The Geometry Effect of Cathode/Anode Areas Ratio on Electrochemical Performance of Button Fuel Cell Using Mixed Conducting Materials

energies

Article The Geometry Effect of Cathode/Anode Areas Ratio on Electrochemical Performance of Button Fuel Cell Using Mixed Conducting Materials

Daifen Chen 1, Biao Hu 1, Kai Ding 1, Cheng Yan 1 and Liu Lu 2,* 1 School of Energy and Power, Jiangsu University of Science and Technology, Zhenjiang 212003, China; [email protected] (D.C.); [email protected] (B.H.); [email protected] (K.D.); [email protected] (C.Y.) 2 School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, China * Correspondence: [email protected] or [email protected]

Received: 10 June 2018; Accepted: 17 July 2018; Published: 18 July 2018

Abstract: Intermediate temperature (IT) fuel cells using mixed conducting materials have been reported by many researchers by adopting different compositions, microstructures, manufacture processes and testing conditions. Most iop-Vop relationships of these button electrochemical devices are experimentally achieved based on anode or cathode surface area (i.e., Aan 6= Aca). In this paper, a 3D multi-physics model for a typical IT solid oxide fuel cell (SOFC) that carefully considers detail electrochemical reaction, electric leakage, and e−, ion and gas transporting coupling processes has been developed and veriﬁed to study the effect of Aca/Aan on button cell iop-Vop performance. The result shows that the over zone of the larger electrode can enhance charges and gas transport capacities within a limited scale of only 0.03 cm. The over electrode zone exceed this width would be inactive. Thus, the active zone of button fuel cell is restricted within the smaller electrode area min(Aan, Aca) due to the relative large disc radius and thin component layer. For a speciﬁed Vop, evaluating the responded iop by dividing output current Iop with min(Aan, Aca) for a larger value is reasonable to present real performance in the current device scale of cm. However, while the geometry of button cells or other electrochemical devices approach the scale less than 100 µm, the effect of over electrode zone on electrochemical performance should not be ignored.

Keywords: electrode areas ratio effect; electrochemical performance; mixed conducting material; multi-physics numerical modeling

1. Introduction In the past decade, low cost, clean and high efficiency energy conversation and storage devices, such as fuel cells [1], batteries [2] and super-capacitors [3], have been receiving more and more attentions. Solid oxide fuel cell (SOFC) has being recognized as a promising energy conservation device due to its efficiency [4] and capability to work with various fuels [5]. As high operating temperature might cause strict material compatibility constraints [6] and operational complexity [7], attentions have being devoted to the development of intermediate temperature (IT-) SOFC components (i.e., 350–650 ◦C) [8]. The key obstacles for reducing SOFC operation temperature are attributed to the insufficient activities of conventional cathode materials and low ionic conductivities of traditional electrolyte materials [9] (i.e., YSZ) in this temperature regime. Thus, mixed ion/e- conducting (MIEC) electrode materials [10] and alternative electrolyte materials [11] have received great attention for their potential applications in IT-SOFCs. Z. Shao et al. reported a mixed conducting Ba0.5Sr0.5Co0.8Fe0.2O3-δ as a potential cathode material, which can conduct both electron and O2− charges [10] Then, an in-situ photoelectron spectroscopy method was proposed to investigate the electrochemically active region

Energies 2018, 11, 1875; doi:10.3390/en11071875 www.mdpi.com/journal/energies Energies 2018, 11, 1875 2 of 16

within mixed conducting CeO2−x electrode [12]. S. Wang et al. compared the performances of various LSCF-based cathodes and found that LSCF-SDC exhibited a larger activation overpotential than did the single-phase LSCF cathode [13]. More interestingly, the LSM-coated LSCF composite electrode was reported to exhibit a lower activation overpotential compared with that in a pure LSCF cathode [13]. Furthermore, proton conducting oxides [14], such as BaZr0.7Pr0.1Y0.2O3_d [15] and BaZr0.1Ce0.7Y0.2O3_d [16] were also greatly invented to be used in IT-SOFCs because of their low activation energy and high ionic conductivity around IT-range. Generally, the electrochemical reaction processed within an IT-SOFC using mixed conducting materials or proton conducting oxides are very different from those using conventional composite electrodes [17]. Taking the cathode of LSCF-SDC/SDC/NI-SDC IT-SOFC using mixed conducting materials as an example [13], the electrochemically active sites not only can be taken placed around the percolated three phase boundary sites (i.e., LSCF-SDC-pores and LSCF-dense electrolyte interfaces), but also can happen around the percolated double phase boundary sites (e.g., LSCF-pore) [18]. Up to now, many IT-SOFC button cells using the mixed conducting materials have been reported by many researchers by adopting different compositions (or materials), volume fractions, microstructure parameters, manufacture processes, operating conditions and different cell geometry sizes. It is interesting to note that different anode and cathode surface areas (i.e., different discs radii) were chosen during button cell fabricating and measuring. For a specified output voltage Vop, the corresponding output electric current density iop was always evaluated by dividing output current Iop with the relative smaller surface area between anode Aan and cathode Aca. This may lead to higher electrochemical performance results based on follow consideration. Generally, the performance of IT-SOFC button cell is a trade-off of electrochemical reacting, gas transporting, e− and ionic conducing and their mutual coupling processes. The over zone of larger electrode can enhance charges and gases transport capacities within the button cells in a proper zone; and this may affect the Vop-iop performance measuring result. Thus, it is important to study the sensitivity of the SOFC button cell performance on different cathode and anode surface areas ratios. 3D multi-physics coupling numerical modeling is generally agreed to be an economic, valid and time saving approach for working detail investing [19], parameters-performance studying [20], geometric optimizing [21] and system operation optimizing [22]. In this paper, a 3D multi-physics model which carefully considers detail electrochemical reaction, electric leakage, and e−, ion and gas transporting coupling processes within a typical IT-SOFC button cell is developed and verified. Then, the influences of different cathode/anode area ratios Aca/Aan on the button cell Vop-iop performances are carefully investigated, while different micro-structure parameters, electrode properties, component thicknesses and exchange current densities of reaction interfaces are varied within reasonable value ranges. The study results can help us achieve the valid affecting zone of the relative larger electrode; and assess the rationality that evaluating responded iop by dividing Iop with min(Aan, Aca) for a larger value, while Vop is specified. The achieved conclusions would provide good references for understanding the geometric effects of cathode/anode cross sections relationship on electrochemical performance of IT-SOFC button cell and similar electrochemical devices.

