On the diffusion in porous electrodes of SOFCs

Chemical Engineering Science 69 (2012) 571–577

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On the diffusion in porous electrodes of SOFCs Ioannis K. Kookos n Department of Chemical Engineering, University of Patras, 26504 Rio, Patras, Greece

a r t i c l e i n f o

abstract

Article history: Received 26 July 2011 Received in revised form 7 November 2011 Accepted 10 November 2011 Available online 19 November 2011

The need to develop mathematical models that are as simple as possible but yet encompass all major features of the underlying physical and chemical phenomena in a system under study is a central issue in engineering science and technology. In the macro and meso-scale modeling of solid oxide fuel cells there is a need to develop accurate expressions that relate the composition in the fuel and air channels with the composition in the three phase boundary where all important electrochemical reactions are taking place. In this work the available models are reviewed and some new results are presented and discussed. & 2011 Elsevier Ltd. All rights reserved.

Keywords: SOFC Electrodes Mathematical modeling Diffusion Mass transfer Porous media

1. Introduction Fuel cells have been identified as a promising alternative to today’s energy conversion technologies, which are bounded by low thermodynamic efficiency limits and high environmental footprints. The increasing energy prices of the quickly depleting fossil fuels that are concentrated in certain geographical areas have lead many on a quest that could change the way energy conversion systems are designed and operated. Today the problem of improving the design of fuel cell systems is attacked from every conceivable angle ranging from micro-scale materials properties determination and design to holistic (systems) considerations. The elusive target of directly connecting micro-scale properties and designs to fuel cells macroscale properties and performance is achieved at a rate much slower than it was anticipated 30–40 years ago. Therefore, mathematical techniques and macroscopic models will continue to rely on efficient experimental techniques for extracting the information necessary to develop their predicting capabilities and this is not expected to change dramatically in the years to come. Key to the success or failure of any modeling exercise is to identify quickly but systematically the simplest possible mathematical model that encompasses the main features of the physical system under study at the level which this is necessary. In the fuel cells modeling research two clear trends can be identified from the careful examination of some excellent reviews

n

Tel.:þ 30 2610 969 567. E-mail address: [email protected]

0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2011.11.013

that have been conducted recently (Bhattacharyya and Rengaswamy, 2009; Hajimolana et al., 2011): one trend is to develop the simplest possible models of fuel cells that can easily be incorporated into general process models (with the aim to study their integrated design) while the other trend is directed towards the incorporation of detailed chemical and electrochemical kinetics and materials micro-structural properties into meso-scale models (with the aim to guide material selection and design). These trends are clearly diverging however they aim at satisfying different needs and they are totally justified. This work aims to review the available one dimensional models of mass transfer into solid oxide fuel cell (SOFC) porous electrodes, to present them in a unified framework and to develop some new analytic solutions for conditions commonly used in experiment or commercial SOFCs. These models are normally used in system’s level models with the aim of connecting directly the reactant and product compositions across the electrolyte and thus relate directly the fuel or air channel compositions with the three phase boundary (TPB) compositions (Kim et al., 1999; Jiang and Virkar, 2003; Suwanwarangkul et al., 2003; Costamagna et al., 2004; Iora et al., 2005; Cresswell and Metcalfe, 2006; Georgis et al., 2011; He et al., 2010). The purpose is self-evident and is related to the complexity of the underlying full scale models and the resulting computational requirements (Tseronis et al., 2008, 2011; Xie and Xue, 2009; Chen et al., 2010). Finally, the use of Graham’s law in the models of SOFC electrodes is discussed extensively and the inconsistency between Graham’s law and constraints that emanate from the reaction stoichiometry are analyzed.

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I.K. Kookos / Chemical Engineering Science 69 (2012) 571–577

2. Models of multicomponent mass transfer in porous media

ui ¼ 

2.1. The Maxwell–Stefan model (MSM)

Ni f CDef i,NC

,

i ¼ 1,2,. . .,NC1

ð7cÞ

Eq. (6) can be solved analytically to yield (Taylor and Krishna, 1993)

The model of Maxwell and Stefan can be applied to any multicomponent gas system that is devoid of walls (i.e. the gas is unconfined) and its mathematical form in 1-D for ideal gas mixtures is the following (Jackson, 1977; Cunningham and Williams, 1980; Taylor and Krishna, 1993; Krishna and Wesselingh, 1997):

when Eqs. (5) is applied to a binary mixture of components A and B then we obtain the following equation for component A:

