths of 0.5, 1.0, and 1.5 mm was 71.5, 70.4 and 66.1 %, respectively. The maximum variation of the efficiency with different electrode thicknesses (2.8% for 1.0-mm rib width) was 3.3 % for 0.5-mm rib width and 3.8 % for 1.5-mm rib width. Due to the alleviation effect on the reactant depletion under the rib, the FBL or ABL thickness for the maximum efficiency increased with enlarging the rib width. In addition, with a 0.5-mm rib width, the 800-μmthick FBL demonstrated 1.4 % higher efficiency of the RSOC than the 800-μm-thick ABL. From the results in Fig. 8, it can be seen that decreasing the rib size enhances the RSOC efficiency, but at same time the electrode thickness becomes the most dominant factor for the mass transport resistance. 21
However, it should be also taken into account that the small rib size could lead to severe contact resistance problems. As can be seen from Fig. 9, the optimal rib width increased with higher contact resistance, and for the contact resistance of 0.05 Ω cm2, the 0.5-mm rib width case exhibited 7.2 % lower efficiency of the RSOC than the 1.0-mm rib width case. Thus, it means that the rib size has the greatest impact on the performance improvement for the RSOC with considerable contact resistance.
3.4. Effect of fuel-inlet composition on cell performance The fuel-inlet composition may be varied for different operating conditions and control strategies in the RSOC operations. In this section, the fuel composition was investigated, as it determines the theoretical potential for electrolysis or fuel cells and thus changes the overall RSOC performance. Specifically, the molar fractions of H2O and H2 were set at 33, 50, and 67 % for the SOEC and SOFC modes, respectively. Fig. 10 displays the current densities obtained at various fuel compositions and thicknesses of the backing layers. Although the results are similar to the variations in performance with different the rib sizes (Fig. 6), the fuel composition had less impact on the magnitude of the current density in the SOFC operating mode, where the O2 diffusion is a limiting factor. On the other hand, the current density of the SOEC operating mode was significantly dependent on the reactant fraction. In addition, for the two operating modes, the varying fuel composition was ineffective in determining the optimal thickness of the backing layers in the light of maximum current density. This tendency can be also seen in the round-trip efficiency of the RSOC with respect to the fuel 22
composition and the thickness of backing layers at 0.8 A cm -2 (Fig. 11). In the composition range studied, the backing layer thickness for the maximum efficiency remained constant, and the difference between the average efficiency in each fuel composition was up to 0.5 %.
4. Conclusions A mathematical model, taking into account the electrochemical kinetics and gas diffusion mechanisms in both the SOEC and the SOFC operating modes, has been developed to investigate the effects of several electrode design parameters on the RSOC performance. The numerical solution was obtained using the user-define functions written in the computer language as part of the work. Then, the simulated results were compared with the measured data sets reported elsewhere [19]. The developed model enabled practical prediction on the cell performance with respect to the thickness of the electrode backing layer, the interconnector rib size, and the fuel composition, also evaluating the limitation of mass transport at the same time. In the detailed analysis, the thickness of the electrode backing layer restricted the performance of the SOEC with a thick FBL and that of the SOFC with a thick ABL. For the overall performance of RSOC, the fuel electrode-supported cell exhibited better round-trip efficiency (1.1 % higher at 0.8 A cm -2) than the air electrode-supported cell due to the severe performance degradation in the SOFC operation with the thick ABL. Furthermore, from the comparison of cell configurations, it was found that for the electrode-supported cell, the thickness of the thin electrode should be optimized to 23
enhance the RSOC performance. Meanwhile, the decrease in the current density was observed upon enlarging the rib width, thus increasing the optimal thickness of thin electrode layer. In particular, our analysis of the reactant distribution revealed that an electrode that is too thin will accelerate the depletion of the reactants under the rib. Additionally, when the rib width was varied from 1.5 to 0.5 mm, the average round-trip efficiency increased by 5.4 %, while the maximum variation of the efficiency with backing layer thickness of 50-800 μm was between 3.8 to 2.8 %. Therefore, although the rib size should be preferentially reduced to achieve high efficiency, the electrode thickness also has to be optimized for further improvement (e.g. even with a 0.5-mm rib width, the fuel electrodesupported cell demonstrated 1.4 % higher efficiency than the air electrode-supported cell). However, it was also observed that the small rib size has the greatest impact on the performance degradation of the RSOC with considerable contact resistance (e.g. for the contact resistance of 0.05 Ω cm2, the 0.5-mm rib width case exhibited 7.2 % lower efficiency than the 1.0-mm rib width case). On the other hand, compared with the case of SOFC, the fuel-inlet composition had more of an effect on improving the SOEC performance upon increasing the reactant fraction, but it was not influential in determining the electrode thickness for the best efficiency. It means that the once-optimized geometric parameters are valid for the range of the considered fuel compositions.
Acknowledgements This research was supported by the Commercializations Promotion Agency (2015K000131) for R&D Outcomes (COMPA) funded by the Ministry of Science, ICT and Future Planning (MSIP). 24
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28
Fig. 1. Schematic representations of (a) a planar RSOC and (b) a cross-sectional domain for calculation of the model.
29
Fig. 2. Grid dependency test results for the model domain at 0.4 V
30
Fig. 3. Validation of the model results with the experimental data in reference [19].
31
Fig. 4. I–V curves predicted by the model for various thicknesses of (a) FBL and (b) ABL in both the SOEC and the SOFC operating modes.
32
Fig. 5. Comparison between the round-trip efficiencies (a) with various thicknesses of the FBL and ABL, (b) with constant total backing layer thickness of 900 μm.
33
Fig. 6. Effect of rib size on current densities at 0.6 V for the SOFC and 1.4 V for the SOEC operating mode, predicted upon varying thicknesses of the (a) FBL and (b) ABL.
34
Fig. 7. H2O molar concentration (mol m-3) profiles in fuel electrodes operated at 1.4 V at FBL thicknesses of (a) 50 and (b) 200 μm; distribution of fuel electrode overpotential (V) operated at 1.4 V at FBL thicknesses of (c) 50 and (d) 200 μm.
35
Fig. 8. The round-trip efficiencies with various backing layer thicknesses and rib sizes at 0.8 A cm -2.
36
Fig. 9. The round-trip efficiencies with various contact resistances and rib sizes at 0.8 A cm -2.
