Mass transfer in hydrogen-fed anode-supported SOFCs

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Mass transfer in hydrogen-fed anode-supported SOFCs M. Garcı´a-Camprubı´, N. Fueyo* Fluid Mechanics Group (University of Zaragoza) and LITEC (CSIC), Marı´a de Luna 3, 50018 Zaragoza, Spain

article info

abstract

Article history:

A comprehensive numerical study of the mass-transfer mechanisms inside the anode of

Received 10 December 2009

a hydrogen-fed Solid-Oxide Fuel-Cell (SOFC) is presented. The study is based on a detailed

Received in revised form

mathematical model of the channel and anode, using for the latter the equations of the

12 April 2010

Dusty Gas Model (DGM) to describe mass transfer in the porous medium.

Accepted 14 April 2010

The physical meaning and relative importance of the fluxes present in the DGM have been

Available online 21 May 2010

researched; the outcome is a better understanding of the physical phenomena involved in the several species transfer to and from the reaction layer. The relevance of the convective

Keywords:

flux and the contribution of the pressure-driven DGM one in the SOFC anode has been

Hydrogen

numerically and analytically estimated for the most common SOFC operating conditions; it

SOFC

has been found that these fluxes may not be negligible in many cases.

Mass transfer

ª 2010 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved.

Numerical analysis Dusty Gas Model

1.

Introduction

The theoretical, numerical and experimental research of SolidOxide Fuel-Cell (SOFC) systems is gaining momentum in recent years, due to their promising expectations for clean electricity generation with high efficiencies [1,2]. Among the different SOFC configurations, anode-supported SOFCs often offer the advantage of their suitability to operate at lower temperatures. However, due to their thicker anode, important potential losses may appear, caused by the mass-transfer resistance. This resistance hinders the supply of the reactant gases to the reaction sites and may become large enough to prevent the electrochemical reaction from taking place. Thus, while anode-supported designs are often the preferred configuration for SOFCs, they can benefit from a substantial research effort geared towards improving their performance, e.g. by minimizing the concentration resistance on the anode side. According to the Nernst equation [3], the concentration overpotential depends on: the feed gas composition ðxain Þ, the temperature (T) and the gas composition at the reaction sites

ðxarw Þ. Among these parameters, the last two are difficult to measure experimentally. Thus, the mathematical modeling of heat and mass transfer plays an important role in improving the performance of anode-supported SOFCs. Global mass transfer through a SOFC porous anode is the result of a combination of up to three different mass-transfer mechanisms: (i) multicomponent bulk diffusion, (ii) Knudsen diffusion, (iii) viscous flow [4]. As reported by Krishna and Wesselingh [5], the most convenient approach to modeling combined bulk and Knudsen diffusion in porous media is the Dusty Gas Model [6], which may also take into account viscous flow. Suwanwarangkul et al. [7] confined their study to the SOFC anode, and made an analytical comparison of the three most commonly-used mass transfer models (extended Fick’s model, Dusty Gas Model, Stefan-Maxell model) assuming a constant pressure through the porous medium; as a result they also recommended the use of the Dusty Gas Model for the commonly-prevailing SOFC operating conditions. Recently, Pisani [4] stated the lack of a rigorous justification when Suwanwarangkul et al. [7] neglected the convective transport.

* Corresponding author. Tel.: þ34 976762153; fax: þ34 976761882. E-mail address: [email protected] (N. Fueyo). 0360-3199/$ e see front matter ª 2010 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2010.04.085

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Thus, Pisani [4] extended the study by Suwanwarangkul et al. [7] and provided an analytical analysis of the maximum error made in neglecting the convective term. He concluded that the convective term is negligible when the molecular weights of the species in the gas mixture are very similar to each other; in the opposite limit of very different weights, the convective term cannot be neglected. In the trail of these findings, the present authors implemented a model and algorithm for the numerical simulation of multicomponent mass transfer in solid-oxide fuel-cells, based on the Dusty Gas Model but without neglecting any of the three possible mass transport mechanisms in the porous medium. The model has been previously validated with measured concentration overpotential data found in the literature [8,9]; more details can be found in Garcı´a-Camprubı´ et al. [10]. In this paper, we present a comprehensive numerical study of the mass-transfer mechanisms inside the SOFC anode, and a supporting analytical analysis. For many experimental setups, and for fundamental studies, the temperature field inside the cell can be regarded as nearly constant, and thus isothermal conditions may be assumed; we have done so in this work. The importance and the physical meaning of the fluxes present in the Dusty Gas Model are investigated to gain a better understanding of the physical phenomena involved in the transfer of the species to and from the reaction layer. The convective flow in the SOFC anode will be numerically and analytically calculated to assess whether it can be neglected.

