Elementary reaction modeling and experimental characterization of solid oxide direct carbon-assisted steam electrolysis cells

Solid State Ionics 295 (2016) 78–89

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Elementary reaction modeling and experimental characterization of solid oxide direct carbon-assisted steam electrolysis cells Yiyang Wu, Yixiang Shi ⁎, Yu Luo, Ningsheng Cai Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua University, Beijing 100084, China

a r t i c l e

i n f o

Article history: Received 2 June 2016 Received in revised form 11 July 2016 Accepted 11 July 2016 Available online xxxx Keywords: Solid oxide direct carbon-assisted electrolysis cell (SO-DCEC) Cell performance Heterogeneous chemistry Elementary reaction model Carbon monoxide

a b s t r a c t A detailed one-dimension mechanistic model for solid oxide direct carbon-assisted steam electrolysis cell (SODCEC) is well developed by considering heterogeneous elementary reactions in both the carbon bed and the cell, coupling with mass and charge transfer processes. The model is calibrated and validated by experimental data from a button cell test with different anode carrier gases at 800 °C.·The experimental and modeling results show that using CO2 instead of argon gas as the carrier gas benefits the cell performance. A mismatching between CO production in the carbon bed, its diffusion from the carbon bed to the cell and its consumption in the cell results in different transition zones in the cell polarization curve. A concept of producing H2 and CO simultaneously in the SO-DCEC system is proposed, and analyses indicate that working voltages of the cell and the carbon bed height greatly influence the system performance and production rates of gaseous fuels. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Currently, the demand for hydrocarbon fuels keeps increasing especially in developing countries like China and India, which challenges the truth that the supply of commercially usable hydrocarbon resources is limited in the world. Moreover, environmental issues such as global warming and air pollution caused by the traditional utilization of hydrocarbon fuels become more and more severe. To address these two severe problems, hydrogen is an ideal substitution for hydrocarbon fuels as a potential fuel and an energy storage medium [1,2], for reasons that: (1) Hydrogen is an “infinite” source of energy which can be generated from other renewable sources, although no direct natural source exists for it. (2) Hydrogen is environmentally friendly since it produces water only as the combustion and oxidation product. The development of efficient systems for hydrogen production is a crucial issue before the widely use of hydrogen. Most of hydrogen is presently produced by the steam reforming of methane, however this approach has the drawback that it cannot employ renewable energies like solar or wind energy which is totally carbon-free. Also, its reaction device is so large that it is not particularly amenable to a reliable and highly distributed hydrogen supply infrastructure [3]. Another alternative method is producing hydrogen by water electrolysis, which produces a high purity hydrogen containing no CO unlike steam reforming and also suitable for distributed generation. Hydrogen generation by high-temperature electrolysis of water vapor using solid oxide ⁎ Corresponding author. E-mail address: [email protected] (Y. Shi).

http://dx.doi.org/10.1016/j.ssi.2016.07.003 0167-2738/© 2016 Elsevier B.V. All rights reserved.

electrolysis cells (SOEC) has been demonstrated to be a high-efficiency method [4–7]. The main disadvantage of electrolysis is the high electricity consumption. For the reason that electricity is an high quality and expensive form of energy, the cost of hydrogen production by electrolysis is as expensive as 2–3 times of that by steam reforming of methane [4]. And in essence, the electricity consumed mainly comes from the grid which produced from the fossil fuels, that means the steam electrolysis process actually is not absolutely carbon-free. Recently, there has been renewed interest in using solid oxide fuelassisted electrolysis cell (SOFEC) for electrolytic production of hydrogen. In SOFEC steam is fed into the cathode just as in SOEC, but fuels (like carbon, CO, CH4, biomass and other hydrocarbon fuels) are sent into the anode to facilitate the removal of oxygen from steam. It is obviously that adding assisting fuels significantly decreases the energy demand based on thermodynamics principle. Therefore, large amounts of electric power can be saved when the cell works in the SOFEC mode [8]. Several studies have demonstrated the use of anodic gaseous fuels in assisting steam electrolysis both experimentally and numerically [2–4,9]. Experiments performed on single cells by Martinez-Frias et al. [4] showed a voltage reduction of as much as 1 V when compared to the SOEC by natural gas-assisted electrolysis, and the system efficiency was up to 70% with respect to primary energy incorporating the cell with a heat recovery system. Using carbon or carbonaceous solids for carbon-assisted hydrogen production offers several advantages over those gaseous fuel-assisted performance such as: the abundance of cheaper and low-quality fuel sources including coal and biomass, high theoretical efficiency, and high CO2 emission reduction potential. Experiments done on a SOFEC

Y. Wu et al. / Solid State Ionics 295 (2016) 78–89

with a single carbon bed at the anode by Lee et al. [1] confirmed the concept of spontaneous hydrogen production and cogeneration of electricity. Xu et al. [10] developed a 2D DC-SOEC numerical model for syngas production at two different electrodes with easy control of H2/CO ratio, which is helpful for subsequent process to synthesize other chemicals from syngas. The performance of SOFEC with carbon as the assistant fuel relies on the coupling of carbon gasification and electrochemical reactions. The typical operating temperature of the carbon bed as well as the cell ranges from 600 to 1000 °C [11]. The cell performance decays seriously when the temperature is above 800 °C. However, the reaction rate of carbon gasification strongly depends on temperature, which is relatively slow below 800 °C [12]. Hence accelerating the reaction rate of carbon gasification to match the optimal operating temperature of the cell is crucial for improve the performance of the cell. It has been demonstrated that alkali metals are effective catalysts for carbon gasification [12–16], which are widely used for reducing the gasification reaction temperature in coal conversion processes. Li et al. [16] experimentally proved the improvement of the SO-DCFC performance by carbon catalytic gasification. To gain insight into the complex physical phenomena governing the cell performance, it is requisite to establish validated mechanism models for cell design and operating condition optimization. Numerous SOFC (or SOEC) models concerning gas transport phenomena, ionic and electronic conduction, and electrochemical processes have been reported in literatures for syngas and methane [2,17–21]. Zhao et al. [22] developed the SO-DCFC model with carbon fuel from CH4 cracking. Alexander et al. [23] and Yu et al. [24] developed detailed mechanism models coupling the carbon bed, fuel cell reaction and transport processes, which are based on biomass and carbon black as the fuels respectively. In these two models, the detailed carbon gasification kinetics and the diffusion processes within the carbon bed are carefully considered.