NC X yi Nj yj Ni dyi ¼ dz CDi,j j¼1

dyA y NB yB NA ¼ A f dz CDef A,B

ð1Þ

where yi is the mole fraction of component i, Ni is the total molar flux of component i, C is the total concentration (C¼P/RT) and Di,j is the Maxwell–Stefan diffusion coefficient. The MSM is strictly applicable under constant pressure and temperature and unconfined gas mixtures. In the cases where the gas mixture is transported in porous media, the diffusion coefficient is replaced by an effective diffusion coefficient i ¼ 1,2,. . .,NC

The effective coefficient is defined as (Krishna and Wesselingh, 1997)

e D ð3Þ t i,j where e is the porosity and t the tortuosity of the pores which are f Def ¼ i,j

considered uniform. Adding Eq. (2) for all components results in the following equation:

i¼1

i

dz

¼0

ð4Þ

which is consistent with the condition that the sum of mole fractions is constant and equals to 1. In addition, Eq. (4) shows that the differential equation (2) are dependent and therefore the following formulation is used instead of (2) NC X yi Nj yj Ni dyi ¼ , f dz CDef j¼1 i,j NC X

i ¼ 1,2,. . .,NC1

yi 1 ¼ 0

ð5aÞ

ð5bÞ

i¼1

This is a system of differential and algebraic equations (DAE) with NC  1 differential variables (y1, y2, y, yNC  1) and NCþ1 algebraic variables (yNC, N1, N2, y, NNC). However, the formulation involves NC  1 differential equations and only 1 algebraic equation. In order to be able to solve the model we need to develop NC additional algebraic equations that involve the total molar fluxes. These additional constraints emanate from the nature of the particular mass transfer problem under investigation (Taylor and Krishna, 1993). If we assume that the total molar fluxes are known then Eqs. (5) can be written in the form dy ¼ Uy þ u dz

ð6Þ T

where y ¼[y1 y2 y yNC  1] and (Krishna and Wesselingh, 1997) 0 1 1 1 ð7aÞ Uij ¼ Ni @ ef f  ef f A, i,j ¼ 1,2,. . .,NC1, i a j CDi,j CDi,NC

Uii ¼

Ni f CDef i,NC

þ

NC X

Nj

ef f j ¼ 1 CDi,j

jai

,

i ¼ 1,2,. . .,NC1

ð7bÞ

ð9Þ

dyA NA ¼ dz CDef f

ð10Þ

A,B

which has the solution

ð2Þ

i,j

ð8Þ

If we further assume equimolar counter-diffusion (NA ¼  NB) then Eq. (9) simplifies to

yA ðzÞ ¼ yA ð0Þ

NC X yi Nj yj Ni dyi ¼ , dz CDef f j¼1

NC X dy

yðzÞ ¼ eUz yð0Þ þðeUz IÞU1 u

NA f CDef A,B

z

ð11Þ

For the cases where pure hydrogen or humidified hydrogen is fed to the fuel channel of a SOFC a binary mixture of H2/H2O is formed in the anode and using NH2 ¼  NH2O ¼j/2F Eq. (11) becomes (Eguchi et al., 1996; Jiang and Ramprakash, 1999; Kim et al., 1999; Suwanwarangkul et al., 2003; Jiang and Virkar, 2003) yH2 ðzÞ ¼ yH2 ð0Þ

j RT 2F PDef f

z

ð12Þ

H2 ,H2 O

where j is the current density, F the Faraday constant, P the system pressure and T the temperature. For a ternary mixture of H2/H2O/I, where I is an inert component (such as Ar, N2 or He), with equimolar counter-diffusion (NH2 ¼  NH20, NI ¼0), matrix U is singular and the general solution (8) is not useful. In this case we can write Eq. (5a) for the inert component I 0 1 N H2 @ 1 dlnyI 1 A ð13Þ ¼  ef f dz C Def f D I,H2

I,H2 O

and the solution is (Jiang and Virkar, 2003) 2 0 1 3 j RT 1 1 @  ef f Az5 yI ðzÞ ¼ yI ð0Þ exp4 2F P Def f DI,H2 I,H2 O

ð14Þ

If we now write Eq. (5a) for hydrogen we obtain dyH2 N H2 ¼ dz CDef f ðzÞ

ð15Þ

H2

where f Def H2 ðzÞ ¼

f ef f Def H2 ,I DH2 ,H2 O f ef f ef f Def H2 ,H2 O þ ðDH2 ,I DH2 ,H2 O Þð1yI ðzÞÞ

