A generalized mathematical model to study gas transport in PEMFC porous media

International Journal of Heat and Mass Transfer 58 (2013) 70–79

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

A generalized mathematical model to study gas transport in PEMFC porous media Lalit M. Pant a,b, Sushanta K. Mitra a, Marc Secanell b,⇑ a b

Micro and Nano Scale Transport Laboratory, Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G8 Energy Systems Design Laboratory, Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G8

a r t i c l e

i n f o

Article history: Received 25 June 2012 Received in revised form 31 October 2012 Accepted 5 November 2012

Keywords: PEMFC Mass transport Knudsen diffusion Multicomponent Diffusion slip

a b s t r a c t A new mass transport model named modified binary friction model (MBFM) is proposed. The new mass transport model is based on the previously developed binary friction model (BFM) and is capable of predicting mass transport in viscous and transitional Knudsen regimes. Macro homogeneous mass transport models used in porous media are reviewed. The shortcomings of each model are discussed, and a comparison of the new mass transport model with existing models such as Fick’s, dusty gas model and BFM is presented. It is found that the choice of transport model can significantly impact the predictions of mass transport. The transport predictions of the new model are significantly different compared to existing models for flow in Knudsen region and for mixtures containing gases with large difference in molecular weights. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction During the last decade, polymer electrolyte fuel cells (PEMFCs) have emerged as promising energy conversion devices for power electronics, backup power units, domestic co-generation of electricity and hot water, and as a replacement for the internal combustion engine in automobiles [1]. Fuel cells are highly energy efficient and low noise energy conversion devices that, when fuelled with hydrogen from either nuclear or renewable sources, provide electrical energy while producing zero particulate, nitrogen and sulphur oxide, and green house gas (GHG) emissions. The major challenges facing this technology are the high costs associated with the platinum catalyst used in PEMFC electrodes [2,3]. Mass transport in PEMFC electrodes has to be well understood in order to optimize the catalyst amount and its distribution inside the electrode. PEMFC electrodes are usually a carbon based porous media. Improving mass transport in the porous media would allow fuel cells to operate at much higher current densities, thereby reducing the amount of necessary catalyst [2,3]. To better understand and optimize mass transport processes in porous media of a PEM fuel cell, a reliable and accurate mass transport model is required. The merit of a mass transport model is in its accurate accounting of all driving and frictional forces encountered during transport of fuel and oxidants in fuel cell. The driving force is the chemical potential gradient and the frictional forces are interspecies friction, ⇑ Corresponding author. Tel.: +1 780 492 6961; fax: +1 780 492 2200. E-mail address: [email protected] (M. Secanell). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.11.023

viscous friction and wall collisions (Knudsen friction) [4]. The various frictional forces are dominant in different transport regimes and in the transition regions between them [5]. The general framework to model mass transport is based on combination of momentum balance for mixture and Maxwell–Stefan equations for species [6–8]. In the case of extremely dilute mixtures and in absence of external forces, the Maxwell–Stefan equation reduces to the well known Fick’s law [9,10]. Fick’s law has been extensively used for mass transport studies. However, its application is limited to binary dilute mixtures hence it is not useful for multicomponent species transport in PEMFCs. For better understanding of mass transport in porous media, the effect of the medium has to be accounted for accurately. The importance of porous media structure such as pore size on mass transport has been presented in literature. Recently Pant et al. [11] showed the importance of Knudsen number on the transport in PEMFC electrodes. It was shown that mass transport in micro porous layers (MPLs), which is usually considered to be in continuum region, is actually in the transition region between continuum and Knudsen. Most PEMFC models in the literature have used the Maxwell– Stefan equations and Navier–Stokes equation for modelling gas transport. Recently two articles have presented a comparison of several mass transport models in solid oxide fuel cells [12,13]. These studies have usually focused on Maxwell–Stefan equation, dusty gas model (DGM) and binary friction model (BFM). However, these studies have not highlighted the key differences between the models, especially from the point of view of physical consistency. The use of these models in porous media usually requires estimation of some fitting parameters like tortuosity. Such adjustment

L.M. Pant et al. / International Journal of Heat and Mass Transfer 58 (2013) 70–79

