Keywords

1 Introduction

Cracks formation on CLs during the operational process mostly bring unexpected defects, mostly regarded as degradations and failures. These defects occur at the crack formation reign, including crack propagation to membrane pinholes, increasing in-plane (IP) resistance (RIP) of CLs, formation of flooding areas and catalyst degradation [1]. Some operation conditions are prone to crack initiation, such as freeze-thaw (F-T) cycles [2], dry-wet cycles [3], cyclic loadings [4], and ionomer degradation [5]. However, some work has claimed the mass transfer enhancement by the cracks on CL [6,7,8].

In this work, to understand the crack effect on the PEMFC catalyst layer IP diffusivity, a lattice Boltzmann method D2Q9 model was employed. The normalized concentration of air in the cracked CL was acquired, and the effective diffusivity of the complex crack zone was calculated by the flux field. This model was validated by the analytical result and shows relatively high accuracy. The effects on some key parameters like crack length, crack width, crack quantity and crack shape were carried out.

2 Random Crack Domain Regeneration

A processed base crack regeneration method was employed to obtain a computational field. The details of this method were described in our previous work [9]. The circle cracks were introduced in this simulation because some former work showed a significant enhancement in mass transfer using this CL structure [10]. The non-crack area (NCA), the intact porous catalyst layer, and the crack area (CA), the chapped crack, were defined in the computational domain. These two different regions have different properties on diffusion.

3 Physical and Mathematical Model

3.1 Physical Description

In a cracked CL, the effective diffusivity Deff is determined by the NCA phase and CA phase properties. Although the exact value of the Deff is hard to estimate, its value is in the range of DCA and DNCA.

$$ \begin{array}{*{20}c} {D^{eff} \in \left[ {D_{CA} ,{ }D_{NCA} } \right] } \\ \end{array} $$
(1)

The NCA and CA phase in this research are porous CL structure and air, respectively. The mass diffusivity \(D _{O_{2}}^{p}\) in the nanostructure is reported to be sensitive to the porous CL structure due to the effect of Knudsen diffusion [11], which is shown as

$$ \begin{array}{*{20}c} {\frac{1}{{D _{O_{2}}^{p} }} = \frac{1}{{D_{Kn} }} + \frac{1}{{D_{O_{2}} }}} \\ \end{array} $$
(2)

The \(D_{O_{2}}\) is oxygen diffusivity is equal to the DNCA, and the \(D_{Kn}\) is Knudsen diffusivity. These values can be calculated as

$$ \begin{array}{*{20}c} {D_{O_{2}} = 0.22 \times 10^{ - 4} \frac{{(T/293.2{\text{K}})^{1.5} }}{{p/1\,{\text{atm}}}}} \\ \end{array} $$
(3)
$$ \begin{array}{*{20}c} {D_{Kn} = 4850d_{p} \sqrt{\frac{T}{M}} } \\ \end{array} $$
(4)

where p is the absolute pressure in atm, and is set to 2.5 atm (absolute pressure). T is temperature in K, and is set to 353 K. M is the molecular weight of oxygen in g/mol, and is set to 32 g/mol. dp is the characteristic pore size in cm [11]. The mean dp is used in the modeling and an empirical correlation of the mean dp is given as

$$ \begin{array}{*{20}c} {d_{p} = 140.50\varepsilon^{2} + 68.35\varepsilon + 38.08} \\ \end{array} $$
(5)

To obtain the diffusivity DNCA in the NCA media, a Bruggeman-like power law is employed as [11]

$$ \begin{array}{*{20}c} {D_{NCA} = 1.07 \varepsilon^{1.75} \times D_{O_{2}}^{p} } \\ \end{array} $$

Combining Eq. (2)–(6), the diffusivity of NCA media at different porosities are acquired. In this research, the porosity is in 0.25~0.45.

3.2 Governing Equations

A 2D model is employed for the IP mass transfer analysis. Figure 1 illustrates the computational domain of a cracked CL, and the governing equations for mass diffusion can be expressed as following equations,

$$ \begin{array}{*{20}c} {\frac{\partial C}{{\partial t}} = D_{NCA} \left( {\frac{{\partial^{2} C}}{{\partial x^{2} }} + \frac{{\partial^{2} C}}{{\partial y^{2} }}} \right)} \\ \end{array} $$
(7a)
$$ \begin{array}{*{20}c} {\frac{\partial C}{{\partial t}} = D_{CA} \left( {\frac{{\partial^{2} C}}{{\partial x^{2} }} + \frac{{\partial^{2} C}}{{\partial y^{2} }}} \right) } \\ \end{array} $$
(7b)

The boundary conditions are described as

$$ \begin{array}{*{20}c} {\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {C = C_{1} ,{ }} & {y = L,{ }0 \le x \le L} \\ \end{array} } \\ { \begin{array}{*{20}c} {C = C_{2} ,{ }} & {y = 0,{ }0 \le x \le L} \\ \end{array} } \\ {f_{1,5,8} \left. \right|_{x = 0} = f_{1,5,8} \left. \right|_{x = L} } \\ {f_{3,6,7} \left. \right|_{x = L} = f_{3,6,7} \left. \right|_{x = 0} } \\ \end{array} } \right. } \\ \end{array} $$
(8)

where C is the concentration. The top and bottom boundary are maintained at \(C_{1}\) and \(C_{2}\) using famous Zou-he boundary conditions [12]. To ensure stability of the numerical method, a small difference which is 0.01 is set between these two boundaries. The periodic boundary is set at the left and right boundaries as previous works [13].