ð16Þ

f ef f If we assume that Def H2 ðzÞ  DH2 ð0Þ then Eq. (15) has the following solution (Jiang and Virkar, 2003):

yH2 ðzÞ ¼ yH2 ð0Þ

j RT z 2F PDef f ð0Þ

ð17Þ

H2

Eq. (17) is similar to Eq. (12) which applies for the binary H2/H2O system. Eq. (17) also simplifies to (12) when there is no inert component present in the system. In the more general case the variation of the inert composition can be significant and results obtained by Eq. (17) can be erroneous. In this case we first write Eq. (15) in the following

I.K. Kookos / Chemical Engineering Science 69 (2012) 571–577

form (just to clarify the derivation of the final result): 0 1 C 1 1 1 AyI ð0Þ dy ¼ ef f dzþ @ ef f   f N H2 H2 DH2 ,H2 O DH2 ,H2 O Def H2 ,I 2 0 1 3 N 1 1 H 2 @ exp4  ef f Az5 dz C Def f D

1 0.9 0.8 0.7 ð18Þ

yI yI (0)

I,H2

I,H2 O

j RT z 2F PDef f ðzÞ

=

Eq. (18) has the following solution: yH2 ðzÞ ¼ yH2 ð0Þ

0.5 0.4 0.3 0.2

where 1

0.6

ð19Þ

H2

f Def H2 ðzÞ

573

¼

0

1 f Def H2 ,H2 O

þ@

1 f Def H2 ,I



1

1 f Def H2 ,H2 O

AyI ðzÞ

0.1 ð20Þ

and yI ðzÞ is the average mole fraction of the inert component defined as Rz y ðzÞ dz ð1ecz Þ ð21Þ ¼ yI ð0Þ yI ðzÞ ¼ 0 I cz z

0 10-2

10-1

100  lα

101

102

Fig. 1. The variation of the variable Z as a function of the SOFC operating parameters.

where 0 j RT @ 1 c¼ 2F P Def f

1 1 A  ef f DI,H2 I,H2 O

theory of gases ð22Þ f Def i,K ¼

when yI ðzÞ ¼ yI ð0Þ Eq. (20) simplifies to Eq. (16) and Eq. (19) to Eq. (17). Furthermore, if yI ðzÞ ¼ yI ð0Þ ¼ 0, Eq. (19) simplifies to Eq. (12), as expected. It should be noted that Eqs. (19)–(21), to the best of our knowledge, have not been presented before. The usefulness of Eq. (21) stems from the fact that it offers the means to investigate in a simple manner whether the use of the simplified Eq. (17) will result in significant errors in the calculation of the concentration of hydrogen at the three phase boundary (TPB). To this end we define the following variable:

Z¼

yI 1ec‘a ¼ yI ð0Þ c‘a

ð23Þ

c‘a o 0:1

ð24Þ

then the simplified Eq. (17) can be used. If the above condition is not satisfied then the exact Eq. (19) should be used. 2.2. The Maxwell–Stefan–Knudsen model (MSKM) The MSKM is an extension of the MSM and is useful when the pore radius is comparable to the mean free path of the gas. Then Maxwell–Stefan–Knudsen model (MSKM) can be derived from the MSM by considering an additional giant and motionless gas component to account for the interactions between the gas mixture and the walls (Jackson, 1977; Cunningham and Williams, 1980). The final equation has the following form: NC X yj Ni yi N j dyi N ¼ þ efi f , ef f dz Di,j Di,K j¼1

1=2

rp

ð26Þ

where MWi is the molecular weight if component i and rp is the mean pore radius. An interesting feature of the MSKM is derived by adding all Eqs. (25)



NC X 1 dP Ni ¼ ef f RT dz D i ¼ 1 i,K

ð27Þ

For isobaric conditions and using Eq. (26) it follows that NC pffiffiffiffiffiffiffiffiffiffiffi X MWi N i ¼ 0

ð28Þ

i¼1

from which it follows that Z equals to the ratio of the average inert mole fraction within the porous anode to the mole fraction of inert in the fuel channel. ‘a is the thickness of the anode (thickness of the diffusion path). The variation of Z as a function of the anode parameters is shown in Fig. 1. From the figure it follows that if

C



e 2 8RT t 3 pMWi

i ¼ 1,2,. . .,NC

ð25Þ

f is the effective Knudsen diffusion coefficient which is Def i,K calculated by the following equation derived from the kinetic