parameters make it difficult to clearly estimate the accuracy of each model. Dusty gas model (DGM) has been extensively used for multicomponent mass transport studies in porous media [14]. The basic assumption of DGM is that the flux values in the Maxwell–Stefan equation are diffusive only, and can be extended to include viscous effects. Interspecies terms are summed over all species including the porous media. A modelling technique parallel to the DGM has been used in the literature to model gas flow in porous media [15]. Recently, Kerkhof pointed out that the derivation of the DGM might be erroneous [16]. Some of the inconsistencies of DGM are discussed in detail at a later stage in this article. A multicomponent mass transport model for membranes has been presented by Lightfoot [17]. The model is based on irreversible thermodynamics approach and it accounts for the wall effects (Knudsen and viscous). The Lightfoot equation is theoretically consistent, but the membrane friction parameter is not well defined. Continuing on the work of Lightfoot, a new model named binary friction model (BFM) has been proposed recently by Kerkhof to eliminate the inconsistencies of the DGM [16]. The BFM is based on Lightfoot equation, which has been augmented by evaluating the viscous and Knudsen friction parameters. The BFM evaluates the friction parameters in viscous and Knudsen regimes and eliminates the flux splitting analogy of the DGM. The BFM has been used to model transport phenomena in polymer electrolyte membranes [18]. The BFM approach provides a physically consistent framework but its derivation assumes mass average velocity of species to be the same as the molar averaged velocity. A detailed discussion on the assumptions of BFM is presented later in this article. Another issue with the aforementioned models is the inability to account for diffusion slip. It is known in the literature, that a concentration gradient parallel to the wall in mixtures induces a non-zero velocity at the wall boundary [19]. Hence the conventional no-slip boundary condition can lead to wrong prediction of the fluxes. Very recently a new model has been proposed, which takes into account the slip at the wall [20]. The model uses Maxwell–Stefan equations for mass transfer and a Poiseuille type of momentum balance equation in a cylindrical pore. The diffusion slip is applied on the solution of Poiseuille equation and the net modified flux is obtained [20]. The model has resolved many issues with previous models, however the decoupling of convection from the mass transfer can lead to some inconsistencies. It should be noted that a momentum balance is still required, even if the transport is completely diffusive. Based on the literature review, it is observed that transport models available in literature have several theoretical inconsistencies and inherent assumptions that limit their applicability. This article presents a new mass transport model called modified binary friction model (MBFM) which simultaneously accounts for interspecies, viscous and Knudsen friction in porous media. The model can be used to study multicomponent mass transport in porous media of PEM fuel cells and other porous materials. Section 2 presents a review of previous mass transport models, e.g Fick’s law, DGM and BFM and a discussion on their limitations. Section 3 presents the derivation of the modified binary friction model followed by its extension for porous media in Section 3.1. Section 4 presents some validation studies for MBFM. Section 5.1 presents the comparison of different mass transport models for studies in PEMFC porous media followed by a parametric analysis. Finally Section 6 summarizes and concludes the article.

2. Review of mass transport models This section gives a brief description of several models available in the literature, followed by an analysis of each of the models

71

describing the salient features and shortcomings of each model. The models are presented in the same order, as they were proposed in literature, showing the developments in theories of mass transport over the decades. The various fluxes used in the formulations are defined in Appendix A. 2.1. Fick’s and Darcy’s law For porous media, the effective form of Fick’s law is given as follows: eff  neff i ¼ qi v  Dij $qi

ð1Þ

where neff i is the effective mass flux, qi is density of species i and the effective diffusivity of a porous media is given as [21,22]:

 s

Deff Dij ij ¼

ð2Þ

where Dij is the molecular diffusion coefficient and s is the tortuosity of porous media. The mass average velocity v⁄ in Eq. (1) across the porous media is given by Darcy’s law:

v ¼ 

Bv

g

$p

ð3Þ

where Bv is the permeability of the porous media and g is the mixture viscosity. Even though Fick’s law is easy to implement, it is very restrictive in its use. As this law is valid only for binary mixtures or infinitely dilute solutions, it is not useful for multicomponent mixtures. Also, the Knudsen effects are not accounted for, which can be quite significant for small pore radius, such as in micro porous layers and catalyst layers of a PEM fuel cell, where the pore sizes are in range 10–200 nm [23]. 2.2. Dusty gas model The dusty gas model (DGM) has been proposed for multicomponent mass transport in porous media [14]. In the development of the theory, the molecular and Knudsen diffusion fluxes are assumed to be in series. These two effects are then parallel to viscous and surface fluxes [14]. In the absence of external forces, and for an isothermal ideal gas, the dusty gas model can be simplified to the following expression [16]: n n X X xi NDj  xj NDi xi NDj  xj NDi 1 cd NDi ND $T pi ¼  ¼  i0 0 0 2 0 pt ct Dij ct Did ct Dij ct Dim j¼1 j¼1

ð4Þ

where NDi is the diffusive flux of species i, cd is the concentration of dust molecules and xi, xj are mole fractions. The parameters D0ij and D0im are the molecular and Knudsen diffusion coefficients with second order corrections [14]. It can be seen that the flux term consists of only diffusive fluxes in the DGM. The second term on right hand side is the wall friction, which is obtained by including porous media in the force balance, by considering it as a collection of dust molecules fixed in space. During force balance, the diffusive dust   flux is assumed to be zero NDd ¼ 0 . This is incorrect as it should be the total flux of dust which should be zero, as explained in detail at a later stage in this section. The assumption in the derivation of the DGM is that the net flux can be split up into diffusive and viscous flux, where the viscous flux, Nvisc is given by Darcy’s law in porous media. The second order corrections in the diffusion coefficients are negligible so D0 can be replaced by D [14]. Taking into account the above-mentioned simplifications, and using effective properties for porous media, the final DGM equation is given as follows.