37
Fig. 10. Effect of fuel gas composition on current densities at 0.6 V for the SOFC and 1.4 V for the SOEC operating mode, predicted upon varying thicknesses of the (a) FBL and (b) ABL
38
Fig. 11. The round-trip efficiencies with various backing layer thicknesses and fuel-inlet compositions at 0.8 A cm -2.
39
Table 1. Boundary conditions for species, pressure, electrode/electrolyte potentials and energy. Species i
Pressure
Electrode potential
Electrolyte potential
Energy
FBL–fuel gas channel 𝑃 Ci (= xi) 𝑅𝑇 interface
P
Non-flux
Non-flux
T
FBL–interconnector interface
Non-flux
Non-flux
0
Non-flux
T
FFL–YSZ layer interface
Non-flux
Non-flux
Non-flux
–
–
AFL–GDC layer interface
Non-flux
Non-flux
Non-flux
–
–
ABL–air gas channel interface
Ci (=
P
Non-flux
Non-flux
T
ABL– interconnector interface
Non-flux
Non-flux
V (0.4-1.6)
Non-flux
T
𝑃 𝑅𝑇
x i)
40
Table 2. Operating conditions, material properties, and geometric information based on the results described in references [19, 25]. Parameter
Value
Operating temperature, T (K)
1023
Operating pressure , P (atm) Fuel-inlet mole fraction, xH2 , xH2O
1
Air-inlet mole fraction, xO2, xN2
0.21, 0.79
Ni conductivity, κs,Ni (S m )
3.27×106 -1065.3T [38]
YSZ conductivity, κl,YSZ (S m-1)
3.34×104 e-10300/T [39]
GDC conductivity, κl,GDC (S m-1)
3.5×103 e-6471/T [40]
LSC conductivity, κs,LSC , κl,LSC (S m-1)
2.45×105 -97.5T [41], 2×10-13 e0.0245T [42]
FBL thickness, tFBL (m)
800×10-6
FFL thickness, tFFL (m)
10×10-6
YSZ layer thickness, tYSZ (m)
10×10-6
GDC layer thickness, tGDC (m)
5×10-6
AFL thickness, tAFL (m)
15×10-6
ABL thickness, tABL (m)
20×10-6
Gas channel width, lch (m)
1×10-3
Rib width, lrib (m)
1×10-3
FBL porosity/tortuosity, ε/τ
high- porosity: 0.38/5; low-porosity:0.24/9
FFL porosity/tortuosity, ε/τ
0.24/3
AFL porosity/tortuosity, ε/τ
0.3/3
ABL porosity/tortuosity, ε/τ
0.3/3
FBL pore diameter, dp (m)
high- porosity: 4.8×10-6; low-porosity: 1.4×10-6
FFL pore diameter, dp (m)
0.7×10-6
AFL pore diameter, dp (m)
0.3×10-6
ABL pore diameter, dp (m)
0.4×10-6
Contact angle between electron and ion conducting particles, θ (º)
15
Radius of electron-conducting particle, rel (m)
0.5×10-6
Permeability of electrodes, K (m2)
(2𝑟𝑒𝑙 )2 𝜀 3
SOEC:0.67,0.33; SOFC:0.97,0.03
-1
72𝜏(1−𝜀)2
41
[43]
Table 3. Electrochemical parameters fitted to the experimental data in reference [19]. Parameter
Value
Activation energies for exchange current
120000, 120000
air -1 density, Efuel act , Eact (J mol ) Exponential constants for concentration dependence, a, b, c
0.11, 0.67, 0.25
Pre-exponential factors for exchange current 1×1011, 1×1011 density, γfuel, γair Backward and forward charge transfer coefficients in fuel electrode,
αfuel b ,
1, 0.6
αfuel f
Backward and forward charge transfer coefficients in air electrode,
αair b ,
0.15, 0.45
αair f
42
S0013-4686(17)30968-4 http://dx.doi.org/doi:10.1016/j.electacta.2017.04.174 EA 29432
To appear in:
Electrochimica Acta
Received date: Revised date: Accepted date:
22-12-2016 10-4-2017 30-4-2017
Please cite this article as: Seoung-Ju Lee, Chi-Young Jung, Sung-Chul Yi, Computational analysis on the electrode geometric parameters for the reversible solid oxide cells, Electrochimica Actahttp://dx.doi.org/10.1016/j.electacta.2017.04.174 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Computational analysis on the electrode geometric parameters for the reversible solid oxide cells
Seoung-Ju Leea, Chi-Young Jungb, Sung-Chul Yia,c*
a
Department of Hydrogen and Fuel Cell Technology, Hanyang University, 222 Wangsimni-ro,
Seongdong-gu, Seoul 133-791, Republic of Korea b
Hydrogen and Fuel Cell Center, Korea Institute of Energy Research (KIER), 20-41,
Sinjaesaengeneogi-ro, Haseo-myeon, Jellabuk-do 56332, Republic of Korea c
Department of Chemical Engineering, Hanyang University, 222 Wangsimni-ro,
Seongdong-gu, Seoul 133-791, Republic of Korea
* Corresponding author. Tel.: +82-2-2220-0481; fax: +82-2-2298-5147. E-mail address: [email protected]
Abstract
Increasing global energy demands have been accelerating the research and development of reversible electrochemical systems that can realize an efficient use of the intermittent renewable energy 1
resources. This paper thus describes a numerical investigation of reversible solid oxide cells (RSOCs), for their high energy efficiency delivered from the high operating temperatures ranging from 600 to 1000 °C. Unlike the previous studies, a model-based strategy is applied for the simultaneous integration of different operating modes (namely, fuel cell and electrolysis cell modes) to enable more realistic predictions on the trade-off behavior of the effects of electrode design parameters on the cell performance. This approach was taken to investigate the effects of various geometric designs and operating parameters (electrode backing layer thickness; interconnector rib size; fuel gas composition) on the current-potential characteristic and the round-trip efficiency. The cell performance was significantly affected by the rib size, particularly when the backing layer was thin, because of the uneven distribution of the reactant species. Overall, this study provides insights into key geometric design parameters that dominate the performance of dual-mode RSOCs.