2. Method and Theory Taking a typical anode-supported LSCF-SDC/SDC/Ni-SDC IT-SOFC button cell in Figure1a as an example, the relevant structure and geometry sizes of the distinct four different cell layers from the experiment report is illustrated. As shown in Figure1b, the multi-physics working processes within these IT-SOFC button cells are complicated even in hydrogen fuel case. Oxygen within the air should be transported to the percolated LSCF-SDC-pores three phase boundaries (TPBs) or LSCF-pores double phase boundaries (DPBs) in the cathode side. ‘Percolated’ here is deﬁned as a continuous connection through the entire electrode structure. At these places, O2 will react with the electrons transported by the electronic conducting paths, such as percolated LSCF network and the external current circuit or dense electrolyte that presences electronic conducting capability (e.g., SDC and CGO). The produced Energies 2018, 11, 1875 3 of 16 Energies 2018, 11, x FOR PEER REVIEW 3 of 16

2− 2− andO CGO).will be The conducted produced to theO2− percolatedwill be conducted Ni-SDC-pore to the TPBspercolated in anode Ni-SDC-pore side through TPBs O in conductinganode side 2− throughnetwork O which2− conducting is constructed network by bothwhich LSCF- is constructed and SDC-particles by both andLSCF- dense and electrolyte. SDC-particles These and O densewill electrolyte.react with theThese fuels O2 diffused− will react through with the porous fuels anode. diffused Most thro ofugh the porous produced anode. electrons Most of will the be produced circuited electronsback to cathode will be reactioncircuited sites back through to cathode the reaction external sites current through circuit. the But external part of current the produced circuit. electronsBut part ofwill the be produced conducted electrons from anode will to be cathode conducted side throughfrom anode dense to electrolytecathode side directly through due dense to the presenceelectrolyte of directlyelectronic due conducting to the presence property of electronic of electrolyte conducti material.ng property These electric of electrolyte currents material. are considered These electric as idle currentswork and are cause considered complex relationshipsas idle work among and microstructurecause complex parameters, relationships effective among electrode microstructure properties parameters,and multi-physics effective calculating electrode processes. properties and multi-physics calculating processes.

(a) (b)

Figure 1. (a) The sketch figure of a typical LSCF-SDC/SDC/Ni-SDC IT-SOFC, (b) the corresponding Figure 1. (a) The sketch ﬁgure of a typical LSCF-SDC/SDC/Ni-SDC IT-SOFC, (b) the corresponding multi-physics working processes within it. multi-physics working processes within it.

As proposed in our previous paper [23], the characteristic properties of each SOFC component layer Ascan proposed be evaluated in our previousby the generalized paper [23], the percolation characteristic micro-model properties based of each on SOFC the componentthickness, composition,layer can be evaluated and microstructure by the generalized parameters percolation of each micro-model component based layers. on the Taking thickness, the composition, LSCF-SDC compositeand microstructure cathode as parameters an example, ofeach the potential component electr layers.ochemical Taking active the LSCF-SDC sites consists composite of the percolated cathode as LSCF-SDC-poresan example, the potential TPBs, percolated electrochemical LSCF-dense active sites electrolyte consists ofinterfaces the percolated and percolated LSCF-SDC-pores LSCF-pores TPBs, DPBpercolated surface LSCF-dense sites (illustrated electrolyte in Figure interfaces 1b). and percolated LSCF-pores DPB surface sites (illustrated in FigureThe percolated1b). LSCF-SDC-pores TPBs per unit volume can be evaluated as The percolated LSCF-SDC-pores TPBs per unit volume can be evaluated as 2- λ VV= γ eO TPB, per LSCF,, SDCnZ LSCF LSCF SDC PP LSCF SDC (1) V V e O2− λTPB, per = γLSCF,SDCnLSCFZLSCF, SDCPLSCFPSDC (1) γ ==πθ2 where subscript ‘per’ is used to represent ‘Percolated’. LSCF,SDC rrcc( min( r LSCF , r SDC ) sin ) is the where subscript ‘per’ is used to represent ‘Percolated’. γ = πr2 (r = min(r , r ) sin θ) is electrochemical reaction site per contact between LSCF-LSCF, and SDC SDC-particlesc c (explainedLSCF inSDC Figure 2a), the electrochemical reaction site per contact between LSCF- and SDC-particles (explained in Figure2a),2- nrV =−(1φψ ) / (4 π3 / 3) P e O kkkV g 3 is number of k-particles per unit volume within electrode. LSCF ande PSDC nk = (1 − φg)ψk/(4πrk /3) is number of k-particles per unit volume within electrode. PLSCF and areO2 −the probabilities of relevant particles belonging to the percolated electron and oxygen ion PSDC are the probabilities of relevant particles belonging to the percolated electron and oxygen ion conductingconducting paths,paths, respectively.respectively. Both Both LSCF- LSCF- and and SDC-particles SDC-particles contribute contribute to the to O the2− conductingO2− conducting path, 2− 2-2− 2- O OOO − thus P = P === 1. The probability of LSCF-particle belonging to percolated e − conducting path, thusSDC PPSDCLSCF LSCF 1. The probability of LSCF-particle belonging to percolated e conducting network can be estimated by Reference [16] network can be estimated by Reference [16]

3.73.7 e 4.2364.236−−ZZLSCF, LSCF P e= 1 − LSCF, LSCF LSCFP =−1 LSCF 2.472 2.472 Zk,` is the number of contacts between k-particle and all of its neighboring `-particles Zk , is the number of contacts between k-particle and all of its neighboring  -particles ψ /r Z = 0.5(1 + r2/r2)Z ` ` k,` k ` Mψ / r =+22  ZrrZkk,0.5(1 / ) ∑M ψk/rk k=1 ψ  kk/ r k =1 where ψk and rk are the corresponding solid volume fraction and radius of k-particles. φg is the porosity of compositeψ electrode. φ where k and rk are the corresponding solid volume fraction and radius of k -particles. g is the porosity of composite electrode.

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(a) (b)

Figure 2. (a) Sketch of electrochemical reaction site per contact between LSCF- and SDC-particles Figure 2. (a) Sketch of electrochemical reaction site per contact between LSCF- and SDC-particles γ ; (b) illustrated of the exposed surface area of per LSCF-particle s . γLSCF,LSCF,SDC SDC;(b) illustrated of the exposed surface area of per LSCF-particle seses.

The percolated LSCF-pores DPB surface sites per unit volume can be evaluated basing on the cathode microstructure parameters by [[23]23]

2- VeOV V SsV= nPPe O2− , S LSCF,= pern sLSCFesP es LSCFP LSCF, LSCF, per LSCF LSCF LSCF (2)(2) sr=2 2π 2 [2−− (1 cosθθ )ZZ −− (1 cos ) ] ses = 2esπrLSCF[2LSCF− (1 − cos θLSCF LSCF)ZLSCF, LSCF LSCF, LSCF− (1 − cos θ LSCFLSCF) LSCFZLSCF,, SDC SDC] as illustrated in Figure2 2b,b, the the exposed exposed surface surface area area of of each each LSCF-particle LSCF-particles essesshould should be be estimated estimated by bysubtracting subtracting the the overlap overlap parts parts of neighboringof neighboring particles particles from from spherical spherical surface surface area. area. Similarly, the percolated LSCF-dense electrolyte interfaces per unit electrolyte surface area can be Similarly, the percolated LSCF-dense electrolyte interfaces per unit electrolyte surface area can estimated by [23] be estimated by [23] S S e λTPB,per = γLSCF, ele nLSCFPLSCF (3) λ S = γ S e S 2 TPB,per LSCF ,ele nP LSCF LSCF (3) where nLSCF = (1 − φg)ψLSCF/(2πrLSCF/3) is LSCF-particles number per unit dense electrolyte surface.