the species in the mixture. In order to avoid having to use a mass fraction normalization while ensuring that all mass fractions consistently add up to one, the multicomponent diffusion is modeled as [11]: ! J a ¼ rDam Vya þ ya rSb Dbm Vyb

where Dam is the diffusion coefficient of species a in the gas mixture.

2.2.

The electrode model

In the electrode domain, the momentum-conservation equation is formulated in the form of Darcy’s Law: B0 ! v ¼  Vp m

(5)

where m is fluid viscosity, B0 is the porous medium permeability-coefficient and ! v represents the superficial permeation velocity. Pressure, in turn, is calculated from the mixture composition using the equation: p¼

X

a pa

(6)

where pa is the partial pressure of species a. The equation for the conservation of chemical species a is based on the Dusty Gas Model, and can be written, after some manipulation (see [10]) as:
p 
N  dð3Xa Þ y a Xa þ V$ ! y a Xa ¼ 0  V$ðGa VXa Þ þ V$ ! dt

2.

Mathematical model

The analysis presented in this paper is conducted using a comprehensive model of the flow and mass transfer in SOFCs, developed by the authors and reported in detail in [10]. The model considers two separate domains (viz. the channel and the electrode), for which the relevant transport equations are solved separately and linked through boundary conditions.

2.1.

(7)

where 3 is the medium porosity, Xa is the molar density of p N y a; ! y a are the Dusty Gas Model parameters. species a, and Ga ; ! Thus, Ga represents the diffusion coefficient of species a for the global molecular diffusion through the porous matrix: Ga ¼ RT p

P b



1



Xb Deff ab

(8) þ D1eff Ka

p The velocity ! y a is an effective superficial velocity caused by the pressure gradient in the porous medium:

Channel model

The set of equations solved to model the channel are continuity, momentum and species conservation-equations. The continuity equation solved is; dr þ V$ðr! vÞ ¼ 0 dt

(4)

B0 Vp Ga p ! y a ¼ Ga eff ¼ eff ! v DKa m DKa

N And, finally, ! y a represents the velocity of species a induced by the flow of the other species:

(1)

where r is the fluid density and ! v is the fluid velocity-vector in the channel. The momentum-conservation equations are, in vector form: ! ! dðr! vÞ (2) þ V$ðr! v! v Þ  V$ s0 ¼ Vp dt ! !0 where s is the viscous stress tensor and p is the pressure. The equation for the conservation of the chemical species a is written as:

 d rya ! þ V$ rya ! v þ V$ J a ¼ 0 (3) dt ! where ya and J a are the mass fraction and the mass diffusive flow of species a, respectively. This equation is solved for all

(9)

RT N ! y a ¼ Ga p

" # X ! Nb b

Deff ab

(10)

Therefore, in the Dusty Gas Model the global molar flux of ! a given species a through the anode, N a , can be expressed (from equation [7]), as the effect of three distinct contributions:
p 
!N  ! y a Xa þ y a Xa (11) N a ¼ ðGa VXa Þ þ ! |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} N !G !p ! Na Na Na G ! !p !N The component fluxes N a ; N a ; N a , whose physical meaning has been indicated above, will be from now on referred to collectively as the DGM fluxes. From their definitions, they are respectively due to: Knudsen and binary diffusion; pressure gradient; and the molar fluxes of the other species.

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! The total flux N a can be further regarded as the sum of two additive contributions: the diffusive flow, including ordinary and Knudsen diffusion, and the convective flow [6]: !d !c ! Na ¼ Na þ Na

(12) !c ! The convective flux in the electrode is given by N a ¼ v Xa , v are computed from equations [7] and [5] where Xa and ! respectively. The diffusive flux can be calculated from equations [12] and [11] by substracting the convective flux from the !d !G !p !N !c total one: N a ¼ N a þ N a þ N a  N a . These fluxes will be further investigated, for a practical configuration, in the remainder of this paper.