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n x Neff  x Neff X xi Neff Neff i j j i 1 v isc i $T p i ¼  þ eff eff pt ct Deff ct DKi ct DKi j¼1 ij

ð5Þ

The detailed derivation of DGM can be found in the literature [14,16]. The DGM has been recently questioned for some inherent inconsistencies in the derivation [16]. It has been pointed that the fluxes in the force balance already contain the viscous part and thus accounting for them again is incorrect [16]. The main error in the derivation occurs during summing of all the Maxwell–Stefan forces including the dust molecules. The sum of the Maxwell–Stefan terms can be shown as follows: nþ1 X xi NDj  xj NDi j¼1

ct D0ij

¼

n X xi NDj  xj NDi j¼1

ct D0ij

þ

xi NDd  xd NDi ct D0id

ð6Þ

NDd

where is the diffusive flux of dust, which is assumed to be zero in the DGM derivation. However, for a stationary frame of reference it can be said that the net flux of dust molecules will be zero (i.e. Nd = 0), as they are fixed. If the flux splitting analogy is used, the following relation is obtained between diffusive and net flux of dust:

Nd ¼

NDd

þ xd Nv isc

ð7Þ

Due to the fixed dust molecules, Nd will be zero. The mixture viscous flux Nvisc will have a finite value for a pressure gradient. Therefore, NDd can not be zero at the same time, as it would imply xd = 0, which in turn indicates a zero membrane friction. Due to these errors in derivation, the DGM equations are not self consistent. This can be shown by summing all the equations over all the components, and obtaining an expression for net pressure gradient. For a set of correct equations, the summation should give the Darcy’s equation for net pressure gradient. However a different equation is obtained in this case, which is neither similar nor can be converted to Darcy’s equation. Kerkhof [16] have also observed similar discrepancies in the DGM, and have proposed a new mass transport model for multicomponent mixture. 2.3. Binary friction model The binary friction model (BFM) is based on Lightfoot equation, which is obtained from irreversible thermodynamic considerations of the system. The Lightfoot equation is given as follows [16,17]: n X xi cV q xi N j  xj N i $T;p li þ i i $p  i Fi ¼  rim Ni RT ct RT ct RT ct Dij j¼1

ð8Þ

where li is the chemical potential of species i, V i is the partial molar volume of species i, Fi is the external force on species i and the porous media friction term rim represents the momentum loss due to interaction of species with porous media. For BFM, external forces are assumed to be absent, gases are considered to be ideal and the fluxes are assumed to be per unit area of pore along the pore path. With these assumptions, Eq. (8) is reduced to [16]: n X x 1 ðxi Nj  xj Ni Þ $xi þ i $pt  $pi  RT  r im Ni pt pt Dij pt j¼1

ð9Þ

Summing up over all the components, the interspecies friction term cancels out and the following equation is obtained for the mixture. A new friction parameter bim is also defined: n n X X 1 $pt ¼  r im Ni   bim v i pt i¼1 i¼1

ð10Þ

To determine the porous media friction parameter, two cases; viz. Knudsen and viscous flow are considered [16]. In the Knudsen region, interspecies interaction is neglected and Eq. (9) can be rewritten as follows to describe the flow:

1 RT N $pi ¼  K i pt Di pt

ð11Þ

which gives:

bKim ¼

xi

ð12Þ

DKi

where DKi is the Knudsen diffusion coefficient of species i given as follows [24]:

DKi ¼ 0:89DKi0  1=2 2 8RT DKi0 ¼ r 0 3 pMi where r0 is the pore radius, R is universal gas constant, T is temperature and Mi is molecular weight of species i. In the continuum region, the flow is assumed to be purely viscous. The flow is described by the following transformation of Darcy’s law [16]:

1 RT g $pt ¼  2 Nt pt pt B0

ð13Þ

where B0 is permeability of the pore, and not of the porous media. Note that, the Darcy’s equation gives mass averaged velocity and hence should not be used to estimate the molar flux as done in Eq. (13). The proposed new model will oversee this assumption. The detailed description of this error is given in next section. It is assumed that for all the components, the local and pore averaged velocities are equal. With this assumption, Eq. (10) can be rewritten as follows: n n n X X 1 RT X $pt ¼  bvim v i ¼ v bvim ¼  bv Nt pt pt i¼1 im i¼1 i¼1

ð14Þ

It should be noted here that the viscous parameter is obtained with an inherent assumption of the pore average velocity being equal to the local velocity (i. e., v = vi) in Eq. (14). Also, all the components velocities are assumed to be equal. This assumption is very restrictive and should be eliminated in order to achieve a robust model as discussed later. It was suggested that, the interaction parameter bvim can be written in terms of geometry dependent part and component dependent part as follows [16]:

bvim ¼

1 1 ji /i ¼ ji xi B0 B0

ðIn this caseÞ

ð15Þ

where ji is partial viscosity. Comparing this with Eq. (13), the following relation is obtained: n X

g

i¼1

pt

ji xi ¼

ð16Þ

where the mixture viscosity g is given by Chapman–Enskog theory [25]:

g¼

n X i¼1

xi g0 Pn i j¼1 xj nij

ð17Þ

Hence the partial viscosity ji can be given as follows:

ji ¼

g0 1 P i pt nj¼1 xj nij

ð18Þ

where g0 is the pure component viscosity and nij is Lennard–Zones interaction parameter. It is assumed that the two frictions (i.e. wall friction and viscous friction) work in parallel and thus the flow resistances can be added in parallel.