Keywords: Solid oxide cells; Reversible operation; Model; Optimization
Nomenclature a Av b c C
Exponential constant Active surface area per unit volume (m2 m-3) Exponential constant Exponential constant Molar concentration (mol m-3)
Cp dp D
Heat capacity (J kg -1K-1) Pore diameter (m) Diffusion coefficient (m2 s-1) 2
E F i0 I
Energy (J mol-1) or electric potential (V) Faraday constant (96485 C mol-1) Exchange current density (A m-2) Current density (A m-2)
k K l M ne nel nt p P r
Thermal conductivity (W m-1 K-1) permeability (m2) length (m) Molecular weight (kg mol-1) Number of electrons transferred per reaction Number fraction of electron-conduction particles Total number of particles in unit volume (m-3) Probability of percolation Pressure (atm) Radius (m)
R Rcon S t T
Universal gas constant (8.314 J mol-1 K-1) Contact resistance (Ω cm2) Source term (mol m-3 s-1 or kg m-3 s-1 or J m-3 s-1) Thickness (m) Temperature (K)
u V w x Z
velocity (m s-1) Cell potential (V) Characteristic length (Å) Mole fraction Average coordination number
Greek letters α γ ε η θ κ λ μ
Charge transfer coefficient Pre-exponential factor Porosity Overpotential (V) Contact angle between electron and ion-conducting particles (°) Charge conductivity (S m-1) Round-trip efficiency viscosity (kg m-1 s-1)
ξ ρ τ φ
Stoichiometric coefficient density (kg m-3) Tortuosity Volume fraction of electron-conducting particles 3
Φ ΩD
Charge potential (V) Dimensionless collision integral
Subscripts act ABL AFL b ch el eq f FBL
Activation Air electrode backing layer Air electrode functional layer Backward reaction Gas Channel Electron-conducting particles Equilibrium Forward reaction Fuel electrode backing layer
FFL GDC H2 H2O i
Fuel electrode functional layer GDC layer Hydrogen Steam Species or phase
io j K l m ma N2 O2 rib s SOEC SOFC YSZ
Ion-conducting particles Species Knudsen diffusion electrolyte phase , electrolyte material Species mass Nitrogen Oxygen Interconnector rib Solid phase, electrode material SOEC operation SOFC operation YSZ layer
Superscripts air eff emf fuel
Air electrode Effective Electromotive force Fuel electrode 4
1. Introduction
Renewable energy resources, including solar, wind, tidal, and biomass, are subjects of on-going research in the field of sustainable development. They can replace conventional fossil fueled engines with more eco-friendly ones. However, their intermittency produces undesirable time lags between demand and supply. It requires an energy storage device in which excess energy is stored in the form of an energy carrier and released as needed. Among various candidates for energy storage, solid oxide electrolysis cells (SOECs) and solid oxide fuel cells (SOFCs) have gained popularity for fuel flexibility, contaminant tolerance and fast electrochemical reaction kinetics due to operating temperatures ranging from 600 to 1000 °C [1,2,3,4]. Typically, solid oxide cells consist of oxide ion-conducting electrolytes sandwiched between porous fuel and air electrodes, flow fields, and bipolar plates. The electrodes play an important role in promoting the kinetics of the electrochemical reactions, and they are used as the mechanical supports for the cell assembly. Thus, the structural design parameters of the electrodes – such as the electrode thickness and the rib size (interconnector–electrode interface) – has to be optimized in order to achieve superior cell performance by ensuring sufficient mass transport of reactant species and reaction sites. In order to reduce the trial-and-error efforts in finding the optimal electrode design, a number of researchers have investigated the effects of various design parameters on the performance of SOECs and SOFCs, individually, using the model-based approaches. Kenney and 5
Karan emphasized the need for optimization of electrode microstructure and geometry by estimating their impact on mass transport and reaction characteristics in SOFC cathode [5]. Subsequently, Ni et al. used a mathematical model to evaluate the effect of pore structural parameters of electrode on the overpotentials of SOEC [6], and Shi and Xue developed a two-dimensional (2-D) numerical model to predict the performance of bi-electrode supported SOFC with various pore distributions of electrodes [7]. In terms of stack operation, Laurencin et al. assembled an in-house multiphysics model to study the effects of cell structural design parameters and operating conditions on irreversible losses in SOECs [8]. Hosseini et al. developed a model to optimize the macro- and micro-structural parameters of SOFCs, including the thickness and porosity of the diffusion layer [9]. The optimization of rib size in SOFCs was performed using numerical approach [10,11,12]. In addition, the characteristics of the SOFC electrodes were been explored in detail, estimating the electrochemical active thickness, which means electrochemically active region in the electrodes [13,14,15,16]. For SOECs, the competitive effects of electrochemical reactions and chemical reactions were investigated by varying the thickness, porosity, particle size and ionic conductivity of electrodes [17, 18]. On the other hand, reversible solid oxide cells (RSOCs) are still at an early stage of development [19,20,21,22,23]. The ability to switch between electrolysis cell and fuel cell modes is one of the foremost features that facilitate storing/generating energy in a cost-effective manner. However, the optimized parameters on the designs of SOECs or SOFCs, solely, would not 6
necessarily result in high performance of RSOCs because these devices differ in electrode kinetics, gas environment, heat generation, and chemical stability. In order to address this issue, Njodzefon et al. developed a zero-dimensional RSOC model to predict the current–potential characteristics at various operating temperatures [24]. Inspired by their results, Li et al. developed a theoretical model coupled with heterogeneous elementary reaction paths to enable realistic prediction on the RSOC performance with different air electrode thicknesses and microstructural features [25]. Energy storage system was designed later using RSOC to examine the influence of operating temperature, pressure and fuel utilization on the system efficiency [26]. Though some of the geometric parameters of the electrodes were addressed in [25, 27], most of the existing studies have been focused on the influence of the operating conditions on the RSOC performance. In particular, there is lack of information on the optimization of rib size considering simultaneously both the SOEC and SOFC operations. Therefore, it is still unknown and now under active debate whether all the electrode design parameters are significant in improving the overall performance of RSOC. In this work, we report a 2-D model to evaluate the trade-off between various design parameters for the individual SOEC and SOFC operations, thus the RSOC performance. The proposed model takes into account both the interconnector rib and the gas flow channel at external boundaries. The effects of the electrode backing layer thickness, the rib size, and the fuel gas composition on the performance of RSOCs were investigated with consideration of suitable conditions for their dual operations. In addition, the distribution of reactant is approximated as an indicator for the cell 7
polarization during the RSOC operations.
