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Author's personal copy ARTICLE IN PRESS Chemical Engineering Science 65 (2010) 931–937
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Experimental measurements of effective diffusion coefficient of oxygen– nitrogen mixture in PEM fuel cell diffusion media Nada Zamel a, Nelson G.C. Astrath b, Xianguo Li a,, Jun Shen b, Jianqin Zhou b, Francine B.G. Astrath b, Haijiang Wang b, Zhong-Sheng Liu b a b
20/20 Laboratory for Fuel Cells & Green Energy RD and D, Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON, Canada Institute for Fuel Cell Innovation, National Research Council, Vancouver, BC, Canada
a r t i c l e in fo
abstract
Article history: Received 9 July 2009 Received in revised form 11 September 2009 Accepted 17 September 2009 Available online 22 September 2009
PEM fuel cells are increasingly designed to operate at high current densities. At these densities, mass transport limitations become very significant, but they are not well understood, with many modeling studies but few experimental observations. The use of accurate transport coefficients to simulate the mass transport at high current densities is crucial. In this study, experimental measurements have been carried out to determine the effective diffusion coefficient in the carbon paper gas diffusion layer that is commonly used in PEM fuel cells. It was found that almost all the existing theoretical models significantly overpredict the effective diffusion coefficient by as much as 4–5 times; thus, underestimating the transport limitations considerably. Further, the effects of temperature, Teflon treatment for hydrophobicity and porosity on the effective diffusion coefficient were investigated. It was found that temperature does not affect the overall diffusibility of the gas. The diffusibility is decreased with the increase of Teflon treatment and decrease in porosity. Further work on better understanding the diffusion process in the gas diffusion layer is under way. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Diffusion Porous media Electrochemistry Energy Fuel cell Diffusion coefficient Carbon paper Gas diffusion layer Loschmidt cell
1. Introduction Polymer electrolyte membrane (PEM) fuel cells are used as alternative, green energy technologies. They are highly efficient and result in zero emissions during their operation cycle. PEM fuel cells are increasingly designed to operate at high current densities ( 2 A/cm 2 or even higher) to ensure their competitiveness with internal combustion engine. At these high current densities, however, losses due to mass transport become very significant. Hence, a full understanding of these mass transport limitations is crucial to obtain the proper design of the cell, and numerical modeling is often used to gain this understanding due to a lack of and difficulty in the measurements. Numerical models are used to simulate the transport of gases, electrons, protons, liquid water and heat through the different components of the fuel cell using the conservation equations (mass, momentum, species, charge and energy). Due to the porous nature and the complexity of reconstructing the real geometry of the gas diffusion layer (GDL), catalyst layer and electrolyte membrane, the transport
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coefficients must be adjusted to take the geometry into account. The adjusted transport coefficients are normally referred to as the effective transport coefficients and are less than that of their bulk counterparts. The focus of the present study is on the gas diffusion in the GDL. Carbon paper, shown in Fig. 1, is widely used as the material for the GDL in PEM fuel cells due to its chemical and mechanical durability. It is a good electron conductor and provides free paths for gas and liquid water transport. Mathematically, Fick’s law can be used to represent the mass diffusion in the GDL as follows
e
@Ci ¼ Deff r 2 Ci i @t
ð1Þ
where e is the porosity of the layer, Ci is the concentration of the is the effective diffusion coefficient of gaseous species i and Deff i species i and is usually correlated in the form of bulk f ðeÞgðsÞ Deff i ¼ Di
Dbulk i
ð2Þ
where is the bulk diffusion coefficient of species i and s is the liquid water saturation present in the layer. As it can be seen, the effective diffusion coefficient is a function of the porosity of materials and the liquid water saturation. Many correlations for these functions are found in literature and are
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Fig. 1. Scanning electron microscope image of carbon paper—TORAY (TPGH-120). Fig. 2. Diffusibility predicted by the existing models in literature versus the porosity of the carbon paper gas diffusion layer. Table 1 Summary of available models that have been frequently applied to and used for the determination of the effective diffusion coefficient in the carbon paper gas diffusion layer in PEM fuel cells. References
Model
Type
Bruggeman (1935)
Deff ¼ e1:5 Dbulk i i
Effective medium approximation Effective medium approximation Percolation theory
Neale and Nader (1973) Tomadakis and Sotirchos (1993) Nam and Kaviany (2003) Das et al. (2009)
Deff i Deff i Deff i Deff i
2e ¼ Dbulk 3e i e 0:037 0:661 bulk ¼e Di 1 0:037 0:785 e 0:11 ¼e Dbulk i 1 0:11 3ð1 eÞ ¼ 1 Dbulk i 3e
Percolation theory Effective medium approximation
