Transport Equations

In the following, transport equations for mass and energy are formulated for (M). The charge transport is expressed in terms of the proton flux. Mass transport is described using the Maxwell-Stefan approach, as has already been done for the diffusion layers. But in the membrane, as described in the previous chapters, a more complex formulation of the driving forces has to be chosen. First, the migration term has to be included, as one of the mobile species (protons) is charged and an electric field is present within the membrane. Second, the diffusive term has to account for the highly non-ideal behaviour of the mobile species within the membrane pores, i.e. the gradients of the chemical potentials have to be used as driving force, which are equal to the gradients in the species activities (calculated using the presented

128 6 Model Formulation and Simulations Flory-Huggins approach). The pressure-dependency of the chemical potentials can be neglected as this is an incompressible (liquid) system. Also, as discussed in the introduction of chapter 6.9, viscous flow due to pressure differences over the membrane can be neglected.

All this leaves the following form of the Maxwell-Stefan equations for the mobile species (j = H⁺, H2O, CH3OH):

c^M_j 1 a^M_j

a^M_j

z z^*_jc^M_j F R T^M

M

z =

i j

x_i^Mn^M_j x^M_j n_i^M D_ij^{M , eff}

n^M_j

D^{M , eff}_{j M} (6-102) In eq.(6-102), as superficial viscous flow due to pressure gradients is not accounted for, the total molar flux densities ni appear in the friction terms on the right hand side of the equation.

Three of the six binary diffusion coefficients D_ij^{M , eff} are taken from the literature [64], the others are estimated (see appendix chapter 9.1.8 for details).

As the flux density of protons n_H^M+ is given by the electric current density i^cellusing Faraday's law

n_H^M+=i^M F =i_cell

F , (6-103)

only two unknown fluxes have to be determined (for water and methanol).

As presented for (AD) and (CD), the flux-implicit transport equations,eq.(6-102) , can be transformed into a flux-explicit form by formulating them as vector equations. But here, for only two mobile species, it is also possible to easily obtain an explicit formulation for the flux densities by simple rearranging.

The Maxwell-Stefan equations for water and methanol are:

c_H^M_2O a_H^M_2O

a_H^M_2O

z = x_H^M+n_H^M_2O x_H^M_2On_H^M+ D_H⁺_{, H}

2O M , eff

x_CH^M _3OHn_H^M_2O x_H^M_2On_CH^M _3OH D_H^{M , eff}_{2O , CH3OH}

n_H^M_2O D_H^{M , eff}_{2O , M}

LH2O (6-104)

c_CH^M _3OH a_CH^M _3OH

a_CH^M _3OH

z = x_H^M+n_CH^M _3OH x_CH^M _3OHn_H^M+ D_H⁺_{, CH}

3OH

M , eff

x_H^M_2On_CH^M _3OH x_CH^M _3OH n_H^M_2O D_H^{M , eff}_{2O , CH3OH}

n_CH^M _3OH D_CH^{M , eff}_{3OH , M}

LCH3OH (6-105)

6 Model Formulation and Simulations 129 One can see that the middle terms on the right hand sides are negative identical, so adding up both equations eliminates those terms, and by simple rearranging one obtains the methanol flux density:

n_CH

3OH

M =L_H2O L_CH3OH n_H⁺L₁ n_H2OL₂

L₃ (6-106)

with L₁= x_H

2O

D_H⁺_{, H}

2O eff

x_CH

3OH

D_H⁺_{, CH}

3OH

eff , (6-107)

L₂= x_H⁺ D_H⁺_{, H}

2O eff

1

D_H^eff_{2O , M} and (6-108)

L₃= x_H⁺ D_H⁺_{, CH}

3OH

eff

1

D_CH^eff _{3OH , M} . (6-109)

Inserting this into eq.(6-104), one obtains the water flux density:

n_H

2O

M =

L_H2O n_H⁺ L₄ L₁L₆

L₃ L₂ L₃ L₆ L₃ L₅ L₂L₆

L₃

(6-110)

with L₄= x_H

2O

D_H⁺_{, H}

2O

eff , (6-111)

L₅= x_H⁺ D_H⁺_{, H}

2O eff

x_CH

3OH

D_H

2O , CH3OH eff

1 D_H

2O , M

eff and (6-112)

L₆= x_H

2O

D_H^eff_{2O , CH3O H} . (6-113)

With these equations, all necessary transport equations are available. What remains yet unknown is the electric potential gradient ^M/ z in the membrane material, due to the transport resistance to the proton flux (“Ohmic drop” over membrane). It is calculated from the Maxwell-Stefan transport equation for the protons:

c_H^M+ a_H^M+

a_H^M+ z

c_H^M+F RT^M

M

z = x_H^M_2On^M_H⁺ x_H^M+n_H^M_2O D_H⁺_{, H}

2O M , eff

x_CH^M _{3O H}n_H^M+ x_H^M+n_CH^M _{3O H} D_H⁺_{, CH}

3O H M , eff

n_H^M+ D_H^{M , eff+}_{, M} .

L_H⁺ (6-114)

130 6 Model Formulation and Simulations Rearranging yields the electrical potential gradient in the membrane:

M

z = R T^M

c_H^M+F L_H⁺ n_H⁺L₇ n_H

2OL₈ n_CH

3OHL₉ (6-115)

with L₇= x_H

2O

D_H⁺_{, H}

2O eff

x_CH

3OH

D_H⁺_{, CH}

3OH eff

1

D_H^eff+_{, M} , (6-116) L₈= x_H⁺

D_H⁺_{, H}

2O

eff and (6-117)

L₉= x_H⁺ D_H⁺_{, CH}

3OH

eff . (6-118)

The problem with eq.(6-115) is, that for the protons no Flory-Huggins non-ideality parameters are known. Therefore, as a first approach, the activity of protons is approximated by the mole fraction of protons in the pore liquid, x_H^M+.