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From the discussion above, it is clear that mechanism models for SOFEC with gaseous fuels at the anode and SO-DCFC with carbon gasification have been well studied separately. And only small number of experiments have been conducted on SOFEC with carbonaceous fuels at the anode. A validated mechanism model with experiments is helpful to understand the intricate phenomena in SO-DCEC performance coupling with the carbon bed gasification, as experimental studies on the SO-DCEC are rather complex, expensive and timeconsuming. In this paper, a comprehensive elementary reaction kinetic model of SO-DCEC coupled with potassium catalytic gasification of a single carbon bed is developed. This model fully considers the coupling of electrochemical reactions, charge transport and mass transport processes within the cell and the thermochemical processes of the carbon bed. The modeling results are validated by experimental data obtained for an SO-DCEC button cell with different anodic carrier gases at the setting operating temperature of 800 °C. Then the anodic reaction mechanism is carefully investigated based on the experimental and numerical results. After that, the effects of the operating temperatures, carrier gas composition and carbon bed properties on the SO-DCEC performance optimization are systematically explored. 2. Model development 2.1. Model assumption and geometry The model assumptions are listed as follows: (1) The gases in the computational domain are ideal gases; (2) The temperature within the carbon bed and the cell is uniform. All parameters are determined under the selected temperature; (3) The electrodes are isotropic materials with porous and stable microstructures. And the distribution of electronic conductors and

Fig. 1. Model structures, calculation domains and boundaries of fuel-assisted steam electrolysis.

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Y. Wu et al. / Solid State Ionics 295 (2016) 78–89

ionic conductors in electrodes are assumed to be continuous and uniform, respectively; (4) Molecule diffusion and Knudsen diffusion within the porous electrodes are taken into account, and ignoring the Darcy flow induced by pressure gradient. (5) The carbon gasification mechanism and the electrochemical reaction mechanism are modeled using a set of elementary reactions that represent chemical reactivity at the molecular scale. And the heterogeneous elementary reactions in anode are assumed to only take place on the Ni surface. Which means that Ni is used as a catalyst, and adsorbates are uniformly distributed on the Ni surface; (6) All surface elements in anode are electrically neutral. The charge transfer reactions take place at the three phase boundary(TPB) in one step as kct; f

O2− ðYSZÞ þ ðNiÞ ⇄ OðNiÞ þ ðYSZÞ þ 2e− kct;b

Based on the button cell tested in our group, a simplified onedimensional (1D) model along the cell thickness direction is adopted. The corresponding model geometry, computational domain and boundaries are shown in Fig. 1.

2.2. Governing equations 2.2.1. The SO-DCEC anode mechanism A methane heterogeneous SOFC anodic reactions mechanism using nickel as the catalyst was suggested by Hecht et al. [17], Janardhanan et al. [18], which involving methane-reforming, water-gas shift, gas adsorption/desorption and the Boudouard reactions. On the basis that there is not any elements of hydrocarbons in the porous anode of SODCEC, a simplified reaction mechanism is applied [20,24]. The elementary reactions and corresponding kinetic data are listed in Table 1.

2.2.2. Carbon bed modeling To model the carbon bed chemistry, the Boudouard mechanism (C + CO2 → 2CO) and the carbon oxidation mechanism put forth by Ma [25] are used. Along with the heterogeneous potassium catalytic carbon gasification mechanism from literature [12,26], the overall reactions can be modeled using a eleven elementary reaction mechanism, as showed in Table 1. The elementary reactions R1-R8 are the carbon gasification and oxidation mechanism without a catalyst, corresponding kinetic data are partly taken from [24,25,27–30]. In the reactions, Cf denotes a free carbon site available for adsorption; O(C) and CO(C) denote an adsorbed oxygen atom and an adsorbed CO molecule as you'd expected; and Cb denotes a bulk carbon site, which is assumed to have unit activity when determining reaction rates. The remaining reactions

Table 1 Heterogeneous reaction mechanism for carbon gasification and mechanism on Ni-based catalysts. Reaction Carbon gasification and oxidation mechanism Elementary reactions without catalyst CO2 + Cf → CO + O(C) R1f CO + O(C) → CO2 + Cf R1r R2 Cb + O(C) → CO + Cf R3 Cb + CO2 + O(C) → 2CO + O(C) Cf + CO → CO(C) R4f r CO(C) → Cf + CO R4 R5 CO + CO(C) → Cf + CO2 R6 2Cf + O2 → O(C) + CO R7 Cb + Cf + O(C) + O2 → Cf + O(C) + CO2 R8 Cb + Cf + O(C) + O2 → 2O(C) + CO Potassium catalytic elementary reactions CO2 + [KO] → CO + O[KO] R9f CO + O[KO] → CO2 + [KO] R9r Cf + O[KO] → O[KO](C) R10f O[KO](C) → Cf + O[KO] R10r R11 Cb + O[KO](C) → CO + Cf + [KO]

A (cm, mol, s)a

σb

E (kJ mol−1)a

5 × 10−3 108 1 × 10+13 1 × 10−4 0.89 1 × 10+13 1.01 × 10+07 3.87 × 10+04 1.18 × 10+09 3.74 × 10+16

– – 28 – – 53 – – – –

185 89.7 375 58 148 455 262 60 120 250

7.572

– – – – –

245

9.15 × 10+12 3.74 × 10

16

190.3 250

Adsorption and desorption reactions on Ni-based catalysts 1f 1r 2f 2r 3f 3r

O2 + Ni(s) + Ni(s) → O(s) + O(s) O(s) + O(s) → O2 + Ni(s) + Ni(s) CO2 + Ni(s) → CO2(s) CO2(s) → Ni(s) + CO2 CO + Ni(s) → CO(s) CO(s) → CO + Ni(s)

nc – 0.0 – – – 0.0 −50.0

E (kJ mol−1)a 0 474.95 0 25.98 0 111.27

5.2 × 10+23 1.354 × 10+22 −50.0

– −3

148.1 116.12

2.000 × 10+19 −50.0

0.0

123.60

4.653 × 10+23

−1.0

89.32

A (cm, mol, s)cd 1 × 10−2 4.283 × 10+23 1 × 10−05 6.447 × 10+07 5 × 10−01 3.563 × 10+11 θCOðsÞ d

Surface reactions 4f 4r

C(s) + O(s) → CO(s) + Ni(s) CO(s) + Ni(s) → C(s) + O(s)

5f

θCOðsÞ d CO(s) + O(s) → CO2(s) + Ni(s)