listed in Table 1. In this study, the presence of liquid water in the GDL is not considered. Thus, the function due to liquid water saturation is ignored. The differences between these models shown in Table 1 are obvious. In order to better comprehend these differences and their effect on the overall diffusibility of the gas, Fig. 2 was constructed. It is seen that the calculated diffusibility, Q, which is defined as the ratio of the effective to the bulk diffusion coefficient and is highly dependent on the model used especially at low porosity values. ! Deff i ð3Þ Q ¼ bulk Di The porosity of the carbon paper GDL, once the PEM fuel cell, is assembled (under cell or stack compression force), is in the range of 0.4–0.6. At these porosity values and a high current density, the results of the mass transport limitations in the numerical modeling/simulation of PEM fuel cell performance become very sensitive to the effective diffusion coefficient used. Thus, it is crucial to develop an accurate correlation/expression for the effective diffusion coefficient, which properly takes into account the geometry of the carbon paper GDL involved. However, such an accurate correlation does not exist in literature due to a lack of experimental data for the effective diffusion coefficient in carbon paper. The existing models for the effective diffusion coefficient
shown in Table 1 were all originally developed for porous media made of spherical particles; whereas the carbon paper GDL is made of cylindrical carbon fibers arranged randomly with carbon binder dispersed as shown in Fig. 1. At present, only some limited experimental measurements of the effective diffusion coefficient in carbon paper have been reported in literature (Baker et al., ¨ 2006; Fluckiger et al., 2008; Kramer et al., 2008; Stumper et al., 2005; Ye and Wang, 2007). In this study, a Loschmidt cell apparatus is used to measure the effective diffusion coefficient in TORAY carbon paper samples. The effects of the operating temperature, Teflon treatment and porosity on the diffusibility are investigated. Finally, a comparison between the determined and the predicted diffusibility is made.
2. Experimental 2.1. Experimental apparatus A Loschmidt cell consisting of two chambers was used as the experimental apparatus in this study as shown in Fig. 3. This cell has been developed at NRC-Institute for Fuel Cell Innovation (IFCI); (see references Astrath et al., 2009; Rohling et al., 2007 for more details). The cell consists of a top (a) and bottom (b) chambers with the interior length and diameter of each chamber as 177.5 and 20.6 mm, respectively. The chambers can be connected, position (5a), or separated, position (5b), by a ball valve (5). The ball valve is considered as part of the bottom chamber (Apollo, Model 86-104-49). The upper side of the ball valve marks the middle of the diffusion cell and is represented as z= 0 on the coordinate system. Two mass flow controllers (Omega, Model FMA-5508) with a flow capacity of 0–500 mL/min are connected to the inlets (1 and 2) to control the gas flow rate when the chambers are filled with the gases. An oxygen sensor (Ocean Optics FOXY-AL300) is used to measure the diffusion of an oxygen mixture. Its 300 mm in diameter aluminum jacketed optical fiber probe is installed in the top chamber at a position of z =19 mm, represented as (6) in the figure. On the tip of the optical fiber probe, ruthenium complex in a sol–gel substrate is applied. The probe is connected to an excitation source and a spectrometer (Ocean Optics S2000-FL) by a bifurcated optical fiber. The
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Fig. 4. Sample data fitting—the concentration evolution of oxygen in nitrogen at 79 1C determined by oxygen sensor with 0% relative humidity (open circles). The continuous line represents the best curve fit using Eq. (5) to the experimental data.
2.2. Data analysis
Fig. 3. Schematic diagram of the diffusion cell—1: gas inlet 1; 2: gas inlet 2; 3 and 4: outlets; 5: a ball valve; 5(a): open position of valve; 5(b): closed position of valve; 6: oxygen sensor; 7 and 8: humidity sensors.
spectrometer is connected by the computer via a USB A/D converter (Ocean Optics ADC 1000). The response time of the oxygen sensor was around 1 s and the accuracy is 1% of full range for 0–100% (mole percent). For measurements involving water vapor in the mixture, a relative humidity sensor (Sensirion SHT75) is used. A humidifier system is coupled with the supplied gases at both inlets (1 and 2). The system (TesSol FCTS BH 500) is used to control the relative humidity in the range of 0–90%. The relative humidity in both chambers is monitored using two relative humidity sensors (7 and 8). The oxygen and relative humidity sensors were used to measure the change of oxygen and water vapor concentration with time. The system is built so that the diffusion process of gases would follow the one-dimensional Fick’s law of diffusion. The validity of this assumption was evaluated in detail in earlier studies (Rohling et al., 2006, 2007). Thus, the diffusion process in the Loschmidt cell is governed by the following equation
In order to obtain the equivalent diffusion coefficient, the measured concentration over time was fitted to Eq. (5) as shown in Fig. 4. The presence of t0 should be pointed out here. The diffusion process was delayed by about 125 s in order to ensure the accuracy of the measurements. After obtaining the equivalent diffusion coefficient, the resistance network shown in Fig. 5 was used to find the effective diffusion coefficient in the carbon paper GDL. From this network, it is found that the equivalent resistance, Req , is due to the diffusion in the sample and in the chamber and is obtained by Req ¼
@Ci @ Ci ¼ Deq i @z2 @t
Rbulk ¼
Cb z B C Ci ¼ i erfc@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 2 ðt t ÞDeq 0
ð5Þ
i
where Cib is the initial concentration of species i in the bottom chamber and t0 is the time delay at which diffusion commences.