3.

Fig. 1 e Domain considered: fuel channel and anode (not to scale).

though boundary conditions. Further details of the algorithm are given in [10].

Numerical details 4.

To facilitate the analysis and interpretation of the results, the present study is applied to a planar cell with a constant crosssection. The cell geometry and the reference operating conditions, listed in Table 1, are in accordance with typical values for a SOFC. Fig. 1 shows the domain considered; this is a two-dimensional one, consisting of two separate subdomains: a planarSOFC fuel channel and its corresponding anode. To solve the set of equations given in Section 2 for each subdomain, a numerical algorithm was implemented using OpenFOAM, an open-source CFD-platform based on the Finite Volume Method (FVM) [12,13]. The FVM discretization is achieved using two adjacent structured hexahedral meshes. The channel grid is made up of 5000 cells (200  25  1), while the anode grid is comprised of 1000 cells (100  10  1). The most relevant boundary conditions are: (i) The mass fractions and the velocity are known at the channel inlet; (ii) At the channel outlet, only the pressure is specified; (iii) At the electrode reaction-wall (rw), there is a flux of reactants and products due to the superficial electrochemical reaction. Their ! molar fluxes N arw are given by Faraday’s law and presumed constant along the wall; (iv) At the electrode-channel interface-boundary, the channel and electrode fields are coupled

Table 1 e Parameter and operating conditions for the reference cell. Anode Thickness Length Porosity Mean pore diameter Tortuosity Permeability

400 mm 1 cm 40% 1 mm 3 4.16 1015 m2

Channel Thickness Length

1.2 mm 20 mm

Operating conditions Temperature Pressure Fuel-stream inlet-velocity Fuel-stream hydrogen-content Fuel-stream water-vapor content

900  C 1 atm 0.8 m s1 50 vol% 50 vol%

Study of the mass-transfer mechanisms

In this section, we analyze the mass-transfer mechanisms in the anode by examining the relative contribution of each of the fluxes identified in Subsection 2.2.

4.1.

DGM fluxes

The DGM fluxes, Equation (11), are directly computed in the model presented in Section 2.2. Fig. 2 shows the evolution of the DGM fluxes of hydrogen and water at the transversal centerline in the anode (line AA0 in Fig. 1), from the interface with the channel (left abscissa in the plot) to the interface with the electrolyte (right abscissa). The calculations are made for a typical cell (Table 1) operating at a constant intensity I ¼ 1 A cm2, and with an inlet composition xH2 ¼ 0:5 and xH2 O ¼ 0:5. The fluxes have been made non-dimensional with the hydrogen molar flux at the electrodeeelectrolyte interface, and positive values indicate flow towards the electrolyte (i.e. towards the reaction plane), while negative ones are towards the channel (i.e. away from the reaction plane). From the results, it may be concluded that the diffusive DGM !G flux, N a , is the dominant one for both the hydrogen and the water flows. With the sign convention indicated above, it is of course positive for hydrogen (towards the reaction sites) and negative for the water (away from the reaction sites). The !p pressure-gradient DGM flux N a is slightly negative in both cases, and thus it hinders the supply of the reactants to the reaction sites, while it favors the evacuation of the products !N to the channel. The flux N a , which is due to the molar fluxes of the other species, is in either case against the total flow of the respective species, and hence it hampers the supply of the reactant to the reaction zone and the evacuation of the products. Fig. 3 shows (symbols) how the hydrogen DGM fluxes change with the gas composition and current density, along the reaction plane (line BB0 in Fig. 1). For low current densities !G and high molar fractions, the diffusive DGM flux ðN H2 Þ may be up to 250% of the total molar flux to balance the flux induced by the counterflowing products. However, the relevance of !N N H2 decreases as species depletion ðxH2 /0Þ is approached (high current densities or low species concentration), and then the diffusive DGM flux is the main mechanism of mass transport.