1 1 1 DK B0 ¼ K þ v ¼ i þ bim bim bim xi ji xi

ð19Þ

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A new parameter fim is defined for convenience as follows:

fim

b ¼ im xi

 1 B0 ¼ DKi þ

rKim ¼ ð20Þ

ji

The final binary friction model is [16]:

 1 n X 1 ðx N  xj Ni Þ B0 RT $Tpi ¼ RT Uij i j  DKi þ Ni pt p pt D j ij i t j¼1

ð21Þ

where the parameter Uij is defined as zero in Knudsen region and as unity in continuum region. For porous media, the equation is modified to account for porosity and tortuosity of porous media. For a porous media, effective properties are used and the following expression is obtained for effective flux

2





3

eff n xi Neff X j  xj Ni s 5 $Tpi ¼ RT 4 Uij  fim Neff i D  ij j¼1

ð22Þ

In summary, the BFM presents a new framework for mass transport based on the Lightfoot equation, that can account for interspecies friction, viscous friction and Knudsen friction. Even though the framework is consistent, there are several shortcomings due to the assumption made during its derivation.  All the component velocities are assumed to be equal to the mixture velocity in order to obtain Eq. (18). For near pure diffusion cases, the assumption will lead to erroneous results. It should be noted that the solution of BFM will give different velocities for different species, which is incoherent with the assumption made to derive the very same equation.  The expression in Eq. (13) can only be obtained, under the assumption of mass average velocity being equal to molar average velocity. For gas mixtures with very different molecular weights, the mass average velocity and molar average velocity will be significantly different.  The momentum balance of the mixture is based on Darcy’s equation, which assumes a no-slip boundary condition at wall [26]. It has been known in the literature that a diffusion slip exists for multicomponent mixture flows [19,20]. The no-slip assumption can lead to erroneous estimations in multi-component mixtures. 3. Development of a modified binary friction model Due to the inherent inconsistencies in the assumptions of BFM, a new approach to determine the friction parameters have been presented, which is in line with the approach of BFM. The starting point for the new approach is again the Lightfoot equation under the assumption of ideal gas and zero external forces. For a one dimensional case along the pore, the Lightfoot equation is given as follows: n X ðxi Nj  xj Ni Þ

dxi xi dpt 1 dpi þ  ¼ RT pt df df pt df j¼1

pt Dij

 r im Ni

RT 1 DKi pt

ð25Þ

where DKi is the Knudsen diffusivity in the pore. For the continuum region, it is assumed that the mixture flow can be approximated as a pure viscous flow and can be found out by a Poiseuille approximation in a pore. In classical fluid mechanics, the axial velocity profile in a cylindrical pore is given as follows [27].

v f ðrÞ ¼ v f ðr0 Þ þ



r20  r2 4g

  dp  t df

ð26Þ

where r0 is the pore radius and g is the mixture viscosity. It should be noted that the equation is based on mass averaged velocity of the mixture. From Eq. (26), the pore averaged mass flux can be obtained by multiplication with density and integrating over the radius:

nf ¼ nf ðr 0 Þ þ



qr20 dp  t 8g df

 ð27Þ

For a non-cylindrical pore, the parameter r20 =8 can be replaced with a geometrical parameter B0 which can be interpreted as the pore permeability. The parameter nf(r0) is the flux due to slip at the pore wall, which is usually taken as zero. However, for continuum flow of multicomponent mixtures, a diffusive slip is present which is given as [19,20]:

Pn M1=2 nD nf ðr 0 Þ ¼  Pni¼1 i 1=2i j¼1 xj M j

ð28Þ

where Mi is the molecular weight, nDi is the diffusive mass flux and xi is the mass fraction of species i. Using the latter equation, Eq. (27) can be rewritten as:

Pn M 1=2 nD qB0 dpt nf ¼ Pni¼1 i 1=2i  g df j¼1 xj M j

ð29Þ

The diffusive flux, nDi can be given as:

nDi ¼ ni  xi n

ð30Þ

where ni is the total mass flux of species i and n is net mass averaged flux of the mixture. Substituting Eq. (30) in Eq. (29) and rearranging, we get:

Pn 1=2 ni dpt g i¼1 M i ¼ Pn qB0 j¼1 xj M1=2 df

ð31Þ

j

Eq. (31) can be further rearranged as follows:

dpt ¼ df

1=2 g ni Þ i¼1 qB0 ðM i Pn 1=2 M j¼1 j j

x

Pn ¼

Pn

Pn

¼

1=2 g M i ci i¼1 qB0 ðM i Pn 1=2 M j¼1 j j

x

g

i¼1 qB0

Pn

  M i1=2 qi v i

j¼1

viÞ

xj M1=2 j gM1=2 i

Xn ¼  i¼1

qB0

ð23Þ

n X

x

Ni

ð32Þ

1=2 j Mj

j¼1

Xn gM1=2 1 dpt i Ni ¼  i¼1 Pn pt df pt qB0 j¼1 xj M 1=2

where the direction f is along the pore. Summing over all the species, the following equation is obtained:

)

X 1 dpt ¼ r im Ni pt df

Comparing Eq. (33) with Eq. (24) and rearranging, we get the following expression for friction parameter in continuum region:

ð24Þ

The friction factor rim in Knudsen region is derived in the same way as of BFM. The interspecies terms in Eq. (23) are not applicable and remaining equation is compared with Knudsen equation. The friction parameter in the Knudsen region is obtained as follows:

ð33Þ

j

1=2

rvim ¼

gMi 1 RT Pn 1=2 B0 pt j¼1 pj M j

For convenience, a new parameter vi is defined as follows:

ð34Þ

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L.M. Pant et al. / International Journal of Heat and Mass Transfer 58 (2013) 70–79

gM1=2 vi ¼ Pn i 1=2

where the mixture viscosity g can be given by Chapman–Enskog theory as follows [25]: n X j¼1

xj g0j Pn k¼1 xk njk

ð36Þ

Hence, parameter vi is given as follows:

"

vi ¼ M1=2 Pn i

#" n X

1

1=2 j¼1 pj M j

j¼1

xj g0j Pn k¼1 xk njk

#

χ

5

κO

O

2

2

4 3 2 0

0.2

0.4 0.6 Mole Fraction of O

0.8

1

2

1 RT r vim ¼ vi

ð38Þ

B0 pt

Now assuming that the Knudsen and viscous friction act in parallel, the net friction parameter can be calculated as follows:

 1 1 1 1 p B0 pt RT B0 ¼ K þ v ¼ DKi t þ DKi þ ) r im ¼ r im r im rim pt RT vi RT vi

ð39Þ

Fig. 2. The comparison of O2 viscous friction parameter used in BFM and MBFM in a H2–O2 binary mixture

is to use two porous media parameters: porosity,

 and tortuosity,

s. The porosity is a scaler defined as the void volume fraction and the tortuosity is a tensor, which can be defined by the following equation:

s ¼ ½rf½I

So the final modified binary friction model (MBFM) can be given as follows:

   1 n  X pi Nj  pj Ni dpi  B0  RT DKi þ ¼ RT Ni  df T pt Dij vi j¼1

ð40Þ

By taking a closer look at Eqs. (40) and (21), it can be seen that the only difference between the BFM and the MBFM is in the viscous friction term. The parameters ji in Eq. (21) and vi in Eq. (40) are functions of mole fraction for a given mixture. Figs. 1 and 2 show the difference between ji (used in BFM) and vi (used in the MBFM) for a binary mixture of O2–N2 and O2–H2 respectively at a total pressure of 107 kPa. It can be seen that the parameters are not significantly different for O2–N2 but are significantly different for O2– H2. It appears that the assumptions made in the derivation of BFM have compensated for not having taken diffusion slip effects into account for the case of O2–N2. However, for gases with very different molecular masses, the transport predictions will be significantly different for BFM and MBFM in viscous region. 3.1. Application of MBFM in porous media So far, the development of MBFM has been a generalized approach with minimal assumptions of ideal gas and an isothermal environment. Eq. (40) represents transport of species across a pore and hence can only be used to model mass transport in capillaries. The transition from the pore scale formulation to an effective formulation is not exact and straightforward. A simple approach

Parameter Value × 1010

6

1

ð37Þ

Therefore, the viscous friction parameter can be given as:

ð41Þ

For a direction z it can be given as [20,28]:

szz ¼

df dz

ð42Þ

where f is direction along the pore and z corresponds to the normal of the porous media surface over which effective flux is evaluated. For an orthogonal co-ordinate system, non-diagonal components  will be zero. of s The effective flux on a porous media surface can be given as follows [28]:

Neff ¼ ½s1 N

ð43Þ

The volume fraction  accounts for the unavailable area for mass transport or unavailable volume for the species. The tortuosity accounts for the higher residence time of species inside porous media compared to a straight pore [28]. Hence the effective flux is less than what it would have been if there was no porous media, as  < 1 and ksk > 1. Using the definition of effective flux and tortuosity, Eq. (40), which is along the pore direction f, can be transformed along the porous media direction z by using chain rule of differentiation as follows:

   dpi  dpi  dz dpi  1 ¼ ¼ df T dz T df dz T szz   1 n  X pi Nj  pj Ni B0  RT DKi þ ¼ RT Ni pt Dij vi j¼1

ð44Þ

Now using Eq. (43), the flux along the pore axis can be converted to flux along porous media. With this, Eq. (44) can be rearranged as follows:

2

!  eff n X pi Neff dpi  j  pj N i  RT ¼ RT dz T pt s2 Dij j¼1 zz

1.5 1

χ

O

2

O

2

0

0.2

0.4 0.6 Mole Fraction of O2

0.8



 K s B0 D þ s2zz i vi 2 zz

!1 Neff i

ð45Þ

The effective diffusivity, effective Knudsen diffusivity and permeability in a porous media are tensors, which are respectively given as follows:

κ

0.5 0

Parameter Value × 10

10

j¼1 pj M j

g¼

7

ð35Þ

1

Fig. 1. The comparison of O2 viscous friction parameter used in BFM and MBFM in a O2–N2 binary mixture.