zl Dbulk Ac i
ð7Þ
where l is the thickness of the sample. Similarly, the resistance due to the diffusion in the sample, Reff , is found by
ð4Þ
where Deq is the so-called equivalent diffusion coefficient of i species i, z is the spatial dimension and the general solution to this one-dimensional diffusion process is (Crank, 1975) 0 1
ð6Þ
where z is the distance from the zero axis to the oxygen sensor and Ac is the cross sectional area of the interior of the chamber, which is available for diffusion. The resistance due to the diffusion in the chamber is denoted by Rbulk and is calculated below
Reff ¼
2
z Ac Deq i
l Deff Ac i
ð8Þ
Combining Eqs. (6–8), the equivalent resistance becomes Req ¼
z zl l ¼ bulk þ eff Ac Deq Di Ac Di Ac i
ð9Þ
From Eq. (9), the effective diffusion coefficient, Deff , can then be i obtained as ¼ Deff i
l z zl bulk Deq D i i
ð10Þ
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The experimental conditions of this study are summarized as follows: 1. The effect of the temperature of an oxygen–nitrogen mixture on the effective diffusion coefficient of oxygen in nitrogen was investigated for a temperature range of 25–80 1C. The sample used is a TORAY carbon paper (TPGH-120) with no Teflon treatment and a thickness of 370 mm. 2. The effect of Teflon treatment on the effective diffusion coefficient of oxygen–water vapor in nitrogen–water vapor was determined for a Teflon treatment range of 0–40% by weight. The sample used is a TORAY carbon paper (TPGH-120) with variable Teflon treatment percentage and a thickness of 370 mm. All the measurements were made under atmospheric pressure.
3. Results and discussion 3.1. Effect of temperature The effect of temperature on the effective diffusion coefficient of gases in carbon paper is of great importance since PEM fuel cells can operate on temperatures in the range of 25–120 1C. To study this effect, the effective diffusion coefficient of oxygen in nitrogen was determined for different temperatures. The carbon paper sample used is TORAY carbon paper (TPGH-120) with a thickness of 370 mm, a 0% PTFE content (no wet-proofing) and a porosity of 75.5%. The mixture had 0% relative humidity and was fed at a pressure of 1 atm. The measurements of the effective diffusion coefficient made are given in Fig. 6. In this figure, the bulk diffusion coefficient at different temperatures (also determined in this study) is also plotted. The bulk diffusion coefficient was determined at the beginning of the test to ensure that the system is functioning properly. It is evident that due to the structure of the carbon paper, the diffusion coefficient decreases considerably. The determined bulk diffusion coefficient agrees with the theory developed by Fuller et al. (1966), which evaluates DAB as
Fig. 5. Resistance network due to diffusion in the chamber and the sample—Req is the equivalent resistance, Reff is the resistance due to the diffusion in the sample and Rbulk is the resistance due to the diffusion in the chamber.
DAB ¼
1:00 103 T 1:75 ð1=MA þ 1=MB Þ1=2 P 1=3 P 1=3 P½ð uÞA þ ð uÞB 2
ð11Þ
2.3. Accuracy of measurements In order to ensure the accuracy and repeatability of the measurements a total of 60 repeated measurements at different days and different times of the day were made for each measurement condition (or data point). The accuracy of the measurements was then evaluated using the standard deviation of the measurements with a 95% confidence interval. The determined effective diffusion coefficients (data points) reported in this study are then the average of all the repeated measurements within 95% confidence levels.
2.4. Experimental conditions In this study, the effect of the temperature and Teflon treatment on the effective diffusion coefficient is determined.
Fig. 6. Effect of temperature on the bulk and effective diffusion coefficients of oxygen–nitrogen mixture—error bars are shown in the figure.