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α = H2

α = H 2O

1.5

Nα/NH p

1.5 2

Nα /NH 2 NαN/NH 2

1

NαΓ/ NH d

1

2

Nα /NH 2 Nαc/NH 2

0.5

0

0.5

0

-0.5

-0.5

-1

-1

-1.5 Channel

Electrolyte

-1.5 Electrolyte

Channel

Anode depth →

Anode depth →

Fig. 2 e Normalized DGM fluxes through the anode, along the AA0 line in Fig. 1, for the cell described in Table 1 operating at I [ 1 A cmL2.

4.2.

Convective vs diffusive flux

Previous published studies of mass transfer in SOFC have often either assumed [7,14,15] or concluded [9] that the convective flux in the SOFC anode is negligible compared with the diffusive one. Our numerical analysis indicates that the convective flux may play a non-negligible role in the overall mass transfer in the cell. Thus Fig. 2 shows the convective and diffusive fluxes for the case given in Table 1; while it is clear that diffusion dominates, it is debatable whether convection is small enough to be neglected. Further, Fig. 4 shows contours of the contribution of the convective flux to the total flux in the anode, for both hydrogen and water vapor, and for three different inletfuel compositions. The numerical simulations thus indicate that the convective flux may account for up to approximately 18% of the global molar flux. The importance of convection in the porous anode for any fuel composition, with or without the presence of an inert species, can be investigated analytically with the simplified model presented below. Consistently with the equations presented above, this analysis is based on the Dusty Gas Model [6], the standard form of which is: VXa ¼

! ! RT X N b Xa  N a Xb p

b

Deff ab

! Na Xa B0  eff  eff Vp Dka Dka m

Vp ¼  1þ

DKa ¼

(13)

(14)

dp pffiffiffiffiffiffiffiffi C1 Wa

(15)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here, C1 is defined as C1 ¼ 3 p=8RT: If Equation (14) is used for replacing the pressure gradient in Equation (5), then the !c v may be expressed as a function of convective flux N a ¼ Xa ! the species global flux: s C X! pffiffiffiffiffiffiffiffi Bo !c 3dp 1 N a ¼ pxa N a Wa
pffiffiffiffiffiffiffiffi P m 1 þ B3do s pCm1 a xa Wa a p

(16)

The global fluxes of the three possible species considered in this work for a hydrogen-fed anode (H2, H2O and an inert, e.g. ! ! !global ! ! ! ¼ N H2 ¼ N H2 O ; Ar) are N H2 ¼ N H2 O and N Ar ¼ 0. Let N then the relation between the convective and global fluxes is defined by the following expression (with vectors replaced with scalars since we assume the flow to be one-directional in this simplified study): Nca Nglobal

P By definition, Xa ¼ pxa/(RT ) and a xa ¼ 1. Therefore, the sum, extended to all the species a in the mixture, of the molar density gradient given by Equation (13) results in the following expression for the pressure gradient under isothermal conditions: s RTC1 X! pffiffiffiffiffiffiffiffi 3dp N a Wa P
pffiffiffiffiffiffiffiffi Bo s pC1 W x a a a a 3dp m

where the effective Knudsen diffusion coefficient Deff Ka has 3 ¼ D and the Knudsen already been replaced with Deff Ka Ka s diffusion coefficient DKa is given by its constitutive law from the kinetic theory of gases [16]:

¼

Bo s pxa C1 3dp

mþ

Bo s pC1 3dp

P
pffiffiffiffiffiffiffiffi a xa Wa

pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi WH2  WH2 O

(17)

Considering the classical Poiseuille-flow equation (see, e.g., reference [17]), the permeability is given by: Bo ¼

3d2p

(18)

32s

Inserting Equation (18) into Equation (17), we have: Nca global N

¼

pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi dp pxa C1 WH2  WH2 O P
pffiffiffiffiffiffiffiffiffiffiffiffi 32m þ dp pC1 a Wa xa

(19)