γ S= 2πr sin θ is the2 electrochemical reaction site per connect between an LSCF-particle and whereLSCF, elenr=−(1LSCFφψ ) / (2 π / 3) is LSCF-particles number per unit dense electrolyte surface. a dense electrolyte.LSCFg LSCF LSCF γ = πθ LSCF,eleMore2sinr details LSCF about is the otherelectrochemical effectiveelectrode reaction site properties per connect calculating, between such an LSCF-particle as, the percolated and a 2− − denseNi-SDC-pore electrolyte. TPBs, effective O and e electric conductivities, hydraulic radius of the porous electrodesMore anddetails so on about could other also beeneffective found electrode in our previous properties paper calculating, on percolation such theory as, the for detailspercolated [23]. Ni-SDC-poreCombing with TPBs, the electrode effective microstructure O2− and e− electric parameters conductivities, of LSCF-SDC/SDC/Ni-SDC hydraulic radius of IT-button the porous cell electrodesin experiment and processso on could [13], thealso corresponding been found in effectiveour previous characteristic paper on properties percolation of theory each layers for details of the [23].above Combing button cell with are the estimated electrode and microstructure provided in Supplementary parameters of Materials. LSCF-SDC/SDC/Ni-SDC Based on these properties,IT-button cellthe multi-scalein experiment predictive process model [13], the that corresponding comprehensive effe considersctive characteristic the special characteristicsproperties of each of the layers typical of themixed above conducting button SOFCscell are is estimated developed and to study provided the geometric in Supplementary effects of cathode/anode Materials. Based cross on sections these properties,relationship the on multi-scale electrochemical predictive performance. model that comprehensive considers the special characteristics − 2− of theAccording typical mixed to anodic conducting e -O SOFCscharge is transferdevelope reaction,d to study the the electrochemical geometric effects energy of cathode/anode relationship at crossanode sections active sites, relationship percolated on Ni-SDC-poreelectrochemical TPBs performance. (shown in Figure1b), can be expressed as According to anodic e−-O2− charge transfer− reaction, the electrochemical− energy relationship at H (g) + O2 (SDC) → H O(g) + 2e (Ni) (4a) anode active sites, percolated Ni-SDC-pore2 TPBs (shown2 in Figure 1b), can be expressed as

−2- - µH2 + µ 2 − 2FΦ→2− ≥ µH2O − 2FΦe (4b) H22 (g)+OO (SDC)O H O(g)+2e (Ni) (4a) st st where µα = µα + RT ln pα is chemical potential of reactant α at local reaction sites. µα is chemical stμμ+−Φ≥−Φ μ potential at standard condition p = 1 atm.2-T 22andFFpα 2- are the local temperature and partial pressure(4b) of HHOe22OO species α, respectively. F is Faraday constant. Φ and Φ are the local electrical potentials of O2− O2− e − μμ=+st α μ st whereand e conductingααRT phases,ln p α respectively.is chemical potential of reactant at local reaction sites. α is

‘=’ in Equation (4b) represents the energyst equilibrium state at local place. In this case, the local chemical potential at standard condition p = 1 atm. T and p are the local temperature and electromotive force based on local working condition instead of theα open circuit condition can be got α Φ Φ partialas [18] pressure of species , respectively. F is Faraday constant. 2− and e are the local eq eq eq O 2− E =− Φ − Φ = (µ + µ 2− − µ )/2F electrical potentials of O andan e conductingO2− e phases,H2 respectively.O H 2O

Energies 2018, 11, 1875 5 of 16

‘>’ in Equation (4b) is essential to process forward reaction with electric current produced. Thus, the activation overpotential is called as the ionic-electronic voltage difference shifted from the local an eq electromotive force ηact = Ean − (ΦO2− − Φe). Similarly, the cathodic e−-O2− charge transfer reaction and electrochemical energy relationship in local active sites (i.e., percolated LSCF-pore DPBs or LSCF-SDC-pore TPBs in Figure1b) are

− 2− 0.5O2(g) + 2e (LSCF) → O (SDC or LSCF) (5a)

0.5µ − 2FΦ ≥ µ 2− − 2FΦ − (5b) O2 e O O2 The corresponding electromotive force of equilibrium state at local cathode active sites and the eq activation overpotential shifted from this Eca are

ca 1 eq η = µ − 2µ 2− − (Φ − Φ − ) = E − (Φ − Φ − ) (6) act 4F O2 O e O2 ca e O2

Then, the relation between e−-O2− charge transfer rate per unit TPBs and the activation overpotential can be evaluated by empirical Butler-Volmer equation

2α F 2β F j = j exp f η − exp − r η (7) TPB TPB,0 RT act RT act

αf (or βr) is forward (or reverse) reaction symmetric factor. Local exchange current per unit TPB length at anode and cathode sides can be respectively estimated by Reference [24] an an EH2 1 1 pH2 jTPB,0 = jTPB,0,ref exp − − (8a) R T Tref p0

!0.25 E 1 1 p ca = ca − O2 − O2 jTPB,0 jTPB,0,ref exp 0 (8b) R T Tref p O2 where EH2 and EO2 are activation energies for H2 oxidation and O2 reduction reactions, respectively. 0 jTPB, 0,ref is assigned empirically based on experiment at reference Tref. pα is partial pressure of species α at open circuit state. Thus, the volumetric current sources for the transfer of e−-O2− electric charges around the TPBs are iV = j λV in A m−3. The area metric current sources over e−O2−,TPB TPB TPB.per electrode/electrolyte interfaces are iS = j λS in A m−2. e−O2−,TPB TPB TPB.per Similarly, the e--O2- charge transfer rate over per LSCF-pore DPB area can be evaluated as [18]

2α F 2β F h i i = i exp LSCF ηca − exp − LSCF ηca , A m−2 (9) LSCF LSCF,0 RT act RT act

0.25 where i = i (p /p0 ) exp(−E /(1/T − 1/T )/R). i is assigned LSCF,0 LSCF,0,ref O2 O2 O2 ref LSCF,0,ref empirically at reference temperature Tref. And the volumetric current sources for the transfer of e−-O2− electric charges around DPBs are iV,ca = i SV in A m−3. e−O2−,LSCF LSCF LSCF, per The above items, expect µO2− , can be resolved by the local independent variables, such as, T, − 2− pα, Φe and ΦO2− . Generally, the constant potential shift does not alter e (or O ) electric potential proﬁles within the electronic (or ionic) conducting phase. To exclude the inﬂuence of µO2− during calculating both ηact and charge transfer rate, local electric potentials Φe and ΦO2− were always shifted by different reference amounts, as reported by D. Jean et al. [25] and S. Liu et al. [26]. However, it is necessary to mention that only limited assumptions reported in the above literatures can be used, while the electronic leaking property in dense SDC electrolyte is considered. Because both Φe and ΦO2− are continuously distributed throughout the whole cell structure (anode, electrolyte and cathode). Energies 2018, 11, 1875 6 of 16

st While keeping Φ as it is, local Φ 2− are shifted by a reference amount as Φˆ 2− = Φ 2− + (µ − e O O O O2 2µO2− )/(4F). Then, the overpotential expresses should be adjusted accordingly RT p an st ˆ H2O ηact = E + (Φe − ΦO2− ) − ln (10a) 2F pH2

cc ˆ RT 1 atm ηact = ΦO2− − Φe − ln (10b) 4F pO2 where Est = µst + 0.5µst − µst /(2F) is the Nernst potential at standard state (1 atm). H2 O2 H2O Combing with the above e−-O2− charge transfer rates within the composite electrodes and electrode/dense electrolyte interfaces, multi-physics model can be completed by further coupling momentum, mass, electronic and ionic electric current conservation equations. The relationship among electric current densities, electric potentials and e−-O2 charge transfer rates can be solved by [18]