2. Model Formulation 2.1. System description and model assumptions A cross-sectional 2-D domain of RSOC was considered in order to elucidate the influence of the interconnector design by computational fluid dynamics (CFD) analysis. As illustrated in Fig. 1, the symmetric domain was based on a planar RSOC design comprising six layers: a Ni-yttria-stabilized zirconia (YSZ) fuel electrode backing layer (FBL); a Ni-YSZ fuel electrode functional layer (FFL); a YSZ electrolyte; a gadolinia-doped ceria (GDC) interdiffusion barrier layer; a La0.8Sr0.2CoO3 (LSC)GDC air electrode functional layer (AFL); and an LSC air electrode backing layer (ABL).
The
electrochemical reactions in SOEC and SOFC operations will most likely occur at the functional layers near the electrode–electrolyte interfaces, as they mainly depend on the ionic charge transport [18]. Considering extension of the reaction zones, the FBL, FFL, AFL, and ABL were defined as the electrochemical reaction zone in the model domain. The major assumptions used to develop the RSOC model are listed below: 1) Steady-state operation; 2) Isothermal condition at the interconnector–electrode and gas channel–electrode interfaces; 3) Ideal gas mixture; 4) Isotropic microstructural and material properties; 8
5) Cell potential ranging from 0.4 to 1.6 V for SOFC and SOEC operation. . 2.2. Mass transport In the porous electrode layers, the fuel and air are transferred as they are produced or consumed by the electrochemical reactions. To describe the mass transport, the governing equation was adopted as: eff ∇ ∙ (𝑢𝐶𝑖 ) = ∇ ∙ (𝐷𝑖,𝑚 ∇𝐶𝑖 ) + 𝑆𝑖
(1)
where Deff i,m , Ci, and Si are the effective diffusion coefficient, molar concentration, and generation/consumption rate of species i, respectively. The fluid velocity in porous media (u) can be written in accordance with Darcy’s law: 𝐾
𝑢 = − ∇𝑃
(2)
𝜇
where K is the permeability of porous media and μ is the viscosity of fluid. In order to calculate the pressure (P) field, the conservation equation was employed as follow [28]: 𝐾
0 = ∇ ∙ (𝜌 ∇𝑃) + 𝑆ma
(3)
𝜇
where ρ is the density of fluid and Sma is the rate of mass generation/consumption due to the electrochemical reactions. Species diffusion in the porous electrodes can proceed through molecular diffusion or Knudsen diffusion, depending on the mean free path of the molecules and the pore size. The mean free path of the gas species was approximately 0.2–0.4 μm in this study; the comparable pore radii (0.25 and 2.4 μm) were considered for the computational analysis. Therefore, the effective diffusion coefficient 9
was defined, involving the two parallel diffusion mechanisms, as: 1 eff 𝐷𝑖,𝑚
𝜏
1
= ( 𝜀 𝐷
𝐾,𝑖
+
1
8𝑅𝑇
3
𝜋𝑀𝑖
𝐷𝐾,𝑖 = 𝑑p √ 𝐷𝑖𝑗 =
1 𝐷𝑖𝑗
)
(4) (5)
2.6×10−7 𝑇 1.5
(6)
𝑤𝑖 +𝑤𝑗 2 2 ) 𝛺𝐷,𝑖𝑗 1 1 2 √ + 𝑀𝑖 𝑀𝑗
𝑃(
where Dij, DK,i, and xi are the binary diffusion coefficient, Knudsen diffusion coefficient, mole fraction of species i , respectively; τ is the tortuosity and ε is the porosity; dp, Mi, R, and T are the pore diameter, the molecular weight of species i, the universal gas constant, and the operating temperature, respectively; wi is the characteristic length of species i [29, 30] and ΩD,ij is the dimensionless collision integral [30]. The source term arising from the electrochemical reactions (Si) can be written as: 𝑆𝑖 =
𝜉𝑖 ∇∙𝐼
(7)
𝑛𝑒,𝑖 𝐹
where ξi, ∇ ∙ 𝐼, ne,i, and F are the stoichiometric coefficient of species i, the rate of charge generation/depletion, the number of electrons transferred per electrochemical reaction, and the Faraday constant, respectively. The mass transport was coupled with the electrochemical reaction kinetics via ∇ ∙ 𝐼, which is described in detail in Section 2.3.
2.3. Electrochemistry To calculate the electrode and electrolyte potentials in the two operating modes, the governing equations for charge transport were defined as: 10
eff 0 = ∇ ∙ (𝜅𝑠,𝑖 ∇𝛷𝑠 ) + ∇ ∙ 𝐼𝑠
(8)
eff 0 = ∇ ∙ (𝜅𝑙,𝑖 ∇𝛷𝑙 ) + ∇ ∙ 𝐼𝑙
(9)
where the subscripts s and l indicate the solid phase (electrode material) and electrolyte phase (electrolyte material), respectively; Φ is the potential of the electrode or electrolyte; and I is the current density of the fuel or air electrode. The effective electronic or ionic conductivity ( κeff ) was estimated as [31]: 𝜅𝑠eff = 𝜅𝑠 ( 𝜅𝑙eff = 𝜅𝑙 (
𝜑𝑒𝑙 −0.294
)
𝜑𝑖𝑜 −0.294
)
2
(10)
𝜀 1+1+𝜀−0.294
2
(11)
𝜀 1+ −0.294 1+𝜀
where φ is the volume fraction of electron or ion-conducting particles. The amount of charge generation in the solid and electrolyte phases (∇ ∙ 𝐼𝑖 ) was evaluated using the Butler–Volmer equation: ∇ ∙ 𝐼𝑖fuel = 𝐴v 𝑖0fuel {exp ( ∇ ∙ 𝐼𝑖air = 𝐴v 𝑖0air {exp (
fuel fuel 𝐹𝛼b 𝜂
𝑅𝑇
air air 𝐹𝛼b 𝜂
𝑅𝑇
) − exp (
) − exp (
𝐹𝛼ffuel 𝜂fuel 𝑅𝑇
𝐹𝛼fair 𝜂air 𝑅𝑇
)}
(12)
)}
(13)
where the superscripts “fuel” and “air” indicate the fuel and air electrodes, respectively; i0, αb, and αf are the exchange current density and the backward and forward charge transfer coefficients, respectively. The electrode overpotential (ηi) and i0 were defined as follows [29, 32]: 𝑖 𝜂 𝑖 = 𝛷𝑠 − 𝛷𝑙 − 𝐸eq
(14)
air fuel 𝐸 emf = 𝐸eq − 𝐸eq = 1.253 [V] − 2.452 × 10−4 [V K −1 ]𝑇 −
𝑖0fuel = 𝛾 fuel 𝑥Ha 2 𝑥Hb 2 O exp (−
fuel 𝐸act
𝑅𝑇
)
11
(16)
𝑅𝑇 2𝐹
ln (
𝑃𝐻 2 𝑂
0.5 𝑃𝐻2 𝑃𝑂 2
) (15)
𝑖0air = 𝛾 air 𝑥O𝑐 2 exp (−
air 𝐸act
𝑅𝑇
)
(17)