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xH = 0.25 2

-2

-2

I = 0.25 A cm

-2

I = 0.75 A cm

-2

I = 1.25 A cm

I = 2.95 A cm

2.5

NH p/NH 2

2

2

N

NH /NH

1.5

2

Γ

2

1

NH / NH

0.5

xH

2

2

2

0 -0.5 -1 -1.5

xH = 0.50 2

I = 0.25 A cm-2

I = 0.75 A cm-2

I = 1.25 A cm-2

I = 2.5 A cm-2

I = 4 A cm-2

2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5

xH = 0.75 2

-2

-2

I = 0.25 A cm

-2

-2

I = 0.75 A cm

I = 1.25 A cm

I = 2.5 A cm

I = 0.75 A cm-2

I = 1.25 A cm-2

I = 2.5 A cm-2

-2

I = 4 A cm

2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5

xH = 0.97 2

I = 0.25 A cm-2

I = 4 A cm-2

2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5

0

10 Anode-x → / cm

10 Anode-x → / cm

10 Anode-x → / cm

10 Anode-x → / cm

1 Anode-x → / cm

Fig. 3 e Evolution of the DGM fluxes at the reaction wall (along the BB0 line in Fig. 1), for the cell described in Table 1, operating at four H2:H2O ratios and five current densities (I £ 4 A cmL2).

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Fig. 4 e Contours of convective-to-total-flux ratios (Nca/Na*100) for three different H2:H2O ratios at I [ 1 A cmL2.

where m, the viscosity of the gas mixture, can be estimated from the semi-empirical formula of Wilke: m¼

X

P

ca

fab ¼

xa ma cb xb fab

h i2 1=2
ðWb =Wa Þ1=4 1 þ ma =mb ð8 þ 8Wa =Wb Þ1=2

(20)

(21)

and the viscosity of the species (ma, mb) can be calculated using the Sutherland law, a widely-used expression from the kinetic theory of gases:

ma ¼ moa

T Toa

3=2

Toa þ Sa T þ Sa

(22)

The parameters moa ; Toa ; Sa in this expression are tabulated for the most common gases [18]. The above equations allow us to conclude that the relative importance of the convective flux in the global one depends essentially on four operational parameters: (i) the species molar fractions, xa; (ii) the operating temperature, T; (iii) the operating pressure, p; and (iv) the anode mean pore diameter, dp. Figs. 5 to 9 show, based on equation (19), the contribution (percentage) of the convective flux to the overall flux, Nca/Na. Thus Figs. 5 and 6 show the influence of the fuel gas composition on the convective fluxes of reactant and product, respectively. In both figures, each point in the graph is a possible inlet composition, given by the inlet molar fraction of water vapor, xH2 O , as the abscissa; the inlet molar fraction of hydrogen, xH2 , as the ordinate; and the inert Argon as the

Fig. 5 e Fraction of convective flux in the total molar flux of hydrogen ðNcH2 =NH2  100Þ (thick lines), and fraction of the p pressure-gradient DGM flux in the total molar flux of hydrogen ðNH2 =NH2  100Þ (thin lines with symbols), as a function of the inlet-gas composition (from the analytical study).

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Fig. 6 e Fraction of convective flux in the total molar flux of water vapor ðNcH2 O =NH2 O  100Þ (thick lines), and fraction of the p pressure-gradient DGM flux in the total molar flux of water vapor ðNH2 O =NH2 O  100Þ (thin lines with symbols), as a function of the inlet-gas composition (from the analytical study).

balance xAr ¼ 1  xH2  xH2 O ; the remaining operating parameters are kept constant, with a value equal to that in the reference cell (Table 1). The negative values shown in Fig. 5 indicate that the hydrogen convective and global molar fluxes have opposite directions, and thus convection hinders the supply of the reactant to the reaction sites. The results of the analytical analysis confirm those obtained from the previous numerical simulations; thus, according to these figures, convection may represent up to 16% of the species molar flux, which is in good agreement with the results from the multidimensional calculation shown in Fig. 4. Figs. 7 and 8 show the influence of the other operating parameters, viz. temperature and pressure, on the ratio of

convective to global flux for different fuel concentrations. The pressure and temperature ranges used are those reported as commonplace for SOFC operation [19e21], the fuel-gas mixture consists of H2 and H2O ðxH2 O ¼ 1  xH2 Þ, and the remaining operation parameters are those of the reference cell (Table 1). The results plotted in Figs. 7 and 8 indicate that the importance of the convective flow of H2 increases as the temperature decreases or the pressure increases. These results therefore suggest that convection is not negligible at low operating temperatures or high operating pressures. A sensitivity analysis of the influence of the anode microstructure (in particular, of the pore diameter dp) on the importance of the convective flux has been carried out.