 2 Deff ij ¼ Dij ½s DKi

eff

eff

2 ¼ DKi ½s 2

 Bv ¼ B0 ½s

ð46Þ

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L.M. Pant et al. / International Journal of Heat and Mass Transfer 58 (2013) 70–79

!   1 eff n X pi Neff dpi  Bv zz j  pj N i K eff ¼ RT þ Neff  RT D izz i eff dz T v p D i j¼1 t

ð47Þ

0.2

$T pi ¼ RT

j¼1

pt

i1

"  RT

eff DKi

þ

Bv

vi

−0.1 −0.2 −0.3 He

−0.5

By following similar approach in other two co-ordinates, the vector form of this equation can be given as:

Deff ij

0

−0.4

ijzz

! eff h n X pi Neff j  p j Ni

Ar

0.1 −2 −1 Flux (mol⋅ m ⋅ s )

In literature, usually /s has been used to obtain the effective properties, which is theoretically incorrect [20,28]. For experimental measurements of the effective properties in literature, it does not change the results, as the measured effectiveness parameter was in reality /s2. Now, the z  component of the tensors in Eq. (46) can be replaced in Eq. (45) to obtain the following equation:

#1

2

10

10

3

4

10 Total pressure (Pa)

10

5

Fig. 3. Comparison of experimental flux measurements with MBFM flux predictions for a binary diffusion of He–N2 mixture across capillaries. Lines show MBFM estimation and symbols show experimental data [29]

Neff i ð48Þ

0.2

Ne Ar

0

−2

Flux (mol⋅ m ⋅ s−1)

The modified binary friction model (MBFM) takes into account only three assumptions: isothermal environment, ideal gas behaviour and no external forces. These assumptions can be practically achieved in an experimental setup and thus the MBFM can be considered a realistic model. 4. Validation of the MBFM To verify the accuracy of MBFM, experimental measurements from the literature are analyzed with the new model. The MBFM equations for different species were solved with equations of continuity using MATLAB bvp4c solver. Two different cases are discussed here: the first is of flow in capillaries to validate the generalized MBFM, and second is the study in porous media to validate the effective model. 4.1. Validation of MBFM for capillaries To validate the generalized model, the measurements of binary and ternary diffusion in capillaries done by Remick and Geankoplis [29,30] were modelled. They performed the experiments with capillaries of diameter 39.1 lm and length 96 mm, more details of the experiment can be found in the literature [29,30]. Fig. 3 shows the comparison of MBFM predicted flux with experimental measurements for a binary diffusion of He–N2 across the capillaries. The model shows excellent agreement with the experimental data. Further, using MBFM, the predicted flux ratio of the species for different pressures is NNNHe ¼ 2:6458 for all the 2

−0.2

−0.4

He

−0.6

2

10

3

10 10 Total pressure (Pa)

4

10

5

Fig. 4. Comparison of experimental flux measurements with MBFM flux predictions for a ternary diffusion of He– Ne–Ar mixture across capillaries. Lines show MBFM estimation and symbols show experimental data [30]

ues given by EvansIII et al. [31]. Diffusion of helium and argon under uniform pressure was modelled at different pressures. Fig. 5 shows the comparison of MBFM estimation with experimental results [31]. The model results agree quite well with the experimental results, except in the transition Knudsen region where the errors are higher. Another study by Evans III et al. [32] on convection–diffusion was modelled as well. Figs. 6 and 7 show the comparison of MBFM estimation with experimental results [32] for convection–diffusion experiments. It is seen that MBFM is able to predict the fluxes accurately.

cases. This shows excellent agreement with Graham’s law   qffiffiffiffiffiffiffi MN NHe 2 ¼ 2:6458 . ¼ NN MHe 2

4.2. Validation of MBFM in porous media To study the MBFM in porous media, experimental studies by EvansIII et al. [31,32] on graphite were modelled. The graphite plug was of length 47 mm, with a porosity of 11% and the ratio /s(/s2 in current case) was 1.42  104. The effective Knudsen diffusivity of helium was given as 3.2  108 m2/s and the molecular diffusivity was obtained from Reid et al. [25]. A permeability value of 2.35  1018 m2 was used corresponding to the experimental val-

20 Species flux (mol⋅ m−2⋅ s−1)

Fig. 4 shows the comparison of MBFM estimation of fluxes with experimental measurements for a ternary diffusion of He–Ne–Ar across capillaries. It can be seen that the theoretical results are in excellent agreement with experimental data. It was also observed that removing diffusion slip from consideration causes large errors in the continuum region, where the slip exits.

x 10

−5

He

15

He+Ar

10 5 0

Ar −5 0

1

2 3 4 5 6 7 Average Pressure between channels

(Bar)

8

Fig. 5. Flux of He, Ar and mixture through a graphite septum for different pressures. Lines show MBFM estimation and symbols show experimental data [31]

