Elsevier

Fuel

Volume 199, 1 July 2017, Pages 551-561
Fuel

Full Length Article
Accurate analytical model for determination of effective diffusion coefficient of polymer electrolyte fuel cells by designing compact Loschmidt cells

https://doi.org/10.1016/j.fuel.2017.03.017Get rights and content

Highlights

  • An analytical model is presented for study of diffusion in composite systems.

  • The model can be used to design more compact and cheaper Loschmidt cells.

  • Results are well matched with experimental measurements and finite element method.

  • The conventional models can show poor predictions in later times and short cells in comparison with the new proposed model.

Abstract

Effective diffusion coefficient is an important parameter which needs to be determined in different fields of study, such as cathode catalyst layers of PEM fuel. For this purpose, a Loschmidt diffusion cell can be used. When a porous medium is placed in Loschmidt apparatus, the effective gas diffusion coefficient (EGDC) of this section must be correlated by diffusion coefficient in absence of a porous medium.

In the previous researches studying the Loschmidt diffusion cell, a simplifying infinite-length assumption was used in the analytical solution. Therefore, the solution is only applicable for a short time range, and this can result in high error. In order to overcome this challenge, the length of cell should be quite long. This requirement is not experimentally and economically easy to achieve.

In this study, a new analytical solution is proposed by applying Fick’s second law and separation of variables technique. This model does not use simplifying assumptions such as infinite length and equivalent diffusivity coefficient. The results of the new analytical solution are verified with the experimental measurements, as well as numerical finite element simulation. In order to analyze the reliability of previous methods, a new characteristic time is defined based on diffusion wave propagation in the system. Finally, a sensitivity analysis on thickness of porous media and EGDC is conducted and it is shown that previous models can predict diffusion coefficient with high deviations.

Graphical abstract

Effective diffusion coefficient is an important parameter which needs to be determined in different fields of study, such as cathode catalyst layers of PEM fuel. For this purpose, a Loschmidt diffusion cell can be used. When a porous medium is placed in Loschmidt apparatus, the effective gas diffusion coefficient (EGDC) of this section must be correlated by diffusion coefficient in absence of a porous medium.

In the previous researches studying the Loschmidt diffusion cell, a simplifying infinite-length assumption was used in the analytical solution. Therefore, the solution is only applicable for a short time range, and this can result in high error. In order to overcome this challenge, the length of cell should be quite long. This requirement is not experimentally and economically easy to achieve.

In this study, a new analytical solution is proposed by applying Fick’s second law and separation of variables technique. This model does not use simplifying assumptions such as infinite length and equivalent diffusivity coefficient. The results of the new analytical solution are verified with the experimental measurements, as well as numerical finite element simulation. In order to analyze the reliability of previous methods, a new characteristic time is defined based on diffusion wave propagation in the system. Finally, a sensitivity analysis on thickness of porous media and EGDC is conducted and it is shown that previous models can predict diffusion coefficient with high deviations.

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Introduction

Polymer electrolyte membrane (PEM) fuel cells are considered as innovative, alternative and green energy technologies systems which can be integrated with traditional power sources as new candidates in transportation, stationary power systems, and portable devices [1].

In order to improve the efficiency of PEM fuel cells in comparison with traditional power sources, they are designed to operate at high current densities up to 1.5 A/cm2 with the power in the range of 600 mW/cm2 and even higher [2], [3]. However, high current densities can result in an increase in loss due to mass transport. Therefore, it is essential to study the different parameters of the system such as transport coefficients comprehensively [3].

In order to estimate the binary and effective diffusion coefficients of two gases, based on transient method (time-dependent concentration boundary condition), a Loschmidt diffusion cell can be used [4], [5]. A Loschmidt diffusion cell is an apparatus consisting of two chambers and a sliding gate which connects the top and bottom chambers. A special Oxygen sensor is embedded in top chamber, in order to measure the Oxygen concentration. The chambers are filled with different concentrations of gases, and diffusion takes place upon the removal of the gate. The changes in concentration with time can be used to determine the diffusion coefficient [6], [7].

Several studies have focused on theoretical modeling of binary and effective gas diffusion coefficient through porous media in fuel cells [6], [8]. Göll and Piesche [9] presented a computational model based on a theory of multi component gas transport in porous media, which can be used for investigations in macroscopic scale. In order to validate their model, they simulated an isothermal diffusion problem in a Loschmidt cell by using mass transfer model; however, the results of this model were presented in numerical simulations.