 V,ca V,ca  −(i 2− + i 2− ) in cathode  e−O , TPB e−O ,LSCF eff ˆ 0 in dense electrolyte ∇ · iO2− = ∇ · −σ 2− ∇ΦO2− = (11a) O   iV,an in anode e−O2−  iV, ca + iV, ca in cathode  e−O2−, TPB e−O2−,LSCF eff  ∇ · ie = ∇ · −σe ∇Φe = 0 in dense electrolyte (11b)   −iV, an in anode e−O2− 2− − where iO2− and ie are the O and e electric current densities within the button cell, respectively. σeff and σeff are the effective O2− and e− electric conductivities, respectively. O2− e The dusty gas model is adopted to describe gas transport within porous anode and cathode layers [27] ∇ · Nα = Rα (12a)

N xα Nβ − xβ Nα 1 x B p α + = − (p∇x + x ∇p + α 0 ∇p) (12b) eff eff RT α α eff DαK Dαβ µmixDαK

Nα and xα are molar ﬂux and local molar fraction of species α, respectively. Rα is reaction rate of each species. It can be evaluated through the e−-O2− electric current transfer rates per unit electrode volume as V, ca V, c ) RO = −(i 2− + i 2− )/(4F) 2 e−O , LSCF e−O , TPB in cathode (13a)

RN2 = 0  R = −iV, aa /(2F) H2 − 2−  e O , TPB in anode (13b) R = iV, an /(2F) H2O e−O2−, TPB  The total gas pressure and permittivity within the porous structure can be evaluated by

3 2 c φgrg p = tot , B = (14) RT 0 8τ2 where rg is mean hydraulic pore radius of the speciﬁed porous electrode structure. τ is the corresponding tortuosity of the porous structure [28]. The effective dynamic viscosity of mixture gas µmix can be predicted by ideal gas mixing law [29]

n x µ µ T 1.5 T + S µ = α α , α ≈ 0 (15a) mix ∑ n 0 + α= µα T0 T S 1 ∑ xβΦα,β β=1 Energies 2018, 11, 1875 7 of 16

!−1/2 !1/2 !1/42 1 Mα µα Mα Φα.β = √ 1 + 1 +  (15b) 8 Mβ µβ Mβ where Mα is the molar mass. µα is the dynamic viscosity of species α, which can be evaluated based on Sutherland’s law based on the relevant parameters in Table1.

Table 1. Gas composition and parameters for viscosity calculations by Sutherland’s law.

−6 3 −1 0 −6 −1 −1 Gas να(×10 m mol ) µα ×10 kg m s T0 (K) S (K)

H2 6.12 8.411 273 97 vapor 13.1 11.2 350 1064 O2 16.3 19.19 273 139 N2 18.5 16.63 273 107

The effective Knudsen diffusion coefﬁcient [30] of species α and effective binary diffusion coefﬁcient [31] can be estimated by

s 0.5 −8 1.75 ! φg 2rg 8RT φg 3.24 × 10 T 1 1 Deff = Deff = + αK , αβ 1/3 1/3 (16) τ 3 πMα τ Mα M p(να + νβ ) β where να is diffusion volume of species α, which is collected in Table1.

3. Result and Discussion

Figure3 shows ﬁve calculated iop-Vop curves at different operation T for a LSCF-SDC/SDC/Ni-SDC button cell with the ratio of anode and cathode discs radii around rca/ran = 0.8 cm/1 cm. In other words, the surface areas ratio of anode and cathode is Aca/Aan = 0.64. The corresponding parameters are illustrated in Supplementary Materials. It Is necessary to mention that the deviation of iop-Vop curves between the calculating and experiment results in high current density zone at 700 ◦C was considered to be an error caused by some unknown factors during the testing process based on follow considerations. (i) The sharp drop of iop-Vop curve at high current density zone is considered to be caused by concentration overpotential. However, the limited current density at 700 ◦C smaller than that at 600 ◦C is unreasonable. (ii) These deviations were happened around the up boundary operation zone (up operation temperature and current density zones). The deviation could be caused by abnormal factors. (iii) Good agreements between calculated and experiment results [13] at several other T can well illustrate that the modeling parameters can well represent the electrochemical properties of the button cell; and the cell-level multi-physics model can well describe the working details within it. It should be note that the feature of electronic leakage of dense electrolyte would lead to a sharp decrease of the open circuit voltage (shown in Figure3). Energies 2018, 11, x FOR PEER REVIEW 8 of 16

0.8 cm/1 cm. In other words, the surface areas ratio of anode and cathode is Aca/Aan = 0.64. The corresponding parameters are illustrated in Supplementary Materials. It Is necessary to mention that the deviation of iop-Vop curves between the calculating and experiment results in high current density zone at 700 °C was considered to be an error caused by some unknown factors during the testing process based on follow considerations. (i) The sharp drop of iop-Vop curve at high current density zone is considered to be caused by concentration overpotential. However, the limited current density at 700 °C smaller than that at 600 °C is unreasonable. (ii) These deviations were happened around the up boundary operation zone (up operation temperature and current density zones). The deviation could be caused by abnormal factors. (iii) Good agreements between calculated and experiment results [13] at several other T can well illustrate that the modeling parameters can well represent the electrochemical properties of the button cell; and the cell-level multi-physics model can well describe the working details within it. It should be note that the Energiesfeature2018 of ,electronic11, 1875 leakage of dense electrolyte would lead to a sharp decrease of the open circuit8 of 16 voltage (shown in Figure 3).

Figure 3. Comparison between the numerical modeling and experiment [13] results at various Figure 3. Comparison between the numerical modeling and experiment [13] results at various operating temperatures. operating temperatures.