5r

θCOðsÞ d CO2(s) + Ni(s) → CO(s) + O(s)

a

Arrhenius parameters for the rate constant written as the form: k=Aexp(−E/RT). The activated energy for species in these two reactions are not uniform all through the carbon surface sites due to the local microscopic irregularities of carbon surfaces, so the activated 2 −E 1 þ∞ aÞ Þ  pffiffiffiffiffiffi expð− ðE−E ÞdE. energy for k2 and k4r complies with the normal distribution. That is, k ¼ ∫ 0 A expð 2σ 2 RT 2π σ c Arrhenius parameters for the rate constant written as the form: k = ATn exp(−E/RT). d Coverage dependent activation energy. b

Y. Wu et al. / Solid State Ionics 295 (2016) 78–89

R9-R11 are used to characterize the alkali K-catalysis carbon gasification process, and the data for the catalytic reactions are determined by matching the prediction to the experimental gasification data in our previous work [12]. In catalytic reactions, the active intermediates [KO], O[KO] and O[KO](C) represent potassium clusters, oxygen complexes “dissolved” in potassium clusters and O[KO] complexes adsorbed on the active sites of carbon surface, respectively. Table 2 gives equations for the carbon gasification process, including mass transport and chemical reaction. The specific surface area model is based on the random pore model for describing gasification rates of porous materials [31], and Ψ is the structure parameter denoting the micro-porosity of the char. The gas phase mass conservation coupled with the Darcy's law is also described. And gas species are transported through the carbon bed via convection and diffusion. 2.2.3. Equations for electrochemical processes in the SO-DCEC There are few collisions between gas molecules in the porous electrode since the particle size is the same order of magnitude as the free path of gas molecules. Therefore, chemical reactions between gas species are disregarded. The oxygen reaction process at the cathode is relatively simple in comparison to anodic reactions. Thus, heterogeneous chemistry at the anode surface is carefully considered, as shown in Table 3. As described in the model assumptions, the surface adsorbates are assumed to be uniformly distributed over the Ni surface. By treating the uncovered Ni surface active sites as surface species, heterogeneous reactions can be generally represented as the equation shown in Table 3. Reaction rates depend on the gas species concentration and the surface species concentration, which is usually characterized by surface coverage θk. The coverage θk is the ratio of surface active sites covered by species k to the total Ni-surface active sites (assumed to be conserved [32]). The electrochemical reaction process is accompanied by current transfer and energy conversion from chemical energy to electric energy. Actually, the elementary reactions and charge-transfer chemistry in fuel cells are still a research focus due to the intricate electrochemical processes. Multiple possible reaction paths and mechanism models have been suggested by different researchers [33–36]. To further simplify our modeling process, a one-step charge-transfer reaction is adopted [2,36]: O2− ðYSZÞ þ ðNiÞ⇄OðNiÞ þ ðYSZÞ þ 2e−

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Table 3 Equations for anode heterogeneous chemistry. K g þK s

K g þK s

k¼1

k¼1

∑ ν0k χ k ⇒ ∑ ν ″k χ k

General form of anode Ni/YSZ electrode heterogeneous elementary reactions The net molar production rates for gas species or surface species

K g þK s

N

ν0

s_ k ¼ ∑ðν″ki −ν0ki Þki ∏ ck ki i¼1

The reaction rate constant for the surface reactions and desorption reactions in modified Arrhenius form [18] The reaction rate constant for adsorption reactions [32]

k¼1 K g þK s

μ

Ei ki ck Þ ∏ ck ki expð− εRT Þ ki ¼ Ai T ni expð− RT k¼1

S0

ki ¼ Γ iγ

qffiffiffiffiffiffiffiffi RT 2πW

Ks

γ ¼ ∑ ν0ki k

The initial sticking coefficient

di Þ S0i ¼ ai T bi expð− RT

Table 4 gives electrochemistry equations of the electrode. And governing equations of charge balance and mass balance are summarized in Table 5, which have been described in details in our previous work [19,24].

2.3. Boundary conditions In accordance with operation conditions and model simplifying assumptions, the boundary conditions of the charge, mass and momentum balances' partial differential equations are listed in Table 6. Fig. 1 has schematically illustrated the model computational domain and meanings of corresponding boundaries. The boundary condition “Insulation” indicates that the partial derivative and the fluxes of the variables are all equal to zero at the boundary, and “continuity” denotes continuous fluxes of variables. Vca is the cathode electronic potential which is equals to the cell operation voltage in the calculation on the basis of setting the anode electronic potential as zero. cgas_in and cgas_ca denote the molar concentrations of gaseous species fed at the inlet of the carbon bed and the cathode. Pressure P0 stands for the ambient pressure condition. It should be noted that the boundary conditions for the surface species are “Insulation” at the interface of the carbon bed and the anode, which is different from the continuous conditions for the gaseous species. This is because there is no surface species extending away from the anode.

Table 2 Equations for the carbon gasification and oxidation process. The macro mass balance equation [25] The specific surface area of the char in unit of m2 kg−1 [25] The net rate of carbon atom removal from the bulk carbon bed (in unit of kg m2 s−1) The overall net molar reaction rates of CO and CO2 The changes in absorbed O and CO concentrations with time [28]

Table 4 Electrode electrochemistry.

dxc 1 c ¼ ð1−x ¼ SgC Ri;C RC ¼ − m1c dm dt c Þ dt

SgC ¼ SgC;0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−ψlnð1−xc Þ

kct; f

The charge transfer reaction [2,36]

Ri,C = MC(RR2 + RR3 − RR4 + RR6 + RR7 + RR8 + RR11)

RCO ¼ RR1 þ RR2 þ 2RR3 −RR4 −RR5 þRR6 þ RR8 þ RR9 þ RR11 RCO2 = − RR1 − RR3 + RR5 + RR7 − RR9 d½OðCÞ dt

¼ RR1 −RR2 þ RR6 þ½OðCÞSgC Ri;C ½1−0:5ψðSgC;0 =SgC Þ2 

d½COðCÞ ¼ RR4 −RR5 þ dt ½COðCÞSgC Ri;C ½1−0:5ψðSgC;0 =SgC Þ2  ∂ðρεÞ ∂t

The gas phase mass conservation coupled with Darcy's law [28]

þ ∇  ρu ¼ u ¼ − κη ∇p

The convection-diffusion -reaction equation for gas species through the carbon bed [28]

∂ðci εÞ ∂t

ujz¼0 ¼

isurf 2F

mC SgC V bed ðεþð1−εÞε ðM CO RCO pÞ

The charge transfer reaction rate constants [36]