where Eeq and Eemf represent the equilibrium and electromotive force potentials, respectively; Eact is the activation energy; γ is the pre-exponential factor; and a, b, and c are exponential constants used to consider the concentration dependence. The active surface area per unit volume (Av) was calculated by taking into account the microstructures of the electrodes, and expressed as follows [33]: 𝐴v = πsin2 𝜃𝑟el2 𝑛t 𝑛el (1 − 𝑛el )
𝑍el 𝑍io 𝑍
𝑝el 𝑝io
(18)
where θ, rel, nt, and nel are the contact angle between the electron- and ion-conducting particles, the radius of the electron-conducting particles, the number of particles per volume, and the numberfraction of electron-conducting particles, respectively; Z, Zel, and Zio are the average coordination number and coordination numbers of the electron- and ion-conducting particles, respectively; and pel and pio are the probabilities of the electron- and ion-conducting particles, respectively. The parameters in Eq. (18) were calculated as the following equations: 𝑛t = 4
1−𝜀
𝑛el =
𝜑el
(19)
3 𝑟 π𝑟 3 [𝑛 +(1−𝑛el )(𝑟io ) ] 3 el el el
(20)
3
𝑟
[𝜑el +(1−𝜑el )/(𝑟io ) ] el
𝑍el = 3 +
𝑍−3 𝑟
2
(21)
2
(22)
[𝑛el +(1−𝑛el )(𝑟io ) ] el
𝑟
𝑍io = 3 +
(𝑍−3)(𝑟io )
2
el
𝑟
[𝑛el +(1−𝑛el )(𝑟io ) ] el
0.4 4.236−𝑍el−el 2.5
𝑝el = [1 − (
2.472
)
]
(23)
12
0.4 4.236−𝑍io−io 2.5
𝑝io = [1 − ( 𝑍el−el =
)
2.472 𝑛el 𝑍
]
(24) (25)
2
𝑟
[𝑛el +(1−𝑛el )(𝑟io ) ] el
𝑍io−io =
(1−𝑛el )𝑍
(26)
2
𝑟 [𝑛el ( io ) +(1−𝑛el )] 𝑟 el
In this study, the electron- and ion-conducting particles were assumed to have the same radius and volume fraction.
2.4. Energy transport In the RSOC, the electrochemical kinetic and the transport properties of the materials are dependent on the temperature [34], and the temperature gradient in the cell can result in the cell fracture by thermal stresses. Thus, the thermal management is important to ensure reliable operations with high cell performance. In particular, the system efficiency of the SOEC operation is affected by the cell potential for thermally self-sustaining operation, called as thermoneutral potential [35]. However, the thermal issue, which strongly depends on the gas flow distribution and the operating conditions [36, 37], is beyond the scope of this paper. Hence, the energy balance for maintaining the cell temperature was excluded, and the temperature was assumed constant at the boundaries of the model domain. The energy conservation equation for the system was implemented as: ∇ ∙ (𝜌𝐶p 𝑢𝑇) = ∇ ∙ (𝑘 eff ∇𝑇) + 𝑆𝑇 𝑘 eff = 𝑘f 𝜀 + 𝑘s (1 − 𝜀)
(27) (28)
where Cp is the heat capacity of fluid; kf and ks are the thermal conductivity of the fluid and solid, 13
respectively. The heat generation/depletion term (ST) consists of three sources, that is ohmic resistance, thermodynamic irreversibility, and reversible heat by electrochemical reaction. 𝑆𝑇 = (
𝐼𝑙2
eff +
𝜅𝑙
𝐼𝑠2
𝜅𝑠eff
) + |𝜂𝛻 ∙ 𝐼| +
∇∙𝐼 𝑛𝑒 𝐹
𝑇∆𝑆
(29)
where ∆S is the entropy change for the reaction.
2.5. Model implementation The governing equations for mass, charge and energy conservation were coupled and solved by a CFD software package (FLUENT 6.3.26) with a user-defined function written in the computer language of C++. The numerical calculations were carried out according to the procedure which mitigates the solution convergence problem due to the severe coupling of transport phenomena, by using previous case solutions as initial values. In addition, the grid independence of numerical solution was tested to ensure the accuracy of the simulation results for various electrode geometric parameters. Four different grid sizes, 160×50, 320×100, 480×150 and 640×200 were examined during the test. Fig. 2 shows the comparison of the amount of charge generation at 0.4 V computed based on the four grids. The grid independent solutions were obtained with the finer grids, 480×150 and 640×200. Consequently, 480×150 was selected as an optimum grid density. As for boundary conditions, the YSZ and GDC layers were assumed to separate the fuel and air electrodes completely, conducting oxide ions but not electrons. Although the GDC layer has significant electron conductivity, in particular at very low oxygen pressure, the YSZ layer acts as a 14
barrier to prevent the electron transport between the two electrodes. In addition, the electron transport in the thin GDC layer had nearly no effect on the electrochemical reactions in the model predictions. Thus, for the simplicity of the calculation, non-flux condition was applied for the electrolyte potential at the interfaces between the backing layers and gas channel/interconnector, while Dirichlet boundary condition was chosen for the species, electrode potential and energy, as specified in Table 1. In both the SOEC and SOFC operations, the molar composition was set to 50 % H2O and 50 % H2 at the FBL–fuel gas channel interface, and 21 % O2 and 79 % N2 at the ABL–fuel gas channel interface. The remaining external boundaries were treated as symmetric. The contact resistances on the backing layer–interconnector interfaces were set at 0.005 Ω cm2 for both the fuel and air electrode sides of the cell, to consider their effect on optimizing the rib size. Accordingly, the current density at the rib (irib) can be determined as [31]: 𝑖rib =
∆𝑉
(30)
𝑅𝑐𝑜𝑛
where ∆V is the potential drop at the rib and Rcon is the area specific contact resistance at the rib.