Fig. 7 e Fraction of convective flux in the total molar flux of hydrogen ðNcH2 =NH2  100Þ (thick lines), and fraction of the pressure-gradient DGM flux in the total molar flux of p hydrogen ðNH2 =NH2  100Þ (thin lines with symbols), as a function of the operating temperature and the fuel concentration (from the analytical study).

Fig. 8 e Fraction of convective flux in the total molar flux of hydrogen ðNcH2 =NH2  100Þ (thick lines), and fraction of the pressure-gradient DGM flux in the total molar flux of p hydrogen ðNH2 =NH2  100Þ (thin lines with symbols), as a function of the operating pressure and the fuel concentration (from the analytical study).

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Here, UD is the collision integral and sab is the collision diameter (Lennard-Jones 12-6 potential model)[26]. Equation (24) with Equations (15) and (25) indicate that 0 < Ga/Deff Ka  1, the ratio depending on the species concentration, the operating temperature, the operating pressure and the mean pore size. Thus, from Equation (23): Npa  Nca

Fig. 9 e Fraction of convective flux in the total molar flux of hydrogen ðNcH2 =NH2  100Þ (thick lines), and fraction of the pressure-gradient DGM flux in the total molar flux of p hydrogen ðNH2 =NH2  100Þ (thin lines with symbols), as a function of the anode pore size and the fuel concentration (from the analytical study).

Reported values of pore diameters for SOFC anodes are in the range [0.2, 2.6] mm [9,22e24]. As in the previous sensitivity analysis, there is no inert species in the fuel-gas mixture and the remaining operating parameters are those listed in Table 1. The results are shown in Fig. 9, which indicates that the importance of the convective flux increases with the pore diameter, because the permeability also increases; the relevance of the convective flux again is not negligible for large pore sizes.

4.3. fluxes

The convective and the pressure-gradient DGM

In Subsection 2.2, two distinct (but related) fluxes induced by pressure gradients within the cell electrode have been iden!c !p tified, viz. N a and N a . These are the consequence of two different descriptions of the mass transfer in the electrode, as analyzed in Subsections 4.1 and 4.2. In this Subsection, a relationship between these fluxes is found and an exploration of their relative importance on the global flux is presented. !c From Equations (11), (9) and the definition of N a , given in Section 2.2, the relationship between the convective flux and the pressure-driven DGM flux can be expressed as:

Npa 1 Nca h i ¼ Nglobal DKa P xb þ 1 Nglobal b Dab

(27)

where Nca/Nglobal, DKa and Dab are given by Equations (19), (15) and (25) respectively. The results for hydrogen, in a binary mixture (H2 and H2O), are also plotted (line with symbols) in Figs. 5 to 9. Figs. 5 and 6 show that the magnitude of the pressure-driven DGM flux of hydrogen is, in general, smaller than that of the convective flux. The difference between both fluxes is larger at low hydrogen concentrations, where Npa would be often negligible. Figs. 7 and 8 show that the pressure-driven DGM flux may represent up to 25% of the total molar flux when operating at low temperatures and high fuel concentrations; while it may be up to 45% for high operating pressures and fuel concentrations. This finding is in agreement with Equations (25) and (15), which state that ordinary and Knudsen diffusion are both impeded when operating at low temperatures, while ordinary diffusion is also impeded at high operating pressures. Hence, these analytical results corroborate that convection becomes important under those operating conditions that hinder the diffusive mass transfer. Fig. 9 shows the effect of the electrode microstructure on the pressure-driven fluxes. According to Equation (18), large pore sizes result in high permeabilities. Although large pore sizes also favour Knudsen diffusion (Equation (15)), the diffusion enhancement in electrodes with large pore sizes does not balance that of the pressure-driven fluxes, because the dependence of the Knudsen diffusion coefficient on the pore diameter is weaker than that of the permeability.

Conclusions

(23)

The scaling factor is, according to Equation (8): Ga 1 ¼ P hx i Deff Ka DKa b Dabb þ 1

(24)

where DKa is the Knudsen diffusion coefficient, defined in Equation (15), and Dab is the binary diffusion coefficient of species a in species b, given by Wilke’s empirical correlation [25]: Dab ¼ 2:628$103

Numerical and analytical results, shown in Subsection 4.2, indicate that the convective flux (Nca) is not negligible under certain operating conditions. However, Equation (26) raises the issue of whether the effective-convective DGM flux (Npa) can be neglected. Thus, the analytical study performed in Subsection 4.2 is extended here to assess the relevance of the pressure-driven DGM flux on the global flux of a species a:

5.