Fig. 1
figure 1

Computational domain of cracked CL.

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3.3 2D LBM Model Description

In this research, a two-dimensional (2D) LBM single relaxation time (SRT) model is employed to solve the mass transfer phenomenon in different phases. The density distribution function gα evolution equation for a two-dimensional nine-speed (D2Q9) LBM in both NCA and CA can be given as,

$$ \begin{array}{*{20}c} {g_{\alpha } \left( {{\varvec{r}} + {\varvec{e}}_{\alpha } \delta_{t} , t + \delta_{t} } \right) - g_{\alpha } \left( {{\varvec{r}}, t} \right) = - \frac{1}{\tau }\left[ {g_{\alpha } \left( {{\varvec{r}}, t} \right) - g_{\alpha }^{eq} \left( {{\varvec{r}}, t} \right)} \right]} \\ \end{array} $$
(9)

where r is the position vector, t is the real time, δt is the iteration time step, gαeq is the equilibrium distribution function of the evolution equation.

The gases phase in the NCA is assumed to be static, because the Peclet Number (Pe) in the porous structure is relatively low. Thus, the gαeq shows as [14].

$$ g^{eq} \left( {x, t} \right) = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {\text{i = 0}} \hfill \\ {\frac{{1}}{{6}}C{,}} \hfill & {\text{i = 1 - 4}} \hfill \\ {\frac{{1}}{{{12}}}C{,}} \hfill & {\text{i = 5 - 8}} \hfill \\ \end{array} } \right. $$
(10)

The velocity vector ei for D2Q9 scheme is

$$ {\varvec{e}}_{i} = \left\{ {\begin{array}{*{20}l} {\left( {0, 0} \right)} \hfill & {i = 0} \hfill \\ {{\varvec{e}}\left( {\cos \left[ {\frac{\pi }{2}\left( {i - 1} \right)} \right],\sin \left[ {\frac{\pi }{2}\left( {i - 1} \right)} \right]} \right)} \hfill & {i = 1, 2, 3, 4} \hfill \\ {\sqrt 2 {\varvec{e}}\left( {\cos \left[ {\frac{\pi }{2}\left( {i - 1} \right)} \right],\sin \left[ {\frac{\pi }{2}\left( {i - 1} \right)} \right]} \right)} \hfill & {\sqrt 2 i = 5, 6, 7, 8} \hfill \\ \end{array} } \right. $$
(11)

and τ is the dimensionless relaxation time for NCA and CA, which is determined by the diffusivity coefficient [15],

$$ \begin{array}{*{20}c} {\tau_{CA} = \frac{3}{2} \frac{{D_{CA} }}{{\left( {\rho c} \right)_{CA} \cdot c^{2} \delta_{t} }} + 0.5} \\ \end{array} $$
(12)

and

$$ \begin{array}{*{20}c} {\tau_{NCA} = \frac{3}{2} \frac{{D_{NCA} }}{{\left( {\rho c} \right)_{NCA} \cdot c^{2} \delta_{t} }} + 0.5 } \\ \end{array} $$
(13)

where c is a pseudo sound speed, DCA and DNCA are the diffusivity coefficient in crack area and non-cracked area, respectively. To ensure the continuities of the flux and density distribution functions at the interface of different phases, the \(\rho c\) is set to 1 in this model [16]. Then the calculation of macroscopic quantities of the density and flux can be obtained by

$$ T = \mathop \sum \limits_{i} f_{{i{ }}} ,{ } $$
(14)
$$ J = \left( {\mathop \sum \limits_{i} c_{i} f_{i} } \right)\frac{\tau - 0.5}{\tau }{ }. $$
(15)

At the end of each computational loop, the effective diffusivity coefficient Deff can be obtained by Fick’s law [17, 18]:

$$ D^{eff} = \frac{{L_{x} \smallint JdA}}{{\left( {C_{2} - { }C_{1} { }} \right)\smallint dA}}.{ } $$
(17)

where C2 and C1 are the boundary values of inlet and outlet boundaries, respectively. Lx is the distance between these two boundaries, and A is the computational width of the domain.

The normalized effective diffusivity coefficient \(D^{n}\) is defined as [19],

$$ \begin{array}{*{20}c} {D^{n} = D^{eff} /D_{NCA} . } \\ \end{array} $$
(18)

The DNCA is equal to the \(D^{O_{2}}\), which is the oxygen diffusivity in bulk. This normalized method makes the crack effect obviously on the diffusivity change of the CLs.