For a button cell, the support component layer is always fabricated with a relative lager surface area Forcompared a button with cell, the the measured support componentelectrode [32] layer (i.e is., the always anode fabricated surface area with in a relativecurrent lagerbutton surface cell is area3.14 comparedcm2 and the with corresponding the measured surface electrode area [32 of] (i.e.,measured the anode cathode surface is only area 2 in cm current2). Obviously, button cell for isa 3.14 cm2 and the corresponding surface area of measured cathode is only 2 cm2). Obviously, for a specified output voltage Vop, the responded operating current density iop can be evaluated by two specified output voltage V , the responded operating current density i can be evaluated by two ways, divided the output currentop Iop by Aan for a larger value or Aca for a lowerop value. To improve the ways, divided the output current I by A for a larger value or A for a lower value. To improve performance quality, most of the reportedop aniop-Vop curves of button cellsca were always obtained based the performance quality, most of the reported i -V curves of button cells were always obtained on the relative smaller area between Aan and Aca.op Thus,op it is important to evaluate the influence of the based on the relative smaller area between A and A . Thus, it is important to evaluate the influence over zone from support layer on the experimentan ca measuring and numerical calculating iop-Vop ofperformance the over zone results from for support IT-SOFC layer button on the cell experiment using mixed measuring conducting and material. numerical calculating iop-Vop performance results for IT-SOFC button cell using mixed conducting material. Effect of different Aca/Aan ratio at 700 and 600 °C: Figure 4 compares the Iop-Vop performances of ◦ LSCF-SDC/SDC/Ni-SDCEffect of different Aca /AIT-SOFCan ratio button at 700 cells and 600withC: respectivelyFigure4 compares cathode thearea I2op and-Vop 0.5performances cm2 (labeled 2 ofas LSCF-SDC/SDC/Ni-SDCcells 1 and 2), while kept IT-SOFC the surface button area cells of with support respectively anode as cathode 3.14 cm area2. Generally, 2 and 0.5 cmthe (labeled as cells 1 and 2), while kept the surface area of support anode as 3.14 cm2. Generally, the experimentally measured iop-Vop performances may be obtained by the following two steps. Firstly, experimentally measured i -V performances may be obtained by the following two steps. Firstly, the the responded output currentsop op Iop should be measured while the output voltages Vop are specified. responded output currents I should be measured while the output voltages V are specified. Then, Then, the corresponding outputop current densities iop can be obtained throughop divided Iop by the the corresponding output current densities i can be obtained through divided I by the electrode electrode surface. Table 2 compares the iop-Vopop relations of cells 1 and 2 based onop both anode and surface.cathode Tablecross2 sectioncompares surfaces, the iop -respectively.Vop relations Obviou of cellssly, 1 and using 2 based the onrelative both anodelarger andelectrode cathode surface cross section surfaces, respectively. Obviously, using the relative larger electrode surface (i.e., A in current (i.e., Aca in current anode supported case) means larger iop value. Using the relative lowerca electrode anode supported case) means larger i value. Using the relative lower electrode surface (i.e., A ) surface (i.e., Aan) means lower iop value.op It is interesting to find that there are very similar iop-Vanop means lower i value. It is interesting to find that there are very similar i -V relationships between relationships opbetween cells 1 and 2, while estimated iop based on theop relevantop smaller electrode cells 1 and 2, while estimated i based on the relevant smaller electrode surface area (i.e., A in surface area (i.e., Aca in current case).op There is no obvious difference between the performance resultsca current case). There is no obvious difference between the performance results even at the maximum ◦ power density case. Taking Vop = 0.5 V and T = 700 C as an example, the maximum power densities −2 2 −2 2 are 1.412 W cm for cathode area 0.5 cm and 1.397 W cm for Aca = 2 cm cases, respectively. Energies 2018, 11, x FOR PEER REVIEW 9 of 16 even at the maximum power density case. Taking Vop = 0.5 V and T = 700 °C as an example, the Energiesmaximum2018, power11, 1875 densities are 1.412 W cm−2 for cathode area 0.5 cm2 and 1.397 W cm−2 for Aca = 29 ofcm 162 cases, respectively.

(a) (b)

Figure 4. Effects of different cathode areas on cell, (a) Iop-Vop performances at 700◦ °C; (b) Iop-Vop Figure 4. Effects of different cathode areas on cell, (a) Iop-Vop performances at 700 C; (b) Iop-Vop performances at 600 ◦°C.C.

Thus, it can be concluded that although the over zone of larger electrode is generally considered Thus, it can be concluded that although the over zone of larger electrode is generally considered can enhance charges and gases transport capacities in a proper zone, the active zone of the button can enhance charges and gases transport capacities in a proper zone, the active zone of the button cell will be restricted at the electrode zone with relevant small area (i.e., cathode layer area in current cell will be restricted at the electrode zone with relevant small area (i.e., cathode layer area in current anode support case), instead of the support layer surface areas (i.e., Aan). The influence of the over anode support case), instead of the support layer surface areas (i.e., Aan). The inﬂuence of the over zone of anode surface area on iop-Vop performance would be negligible. Taking the small area zone of anode surface area on iop-Vop performance would be negligible. Taking the small area between between anode and cathode surfaces to calculate iop-Vop performance is more reasonable to indicate anode and cathode surfaces to calculate i -V performance is more reasonable to indicate the the electrochemical properties of the tested opIT-SOFCop button cell. electrochemical properties of the tested IT-SOFC button cell.

Table 2. For a specified Vop, comparing the responded iop that are evaluated by divided Iop with Aan Table 2. For a speciﬁed Vop, comparing the responded iop that are evaluated by divided Iop with Aan for larger value and Aca for lower value, respectively. for larger value and A for lower value, respectively. ca I (A) iop Based Aan (A cm−2) iop Based on Aca (A cm−2) op V −2 −2 Cell 1I (A) Cell 2 Celliop Based1 A Cellan (A 2 cm ) Celliop Based1 on CellAca (A 2 cm ) Vop 0.2 Cell8.139 1 2.047 Cell 2 2.592 Cell 1 0.652 Cell 2 4.069 Cell 1 4.095 Cell 2 0.3 7.585 1.906 2.416 0.607 3.793 3.811 0.2 8.139 2.047 2.592 0.652 4.069 4.095 0.30.4 7.5856.838 1.717 1.906 2.178 2.416 0.547 0.607 3.419 3.793 3.434 3.811 0.40.5 6.8385.587 1.412 1.717 1.779 2.178 0.450 0.547 2.794 3.419 2.824 3.434 0.50.6 5.5873.860 0.971 1.412 1.229 1.779 0.309 0.450 1.930 2.794 1.942 2.824 0.60.7 3.8601.905 0.481 0.971 0.607 1.229 0.153 0.309 0.953 1.930 0.962 1.942 0.7 1.905 0.481 0.607 0.153 0.953 0.962 0.780.78 0.3020.302 0.078 0.078 0.096 0.096 0.025 0.025 0.151 0.151 0.155 0.155 2 2 Aan = 3.14 cm . 2Aca = 2 and 0.5 cm 2 for cell 1 and 2, respectively. Aan = 3.14 cm . Aca = 2 and 0.5 cm for cell 1 and 2, respectively.

Effects of Electrolyte thickness on cell performance: performance: Generally,Generally, too thin electrolyte layer is considered op op as a key factorfactor toto weakenweaken thethe inﬂuenceinfluence ofof thethe overover zonezone ofof largerlarger electrodeelectrode ononi opi --VVop performance. Figure 55 furtherfurther comparescompares fourfour differentdifferent cellcell performancesperformances atat 600600 ◦°CC among the button cells with different combination ofof cathodecathode surfacesurface areasareas (0.5(0.5 andand 22 cmcm22) and electrolyte thicknesses (i.e., 15 and 50 µμmm),), while while keeps keeps the the anode anode surface surface areas areas and and other othe operatingr operating parameters parameters values. values. Obviously, Obviously, we canwe canﬁnd find that that increase increase the densethe dense electrolyte electrolyte thickness thickness in a limitedin a limited value value from from 13 to 13 50 toµm 50would μm would not affect not affectthe above the above conclusions. conclusions. Taking Taking the small the areas small between areas between anode andanode cathode and cathode surfaces surfaces to calculate to calculateiop-Vop performanceiop-Vop performance is more is reasonable more reasonable to indicate to indicate the electrochemical the electrochemical properties properties of the tested of the IT-SOFC tested IT-SOFCbutton cell. button cell.