3. Results and Discussion 3.1. Model validation The flow and the distribution of species along the cell length, which are excluded in this study, can cause significant concentration loss, in particular at high current densities. Thus, the proposed model was validated by comparing its results with the experimental current-potential curves previously 15
reported [19],which were based on button cells under low fuel utilization conditions. Specifically, several electrochemical parameters were fitted to the experimental data for approximation, without making the uncertain assumptions (e.g. charge transfer coefficient being same in both the SOEC and SOFC operations) [32]. To reflect the characteristics of electrochemical reactions in the two operating modes, the charge transfer coefficients (αb, αf) were varied independently, obtaining bestfit curves near the zero-current point. In addition, the exponential constants for concentration dependence (a,b,c) were set to values that minimize the error between the predicted and experimental results at high current region that exhibits different mass transport resistances due to the microstructure of electrodes. Since the pre-exponential factors for exchange current density (γ) had less influence on the slope of fitting curves than the previous parameters, they were estimated by considering the average of absolute errors in overall current range. Table 2 lists the operating conditions, material properties, and geometric information; Table 3 presents the electrochemical parameters in the model validation. As presented in Fig. 3, the simulated results were in good agreement with the experimental data for two different microstructures in the FBL (labelled as highporosity 0.38, low-porosity 0.24), having errors within 3 % in the target range of cell potentials. Thus, the model could be used to capture the effect of mass transport on the cell performance in both the operations. In the following studies, the microstructural parameters were fixed as in the case of the high-porosity 0.38, unless mentioned otherwise.
16
3.2. Cell performance with respect to thickness of electrodes In the intermediate-temperature solid oxide cells, the fuel and air electrode-supported designs have been widely accepted, that the thickness of either fuel or air electrode layer is in between 500 and 1000 μm, whereas the other layers are on the order of tens of micrometers. Because the directions of species transport in SOEC and SOFC operations are opposite to each other as shown in Fig. 1b, the variation of electrode thicknesses leads to different degree of mass transport resistance for each operation. Thus in order to estimate its impact on the RSOC performance, either FBL or ABL thicknesses was varied from 50 to 800 μm, while maintaining the opposite electrode backing layer at 100 μm. Fig. 4 presents the simulated current–potential (I–V) characteristic curves for the various FBL and ABL thicknesses. The species diffusion within the fuel electrode could influence the electrochemical reactions in both the two operating modes because H2O and H2 were the reactants for SOECs and SOFCs, respectively. Nevertheless, similar to the experimental results obtained with different microstructures [19], the performance of the SOEC was more sensitive to its FBL thickness at higher current density, relative to that of the SOFC (Fig. 4a). This was mainly attributed to the concentration loss during SOEC operation, related to the consumption of reactants and the concentration dependence factors in Eqs. (15,16,17). In contrast, the cell performance during SOFC operation was severely degraded, with greater mass transport resistance, upon increasing the ABL thickness, while that during SOEC operation remained constant (Fig. 4b). This was due to O2 being the only reactant in the air electrode for the SOFC, with the diffusion of air being slower than that of 17
H2O/H2. In addition, Fig. 4a reveals that the thinnest FBL (50 μm) did not result in the best cell performance for SOEC mode, despite having the shortest species diffusion path. In fact, the electrodes having thickness smaller than the electrochemical active thickness would reduce the number of active sites involved in the electrochemical reactions [15,16]. Under the conditions examined in this study, most of the electrochemical reactions occurred in the FFL and AFL (>98 % based on simulated result at the most severe condition). Thus, it could be deduced that the uneven distribution of reactant species (H2O) rather than the insufficient active sites was likely the cause of the decrease in the SOEC performance when the thinnest FBL was used. On the other hand, the thinnest FBL exhibited the best SOFC performance due to the higher Knudsen diffusivity of H2 preventing the uneven distribution of reactant. Furthermore, the effect of the thinnest ABL on the SOFC performance degradation at high current density was insignificant compared with that of the thinnest FBL for SOEC mode. This was because the influence of the thinnest ABL on the reactant distribution was reduced, under the operation close to the limiting current density. The overall RSOC performance can be represented by the round-trip efficiency, which is defined as the ratio of the input energy in SOEC operation to the output energy in SOFC operation. Assuming the operating time and the magnitude of the current density are same for the two operations, the round-trip efficiency of the cell (λ) can be calculated as [26]: 𝜆 = 𝑉SOFC /𝑉SOEC
(31)
where V is the operating cell potential. Fig. 5a shows the effect of varying the FBL and ABL 18
thicknesses on the round-trip efficiency at 0.2 to 0.8 A cm -2. The maximum variation of the efficiency with different electrode thicknesses was 0.5% at 0.2 A cm -2, and it reached 2.8 % at 0.8 A cm -2. In the current range studied, the 200-μm-thick FBL or ABL yielded the highest efficiency for each current density. For the thickest backing layers (800 μm), the FBL exhibited higher efficiency (1.1 % higher at 0.8 A cm -2) than the ABL as the current increased. This resulted from the severe performance degradation in the SOFC operation with the thickest ABL as shown in Fig. 4b. In order to evaluate the optimal cell configuration, the round-trip efficiencies were calculated with constant total backing layer thickness of 900 μm, as shown in Fig. 5b. The thin FBL and ABL (100-100 μm case), that can be implemented to metal-supported cells, was also considered as a possible cell configuration in the comparison. Overall, the results in Fig. 5b were consistent with that in Fig. 5a. The fuel electrode-supported cells (thick FBL and thin ABL), which provide sufficient mechanical strength, showed higher efficiency than the air electrode-supported cells (thin FBL and thick ABL) over the whole range of current density. The difference of the efficiency between the 100-100 μm case and 700-200 μm case was only 0.6% at 0.8 A cm -2. Moreover, it is notable that the RSOC performance was most affected by the thinnest electrodes. Therefore, the fuel electrode-supported design is expected to be more suitable for the RSOC operation in practical point of view, and the thickness of the thin electrode should be optimized to enhance the efficiency.