L.M. Pant et al. / International Journal of Heat and Mass Transfer 58 (2013) 70–79

The experimental results of Evans III et al. [31,32] were also modelled by Kerkhof [16] using BFM. They also show a good agreement with the experimental results. One of the reason for this is the coincidental similarity of viscous friction parameter between BFM and MBFM due to assumptions in BFM derivation as discussed earlier. Another reason for the good agreement is the adjustment of parameters done during modelling. Kerkhof [16] has used slightly different values of transport parameters compared to the actual values from experiments, resulting in a better fit than the one which would have been obtained using real parameters. The current validation studies show that MBFM is an accurate and robust mass transport model for capillaries as well as porous media. The accurate accounting of the physical phenomena contributes toward the effectiveness of MBFM. This new improved model is used to study mass transport in PEMFC porous media in next section.

−4

x 10 Species flux (mol⋅ m−2⋅ s−1)

76

1.5 He

1 0.5

He+Ar

0 −0.5

Ar

−1 −1.5 0

0.05

0.1 0.15 Pressure Difference Δ p (Bar)

0.2

Fig. 7. Flux of He, Ar and mixture through a graphite septum for different pressure differences at an average pressure of 2.97 atm. Lines show MBFM estimation and symbols show experimental data [32]

5. Study of mass transport in PEMFC porous media This section presents a study of mass transport in PEMFC porous media. The studies are used to highlight specific differences between previous models and MBFM. It also presents the operating conditions where the MBFM will provide considerably different results from other models. Further, a few parametric studies using MBFM are also presented to highlight the effect of operating parameters on mass transport. The schematic of the problem is shown in Fig. 8. A one-dimensional mass transport study was performed using experimentally obtained transport properties of a SGL SIGRACET 34 BC micro porous layer (MPL) deposited on a SIGRACET 34BA gas diffusion layer (GDL). The simulation parameters used are shown in Table 1. The differential equations of mass transport were solved for all the models on a binary mixture of oxygen and nitrogen (pure oxygen on one side and pure nitrogen on other side of porous media). The total pressure across the porous media is also varied to study convective–diffusive transport in porous media.

5.1. Comparison of different mass transport models for study in PEMFC porous media Fig. 9 shows the comparison of different mass transport model predictions for transport across MPL with the operating conditions given in Table 1. At very low pressure differences (near pure diffusion), all the models have similar predictions. For higher pressure differences, Fick’s law overpredicts the mass transport due to not accounting for interspecies friction forces. The viscous friction

−2 −1 Species flux (mol⋅ m ⋅ s )

x 10

−4

2 1

He He+Ar

0 Ar −1 −2 −0.3

−0.2

−0.1 0 0.1 0.2 Pressure Difference Δ p (Bar)

0.3

0.4

Fig. 6. Flux of He, Ar and mixture through a graphite septum for different pressure differences at an average pressure of 1.49 atm. Lines show MBFM estimation and symbols show experimental data [32]

MPL1

MPL2

Fig. 8. Schematic of the transport problem

parameters v and j for O2–N2 mixture are very close hence the BFM and MBFM have similar predictions. To see the effect of pore radius on the mass transport predictions of each model, a new simulation was carried out with a smaller pore radius of 40 nm and correspondingly smaller permeability of 1.187  1014 m2. All other parameters are the same as in Table 1. Fig. 10 shows the comparison of all the mass transport models. For a pore radius of 40 nm, most of the transport will be in Knudsen region. However, due to not accounting for Knudsen friction, Fick’s law still considers the transport in molecular diffusion region. Therefore it can be seen that Fick’s law overpredicts for the entire pressure difference range. The DGM, due to its derivation errors, highly overestimates the Knudsen (wall) friction in this case and hence underpredicts the net transport considerably. For O2– N2 mixture, BFM and MBFM still show similar predictions with only a slight underprediction for the BFM. To study the effect of species molecular weight, a new simulation was performed with H2 and H2O binary mixture. This mixture is present in the anode of a PEM fuel cell. All the parameters are the same as in Table 1. Fig. 11 shows the comparison of different mass transfer models. It was observed that MBFM satisfied Graham’s law in each of the above cases, while other models were not able to do so.

L.M. Pant et al. / International Journal of Heat and Mass Transfer 58 (2013) 70–79 Table 1 Simulation parameters used for mass transport model comparison. Value

Reference

Thickness [m]

2  63  106

Porosity () Permeability (Bv), [m2]

42.2% 1.187  1013

Tortuosity factor (s2)

0:5 MPL

Experimental measurements [11] Catalogue Experimental measurements [11] Bruggeman correlation

Pore size (r0), [m] Temperature, [°C] Pressure difference, Dp, [kPa]

9

200  10 25 2[0, 20]

150 H Flux (mol⋅ m−2⋅ s−1) 2

Parameter

77

Literature [23,33] Experiment

Ficks & Darcy MBFM BFM DGM

100

50

0

0

5 10 15 Pressure Difference across porous media (kPa)

20

Fig. 11. Prediction of hydrogen flux by different transport models across MPL with varying net pressure difference for a mixture of H2–H2O.