As no superficial convective flow is calculated in the membrane, the question might arise whether electro-osmotic flow is correctly accounted for in the presented model. Basically electro-osmosis is transport of all mobile species in a mixture due to drag exerted on uncharged species by the charged species which move along an electric field. In the presented model, these phenomena are accounted for: Protons move along the electric field (migration) and exert friction on the uncharged species water and methanol (friction terms). One can show that the presented model, with respect to electro-osmotic flow, is equivalent to an often used approach first proposed by S^CHLÖGL [100] for combined electro-osmotic and pressure-driven convective transport:

v= k c_H^M+F

H⁺/H2O

vis z

k_p

H⁺/H2O vis

p

z . (6-119)

In eq.(6-119) the superficial velocity v is the sum of an electro-osmotic term proportional to the gradient of the electric potential φ and one term for pressure-driven flow proportional to the gradient of the pressure p. Both terms feature a permeability parameter k (electrokinetic and hydraulic permeability, kφ and kp [m²], respectively) and the viscosity of the pore fluid

H⁺/H₂O

vis [kg m^-1 s^-1]. Other parameters are the proton concentration in the pores c_H^M+ [mol m^-3] and Faraday's constant F (96485 A s mol^-1).

6 Model Formulation and Simulations 131 As here only electro-osmosis shall be discussed, the pressure-driven term is skipped:

v= k c_H^M+F

H⁺/H2O

vis z = f_Schlögl

z (6-120)

For fully hydrated NAFION^TM at 25°C the parameters can be found in the literature [101]:

k =1.13 10 ¹⁹m² ; c_H⁺=1200 mol m ³ ;

H⁺/H2O

vis =3.353 10 ⁴kg m ¹s ¹ .

With these values, the proportionality factor in S^CHLÖGL'^S equation becomes

f _Schlögl=3.9 10 ⁸ A s²kg ¹ . (6-121)

Formulating the presented Maxwell-Stefan model (eq.(6-102)) for a fully hydrated membrane (i.e. no activity gradients) and only water and protons as mobile species (no methanol), one obtains the two transport equations:

H⁺: ^z_H^{* +c}_H^{M+ F} R T^M

M

z = x_H

2O

M n_H^M+ x_H^M+n_H

2O M

D_H⁺/H₂O eff

n_H^M+ D_H⁺/M

eff and (6-122)

H2O: ^{0= xH+}

M n_H

2O

M x_H

2O M n_H^M+ D_H⁺/H2O

eff

n_H

2O M

D_Heff_2O/M . (6-123)

As only water and protons are assumed to be within the pores, their mole fractions sum up to unity:

x_H^M+ x_H

2O

M =1 . (6-124)

Combining these three equations and rearranging yields an expression for the molar proton flux density as function of the proton concentration and the electric potential gradient:

n_H^M+= 1

H⁺/H2O

c^M_H⁺F R T^M

M

z (6-125)

The binary diffusion coefficients and the proton mole fraction are combined in the parameter

H⁺/H2O= 1 x_H^M+ D_H⁺_/H

2O

x_H^M+ 1 x_H⁺ x_H^M+ D_H⁺/H₂O

D_H

2O/M

D_H⁺/H₂O

1

D_H⁺_/M (6-126)

which has the dimension [s m^-2]. To transform this into a form which is comparable to S^CHLÖGL'^Sequation, the conversion between molar flux densities nⁱand velocity v is performed needing the pore concentration c_H^M+ and the pore volume fraction ^M_pores:

n_H^M+=c_H^M+v=

pores

M c_H^M+v . (6-127)

132 6 Model Formulation and Simulations This yields an expression for the electro-osmotic velocity

v= n_H^M+

pores

M c_H^M+= F

pores M

H⁺/H2OR T

M

z = f_MS

M

z (6-128)

whose structure is equivalent to S^CHLÖGL'^S equation for pure electro-osmotic flow, eq.(6-120).

Using the literature values for the three binary diffusion coefficients [64] (appendix chapter 9.1.8) and the pore volume fraction of fully hydrated NAFION^TM at 25°C, ^M_pores⁼0.47, one obtains a proportionality factor of

f _MS=3.3 10 ⁸A s²kg ¹

which is close to the S^CHLÖGL proportionality factor f^Schlögl, given by eq.(6-121). Obviously the presented Maxwell-Stefan transport model accounts for electro-osmotic transport in a way which in the end yields the same results as the classical approach.

The advantage of the Maxwell-Stefan approach now lies in the following two facts. First, even in multi-component liquid mixtures it accounts for the individual electro-osmotic influence on each mobile species (represented by their binary diffusion coefficients with protons, the pore walls and water as main components). Second, the (strong) temperature and concentration dependence of the S^CHLÖGL-parameter k is automatically accounted for in the Maxwell-Stefan model, as long as the diffusion coefficients are formulated as functions of temperature and water content. Especially the latter advantage is very important, as high gradients in the local relative water content can occur within the membrane.

For the energy balance in (M), transport equations are needed for thermal conduction and convective heat transport. Both are similar to the equations for (AD) and (CD):

Conductive heat flux: q^M= ^{M , eff} T^M

z , (6-129)

Enthalpy flux (convective heat flux): ^eM=

j

n^M_j h_j T^M . (6-130)

The effective thermal conductivity ^M,eff (simplifyingly assumed fully hydrated NAFION^TM) and the specific enthalpies h^j are calculated in the appendix chapters 9.1.3 and 9.1.4, respectively.

Im Dokument Experimental and model-based analysis of the steady-state and dynamic operating behaviour of the direct methanol fuel cell (DMFC) (Seite 145-150)