O2− ðYSZÞ þ ðNiÞ ⇄ OðNiÞ þ ðYSZÞ þ 2e− kct;b

kct; f ¼

O

kct;b ¼

ðM CO2 −M CO Þ=ρ

mC þ ∇  ð−Deff ∇ci þ ci uÞ ¼ SgC V bed ðεþð1−εÞε Ri pÞ ε Deff ¼ τbed Di j

  exp − 2αan Fηan RT   2ð1−α an Þ Fηan exp − RT

c0 ðYSZÞ ðNiÞ

i0;an FSTPB c0OðNiÞ c0ðYSZÞ

The anode electrode ηan = Velec ,an − Vion, an − Vref ,an overpotential The anode current source Qelec, an = − Qion,an = term 2F(kct,fcO2−(YSZ)c(Ni) − kct,rcO(Ni)c(YSZ))STPB ,an The cathode current source Q # " elec;ca ¼ −Q ion;ca ¼ i0;ca STPB;ca   cTPB    term [37] cTPB 2α ca Fηca 2ð1−α ca Þ Fηca H2 H2 O − cbulk exp exp − RT RT cbulk H2

þ M CO2 RCO2 Þ

i0;an FSTPB c0 2−

The cathode electrode overpotential The triple-phase boundary area per unit volume [38] The effective Ni surface area per unit volume [2,37]

H2 O

ηca = Velec, ca − Vion, ca − Vref ,ca STPB ¼ π sin2 θr el 2 nt nel Z el−io P el P io

SNi ¼ πr el 2 nt nel ð4− sin2 θnio Z el Z io =Z− sin2 θnel Z el Z el =ZÞ

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Y. Wu et al. / Solid State Ionics 295 (2016) 78–89

Table 5 Governing equations for charge balance and mass balance.

Table 7 Pore structure parameters in porous electrode [19,24].

Charge balance

Cell layer

Electrode or ionic charge balance equations The ionic charge equation at the anode The electronic charge equation at the anode

∂½C dl STPB ðV i −V j Þ ∂t

The ionic charge equation at the cathode The electronic charge equation at the cathode

∂½C dl;an STPB;ca ðV ion;ca −V elec;ca Þ ∂t

þ ∇  ð−σ

eff

∇V i Þ ¼ Q

∂½C dl;an STPB;an ðV ion;an −V elec;an Þ ∂t

þ ∇  ð−σ eff ion;an ∇V ion;an Þ ¼ Q ion;an

∂½C dl;an STPB;an ðV elec;an −V ion;an Þ ∂t

þ ∇  ð−σ eff elec;an ∇V elec;an Þ

¼ Q elec;an ¼ −Q ion;an

þ ∇  ð−σ eff ion;ca ∇V ion;ca Þ ¼ Q ion;ca

∂½C dl;ca STPB;ca ðV elec;ca −V ion;ca Þ ∂t

þ∇ ¼ Q elec;ca ¼ −Q ion;ca eff ∇(−σion, el ∇Vion,el) = 0

ð−σ eff elec;ca ∇V elec;ca Þ

The ionic charge equation in the electrolyte  0:133  7:79 The cathode exchange i0;ca ¼ βca 10 3 F RT exp − ERTca pca O2 current density [2,39]   2   ca The oxygen pressure at the pH O 2  exp − ERTca pca O2 ¼ pca cathode [2,39] H2 Mass balance The mass balance equation ε ∂ck;g þ ∇ð−Deff ∇c Þ ¼ R k;g k;g k ∂t for gas species at the porous electrode

−1 The effective diffusivity of 1 ¼ þ eff1 Deff k gas species k [19,24] Deff Dk;Kn k;mole # " The effective molecular 1−xk ¼ Deff n diffusion coefficient k;mole eff ∑ j≠k ðx j =Dk; j Þ [19,24] 0:00101εT 1:75 ð1=Mk þ1=M j Þ1=2 The effective binary Deff ¼ τε Dk; j ¼ k; j 1=3 1=3 2 τpðV k þV j Þ molecular diffusion coefficient [19,24,40] qffiffiffiffiffiffiffi The effective Knudsen 8RT ¼ τε Dk;Kn ¼ 2εr Deff 3τ πMk k;Kn diffusion coefficient [19,24,40] The mass balance equation ∂ck;s þ ∇ð−D ∇c Þ ¼ R s k;s k;s ∂t for surface species at the anode K g þK s þ2 0 Nþ2 The source terms of both v eff eff ðν″ki −v0ki Þki ∏ ckki gas and surface species in Rk ¼ S  s_ k ¼ S  ∑ i¼1 k¼1 anode Q The source terms at the RH2 O ¼ −RH2 ¼ elec;ca 2F cathode

2.4. Model parameters The accuracy and reasonability of calculation results is in direct related to the value of model parameters. Table 7 lists the pore structure parameters in porous electrode and the values or expressions of materials properties and other model variables are listed in Table 8, which have been described at length in previous papers [19,24]. Some model parameters are not available from the published literature or by experimentally determined in our group, which then serve as tuning parameters in model calibration and validation. 2.5. Model solution method The commercially available finite element software COMSOL MULTIPHYSICS® 3.5 is employed for solving the model. The button Table 6 Boundary conditions. Boundary

Ionic charge

Electronic charge

Mass balance

Momentum balance

∂Ωac/cb ∂Ωcb/an_sp

– Insulation

– 0

∂Ωan_sp/sn_act ∂Ωan_act/elec ∂Ωelec/ca ∂Ωca/cc

Continuity Continuity Insulation Insulation

Continuity Insulation Insulation Vca

cgas_in Continuity (gas species) Continuity Insulation Insulation cgas_ca

Pressure P0 Inward flux (gas species)/Insulation (surface species) Continuity Insulation Insulation Pressure P0

LTPB_ca Porosity Mean pore STPB_an radius (μm) (m2·m−3) (m·m−2)

Anode support layer 0.335 Anode active layer 0.335 Cathode layer 0.335

2.22 × 105 3.33 × 105 2.66 × 105

0.193 0.129 0.161

SNi (m2·m−3)