The Loschmidt diffusion cell can also be used for determination of the effective diffusion coefficients of cathode catalyst layers of PEM fuel cells. Shen et al. [10] measured the effective diffusion coefficient of dry gas (O2–N2) in gas diffusion layers by an in-house made Loschmidt diffusion cell. In their study, the catalyst layers were deposited on an Al2O3 membrane substrate by an automated spray coater.

Astrath et al. [11] used a Loschmidt diffusion cell to measure the effective O2–N2 diffusion coefficients. They experimented on four types of porous layers, which were made of stainless steel. In this way, both bulk diffusion coefficient and the effective gas diffusion coefficients of the samples were determined. It was found that the obtained values were consistent with the results of the three-dimensional (3D) numerical simulation.

Several experimental measurements have been carried out by Zamel et al. [12] to determine the effective diffusion coefficient of O2-N2 mixture in a Loschmidt cell. In addition, the effects of Teflon treatment, temperature and porosity of GDL on the effective diffusion coefficient were investigated.

Zamel et al. [3] developed an experimental correlation for the EGDC in carbon paper GDL based on a 3D simulation of gas diffusion. The effect of structure on the diffusion coefficient is also considered in their simulation.

A modified Loschmidt cell with dry gas was used by Chan et al. [13] to experimentally determine the effect of micro-porous layer (MPL) on the effective diffusion coefficients. In addition to MPL coating, the effects of Teflon treatment and GDL thickness were also investigated. They reported that the effective diffusion coefficient of the MPL is only about 21% that of its GDL substrate.

Rohling et al. [14] used an in-house made Loschmidt diffusion cell with a photothermal-deflection technique. This cell was employed to measure the effective gas diffusion coefficient of a GDL with a porosity of 70%.

All of the mentioned studies used the “series resistance model” [15], [16] for calculation of the diffusion coefficient. In this study, Fick’s second law is used to describe diffusion in the Loschmidt diffusion cell with a porous medium.

The analytical solution introduced here is verified by both the finite element method and experimental measurements. The new model can overcome the challenges regarding to the length of the cell and measurement time. Therefore, a short and more compact experimental apparatus can be used. Moreover, the results of the new model provide more accurate predictions through the measurement times.

Section snippets

Loschmidt diffusion cell

A schematic diagram of the Loschmidt apparatus is shown in Fig. 1. A Loschmidt diffusion cell consists of two chambers, separated by a flat or ball sliding gate.

First, the top chamber is filled with N2, while the bottom chamber is full of O2. An optic oxygen probe is usually placed in the top chamber. A sample holder is placed between the gate and Oxygen sensor. In order to investigate the EGDC, a porous sample is inserted in the sample holder.

The gap found between the sliding gate and the sample,

Analytical method

In this study a new realistic analytical model was developed for designing more compact Loschmidt diffusion cells. This model can be applied to finite boundary conditions; therefore, it enables a smaller cell length.

C1(z), C2(x) and C3(z) are concentrations distribution in the first chamber, porous region and second chamber, respectively. Concentrations are governed by the following diffusivity equations:Ci(z,t)t=Di2Ci(z,t)z2

Where, i denotes any region, including top chamber, bottom chamber

Numerical method

In order to verify the results of the analytical model, a finite element method was applied to find the concentration distribution throughout the Loschmidt diffusion cell. For this case, one dimensional Fick’s law was used and the domain was divided to finite length and one-dimensional elements. Concentration between two successive nodes (inside an element) can be obtained by averaging node concentrations, based on following relation [23]:C(x,τ)=Ni(x)Ci(τ)+Nj(x)Cj(τ)

Galerkin finite element

Results and discussion

In this section the results of the new analytical solution for the diffusion problem in a Loschmidt cell with a porous layer are compared with the results of experimental measurements obtained by Shen et al. [3] as well as numerical simulations.

In 2011, Shen et al. measured effective diffusion coefficient for air (O2-N2) in cathode catalyst PEM fuel cell using a Loschmidt diffusion cell [3]. They measured results for different catalyst thicknesses varying from 6 to 29 μm.

They used a Loschmidt

Conclusions

In this study, a new analytical solution is presented for diffusion mechanism through a Loschmidt diffusion cell. The results of the new model are verified with experimental measurement data published in literature as well as numerical finite element method. The results of the new model are in great match with experimental and numerical models. The new model is an analytical solution of Fick’s second law of diffusion in a 3-part composite system. This model does not use simplifying assumptions

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