Energies 20182018,, 1111,, 1875x FOR PEER REVIEW 10 of 16 Energies 2018, 11, x FOR PEER REVIEW 10 of 16

Figure 5. The effects of different Aca/Aan on the button cell performances at 600 °C while increases the Figurethickness 5. The of dense effects electrolyte of different from Aca /15Aan to on 50 the μ mbutton. cell performances at 600 °C◦ while increases the Figure 5. The effects of different Aca/Aan on the button cell performances at 600 C while increases the μm thicknessthickness of of dense dense electrolyte electrolyte from from 15 15 to to 50 50 µm. . an Effects of exchange current density: The exchange current density jTPB, 0, ref based on TPBs is an an Effects of exchange current density: The exchange current density anj based on TPBs isan an Effects of exchange current density: The exchange current density jTPB,TPB, 0, 0, ref ref based on TPBs is an important factor to character the electrochemical property of the button cells. A higher jTPB, 0, ref an an important factor to character the electrochemical property of the button cells. A higher jTPB, 0, ref jmeans importantmeans that factor smaller to activationcharacter overpotentialthe electrochemi is needcal edproperty to convert of the same button amount cells. of chargeA higher between TPB, 0, refe− that smaller activation overpotential is needed to convert same amount of charge between e− and O2− 2− − meansand O that electric smaller currents. activation As shown overpotential in Figure is 6a,need fored a to specified convert isameop, Vop amount increases of withcharge the between increasingan e electric currents. As shown in Figure6a, for a speciﬁed iop, Vop increases with the increasing j . an 2− an TPB, 0, ref and O electric currents. As shown in Figure 6a, for a specified aniop, Vop increases with−2 the −increasing2 −3 −3 However,jTPB, 0, ref . However, it is interesting it is interesting to get that to forget all that the for three all the cases three with casesj with =jTPB, 8.0 0, ×ref 10 = 8.0, ×8.0 10×, 108.0 × and10 an TPB, 0, ref an −2 −3 j . However,−4 −4 −1 it−1 is interesting to get that for all the three cases with j = 8.0 × 10 2,2 8.0 × 10 8.0andTPB, ×0, 8.0ref10 × 10A mA m, the, the maximum maximum power power density density differences differences between betweenA AcaTPB,ca= =0, ref0.50.5 andand 22 cmcm casescases are −2 ◦ andlessless 8.0 thanthan × 10 100100−4 A WW m mm−1−,2 ,the ,at at maximum0.4 0.4 V V and and power600 600 °C.C. density Therefore, differences for those between button A cellsca = 0.5 with and exchangeexchange 2 cm2 cases currentcurrent are lessdensities than within100 W reasonablem−2, at 0.4 V range, and the600 effect°C. Therefore, of thethe over for zone those from button larger cells electrode with exchange surface on current button densitiescell performance within reasonable isis insigniﬁcant.insignificant. range, the effect of the over zone from larger electrode surface on button cell performance is insignificant.

(a) (b) (a) (b) Figure 6. The sensitivity of button cell performances on different Aca/Aan: (a) Under various exchange Figure 6. The sensitivity of button cell performances on different Aca/Aan:(a) Under various exchange Figurecurrent 6. densities, The sensitivity (b) while of button different cell component performances support on different cases are A caconsidered./Aan: (a) Under various exchange current densities, (b) while different component support cases are considered. current densities, (b) while different component support cases are considered. Effects of different component support cases: Figure 6b shows the sensitivities of button cell Effects of different component support cases: Figure6b shows the sensitivities of button cell performancesEffects of differenton different component anode/cathode support cases:surface Figure area ratios6b shows at 600 the °C, sensitivities while different of buttoncomponent cell performances on different anode/cathode surface area ratios at 600 ◦C, while different component performancessupport cases on are different adopted. anode/cathode The geometry surface parameters area ratios of atthe 600 anode, °C, while cathode, different electrolyte component and support cases are adopted. The geometry parameters of the anode, cathode, electrolyte and supportcomponent-self-support cases are adopted. button The cellsgeometry are respectivelyparameters oflisted the inanode, Table cathode, 3. Although electrolyte different and component-self-support button cells are respectively listed in Table3. Although different component component-self-supportcomponent support cases button would leadcells toare very respectively different button listed cell in performances, Table 3. Although it should differentbe noted support cases would lead to very different button cell performances, it should be noted that the componentthat the sensitivities support cases of wouldiop-Vop leadperformances to very differ onent different button cellAca /Aperformances,an are still insignificantit should be notedwhile sensitivities of iop-Vop performances on different Aca/Aan are still insigniﬁcant while evaluating iop thatevaluating the sensitivities iop based on of the iop smaller-Vop performances electrode surface on differentarea between Aca/A anodean are and still cathode insignificant layers. while based on the smaller electrode surface area between anode and cathode layers. evaluating iop based on the smaller electrode surface area between anode and cathode layers.

Energies 2018, 11, 1875 11 of 16

Table 3. Geometry parameters of the anode, cathode, electrolyte and component-self-support button cells in the unit of mm.

Item Anode Support Functional Layer Dense Electrolyte Cathode Anode support 460 6 13 40 Cathode support 20 6 13 480 Electrolyte support 20 6 460 40 Energies 2018Self, 11,support x FOR PEER REVIEW 100 100 50 200 11 of 16

Table 3. Geometry parameters of the anode, cathode, electrolyte and component-self-support button cellsThe realin the affection unit of zonemm. of the over electrode surface area: It is theoretically agreed that the over anode (or cathode) surface area due to Aca/Aan 6= 1 will decrease the potential losses of the gas, electron Item Anode Support Functional Layer Dense Electrolyte Cathode and ion transports in the corresponding electrode. The above study results, however, show that the Anode support 460 6 13 40 geometric effect of anode and cathode surface areas ratio on the IT-button cell iop-Vop performance is Cathode support 20 6 13 480 quite limited, while the working parameters vary in a reasonable range. The active zone of the button Electrolyte support 20 6 460 40 cell is restricted within the smaller electrode area zone between anode and cathode (i.e., min (Aca, Aan)). This shouldSelf support be caused by the geometric100 characteristics100 of button cell with relative 50 thin component200 layer. Taking dense electrolyte layer as an example, the length-width ratio between thickness and radius of disc surfaceThe real isaffection 13 µm/1 zone cm of =the 0.0013. over electrode Thus, ﬁnding surface area: out the It is real theoretically inﬂuence widthagreed of that this the over over electrode anode ≠ (orzone cathode) on the buttonsurface cellarea performance due to AAca/1 would an bewill very decrease helpful the to potential understand losses the workingof the gas, properties electron andof IT-SOFCs. ion transports in the corresponding electrode. The above study results, however, show that the ◦ geometricTaking effectT = of 600 anodeC and andV cathodeop = 0.4 su Vrface as an areas example, ratio on the the concentration IT-button cell distributions iop-Vop performance of H2 and is quiteH2O withinlimited, porous while compositethe working anode, parameterscH2 and varycH2 Oin, area reasonable shown in Figurerange.7 .ThecH active2 on anode zone sideof the is −3 −3 button13.563 molcell is m restrictedinitially within and drops the smaller sharply electr alongodez-axis area to zone 6.81 molbetween m anoddue toe and the oxidationcathode (i.e., reaction min