2.76 × 107 3.97 × 106 6.20 × 107 5.96 × 106 3.97 × 107 –

cell performance is predicted by giving the cell operation voltage Vca and gas species concentrations both at the inlet of the carbon bed and the cathode. For the 1D geometry, the ionic current density in the electrolyte can be treat as the average current density at a given cell voltage. In this way, a complete polarization curve can be generated under linear operation voltage sweep. 3. Experiment 3.1. SO-DCEC button cell structure and fabrication An anode-supported SO-DCEC button cell fabricated by the Shanghai Institute of Ceramics, Chinese Academy of Sciences (SICCAS) was used in this research. It consisted of a Ni/YSZ anode support layer (680 μm), a Ni/ScSZ anode active layer(15 μm), a ScSZ electrolyte layer(20 μm) and a lanthanum strontium manganite (LSM)/ScSZ cathode layer(15 μm), as shown in Fig. 1. The diameter of the cathode was 13 mm and diameters of all other parts were 26 mm. Reticular silver layers were deposited by screen-printing on the anode and cathode surfaces for current collection. 3.2. Testing setup and procedure The button cell system experimental setup and an experimental measurement system were built for evaluating the cell performance, which is described in more detail in our previous work [19,24]. The reactant gas fed into the cathode is argon gas served as the carrier gas and steam, which is generated by a water bath. The fraction of steam could be adjusted by varying the carrier gas amount and the temperature of the water bath. Commercial carbon black (Black Pearls 2000, GP 3848, Cabot Corporation, Boston, MA) was used as the fuel replenished in the anode. And the potassium catalyst was added into the carbon black by impregnation. For more detailed description about the preparation of the catalytic carbon see our previous work [12]. Then the treated fuel was spread evenly on the anode surface of the tested cell. Table 8 Properties and parameters for model calculation. Property and parameter

Value or expression

Unit

Ionic conductivity (σion) ScSZ YSZ

6.92 × 104·exp.(−9681/T) 3.34 ×

S·m−1 S·m−1

104·exp.(−10,300/T) Electronic conductivity (σelec) LSM Ni Equivalent ionic conductivity of electrolyte (σelectrolyte) Concentration of oxygen interstitial at the YSZ (cO2‐(YSZ)) Concentration of oxygen vacancies at the YSZ (c(YSZ)) Maximum surface sites density (Γ) [2,19,24] Electric double-layer capacitance(Cdl,an, Cdl,ca) a

4.2 × 107/T exp(−1150/T) 3.27 × 106–1065.3T −3.622 × 10−5T2 +

S·m−1 S·m−1 S·m−1

0.083T − 45.90a 4.45 × 104

mol·m−2

4.65 × 103

mol·m−2

2.6 × 10−5

mol·m−2

27

F·m−2

Experimentally measured by electrochemical impedance spectroscopy (EIS).

Y. Wu et al. / Solid State Ionics 295 (2016) 78–89

Before testing, pure H2 was passed through the chamber for 1 h to fully reduce the anode at 800 °C. The operating temperature was kept constant at 800 °C during the test. Then, two kinds of carrier gas with given flow rates were introduced into the anode and the gas with 20% steam plus 80% Ar flowed through the cathode. 4. Results and discussion 4.1. Model calibration and validation

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Table 9 Model tuning parameters. Parameter

Value

Anode tortuosity, τan Cathode tortuosity, τca Cathode electrochemical kinetics parameter, β(Ω−1·m−2) Anode exchange current density i0_an(A·m−2) Anode charge transfer coefficient, αan Cathode charge transfer coefficient, αca

6.2 3 0.6 1.47 0.35 0.5

The mechanism model of the solid oxide direct carbon fuel cells without a catalyst and the proposed potassium catalytic carbon gasification mechanism (Table 1. R9-R11) have been validated separately in our previous work [12,24]. The experimental and simulation results indicate that the model results fit generally the experimental data well. And as we state in Section 2.2.2, the kinetic data for R9–R11 are obtained by fixing the experimental measurements to the mechanism. The results confirm that the catalytic effect of the potassium salt is capable of accelerating the carbon gasification. For more detailed description of calibration and validation see Ref. [12]. Fig. 2 shows the experimental and simulated polarization curves with Ar and CO2 as carrier gases at constant flow rates of 40 sccm (Standard Cubic Centimeter per Minute). The simulation results agree well with the experimental data, which illustrates the validity of this model for reflecting the true phenomena in SO-DCEC. The determined values of the tuning parameters are listed in Table 9. Among these parameters, the electrode tortuosity is the factor that influences the diffusion and mass transfer in the porous electrode. For porous sintered metal ceramics, the common range of tortuosity is 2–10, and in many cases, it is 2–6 [41]. Therefore, the tuning results of electrode tortuosity is reasonable. When the gas fed into the carbon bed is pure CO2, the current density is larger than that in an Ar environment under the same voltage condition. This is because the CO2 rich atmosphere can promote the CO production reaction via the Boudouard reaction in the carbon bed, then the elevated CO concentration raises the CO consumption electrochemical reaction rate, which improves the cell performance. Thus, it is better to operate with CO2 as the anode carrier gas according to the experimental data. An interesting phenomenon should be noticed is that there exist two general current density transition period in both CO2 and Ar atmosphere. The first transition period is that the cell current density remains nearly the same while the cell voltage falls from 0.5 V to 0.2 V roughly, which reflects the coupling of the carbon bed gasification reactions and the cell electrochemical reactions. The consumption, diffusion and production of CO2 in the whole system play important roles at this

condition. The O2 generation by electrochemical process occupying a leading position instead of the CO consumption electrochemical reaction accounts for the second transition period, where the current density returns to growing at a new constant rate when the cell voltage drops. To better demonstrate these two transitions, Figs. 3 and 4 show the modeling results at the condition that 40sccm CO2 flows into the anode chamber at 800 °C, which is totally the same polarization process as in Fig. 2. Fig. 3 shows that CO concentrations in both the carbon bed and the cell anode decease rapidly when the current density increases, indicating that the CO production rate by carbon gasification cannot satisfy its consumption by electrochemical reaction inside the cell anode. During the first transition period (point B and C in the figure), the gradient of CO concentration in the cell anode is much larger than that in the carbon bed, and it makes clear that the first transition period results from the relatively slow diffusion rate of CO from the carbon bed into the cell anode. This reveals that the gas transport between the carbon bed and the cell anode greatly impacts the cell performance. It can be seen that the CO concentration is nearly zero all through the cell anode from the point D condition to the point E condition. Therefore, the oxygen pumping reaction takes lead at the large current density. Fig. 4 demonstrates the whole process more intuitively. The cell performance is test by linear sweep voltammetry, thus the abscissa value in Fig. 4 and the ordinate value in Fig. 2 are one-to-one correlated. The two transition periods divide the whole process into three zones. In zone I, the current density is incremented from zero, and the consumption rate of CO in the cell anode keeps growing at the same growth trend with the current density while the O2 production rate remains at zero. This zone thus is named as “CO consumption-dominant zone”. When the current density grows into zone II, the CO reaction rate gradually increases to the maximum value at a lower growth rate, and the O2 pumping rate is still near zero, which causes that the current density remains nearly the same while the cell voltage falls, we call this zone as “current density sluggish zone”. In zone III, that is at higher current,

Fig. 2. Simulated and experimental I\ \V curves at different anode gaseous conditions at 800 °C.