Ga !c !p N a ¼ eff N a DKa

(26)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W þW T3 2Wa a Wbb s2ab UD p

(25)

The system of equations given by the Dusty Gas Model to describe the combined mass-transfer mechanisms present in the SOFC anode may be rearranged to obtain a transport equation for the species molar density (Equation (7)), which indicates that mass transfer consists of three distinct contributions, or DGM fluxes. The physical meaning and the relevance of these fluxes has been studied in this paper. It follows that the mass transfer of both species, reactant and product, is dominated by an effective diffusion and is hindered by the counterflow of the other species. For the reactant, an additional resistance to the mass transfer appears due to an effective flow caused by the pressure gradient; this pressure-

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gradient diffusion, on the contrary, benefits the evacuation of the product from the reaction sites. The Dusty Gas Model describes the global mass transfer due to three simultaneous classical mass-transfer phenomena: (i) ordinary diffusion; (ii) Knudsen diffusion; and (iii) convective flux. The latter is often neglected in numerical simulations. The numerical and analytical analysis performed in this paper indicate that it may represent a nonnegligible part of the global flow under certain operating conditions. The analytical procedure detailed in Subsections 4.2 and 4.3, and corroborated by the results from numerical experiments, can be employed to estimate, for a given set of operating conditions, the contribution of the convective flux and the pressure-driven DGM flux to the overall one. From the results of the analytical study, it may be concluded that the magnitude of the pressure-driven DGM flux is generally smaller than that of the convective flux. However, neither of them are negligible for large pore sizes, or when operating at medium to high fuel concentrations, at high pressures or low temperatures; which, in fact, are the desired operating conditions for efficient and economical SOFCs.

Acknowledgements This work is supported by the Science and Innovation Ministry of the Spanish Government (Ministerio de Ciencia e Innovacio´n, Gobierno de Espan˜a) under project ENE2008-06683-C03-03/CON. The authors are grateful to Prof Vı´ctor Orera and Mr Roberto Campana, of University of Zaragoza, for many fruitful discussions; and to Prof Hrvoje Jasak, University of Zagreb, for his invaluable advice on the implementation of the models in OpenFOAM.

Xa xa ya

11559

molar density of species a [k mol m3] molar fraction of species a [ e ] mass fraction of species a [ e ]

Greek symbols a species [ e ] b species [ e ] 3 porosity [ e ] global molecular diffusion including DGM effects Ga [m2 s1], Eq. (7) m dynamic viscosity of the fluid [kg m1 s1] Sutherland-law parameter [kg m1 s1] mo collision integral in the Lennard-Jones 12-6 potential UD model [ e ] r fluid density [kg m3] collision diameter in the Lennard-Jones 12-6 sab _ potential model ½A s ! !0 s N ! ya p ! ya

tortuosity factor [ e ] viscous stress tensor [kg m1 s2] velocity of species a through the porous medium, induced by the other species fluxes, including DGM effects [m s1], Eq. (7) velocity of species a through the porous medium, due to the pressure gradient, including DGM effects [m s1], Eq. (7)

Superscripts c convective d diffusive eff effective global global

Nomenclature references B0 Dab Dam DKa dp I ! Ja ! Na !G Na !p Na !N Na p pa R S T To ! v Wa

permeability [m2] binary diffusion coefficient of species a in b [m2 s1] diffusion coefficient of species a in the gas mixture [m2 s1] Knudsen diffusion coefficient of species a [m2 s1] mean pore diameter [m] current density [A m2] mass diffusive flow of species a [kg m2 s1] molar flux of species a [k mol m2 s1] DGM flux, Eq (11) [k mol m2 s1] DGM flux, Eq (11) [k mol m2 s1] DGM flux, Eq (11) [k mol m2 s1] pressure [Pa] partial pressure of species a [Pa] universal ideal-gas constant [J k mol1 K1] Sutherland-law parameter [K] temperature [K] Sutherland-law parameter [K] fluid velocity [m s1] molecular weight of species a [kg k mol1]

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