(ofAca H, 2A.an In)). contrast, This should the concentration be caused by of the product geometric vapor charcH2acteristicsO on anode of side button increases cell with along relativez axis fromthin −3 component0.42 to 12.4 mollayer. m Taking. Obviously, densec Helectrolyte2 and cH2O layerhave as opposite an example, distribution the leng characteristics.th-width ratio Theoretically, between 2 2 thicknessthe over zoneand ofradius the currentof disc anodesurface surface is 13 μ (i.e.,m/1A cman == 3.140.0013. cm Thus,, Aca =finding 2 cm , out and theAca /realAan influence= 0.64 in widthcurrent of button this over cells) electrode would enhance zone on the the hydrogen button and cell vapor performance transports would within be porous very anodehelpful in to a understandproper width the because working of properties the enlarged of crossIT-SOFCs. section.

Figure 7. The concentration distributions of H2 and H2O within anode at T = 600 °C◦ and Vop = 0.4 V as Figure 7. The concentration distributions of H2 and H2O within anode at T = 600 C and Vop = 0.4 V an example. as an example.

Taking T = 600 °C and Vop = 0.4 V as an example, the concentration distributions of H2 and H2O Figure8a further shows a c distribution within porous cathode. The O concentration is within porous composite anode, cO2 and c , are shown in Figure 7. c on anode2 side is 13.563 consumed from 2.93 to 1.41 mol m−H32 . As shownHO2 in Figure8c, the over zoneH of2 the anode surface area −3 −3 molcan alsom enhanceinitially theand conducting drops sharply capacity along of z electronic-axis to 6.81 current mol densitym duein toy thedirection oxidation due reaction to the enlarged of H2. In contrast, the concentration of product vapor c on anode side increases along z axis from 0.42 cross section. Combining Figures7 and8, we can ﬁndHO2 that the real effective width of the over electrode to 12.4 mol m−3. Obviously, c and c have opposite distribution characteristics. Theoretically, H2 HO2 the over zone of the current anode surface (i.e., Aan = 3.14 cm2, Aca = 2 cm2, and Aca/Aan = 0.64 in current button cells) would enhance the hydrogen and vapor transports within porous anode in a proper width because of the enlarged cross section.

Energies 2018, 11, x FOR PEER REVIEW 12 of 16

Energies 2018, 11, 1875 12 of 16

zone in an IT-SOFC button cell is only in a scale of 0.03 cm. The over electrode zone exceed this width would be inactive. This can well explain that why the IT-SOFC button cells iop-Vop performance is insensitive to the Aca/Aan ratio; and a smaller Aca/Aan may not greatly increase the measured iop-Vop electrochemical quality. Because the button cell is fabricated in a radius scale of rdisc = 1 cm. These real Energieseffective 2018 width, 11, x FOR of the PEER over REVIEW electrode zone reference to the button cell disc overall radius is less than12 of 5%.16

Figure 8. (a,b) c distribution within porous cathode at T = 600 °C and Vop = 0.4 V as an example; O2

Figure 8a further shows a c distribution within porous cathode. The O2 concentration is O2 consumed from 2.93 to 1.41 mol m−3. As shown in Figure 8c, the over zone of the anode surface area can also enhance the conducting capacity of electronic current density in y direction due to the enlarged cross section. Combining Figures 7 and 8, we can find that the real effective width of the over electrode zone in an IT-SOFC button cell is only in a scale of 0.03 cm. The over electrode zone exceed this width would be inactive. This can well explain that why the IT-SOFC button cells iop-Vop performance is insensitive to the Aca/Aan ratio; and a smaller Aca/Aan may not greatly increase the Figure 8. (a,b) c distribution within porous cathode at T = 600◦ °C and Vop = 0.4 V as an example; Figure 8. (a,b) c O2 distribution within porous cathode at T = 600 C and Vop = 0.4 V as an example; (c) measured iop-Vop electrochemicalO2 quality. Because the button cell is fabricated in a radius scale of rdisc ie distribution throughout the whole button cell. = 1 cm.(c) iThesee distribution real effective throughout width the of whole the over button electr cell.ode zone reference to the button cell disc overall radius is less than 5%. To further conﬁrm this conclusion, a button cell in a smaller scale (i.e., r = 0.1 cm, FigureTo further 8a further confirm shows this conclusion, a cO distribution a button cellwithin in a poroussmaller cathode.scale (i.e., The rdisc =O 0.12 concentration cm,disc Aan = 3.14 is × −2 2 2 A −2 = 3.142 × 10 cm ) is developed. Figure9 compares the i -V performances between the buttons 10an cm ) is developed. Figure 9 compares−3 the iop-Vop performancesop op between the buttons cells with consumed from 2.93 to 1.41 mol m . As shown in Figure 8c, the over zone of −the2 anode surface−2 area2 cellsdifferent with cathode/anode different cathode/anode surface areas surface ratios (i.e., areas A ratiosca = 2 × (i.e., 10−2 Aandca =0.5 2 ×× 1010−2 cmand2, Aca0.5/Aan× =10 0.64 cmand, can also enhance the conducting capacity of electronic current density in y direction due to the−2 Aca/Aan = 0.64 and 0.16). The corresponding maximum power densities are 0.9749 and−2 0.8814 W cm enlarged0.16). The cross corresponding section. Combining maximum Figures power 7 densities and 8, we are can 0.9749 find andthat 0.8814the real W effective cm for widthAca/Aan of = 0.64the for Aca/Aan = 0.64 and 0.16 cases, respectively. Obviously, while the geometry of the button cells or overand electrode0.16 cases, zone respectively. in an IT-SOFC Obviously, button cell whil is one lythe in ageometry scale of 0.03 of cm.the The button over electrodecells or zoneother otherelectrochemical electrochemical devices devices approach approach the scale the less scale than less 100 than μm, 100 theµ effectm, the of effect the over of the electrode over electrode zone on exceed this width would be inactive. This can well explain that why the IT-SOFC button cells iop-Vop zonethe electrochemical on the electrochemical performance performance should not shouldbe ignore notd. beTaking ignored. the smaller Taking electrode the smaller surface electrode area to performance is insensitive to the Aca/Aan ratio; and a smaller Aca/Aan may not greatly increase the surfaceevaluate area iop for to evaluatea specifiediop Vforop would a speciﬁed causeV opan wouldimproper cause over an ev improperaluation overof the evaluation electrochemical of the measured iop-Vop electrochemical quality. Because the button cell is fabricated in a radius scale of rdisc electrochemical performance. =performance. 1 cm. These real effective width of the over electrode zone reference to the button cell disc overall radius is less than 5%. To further confirm this conclusion, a button cell in a smaller scale (i.e., rdisc = 0.1 cm, Aan = 3.14 × 10−2 cm2) is developed. Figure 9 compares the iop-Vop performances between the buttons cells with different cathode/anode surface areas ratios (i.e., Aca = 2 × 10−2 and 0.5 × 10−2 cm2, Aca/Aan = 0.64 and 0.16). The corresponding maximum power densities are 0.9749 and 0.8814 W cm−2 for Aca/Aan = 0.64 and 0.16 cases, respectively. Obviously, while the geometry of the button cells or other electrochemical devices approach the scale less than 100 μm, the effect of the over electrode zone on the electrochemical performance should not be ignored. Taking the smaller electrode surface area to evaluate iop for a specified Vop would cause an improper over evaluation of the electrochemical performance.