19
3.3. Effect of rib size on cell performance In the flow field design, the interconnector rib size is crucial for balancing the concentration and ohmic losses as it determines the length and area of transport paths for species and charge. Thus, three different rib widths of 0.5, 1.0, and 1.5 mm were examined to identify their influence on mass and charge transport. The simulations were carried out for the three cases in the SOFC and SOEC operations at 0.6 V and 1.4 V, respectively. As shown in Fig. 6, under both of the operating conditions, the magnitude of the current density increased upon decreasing the rib width, regardless of the thickness of the backing layer, due to the easier diffusion of reactants. In addition, the rib size resulted in more drastic changes in the current density of the SOFC upon varying the FBL thickness and as well as the current density of the SOEC upon varying the ABL thickness. Hence, the thickness of the backing layer at the maximum current density had a tendency to increase upon enlarging the rib width. For the SOEC operation, the optimal FBL thickness for the maximum current density was found to be only 100 μm for a rib width of 0.5 mm, while it increased to approximately 200 μm for a rib width of 1.0 mm. To further discuss the relationship between the optimal electrode thickness and the rib size, the distributions of the H2O concentration and fuel electrode overpotential were analyzed by selecting cases at 1.4 V for the SOEC operation (Fig. 7). As expected, the reactant depletion became more prominent upon enlarging the rib width, due to the increased diffusion distance. Thus, in Fig. 7c, the electrode overpotential was increased specifically at the area under the rib. On the contrary, as shown 20
in Figs. 7a and 7b, it was clear that the thicker FBL alleviated the reactant depletion under the rib. In particular, in the 200-μm-thick FBL having a 0.5-mm rib width (lrib), the H2O distribution was the closest to a one-dimensional profile, meaning that the performance improved as a result of uniform distribution of reactant with higher utilization of the electrode materials [44]. Consequently, for the 200-μm-thick FBL in Fig. 7d, the rib size had little effect on the distribution of the electrode overpotential. However, the benefit from mitigating the reactant depletion under the rib was not observed upon further increasing the thickness of the backing layers as the concentration loss was more dominant. In other words, the influences of the thickness and the rib size on the overall RSOC performance should be simultaneously considered to determine the optimal electrode design. Fig. 8 illustrates the comparison of the round-trip efficiencies of the RSOC at 0.8 A cm -2, according to the variation of the thickness of the backing layers and the rib size. The average of efficiency with respect to the three rib widths of 0.5, 1.0, and 1.5 mm was 71.5, 70.4 and 66.1 %, respectively. The maximum variation of the efficiency with different electrode thicknesses (2.8% for 1.0-mm rib width) was 3.3 % for 0.5-mm rib width and 3.8 % for 1.5-mm rib width. Due to the alleviation effect on the reactant depletion under the rib, the FBL or ABL thickness for the maximum efficiency increased with enlarging the rib width. In addition, with a 0.5-mm rib width, the 800-μmthick FBL demonstrated 1.4 % higher efficiency of the RSOC than the 800-μm-thick ABL. From the results in Fig. 8, it can be seen that decreasing the rib size enhances the RSOC efficiency, but at same time the electrode thickness becomes the most dominant factor for the mass transport resistance. 21
However, it should be also taken into account that the small rib size could lead to severe contact resistance problems. As can be seen from Fig. 9, the optimal rib width increased with higher contact resistance, and for the contact resistance of 0.05 Ω cm2, the 0.5-mm rib width case exhibited 7.2 % lower efficiency of the RSOC than the 1.0-mm rib width case. Thus, it means that the rib size has the greatest impact on the performance improvement for the RSOC with considerable contact resistance.
3.4. Effect of fuel-inlet composition on cell performance The fuel-inlet composition may be varied for different operating conditions and control strategies in the RSOC operations. In this section, the fuel composition was investigated, as it determines the theoretical potential for electrolysis or fuel cells and thus changes the overall RSOC performance. Specifically, the molar fractions of H2O and H2 were set at 33, 50, and 67 % for the SOEC and SOFC modes, respectively. Fig. 10 displays the current densities obtained at various fuel compositions and thicknesses of the backing layers. Although the results are similar to the variations in performance with different the rib sizes (Fig. 6), the fuel composition had less impact on the magnitude of the current density in the SOFC operating mode, where the O2 diffusion is a limiting factor. On the other hand, the current density of the SOEC operating mode was significantly dependent on the reactant fraction. In addition, for the two operating modes, the varying fuel composition was ineffective in determining the optimal thickness of the backing layers in the light of maximum current density. This tendency can be also seen in the round-trip efficiency of the RSOC with respect to the fuel 22
composition and the thickness of backing layers at 0.8 A cm -2 (Fig. 11). In the composition range studied, the backing layer thickness for the maximum efficiency remained constant, and the difference between the average efficiency in each fuel composition was up to 0.5 %.
4. Conclusions A mathematical model, taking into account the electrochemical kinetics and gas diffusion mechanisms in both the SOEC and the SOFC operating modes, has been developed to investigate the effects of several electrode design parameters on the RSOC performance. The numerical solution was obtained using the user-define functions written in the computer language as part of the work. Then, the simulated results were compared with the measured data sets reported elsewhere [19]. The developed model enabled practical prediction on the cell performance with respect to the thickness of the electrode backing layer, the interconnector rib size, and the fuel composition, also evaluating the limitation of mass transport at the same time. In the detailed analysis, the thickness of the electrode backing layer restricted the performance of the SOEC with a thick FBL and that of the SOFC with a thick ABL. For the overall performance of RSOC, the fuel electrode-supported cell exhibited better round-trip efficiency (1.1 % higher at 0.8 A cm -2) than the air electrode-supported cell due to the severe performance degradation in the SOFC operation with the thick ABL. Furthermore, from the comparison of cell configurations, it was found that for the electrode-supported cell, the thickness of the thin electrode should be optimized to 23
enhance the RSOC performance. Meanwhile, the decrease in the current density was observed upon enlarging the rib width, thus increasing the optimal thickness of thin electrode layer. In particular, our analysis of the reactant distribution revealed that an electrode that is too thin will accelerate the depletion of the reactants under the rib. Additionally, when the rib width was varied from 1.5 to 0.5 mm, the average round-trip efficiency increased by 5.4 %, while the maximum variation of the efficiency with backing layer thickness of 50-800 μm was between 3.8 to 2.8 %. Therefore, although the rib size should be preferentially reduced to achieve high efficiency, the electrode thickness also has to be optimized for further improvement (e.g. even with a 0.5-mm rib width, the fuel electrodesupported cell demonstrated 1.4 % higher efficiency than the air electrode-supported cell). However, it was also observed that the small rib size has the greatest impact on the performance degradation of the RSOC with considerable contact resistance (e.g. for the contact resistance of 0.05 Ω cm2, the 0.5-mm rib width case exhibited 7.2 % lower efficiency than the 1.0-mm rib width case). On the other hand, compared with the case of SOFC, the fuel-inlet composition had more of an effect on improving the SOEC performance upon increasing the reactant fraction, but it was not influential in determining the electrode thickness for the best efficiency. It means that the once-optimized geometric parameters are valid for the range of the considered fuel compositions.