4 Results and Discussion

4.1 Model Validation

Two typical structures, the “parallel” and the “series” model which are introduced in the previous work are used to validate the LBM model [20]. In the model validation section, the CR is set to 0.5. The DCA/DNCA value is set to 1–1000. Meanwhile, the domain side length effect is also taken into consideration which is set from 300 to 5000 μm at the resolution of 10 μm. The model validation result is shown in Fig. 2. When the domain side length is small, the error value is big, and the error is sensitive to the ratio between two conductivity values. To balance the computational consumption and an error less than 1%, the domain side length is set to 3000 μm, and the conductivity ratio is limited at 1:100. In another word, the representative element volume (REV) in this case is 3000 μm × 3000 μm.

Fig. 2.
figure 2

Comparison between model prediction value and theoretical value.

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4.2 IP Effective Transfer Properties Influenced by Cracks

Normalized Concentration with Diffusion Time. To evaluate the effect of diffusion time on normalized concentration in the crack area, a bunch of simulations are designed. The porosity of NCA is 0.3, and five different crack quantities are selected from 20 to 180. The triangle crack with 25 μm crack width are reconstructed, and every crack arm length is random from 0 to 200 μm. As shown in Fig. 3, the average temperature at each coordinate with 180 cracks is higher than that of 20 cracks. The DCA is significant than the DNCA, thus a higher CR is good for mass transport in the CLs. After 15e5 lattice steps, the computing converges, but none of the concentration reaches the theoretical linear line. This is because the existence of cracks changes the real tortuosity of the mass transfer. Furthermore, the more crack quantity, the farther the concentration deviates from the theoretical value.

Fig. 3.
figure 3

The normalized concentration distribution with different time steps.

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Normalized Concentration with Crack Shape. To evaluate the effect of crack shape on the on normalized concentration in the crack area, three kinds of domains with different crack shape, which are linear cracks, circle cracks and triangle cracks were employed. The porosity of NCA is 0.3, and the CR were set to 0.15 ± 0.05. The triangle cracks case shows a little higher normalized concentration than that of linear crack and circle crack cases. However, when the calculation converges, all three cases shows similar normalized concentration distribution. In addition, the deviation of the computational value from the theoretical value is caused by the cracks affecting the tortuosity of the computational domain (Fig. 4).

Fig. 4.
figure 4

The normalized concentration distribution of different crack shapes.

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Crack Ratio Effect on Diffusivity. To understand the effect of crack ratio on effective diffusivity, the relationship between crack ratio and normalized diffusivity was carried out. The NCA porosity is set to 0.2 and the DNCA/Dbulk is 0.04111. Different crack width, length and quantity cases are carried out. Meanwhile, the parallel model and series model are also used as the upper and lower limit of the normalized diffusivity. With the CR increase from 0 to 0.15, the normalized diffusivity of crack width case, crack length case and crack quantity case increase from 0.0411 to 0.0682, 0.0780 and 0.0866, respectively. Thus, the crack length shows the highest influence on the IP mass transfer enhancement. The computational model result is between the parallel model and series model results (Fig. 5).

Fig. 5.
figure 5

Diffusivity change caused by crack ratio change.

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NCA Porosity Effect on Diffusivity. A lower porosity will cause a lower DNCA, and the relationship between these two parameters is shown in Eqs. (2)–(6). When the porosities are 0.2, 0.3, 0.4 and 0.5, the DNCA/Dbulk are 0.0411, 0.089, 0.1579 and 0.2452, respectively. And the simulation is carried out for these four cases. Result shows, with the crack ratio increase from 0 to 0.17, the IP diffusivity enhancements of 0.2, 0.3, 0.4 and 0.5 porosity cases are about 96.93%, 63.86%, 43.40% and 30.08%, respectively. Considering the relative low porosity of CL structure, this enhancement shows great potential to enhance the IP mass transportation of oxygen phase and water vapor phase (Fig. 6).

Fig. 6.
figure 6

Diffusivity change caused by NCA porosity change.

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5 Conclusions

In order to understand the crack effect on the PEMFC catalyst layer IP diffusivity, a lattice Boltzmann method D2Q9 model was employed in this research. The normalized concentration of mass transport was acquired, and the effective diffusivity of the complex crack zone was calculated by the flux result. The following conclusions were mainly obtained:

  1. 1.

    The LBM model shows good accuracy when the diffusivity ratio is bigger than 1: 100. The error is below 5% when the domain size is 300 μm × 300 μm, and the error is below 0.5% when the domain size is 3000 μm × 3000 μm.

  2. 2.

    For a triangle crack, the crack length shows a little bit more influence than the crack width and quantity. However, comparing to the parameter including length, width, quantity and shape, the CR plays the dominant role in the diffusivity enhancing.

  3. 3.

    Considering the relative low porosity of CL structure, there is a great potential to enhance the IP mass transportation of oxygen phase and water vapor phase by design the cracked morphology.