Figure 9.9. iiopop--Vopop performancesperformances between between the the buttons buttons cells wi withth different cathode/anodecathode/anode surface areas −2−2 2 2 ratios (i.e., Aan == 3.14 3.14 ×× 1010 cmcm, A,caA/Acaan/ =A 0.64an = and 0.64 0.16). and 0.16).

Figure 9. iop-Vop performances between the buttons cells with different cathode/anode surface areas

ratios (i.e., Aan = 3.14 × 10−2 cm2, Aca/Aan = 0.64 and 0.16).

Energies 2018, 11, 1875 13 of 16

4. Conclusions The comprehensive multi-physics model of IT-SOFC button cells considers special features, such as using mixed conducting materials, electric leakage and complex multi-physics mutual coupling processes have been developed and veriﬁed. The geometry effect of different anode and cathode surface area ratios on iop-Vop performance of IT-SOFC button cells are investigated and many conclusions are reached,

(i). The over zone of the larger electrode can only enhance charges and gas transport capacities within a limited scale of only 0.03 cm, an over electrode zone exceeding this width would be inactive.

(ii). The active zone of button cell is restricted within the smaller electrode area min(Aan, Aca) due to the relatively large disc radius in scale of cm and the thin component layer.

(iii). For a speciﬁed Vop, evaluating the responded iop by dividing output current Iop with min(Aan, Aca) for a larger value is reasonable for presenting the real performance in a current device scale. (iv). While the geometry of button cell or other electrochemical device approaches a scale of less than 100 µm, taking the smaller electrode surface area to evaluate iop for a speciﬁed Vop would cause an improper over evaluating of the electrochemical performance.

Supplementary Materials: The following are available online at http://www.mdpi.com/1996-1073/11/7/1875/ s1. Author Contributions: Data curation, B.H.; Investigation, K.D.; Writing-Original Draft Preparation, C.Y.; Writing-Review and Editing, D.C. and L.L. All authors read and approved the final manuscript. Funding: This research was funded by the National Science Foundation of China (51776092 and 21406095) and the Natural Science Foundation of Jiangsu Province BK20151325. Conflicts of Interest: The authors declare no conflicts of interest.

Nomenclature

2 Aca the cross section area of the cathode layer, m 2 Aan the cross section area of the anode layer, m 2 B0 the ﬂow permeability, m 0 −3 cα the concentration of species α at the channel inlet, mol m DPBs the LSCF-pore double phase boundaries eff 2 −1 Dαβ the effective binary diffusivity, m s eff 2 −1 DαK the effective Knudsen diffusivity of species α, m s Eeq the local equilibrium electric potentials difference at working state, V Est the Nernst potential at the standard state, V

EH2 the activation energy for H2 oxidation reaction, J

EO2 the activation energy for O2 reduction reaction, J F the Faraday constant, C mol−1 GDC the Gd0.1Ce0.9O1.95 − 2− −1 jTPB the local e -O charge transfer rate per unit TPB length, A m −1 jTPB,0 the local exchange transfer current per unit TPB lengths, A m jTPB,0,ref the value assigned empirically based on experiment at reference temperature − −2 ie the local e electric current density, A m 2− −2 iO2− the local O electric current densities, A m −2 iop the output current density, A m Iop the output current, A − 2− −2 iLSCF the e -O charge transfer rate per unit percolated DPB area, A m −2 iLSCF, 0 the local exchange transfer current per unit percolated DPB area, A m iV − 2− −3 e−O2−,TPB the e -O charge transfer rate per unit volume based on percolated TPBs, A m iV − 2− −3 e−O2−,LSCF the e -O charge transfer rate per unit volume based on percolated LSCF-pore DPBs, A m iS − 2− −2 e−O2−, TPB the e -O charge transfer rate per unit dense electrolyte surface, A m Energies 2018, 11, 1875 14 of 16

LSCF the La0.6Sr0.4Co0.2Fe0.8O3−δ LSM the La1-xSrxMnO3 −3 Mα the mole mass of species α, kg mol −2 −1 Nα the molar ﬂux of species α, mol m s V nk the number of k-particles per unit volume S nk the number of k-particles per unit dense electrolyte surface area pα the partial pressure of gas species α at the local reaction sites, atm 0 pα the partial pressure of gas species α in the channel inlet, atm O2− 2− PSDC the probabilities of SDC-particles belonging to percolated O conducting path O2− 2− PLSCF the probabilities of LSCF-particles belonging to percolated O conducting path e − PLSCF the probabilities of LSCF-particles belonging to percolated e conducting path rg the mean hydraulic pore radius of porous electrode structure, m rk the radius of k-particle, m rc the neck radius between two connected particles, m R the universal gas constant, J mol−1 K−1 −3 −1 Rα the sources/leak of species α, mol m s SDC the Sm0.2Ce0.8O2−δ V −1 SLSCF, per the percolated LSCF-pore DPBs per unit volume, m 2 ses the exposed surface area of each LSCF-particle, m T the operating temperature, K TPBs the three phase boundary sites Vop the output voltage at working state, V 3 −1 να the diffusion volume for species α, m mol xα the molar fraction of species α YSZ the yttrium-stabilized zirconia

Zk,l the average number of contacts between k- and all of its neighboring l-particles Z the average coordination number of all particles Greek letters

αf, βr the forward and reverse reaction symmetric factors γLSCF, SDC the 1D circular length per contact between LSCF- and SDC-particles, m γLSCF, ele the 1D circular length per contact between LSCF-particle and the dense electrolyte, m V −2 λTPB,per the percolated TPB length per unit volume, m S −1 λTPB,per the percolated TPB length per dense electrolyte surface area, m φg the porosity of porous structure − Φe the local e electric potential, V 2− ΦO2− the local O electric potential, V ˆ ΦO2− the shift of ΦO2− by a reference amount, V ψk the solid volume fraction of k-particles ηact the local activation overpotential, V θ the smaller contact angle between two particles τ the tortuosity of gas transport path within the porous electrode eff −1 σe the effective electronic conductivity, S m eff 2− −1 σ − the effective O ionic conductivity, S m O2 −1 −1 µmix the viscosity of gas mixture, kg m s −1 µα the mole chemical potential of reactant α, J mol Superscripts and subscripts an anode act activation ca cathode eq equilibrium st standard condition (1 atm) ref reference value Energies 2018, 11, 1875 15 of 16

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