This is the well-known Graham’s law which has been, in the absence of chemical reactions, validated experimentally by a number of researchers (Jackson, 1977; Soukup et al., 2008). In the cases where a chemical reaction occurs the component fluxes are related through the reaction stoichiometry and only the case of isomerization reactions is consistent with Graham’s law. As has been discussed by a number of researchers Graham’s law does not apply in the case of chemical reactions in porous media as chemical reactions induce pressure gradients in the diffusion path invalidating the basic assumption of Graham’s law (Hite and Jackson, 1977; Burghardt and Aerts, 1988). If a chemical reaction is taking place then the induced pressure variation needs to be included in the MSKM which results in the dusty gas model presented in the section that follows. In order to make possible the use of the MSKM for the description of the multicomponent diffusion in porous SOFC anodes, Eq. (25) must be written for NC 1 components only (as the introduction of the equation for the NCth component is equivalent to implicitly imposing Graham’s law). The composition of the NCth component can be derived by the constraint on the summation on the mole fractions. In this case the constraints on the fluxes that stem from the reaction stoichiometry can be used (see Jackson, 1977). This eliminates many of the inconsistency problems that appear when Graham’s law is used. This will be discussed after the introduction of the dusty gas model.

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I.K. Kookos / Chemical Engineering Science 69 (2012) 571–577

When pure hydrogen or humidified hydrogen is fed to the fuel channel then the MSKM, written for hydrogen, takes the form 0 1 N H2 @ 1 1 A dz ð29Þ dyH2 ¼  þ ef f C Def f D H2 ,K

H2 ,H2 O

H2 ,K

The solution of (29) follows immediately 0 1 j RT @ 1 1 A z þ ef f yH2 ðzÞ ¼ yH2 ð0Þ 2F P Def f D H2 ,H2 O

H2 ,I

ð30Þ

H2 ,K

H2 ,K

The solution of Eq. (31) is given by Eq. (19) where 0 1 0 1 1 1 1 A @ 1 1 A @ þ þ ef f  ef f yI ðzÞ ¼ f f f Def Def DH2 ,H2 O Def DH2 ,H2 O H2 ðzÞ H2 ,K H2 ,I

ð32Þ

The approximate solution with constant inert composition that equals the one in the fuel channel has been presented by Jiang and Virkar (2003). As has been shown in the previous section the accuracy of this approximation can be evaluated with the condition given by Eq. (24) or Fig. 1. 2.3. The dusty gas model (DGM) The dusty gas model is an extension of the MSM in which both the gas/wall interactions and the viscous fluxes due to pressure gradient are taken into consideration. The general form of the DGM is the following (Jackson, 1977; Cunningham and Williams, 1980) ! P dyi yi 1 B0 P dP   1þ ef f RT dz RT m dz D

3. Discussion It is important to observe that in the cases of the MSM and MSKM, for which analytic solutions for the distribution of hydrogen mole fraction across the diffusion path have been obtained, the general form of the solution is the same irrespective of the model used yH2 ðzÞ ¼ yH2 ð0Þ

¼

j¼1

f Def ij

þ

Ni

jmax ¼ 2F

f Def iK

i ¼ 1,2,. . .,NC

iK

1 jA nf 2F

ð38Þ

where A is the area of the cell and nf the total inlet molar flow rate of the fuel (see also discussion by Chick et al., 2011). The first set of published experimental data to be considered are the ones presented by Jiang and Virkar (2003). This is one of the most influential and comprehensive studies that have been published on the calculation of the anodic concentration polarization. The study of an anode of thickness la ¼0.1 cm (porosity

iK

Observe that as pressure gradient is different than zero, Graham’s law is not implicit in Eq. (33). Eqs. (33) and (34) cannot be solved analytically and one needs to rely on numerical methods for their solution (see Cunningham and Williams, 1980). For the case of equimolar counter-diffusion of a binary mixture of H2/H2O in porous SOFC anodes the following set of differential equations needs to be solved: 0 1 ! dyH2 P 1 1 B0 dP RT 1 1 A @ þ ¼ N H2 þ yH2 þ ef f RT P Def f m dz P dz Def f D H2 ,K

ð37Þ

Eq. (37) offers the simplest means for establishing the validity of the approach of investigating mass diffusion in porous anodes and has been used extensively in the literature in order to calculate the effective diffusion coefficients. It is important to emphasize at this point that the yH2(0) that appears in Eq. (37) is the hydrogen mole fraction at the fuel channel. Unfortunately, many authors have used the hydrogen concentration in the feed stream yH2,f which, under high utilization, can be significantly different than yH2(0) as they are related through the hydrogen material balance in the fuel channel

ð33Þ

where B0 is the permeability of the porous medium and m is the viscosity of the gas mixture. Adding Eq. (33) for all components results in " !# NC NC X 1 B0 P X yi dP Ni 1þ ¼  ð34Þ ef f RT dz m i ¼ 1 Def f i¼1D