Fig. 3. The distribution of CO throughout the SO-DCEC system with CO2 as the anode carrier gas.

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Y. Wu et al. / Solid State Ionics 295 (2016) 78–89 Table 10 The expressions for energy forms in SOFEC. Energy Q(W·m−2)

Expression

Electricity, Qe Reaction enthalpy, ΔH

iV   0 0 ∑ ni h f ;i ðTÞ−∑ n j h f ; j ðTÞ

Gibson free energy change of formation, ΔG Irreversible heat, Qir Hydrogen, QH2

ΔH − TΔs i(V − VOCV)

Carbon monoxide, QCO Fuel, QC

nCO,netLHVCO nCLHVC

pro

reac

iLHV H2 2F

then the O2 reaction rate keeps growing with the growth of c_Os. And the Z shaped curve doesn't occur. 4.2. The efficiency of the SO-DCEC system Fig. 4. The reaction rates of CO and O2 at the cell anode with CO2 as the anode carrier gas.

the CO reaction rate stays almost at an unchanged value, which is due to the matching of its production in the carbon bed, its diffusion into the cell anode and its consumption in the anode. Meanwhile, the O2 pumping rate starts growing at a constant pace, similar to the growth trend of the current density. This zone is called as “O2 pumpingdominant zone”. Furthermore, according to Fig. 2, we can see that there exists a “Z” shaped curve at the voltage between 0.1 V and 0.2 V for the Ar inlet condition. This phenomenon has been seen during our experiment. Fig. 5 shows that the relatively intricate transition is caused by the O2 pumping reaction. The chemical source term of O2 in the cell anode is written as   RO2 ¼ −SNi  kf 1  ðc NiÞ2  cO2 −kr 1  ðc OsÞ2 ; where SNi denotes the surface area of the Ni-electrode, c_Ni and c_Os denote the concentrations of the uncovered Ni surface and oxygen adsorbed on the Ni surface respectively. c_Os grows because of the decline of CO concentration near the anodic active sites, which increases the reaction rate of O2, then the growing of cO2 lowers the reaction rate of O2, but the c_Ni keeps reducing to a very low value, then the reaction rate returns to growing within the control of c_Os. This evolution process finally gives rise to the small “Z” shaped curve in the I\\V curves. However, when CO2 is used as the carrier gas (Fig. 4), c_Ni stays at a relatively low value always that RO2 can be simplified into RO2 ¼ SNi  kr 1  ðc OsÞ2

The efficiency is a significant concept for all energy conversion devices. In this SO-DCEC system, carbon gasification reactions in the carbon bed are endothermic, while a large amount of energy is transferred into heat because of the voltage loss inside the cell. Theoretically, the heat requirement in the carbon bed can be replenished by the heat released in the cell. Thus, it is necessary to analyze the efficiency of this system. All the expressions for the energy forms and energy efficiencies are listed in Tables 10 and 11 separately. And Fig. 6 depicts efficiencies vs current density curves. When working voltage is greater than zero (in the SOFC mode), part of electrical energy can be generated as well as the generation of gaseous fuels. Based on the thermodynamics and electrochemistry, the ideal power generation efficiency equals to the ratio of the Gibson free energy change ΔG and the enthalpy change ΔH. It should be mentioned here that the integral molar quantities of species used to calculate the enthalpy change are obtained by the source term integral in our model. From Fig. 6, the theoretical power generation efficiency is N90% all through the SOFC mode. While the calculated actual power generation efficiency vs current density curve takes the shape of a bell, which quite resembles the power density curve of the cell. The peak efficiency is 40% when V = 0.4, and when the current keeps growing, the effect of voltage losses reduces the efficiency. The total efficiency of the SO-DCEC system is also considered, which fully takes the heat input, chemical energy of fuels and electrical energy into account. Firstly, the temperature of the system is assumed to be maintained at 800 °C constantly, all the species energy are calculated at 800 °C. Secondly, to be simplified, the irreversible heat caused by voltage losses is considered useless for replenishing the heat demand of the whole system. While if the total heat effect of the system is exothermal, the surplus heat is released to the environment and not considered. Furthermore, CO from the net production in the carbon bed and the cell anode is also considered as useful fuels. The detailed expression is listed in Table 11. From the result curve in Fig. 6, it is found that the shapes of the total efficiency curve and the cell polarization curve are really alike. This similarity is because various energy losses in both the carbon bed and the cell are not taken into account other than voltage losses in the cell. And when V = 0, the total system efficiency is 70%.

Table 11 The expressions for energy efficiencies. Theoretical power generation efficiency



T ΔG ¼ 1− ηelec;th ¼ ΔH

∑ ni s0i ðTÞ−∑ n j s0j ðTÞ pro

reac

0

0

∑ ni h f ;i ðTÞ−∑ n j h f ; j ðTÞ pro

Actual power generation efficiency

Qe ηelec;real ¼ ΔH ¼

SO-DCEC total efficiency ηtotal ¼ Fig. 5. The reaction rates of CO, CO2 and O2 at the anode with Ar as the anode carrier gas.

8 < :

reac

0

iV

0

∑ ni h f ;i ðTÞ−∑ n j h f ; j ðTÞ pro

Q e þQ H2 þQ CO Q C þΔHþiV OCV Q H2 þQ CO Q C þQ e þΔHþiV OCV

reac

ðVN0Þ ðV ≤0Þ

Y. Wu et al. / Solid State Ionics 295 (2016) 78–89

Fig. 6. The power generation efficiency and the total efficiency of the SO-DCEC system.