Acknowledgements This research was supported by the Commercializations Promotion Agency (2015K000131) for R&D Outcomes (COMPA) funded by the Ministry of Science, ICT and Future Planning (MSIP). 24
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28
Fig. 1. Schematic representations of (a) a planar RSOC and (b) a cross-sectional domain for calculation of the model.
29
Fig. 2. Grid dependency test results for the model domain at 0.4 V
30
Fig. 3. Validation of the model results with the experimental data in reference [19].
31
Fig. 4. I–V curves predicted by the model for various thicknesses of (a) FBL and (b) ABL in both the SOEC and the SOFC operating modes.
32
Fig. 5. Comparison between the round-trip efficiencies (a) with various thicknesses of the FBL and ABL, (b) with constant total backing layer thickness of 900 μm.
33
Fig. 6. Effect of rib size on current densities at 0.6 V for the SOFC and 1.4 V for the SOEC operating mode, predicted upon varying thicknesses of the (a) FBL and (b) ABL.
34
Fig. 7. H2O molar concentration (mol m-3) profiles in fuel electrodes operated at 1.4 V at FBL thicknesses of (a) 50 and (b) 200 μm; distribution of fuel electrode overpotential (V) operated at 1.4 V at FBL thicknesses of (c) 50 and (d) 200 μm.
35
Fig. 8. The round-trip efficiencies with various backing layer thicknesses and rib sizes at 0.8 A cm -2.
36
Fig. 9. The round-trip efficiencies with various contact resistances and rib sizes at 0.8 A cm -2.
37
Fig. 10. Effect of fuel gas composition on current densities at 0.6 V for the SOFC and 1.4 V for the SOEC operating mode, predicted upon varying thicknesses of the (a) FBL and (b) ABL
38
Fig. 11. The round-trip efficiencies with various backing layer thicknesses and fuel-inlet compositions at 0.8 A cm -2.
39
Table 1. Boundary conditions for species, pressure, electrode/electrolyte potentials and energy. Species i
Pressure
Electrode potential
Electrolyte potential
Energy
FBL–fuel gas channel 𝑃 Ci (= xi) 𝑅𝑇 interface
P
Non-flux
Non-flux
T
FBL–interconnector interface
Non-flux
Non-flux
0
Non-flux
T
FFL–YSZ layer interface
Non-flux
Non-flux
Non-flux
–
–
AFL–GDC layer interface
Non-flux
Non-flux
Non-flux
–
–
ABL–air gas channel interface
Ci (=
P
Non-flux
Non-flux
T
ABL– interconnector interface
Non-flux
Non-flux
V (0.4-1.6)
Non-flux
T
𝑃 𝑅𝑇
x i)
40
Table 2. Operating conditions, material properties, and geometric information based on the results described in references [19, 25]. Parameter
Value
Operating temperature, T (K)
1023
Operating pressure , P (atm) Fuel-inlet mole fraction, xH2 , xH2O
1
Air-inlet mole fraction, xO2, xN2
0.21, 0.79
Ni conductivity, κs,Ni (S m )
3.27×106 -1065.3T [38]
YSZ conductivity, κl,YSZ (S m-1)
3.34×104 e-10300/T [39]
GDC conductivity, κl,GDC (S m-1)
3.5×103 e-6471/T [40]
LSC conductivity, κs,LSC , κl,LSC (S m-1)
2.45×105 -97.5T [41], 2×10-13 e0.0245T [42]
FBL thickness, tFBL (m)
800×10-6
FFL thickness, tFFL (m)
10×10-6
YSZ layer thickness, tYSZ (m)
10×10-6
GDC layer thickness, tGDC (m)
5×10-6
AFL thickness, tAFL (m)
15×10-6
ABL thickness, tABL (m)
20×10-6
Gas channel width, lch (m)
1×10-3
Rib width, lrib (m)
1×10-3
FBL porosity/tortuosity, ε/τ
high- porosity: 0.38/5; low-porosity:0.24/9
FFL porosity/tortuosity, ε/τ
0.24/3
AFL porosity/tortuosity, ε/τ
0.3/3
ABL porosity/tortuosity, ε/τ
0.3/3
FBL pore diameter, dp (m)
high- porosity: 4.8×10-6; low-porosity: 1.4×10-6
FFL pore diameter, dp (m)
0.7×10-6
AFL pore diameter, dp (m)
0.3×10-6
ABL pore diameter, dp (m)
0.4×10-6
Contact angle between electron and ion conducting particles, θ (º)
15
Radius of electron-conducting particle, rel (m)
0.5×10-6
Permeability of electrodes, K (m2)
(2𝑟𝑒𝑙 )2 𝜀 3
SOEC:0.67,0.33; SOFC:0.97,0.03
-1
72𝜏(1−𝜀)2
41
[43]
Table 3. Electrochemical parameters fitted to the experimental data in reference [19]. Parameter
Value
Activation energies for exchange current
120000, 120000
air -1 density, Efuel act , Eact (J mol ) Exponential constants for concentration dependence, a, b, c
0.11, 0.67, 0.25
Pre-exponential factors for exchange current 1×1011, 1×1011 density, γfuel, γair Backward and forward charge transfer coefficients in fuel electrode,
αfuel b ,
1, 0.6
αfuel f
Backward and forward charge transfer coefficients in air electrode,
αair b ,
0.15, 0.45
αair f
42