ð36Þ

P ef f yH2 ð0Þ D RT ‘a

yH2 ð0Þ ¼ yH2 ,f  ,

j RT z 2F PDef f

where the appropriate form of the effective diffusion coefficient to be used is given in Table 1 where a summary of the results for the MSM and MSKM are presented. It should be observed that for the case of binary mixtures a linear hydrogen composition profile is predicted. For the case of ternary mixtures the profile is again linear when Z E1. When a significant variation of the composition of inert component is predicted then Z o1 and the hydrogen profile will also deviate from linearity. It is also important to note that when hydrogen composition at the TPB approaches zero then the current density will approach the limiting current density

iK

NC X yj N i yi Nj

ð35bÞ

H2 O,K

The equations for the case of the ternary mixture H2/H2O/I are significantly more complex.

f ef f It should be observed that when Def H2 ,H2 O 5 DH2 ,K then (30) simplifies to Eq. (12). In the case of the ternary mixture H2/H2O/I the same approach as in the previous paragraph can be followed. Eq. (25) written for the inert component results in Eq. (13) and therefore the composition distribution of the inert component (given by Eq. (14)) is not affected by the Knudsen diffusion. Eq. (25) written for hydrogen gives 0 1 N H2 @ 1yI yI 1 A dyH2 ¼  dz ð31Þ þ ef f þ ef f C D Def f D H2 ,H2 O

8 2 0 1 39 = dP <1 B 1 1 1 04 þ yH2 @ ef f  ef f A þ ef f 5 :P m DH2 ,K DH2 O,K DH2 O,K ; dz 0 1 RT 1 1 NH2 @ ef f  ef f A ¼ P D D

H2 ,H2 O

H2 ,K

ð35aÞ

Table 1 Summary of the results for the MSM and MSKM. 1=Def f

Binary: H2/H2O

Ternary: H2/H2O/I

  ef f1 yI ð0ÞZ DH ,H O 2 2       MSKM 1 1 þ ef f1 þ ef f1 þ ef1f  ef f1 yI ð0ÞZ f ef f Def D D D D D H2 ,K H ,K H2 ,I H2 ,H2 O H2 ,H2 O H2 ,H2 O 2  cz j RT 1 Z ¼ 1e  ef1f , yH2 ðzÞ ¼ yH2 ð0Þ 2Fj RTef f z ef f cz , Z  1 if cz o0:1, c ¼ 2F P MSM

1 f Def H ,H 2

2O

1 f Def H ,H 2



þ

2O

DI,H

1 f Def H ,I 2

2O

DI,H

2

PD

I.K. Kookos / Chemical Engineering Science 69 (2012) 571–577

100

Fuel utilization (%)

e ¼ 0.54 and tortuosity t ¼6.18) operated at 800 1C and 1 atm is presented and equations for the estimation of the anodic overpotential are developed and tested using experimental data. Nitrogen and Helium are used as inert components in this study. When 50% nitrogen is used as an inert component the limiting current density is 2.65 A cm  2 and the experimentally calculated concentration overpotential is used to estimate the effective diffusion coefficient. Jiang and Virkar used the MSKM under the assumption that the inert component mole fraction is constant across the anode (Eq. (32) has been used with yN2 ðzÞ ¼ yN2 ð0Þ ¼ 0:5). In Fig. 2 the parameter cla and the inert component mole fraction at the TPB as a function of the current density is presented. From this figure it becomes clear that cla is larger than 0.1 even at very small current densities and becomes 0.6 at the limiting current density. The mole fraction of nitrogen at the TPB deviates significantly from its value at the fuel channel and at limiting current density conditions is more than 40% lower than that at the fuel channel. For the case of helium the errors involved are less dramatic as shown in Fig. 3 (42% helium in fuel, anode

575

Exp. Data-Chick et al., (2011) - Calculated values 90

80

70 0

0.5

1

1.5

jmax (A cm−2) Fig. 4. Comparison of experimental data and calculated values of the limiting current density.

Inert Gas: N2

1

0.6

0.5

0.4

yN2 (la)

l a

0.5

0.3

0.2

0 1

0

2

3

j (A cm−2) Fig. 2. Parameter cla and inert component mole fraction at the TPB as a function of the current density for the ternary mixture H2/H2O/N2, T¼ 1073 K, P¼ 1 atm, la ¼0.2 cm.