4.3. Fuel conversion rates analysis in the SO-DCEC system It has been demonstrated in Section 4.1 that the cell performance is greatly influenced by the carbon gasification reactions and the diffusion processes in the carbon bed and the cell anode. Therefore, it is helpful for a better understanding of the whole process by diving deeply into the species conversion both in the carbon bed and the cell anode. Fig. 7 depicts the production or consumption rates of species in the anode chamber vs current density. It can be seen that the cell performance has comparatively less effect on the reaction rates of species in the carbon bed. The production rate of CO in the carbon bed when V = 0.6 (Open Circuit Voltage) is only 4% larger than that at V = − 0.5. The consumption rate of CO2 and the removal rate of the bulk carbon at V = −0.5 vary −23% and 20% respectively when compared to those in the working condition that V = 0.6. The CO net generation of the whole system is defined as the difference between its production in the carbon bed and its consumption in the cell anode. Due to the stable production rate in the carbon bed, the changing trend of the CO net production is controlled by the consumption rate of CO in the cell anode, firstly decreasing rapidly when the current grows and then maintaining at a low value. Fig. 8 shows the production rates of CO in the anode chamber and H2 in the cathode chamber. It can be seen that these two products present opposite trends along with the current enhancement when V N 0. Obviously different molar production ratios of H2 and CO can be obtained by turning voltage. When the system works at the maximum electrical power output condition, the molar production ratio of H2 and CO is

Fig. 7. The production or consumption rates of species in the anode chamber of the SODCEC system.

85

Fig. 8. The net production rates of CO and H2 of the SO-DCEC system.

2:1. While in the half of maximum electricity output, the ratio is 1:2. When V = 0, the ratio comes to 10:1. And by applying voltage, the production rate of H2 far exceeds that of CO. 4.4. The effects of working voltages on SO-DCEC The operating conditions has a significant effect on both the cell performance and productions of gaseous fuels. Figs. 9–11 show the modeling results with the working voltages of 0.3 V/0 V/−0.3 V at the same conditions: 40 sccm CO2 as the anode carrier gas. The net molar production rates of CO and H2 in the system under these voltage conditions are shown in Fig. 9. When working voltages V = 0.3, the hydrogen production rate (or current density) and the CO net production rate all keep dropping with the time, which indicates that the system cannot provide stable gaseous fuels output under this condition, it can be explained by chemical reaction kinetics that the increase of CO concentration in the carbon bed limits its production reversely. When V = 0 and V = − 0.3, production rates of hydrogen stay nearly invariant with time, which is because the system works in “O2 pumping-dominant zone” under these two conditions, while the net productions of CO keep lowering as that at working voltage V = 0.3. Look inside the cell anode, Fig. 10 shows the electronic current density and gaseous CO concentration distributions in the anode. Both decrease notably along the anode thickness direction. When V = 0.3, the CO concentration and current density distributions present the same downward trend, which indicates that the system works in “CO consumption-dominant zone” and there is a mismatch between the low CO diffusion rate and relatively high CO consumption rate in the

Fig. 9. The H2 and CO net production rates of the Carbon-SOFEC system with the working voltages of 0.3 V/0 V/−0.3 V.

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Y. Wu et al. / Solid State Ionics 295 (2016) 78–89

Fig. 10. The electronic current density and CO concentration distributions within the anode with the working voltages of 0.3 V/0 V/−0.3 V.

Fig. 12. The total CO production rate in the carbon bed at the working voltage of 0 V for different carbon bed heights.

cell anode. When V = 0 and V = − 0.3, the CO concentration drops sharply to almost zero firstly (0–100 μm distance from the anode surface), then maintains at an extremely low value in the rest region. The decrease tendency of the current density along the anode can be divided into two parts, the first part falls in step with the CO concentration at the first 100 μm, and the second one drops at a constant rate till the end of the anode, which implies the O2 pumping mechanism dominates the cell performance under this region. To dive deeply, the concentration distributions of surface species in the anode are shown in Fig. 11. O(Ni) is the major surface species, which is two or three orders of magnitude more than (Ni), and CO(Ni) is the minimum one all under these voltages. Further, CO(Ni) and (Ni) decrease with stepping down of working voltages, while O(Ni) varies in the opposite trend when voltages falling down. From the heterogeneous reaction mechanism on the Ni-based catalysts written in Table 1, it can be easily to reason out that the variant trends of these three surface species turn the CarbonSOFEC system from working in “CO consumption-dominant zone” to in “O2 pumping-dominant zone”. From the analysis above, it can be inferred that increasing CO concentration in the cell anode is the key factor of improving the cell performance.

We have demonstrated that CO concentration in both the carbon bed and the cell anode is critical for the cell performance. Therefore, it

can be inferred that the carbon bed height is a key parameter for the system performance when the radius of the carbon bed is fixed at the same value as the cell anode. Figs. 12–15 depict the modeling results with different carbon bed heights under the same working condition: 40 sccm CO2 as the anode carrier gas and working voltage of 0 V. As Fig. 12 shows, the total amount of the CO production(mol·m−2 s−1) in the carbon bed increases monotonically with enlarging the bed height, and the CO production rate stays almost unchanged with the time other than the early transition time under the three bigger bed height conditions. However, the total production amount is not proportional to the carbon bed height. Fig. 13 shows the average volumetric CO production rates (mol·m− 3 s−1) in the same conditions as in Fig. 12, it can be seen that the overall trend of the volumetric CO production rate with the carbon bed height is contrary to the total production rate. Besides, it decreases rapidly at low carbon bed heights while changes relatively slow at high bed heights. The higher CO concentration and lower CO2 concentration in the carbon bed, as well as slower diffusion rates of gases caused by the enlarged carbon bed heights, limit the reaction rates of the carbon gasification reactions, which leads to the decreasing CO production rate volumetrically. We have inferred that the carbon bed height has an effect on the cell performance, the modeling results shown in Fig. 14 validate our deduction. The hydrogen production rate (or current density) increases with the growing of carbon bed height. Taking a closer look, when H = 0.002 m, hydrogen production rate keeps dropping with the time. When H = 0.01 m, 0.02 m and 0.06 m, it falls down with the time at

Fig. 11. The concentration distributions of surface species within the anode with the working voltages of 0.3 V/0 V/−0.3 V.

Fig. 13. The volumetric CO production rate in the carbon bed at the working voltage of 0 V for different carbon bed heights.