2

O Flux (mol⋅ m−2⋅ s−1)

40

Ficks & Darcy MBFM BFM DGM

30 20 10 0

0

5 10 15 Pressure Difference across porous media (kPa)

20

Fig. 9. Prediction of oxygen flux by different transport models across MPL with varying net pressure difference (rp = 200 nm) Fig. 12. Effect of temperature on mass transport of oxygen across MPL.

O2 Flux (mol⋅ m−2⋅ s−1)

5

5.3. Effect of concentration gradient

Ficks & Darcy MBFM BFM DGM

4 3 2 1 0

5 10 15 Pressure Difference across porous media (kPa)

20

To study the effects of concentration gradient, the simulations were carried out at three conditions; pure O2 in side 1 and pure N2 in side 2, 75% O2 and 25% N2 in side 1 and 25% O2 and 75% N2 in side 2, and 50% O2 and 50% N2 on both sides of porous media. All other parameters are the same as in Table 1. Fig. 13 shows the comparisons of flux predictions at different concentration gradients. As the concentration gradient (chemical potential gradient) is reduced, the driving force for transport also reduces and hence the flux of oxygen reduces. For the case of equal composition on both sides, it can be seen that the pure diffusive flux is zero.

Fig. 10. Prediction of oxygen flux by different transport models across MPL with varying net pressure difference for smaller pores (rp = 40 nm).

40

2

To study the effects of temperature on the mass transport predictions using MBFM, the simulations were carried out with MBFM at an operating temperature of 25, 45 and 90 °C. All other parameters remain the same as in Table 1. Fig. 12 shows the transport predictions by MBFM at different temperatures. It can be seen that for near pure diffusion, the fluxes for higher temperatures are higher. However, for high pressure differences, the fluxes are lower for higher temperature. At higher temperatures, the diffusivity values increase and the corresponding molecular and Knudsen diffusion increases. However, with increasing temperature, viscosity of the gas also increases, which reduces the viscous transport. For very high pressure differences, transport is governed by viscous forces and and therefore is lower in this case.

Case 1 30

Case 2

−2

−1

O Flux (mol⋅ m ⋅ s )

5.2. Effect of temperature

Case 3 20

10

0

0

5 10 15 Pressure Difference across porous media (kPa)

20

Fig. 13. Effect of concentration gradient on mass transport of oxygen across MPL. Case 1: pure O2 on side 1 and pure N2 on side 2, Case 2: 75% O2 + 25% N2 on side 1 and 25% O2 + 75% N2 on side 2, Case 3: 50% O2 + 50% N2 on both sides.

78

L.M. Pant et al. / International Journal of Heat and Mass Transfer 58 (2013) 70–79

P qv Pi i q P i ci v i Molar averaged velocity : v ¼ P ci X X qi v i Mass averaged flux : n ¼ ni ¼ X X Molar averaged flux : N ¼ Ni ¼ ci v i X ci Mixture concentration : ct ¼

Mass averaged velocity : 200 nm 120 nm 40 nm

30

−2

−1

O2 Flux (mol⋅ m ⋅ s )

40

20

10

0

0

5 10 15 Pressure Difference across porous media (kPa)

v ¼

ðA:1Þ ðA:2Þ ðA:3Þ ðA:4Þ ðA:5Þ

20

Fig. 14. Effect of pore size on mass transport of oxygen across MPL.

5.4. Effect of pore radius To study the effect of pore radius on mass transport, three cases are considered; r0 = 200 nm, r0 = 120 nm and r0 = 40 nm. The permeability will also be reduced with reduction in pore size by the ratio of r20 . All other parameters are the same as in Table 1. Pure O2 flows on side 1 and pure N2 flows on side 2 of porous media. Fig. 14 shows the flux predictions at different pore radius values. It can be seen that the flux reduces as the pore radius reduces due to to the increase of wall friction (viscous and Knudsen). For the pore size of r0 = 40 nm, the friction is so high that increasing the net pressure difference has a negligible effect on mass transport.

6. Conclusion A new multicomponent mass transport model named modified binary friction model (MBFM) is presented. The new model accounts for all driving and friction forces in porous media and corrects for inconsistencies and limiting assumptions in previous models. The limitations of previous models are illustrated with theoretical as well numerical examples. The new model accounts for diffusion slip in the continuum region, which is often neglected in previous models. The validation studies show that this model is able to reproduce the experimental results in Knudsen, continuum and transition region. A comparison of previous models with the proposed model shows that the conventional models can provide accurate results in a few limiting cases only. The new model is capable of predicting mass transport over a wider range of operating conditions in various types of porous media. The versatility of the MBFM makes it an ideal choice for study of mass transport in fuel cells.

Acknowledgement The authors thank University of Alberta and the Natural Science and Engineering Research Council of Canada for financial assistance. The authors would also like to acknowledge RR Gilpin memorial scholarship for financial assistance.

Appendix A. Definition of fluxes and driving forces This section defines the definition of various velocities and fluxes. Lets qi, ci and vi be the density, concentration and velocity of species i respectively. Then the fluxes and velocities are defined as follows [9]:

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