Inert Gas: He

0.15

0.42

0.4

0.05

0.38

la

yHe (la)

0.1

thickness la ¼0.1 cm, 800 1C, 1 atm, limiting current density 4 A cm  2). The second set of experimental data considered are the ones presented by Chick et al. (2011). In this work the limiting current density is measured experimentally as a function of fuel utilization for the case of a 15% H2, 3% H2O and 82% N2 feed stream in a porous anode with an average pore radius of 0.6 mm, porosity of 0.38 and tortuosity of 2.5 that operates at 750 1C. Using these values it is calculated that cla o0.05 and therefore the MSKM with the constant diffusion coefficient given by Eq. (32) can be used. The fact that the SOFC is operated with high fuel utilization conditions necessitates the use of Eq. (37) together with Eq. (38) to account for the significant decrease in the concentration of hydrogen in the fuel channel. The results of the calculations are presented in Fig. 4 and are compared with the experimental data reported by Chick et al. (2011). The agreement of the experimental and calculated values is more than acceptable as the maximum error is 5.8%. The third study considers the numerical simulation using multidimensional DGM by Tseronis et al. (2008, 2011). In this study numerical simulation is performed for a porous anode of thickness 0.002 m at 1023 K for a binary mixture of H2/H2O in the feed stream and the results are validated with experimental data from the literature (Yakabe et al., 2000). The anode channel is 0.020 m long and the fresh feed is 80% H2. Using an average current density of j ¼15,000 A m  2 a fuel utilization of approximately 72% is achieved resulting in an exit H2 mole fraction of approximately 0.23. The hydrogen mole fraction becomes 0.5 at a distance of approximately 0.008 m from the fuel cell entrance (the parameters of the anode are given in Table 1 of Tseronis et al. (2008)). This mole fraction is used in order to calculate the hydrogen composition across the porous anode. To this end the DGM is solved first using Eqs. (35a) and (35b) written in the form 2 dy 3 H2

AðyH2 ,PÞ4

0

0.36 0

1

2 j (A cm−2)

3

4

Fig. 3. Parameter cla and inert component mole fraction at the TPB as a function of the current density for the ternary mixture H2/H2O/He, T¼ 1073 K, P¼ 1 atm, la ¼0.2 cm.

dz dP dz

5 ¼ bðyH ,PÞ 2

ð39Þ

Then the ode15s solver in MATLAB is used with the appropriate ‘‘mass’’ matrix A. The results are shown in Fig. 5. In Fig. 5(a) the variation of the hydrogen mole fraction is shown and in Fig. 5(b) the deviation of the total pressure from its value in the fuel channel is also shown. The results agree well with the multidimensional DGM and are considered as an accurate representation of the physical problem under study. Then the MSKM with the

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I.K. Kookos / Chemical Engineering Science 69 (2012) 571–577

In this model that last algebraic equation in (40) which is dictated by stoichiometry has been replaced by Eq. (28). The predictions of the model MSKM(2) are also shown in Fig. 5(a). The deviation from the predictions of the DGM is now significant. Using the MSKM(2) will result in predicting much larger hydrogen mole fractions at the TPB and much smaller concentration overpotential when compared to the DGM. It is important at this point to observe that the decision to relate the molar flux of hydrogen to the current density was arbitrary as Graham’s law only requires the incorporation of Eq. (28) in the mathematical formulation. There is no reason to prevent one from relating the molar flux of water to the current density in which case the following model is obtained (model MSKM(3)):

0.5 MSK (2) Graham’s law 0.4

0.3

DGM

yH2

MSK (3) Graham’s law

MSK (1)

0.2

0.1

0 0

0.001 z (m)

0.002

dyH2 yH O NH2 yH2 NH2 O N H2 ¼ 2  ef f dz CD CDef f H2 ,H2 O

H2 ,K

0 ¼ yH2 O þyH2 1 j 0 ¼ N H2 O þ 2F qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ¼ MWH2 N H2 þ MWH2 O N H2 O

ΔP (Pa)

2

1

0 0

0.001 z (m)

0.002

Fig. 5. Hydrogen mole fraction variation (a) and total pressure variation (b) across the porous anode considered in Tseronis et al. (2008).

relative fluxes set by reaction stoichiometry is considered and solved numerically using the following DAE formulation: dyH2 yH O NH2 yH2 NH2 O N H2  ¼ 2 dz CDef f CDef f H2 ,H2 O