4.5. The effects of the carbon bed heights on SO-DCEC

Y. Wu et al. / Solid State Ionics 295 (2016) 78–89

Fig. 14. The H2 production rate in the cathode at the working voltage of 0 V for different carbon bed heights.

first, then comes to a platform that the value stays invariant, which implies that a match between the carbon gasification rates in the carbon bed, the CO diffusion from the carbon bed to the cell anode and the CO consumption rate inside the anode is reached. Keep magnifying the bed height to H = 0.1 m, the variant trend of hydrogen production rate with the time reverses. That means the production of CO in the carbon bed surpasses its consumption in the cell anode, and with a short transient time, it comes to a new balance. The concept of simultaneous production of H2 and CO in the Carbon-SOFEC system has been demonstrated briefly in Section 4.3. Fig. 15 shows the production ratio of these two gaseous fuels with different carbon bed heights. When the electrical energy output of the system is zero (V = 0), the hydrogen production far exceeds the net CO production of the system. By extending the carbon bed heights, the molar ratio of H2 and CO goes down, also, when the carbon bed is high enough, the falling range is smaller with growing the carbon bed heights. 5. Conclusion In this paper, a one-dimension mechanistic model for solid oxide carbon-assisted steam electrolysis cell (SO-DCEC) is well developed by considering heterogeneous elementary reactions in both the carbon bed and the cell, coupling with mass and charge transfer processes. The model is calibrated and validated by experimental data from a button cell test with different anode carrier gases at 800 °C. It is found that using CO2 as the anode carrier gas can get a better cell performance than using argon flowing through the anode chamber. By carefully analysis,

Fig. 15. The molar ratio of H2 and CO production rates at the working voltage of 0 V for different carbon bed heights.

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the production of CO in the carbon bed and its diffusion into the cell anode play an important role in the cell performance, the whole process with increasing current density (or by lowing working voltage) can be divided into three zones, which is “CO consumption-dominant zone”, “current density sluggish zone” and “O2 pumping-dominant zone” successively. The system efficiencies and production rates of gaseous fuels are studied. The system efficiency is dominant by voltage losses of the cell by carefully considering the reaction heat. The exploration work at the gas fuels production shows that the cell performance has relatively little effect on carbon gasification reactions. And the production rates of H2 and CO of the SO-DCEC system show reverse variant trends with increasing current density. In addition, the effect of working voltages and the carbon bed geometry are also studied. It indicates that the lack of CO in the cell anode is worsen under lower working voltages, which transfers the system to work in “O2 pumping-dominant zone”. The carbon bed height is also closely related to the system performance. Higher carbon bed height benefits the integral CO production rate in the carbon bed while limits its production in unit volume, and it improves the cell performance. Furthermore, the molar ratio of H2 and CO for this system keeps decreasing with extending the carbon bed height, but the decreasing trend is slower in higher carbon bed height. Nomenclature Abbreviation SO-DCEC solid oxide direct carbon-assisted electrolysis cell SOEC solid oxide electrolysis cell SOFC solid oxide fuel cell SOFEC solid oxide fuel-assisted electrolysis cell SO-DCFC solid oxide direct carbon-assisted fuel cell TPB triple phase boundary YSZ yttrium stabilized zirconium ScSZ scandium stabilized zirconium LSM lanthanum strontium manganate English letter pre-exponential factor in sticking coefficient expression ai A pre-exponential factor of the Arrhenius form (in terms of cm, mol and s) temperature exponent in sticking coefficient expression bi concentration (mol m−3) ci bulk carbon atom Cb surface carbon site concentration (mol m−2) csites free carbon site Cf O(C) adsorbed oxygen atom species on carbon site CO(C) adsorbed carbon monoxide species on carbon site [KO] the KxOy complex or the potassium cluster O[KO] the oxygen complex “dissolved” in the potassium cluster O[KO][C] the complex of O[KO] adsorbed on the active sites of the carbon surface specific double-layer capacitance (F m−2) Cdl cO2‐(YSZ) the volumetric concentrations of interstitial oxygen in the YSZ ionic conductor (mol·m−3) the volumetric concentrations of interstitial oxygen in the c(YSZ) YSZ ionic conductor (mol·m−3) bulk the concentration of gaseous species i in the bulk (mol·m−3) ci the concentration of gaseous species i at the TPB (mol·m−3) cTPB i activation energy in sticking coefficient expression (J mol−1) di D diffusion coefficient (m2·s−1) E active Energy(J·mol−1) F Faraday constant (96,384C.mol−1) i current density (A·m−2) current density at the interface of carbon bed and cell anode isurf exchange current density (A·m−2) i0 j(s) species of j adsorbed on the surface of Ni

88

k K LTPB mc Mj n nel nio nt N p P Q rel R RRi Ri_C Rk s_ k S0 Seff SgC SgC,0 SNi STPB u Vbed Vj, Vk W xc Z Zel Zio

Y. Wu et al. / Solid State Ionics 295 (2016) 78–89

reaction rate constant(in terms of m, mol and s) number of species length of TPB (m·m−3) weight of the carbon molecule weight (kg mol−1) temperature exponent, fraction number of electronic or ionic conductor particles number of electrons participating in the reaction number of ions participating in the reaction total number of particles per unit volume number of the reactions pressure(Pa) power density(W m−2) source term of charge balance equations (A m−3) mean radius of the electronic conductor particle (m) gas constant (8.314 J mol−1 K−1) reaction rates of reaction i in the carbon bed net rate of the removal of the carbon atoms from the bulk carbon (kg m−2 s−1) source term of mass balance equations (kg m−3 s−1) molar production rate (mol m−2 s−1) initial sticking coefficient Effective reaction area per unit volume (m2·m−3) specific surface area of the activated carbon (m2 kg−1) initial specific surface area of the activated carbon (m2 kg−1) active surface area of Ni per unit volume (m2 m−3) TPB active area per unit volume (m2 m−3) velocity of gaseous species (m s−1) volume of the carbon bed diffusion volume molecular weight of gas species (kg·mol−1) carbon conversion ratio mean coordination number of electron and ionic conductor particles coordination number of electron conductor particles coordination number of ionic conductor particles

Greek letters α charge transfer coefficient cathode electrochemical kinetics parameter βca γ parameter modeling the rate constant from sticking coefficient Γ surface sites density (Ω−1 m−2) ΔS entropy change (J·mol−1·K−1) ε porosity parameter modeling the species coverage εki carbon particle porosity εp η overpotential (V) or efficiency or dynamic viscosity κ permeability (m s−1) θ contact angle between the electronic and ionic conductors particles surface coverage of species k θk parameter modeling the species coverage μki ν' stoichiometric coefficient of the reactants ν" stoichiometric coefficient of the products σ electric conductivity (S·m−1) τ tortuosity χ species symbol ψ structural parameter of carbonaceous fuel ∂Ω computational boundary Subscript ac an b an_act

anode chamber anode backward reaction anode active layer

an_sp ca cb cc ct el elec f g i io ion ir k Kn mole r ref.

anode support layer cathode carbon bed cathode chamber charge transfer electrolyte electronic forward reaction gas phase species reaction index ionic conductor particles ionic irreversible heat species index Knudsen diffusion molecular diffusion reverse reaction reference

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