H2 ,K

0 ¼ yH2 O þ yH2 1 j 2F 0 ¼ N H2 þ N H2 O

ð42Þ

The predictions of the MSKM(3) are also shown in Fig. 5. The deviations from the DGM are now dramatic and the predictions are clearly unacceptable. The evidence is strong enough to suggest that the use of Graham’s law in modeling mass transfer in porous anodes of SOFCs might result in significant errors in the calculations. The use of the MSKM with the fluxes bounded by reaction stoichiometry is not only simpler to describe mathematically (as it is amenable to analytic treatment) but it also results in predictions that are in good agreement with the most accurate model available today which is the DGM. Before closing this section it is important to discuss Fig. 5(b). As is shown in this figure the pressure variation across the porous anode is of the order of few Pascal. An initial examination of this result might result in the erroneous conclusion that a pressure variation of this order is insignificant and of academic interest only. However, as the examination of the literature on the classical Wilke–Kallenbach and Graham’s diffusion cells (Soukup et al., 2008) shows, even a tiny pressure difference can cause permeation fluxes that are comparable to diffusion fluxes invalidating Graham’s law (even in the absence of chemical reaction). For the problem under study in this paper the following conclusion can be drawn: the tiny but finite pressure gradient that is induced in the porous electrode by the chemical reaction, as predicted by the DGM, frees the molar fluxes from the constraints imposed by Graham’s law and allows them to satisfy the reaction stoichiometry and overall material balances.

0 ¼ N H2 

ð40Þ

The results are also shown in Fig. 5(a) as model MSKM(1). The MSKM with the fluxes related by the reaction stoichiometry are in close agreement with the ‘‘exact’’ solution offered by the DGM. In order to investigate the effect of assuming that the ratio of the fluxes is dictated by Graham’s law (Eq. (28)) we consider the following model (MSKM(2)): dyH2 yH O NH2 yH2 NH2 O N H2 ¼ 2  dz CDef f CDef f H2 ,H2 O

H2 ,K

0 ¼ yH2 O þ yH2 1 j 0 ¼ N H2  2F qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ¼ MWH2 NH2 þ MWH2 O N H2 O

ð41Þ

4. Conclusions In this work the most commonly used models for mass transfer in porous electrodes of SOFCs are reviewed and some important new results are presented. More specifically an analytic expression is derived for the diffusion of hydrogen in ternary mixture of H2/ H2O/Inert for the case where the relative molar fluxes are related through reaction stoichiometry. A criterion that is easy to apply (as it requires knowledge of the applied current density and thickness of the porous electrode) is presented in order to calculate whether simplified expressions can be used in order to establish the mole fractions of hydrogen and water at the TPB. This is of particular importance in order to link accurately material balances and mass transfer calculations to electrochemical models in modeling SOFC-based systems. Furthermore, evidence is presented to support the argument that the commonly used Graham’s

I.K. Kookos / Chemical Engineering Science 69 (2012) 571–577

law may result in the introduction of significant errors in the calculation of mole fractions at the TPB. Some numerical and experimental data, taken from the open literature, are used in order to demonstrate the usefulness of the proposed equations and the fundamental limitations of the approaches that are based on the use of Graham’s law.

Nomenclature B0 C D Deff F j la MWi nf Ni P rp R T y y z

permeability of the porous medium concentration, kmol m  3 diffusion coefficient, m2 s  1 effective diffusion coefficient, m2 s  1 Faraday’s constant, 96,487 C/mol electron current density, A m  2 anode thickness, m molecular weight of component I total inlet molar flow rate of the fuel, kmol s  1 total molar flux of component i, kmol m  2 s  1 pressure, Pa mean pore radius, m ideal gas constant temperature, K mole fraction vector of mole fractions distance, m

Greek symbols

e Z m t u U c

porosity parameter defined in Eq. (22) viscosity, N s m  2 tortuosity vector defined in Eq. (7c) matrix defined in Eqs. (7a) and (7b) parameter defined in Eq. (22)

References Bhattacharyya, D., Rengaswamy, R., 2009. A review of solid oxide fuel cell (SOFC) dynamic models. Ind. Eng. Chem. Res. 48 (13), 6068–6086. Burghardt, A., Aerts, J., 1988. Pressure changes during diffusion with chemical reaction in a porous pellet. Chem. Eng. Process.: Process Intensif. 23 (2), 77–87. Chen, D., Bi, W., Kong, W., Lin, Z., 2010. Combined micro-scale and macro-scale modeling of the composite electrode of a solid oxide fuel cell. J. Power Sources 195 (19), 6598–6610.

577

Chick, L.A., Meinhardt, K.D., Simner, S.P., Kirby, B.W., 2011. Factors affecting limiting current in solid oxide fuel cells or debunking the myth of anode diffusion polarization. J. Power Sources 196 (10), 4475–4482. Costamagna, P., Selimovic, A., Del Borghi, M., Agnew, G., 2004. Electrochemical model of the integrated planar solid oxide fuel cell (IP-SOFC). Chem. Eng. J. 102 (1), 61–69. Cresswell, D.L., Metcalfe, I.S., 2006. Energy integration strategies for solid oxide fuel cell systems. Solid State Ionics 177 (19–25), 1905–1910. Cunningham, R.E., Williams, R.J.J., 1980. Diffusion in Gases and Porous Media. Plenum Press, New York. Eguchi, K., Kunisa, Y., Adachi, K., Arai, H., 1996. Effect of anodic concentration overpotential on power characteristics of solid oxide fuel cells. J. Electrochem. Soc. 143 (11), 3699–3703. Georgis, D., Jogwar, S.S., Almansoori, A.S., Daoutidis, P., 2011. Design and control of energy integrated SOFC systems for in situ hydrogen production and power generation. Comput. Chem. Eng. 35 (9), 1691–1704. Hajimolana, S.A., Hussain, M.A., Ashri, W.M., Wan Daud, A., Soroush, M., Shamiri, A., 2011. Mathematical modeling of solid oxide fuel cells: a review. Renewable Sustainable Energy Rev. 15 (4), 1893–1917. He, W., Yoon, K.J., Eriksen, R.S., Gopalan, S., Basu, S.N., Pal, U.B., 2010. Out-of-cell measurements of H2–H2O effective binary diffusivity in the porous anode of solid oxide fuel cells (SOFCs). J. Power Sources 195 (2), 532–535. Hite, R.H., Jackson, R., 1977. Pressure gradients in porous catalyst pellets in the intermediate diffusion regime. Chem. Eng. Sci. 32 (7), 703–709. Iora, P., Aguiar, P., Adjiman, C.S., Brandon, N.P., 2005. Comparison of two IT DIRSOFC models: impact of variable thermodynamic, physical, and flow properties. Steady-state and dynamic analysis. Chem. Eng. Sci. 60 (11), 2963–2975. Jackson, R., 1977. Transport in Porous Catalysts. Elsevier, Amsterdam. Jiang, Y., Virkar, A.V., 2003. Fuel composition and diluent effect on gas transport and performance of anode-supported SOFCs. J. Electrochem. Soc. 150 (7), A942–A951. Jiang, S.P., Ramprakash, Y., 1999. H oxidation on Ni/Y–TZP cermet electrodes—polarisation behaviour. Solid State Ionics 116 (1–2), 145–156. Kim, J.W., Virkar, A.V., Fung, K.Z., Mehta, K., Singhal, S.C., 1999. Polarization effects in intermediate temperature, anode-supported solid oxide fuel cells. J. Electrochem. Soc. 146 (1), 69–78. Krishna, R., Wesselingh, J.A., 1997. The Maxwell–Stefan approach to mass transfer. Chem. Eng. Sci. 52 (6), 861–911. ˇ ´ , O., 2008. Comparison of Wicke–Kallenbach and Soukup, K., Schneider, P., Solcova Graham’s diffusion cells for obtaining transport characteristics of porous solids. Chem. Eng. Sci. 63 (4), 1003–1011. Suwanwarangkul, R., Croiset, E., Fowler, M.W., Douglas, P.L., Entchev, E., Douglas, M.A., 2003. Performance comparison of Fick’s, dusty-gas and Stefan–Maxwell models to predict the concentration overpotential of a SOFC anode. J. Power Sources 122 (1), 9–18. Taylor, R., Krishna, R., 1993. Multicomponent Mass Transfer. John Wiley & Sons, New York. Tseronis, K., Kookos, I.K., Theodoropoulos, C., 2008. Modelling mass transport in solid oxide fuel cell anodes: a case for a multidimensional dusty gas-based model. Chem. Eng. Sci. 63 (23), 5626–5638. Tseronis, K., Bonis, I., Kookos, I.K., Theodoropoulos, C., 2011. Parametric and transient analysis of non-isothermal, planar solid oxide fuel cells. Int. J. Hydrogen Energy. doi:10.1016/j.ijhydene.2011.09.062. Xie, Y., Xue, X., 2009. Transient modeling of anode-supported solid oxide fuel cells. Int. J. Hydrogen Energy 34 (16), 6882–6891. Yakabe, H., Hishinuma, M., Uratani, M., Matsuzaki, Y., Yasuda, I., 2000. Evaluation and modeling of performance of anode-supported solid oxide fuel cell. J. Power Sources 86 (1), 423–431.