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where DAB is the mass diffusivity of gas species A through B in cm2/s; T is the absolute temperature in K, MA, MB are the molecular weights of A and B, respectively in g/mol, P is the absolute pressure in atm, with the atomic and structural diffusion volume increment u is summed over the atoms, group of atoms, and structural features of each diffusion species. Studying the trend of variations illustrated in Fig. 6, it is interesting to note the difference in the slope of both lines representing the bulk and effective diffusion coefficients determined. It is seen that the effect of temperature on the bulk diffusion coefficient is different from that on the effective diffusion coefficient, or the effective diffusion coefficient seems to increase with temperature more slowly than the bulk temperature. This suggests that the effective diffusion coefficient is not dependent on the temperature of the gas only. To better understand the extent to which temperature affects the overall diffusion process in the carbon paper GDL, the diffusibility (Q) should be investigated. As mentioned earlier, Q is defined as the ratio of the effective to the bulk diffusion coefficient. The determined diffusibility for different temperatures is given in Table 2. From Table 2, it is seen that the temperature has very little effect on the overall diffusibility. Hence, the structure of the carbon paper is the determinant of the magnitude of diffusibility. This finding is consistent with that found by Stumper et al. (2005). In their work, they determined the effect of operating pressure on the diffusibility in carbon paper and found that the operating pressure does not affect the overall diffusibility. Therefore, the temperature dependence of the effective diffusion coefficient originates from the corresponding dependence of the bulk diffusion coefficient on temperature. As pointed out earlier, one of the motivations for the present study is to investigate the accuracy of the available theoretical models in literature (as shown in Table 1) that are often used to predict the effective diffusion coefficient in the carbon paper GDL. To do so, the diffusibility for the TPGH-120 sample is calculated using these models as given in Table 3. The diffusibility is calculated under the dry condition (i.e. the liquid water saturation in the carbon paper GDL is zero) and using a porosity of 75.5%. Comparing the model predicted diffusibility to that determined, it is clear that the existing
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models available in literature overpredict the effective diffusion coefficient significantly, by as much as a factor of 2.5. Hence, the use of these models for the purpose of PEM fuel cell modeling/simulation will result in considerably underestimating the mass transport limitations, especially at high current densities. The difference between the model predicted and the determined values is due to the difference in the geometry involved, as pointed out earlier, all the available models in literature for the effective diffusion coefficient were developed based on spherical particles that form the porous media. In order to account for the geometry effects, some models, such as the Bruggeman (1935) approximation, suggest the use of the tortuosity factor, and others, such as Das et al. (2009), propose the use of a shape factor. In order to predict either the tortuosity or shape factors, which properly describe the geometry of the carbon paper GDL, experimental data for different porosity values are needed.
3.2. Effect of Teflon treatment Liquid water management is a critical issue in PEM fuel cells. To better control the transport of liquid water, especially its removal from the porous GDL, carbon paper is conventionally treated with Teflon. With the application of the Teflon treatment, carbon paper becomes a more hydrophobic material aiding in the water removal process. This addition of material, however, could also change the diffusion process of gases in the carbon paper GDL. In order to understand this change, the effective diffusion coefficient of an oxygen–water vapor in nitrogen–water vapor was determined for the range of 0– 40 wt% Teflon treatment, and the results are reported in Fig. 7. The bulk diffusion coefficient of oxygen–water vapor in nitrogen–water vapor is 0.322 cm2/s and it was also determined by Astrath et al. (2009). The diffusibility decreases as the Teflon treatment percentage is increased. It is believed that this increase is due to the change in porosity. The porosity of the carbon paper GDL is decreased since more solid material, being Teflon, is added; thus, decreasing the volume of pore available for diffusion. The effect of
Table 2 Effect of temperature on the diffusibility. Temperature (1C)
Deff Diffusibility, Q ¼ Dbulk
25 40 50 60 70 80
0.252 0.272 0.270 0.276 0.258 0.281
Table 3 Diffusibility for TORAY TPGH-120 as predicted by the existing effective diffusion coefficient models available in literature as shown in Table 1. Model
Deff Diffusibility, Q ¼ Dbulk
Bruggeman (1935) Neal and Nader (1973) Tomadakis and Sotirchos (1993) Nam and Kaviany (2003) Das et al. (2009)
0.650 0.667 0.615 0.579 0.667
Fig. 7. Effect of Teflon treatment (wt%) on the diffusibility of oxygen–water vapor in nitrogen–water vapor mixture—error bars are shown in the figure.
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porosity can then be investigated as well, as shown in the next sub-section. 3.3. Effect of porosity As mentioned earlier, the addition of Teflon changes the total porosity of the carbon paper GDL. Teflon is added as a percentage by weight; thus, the total porosity can be calculated as follows
es ¼
rc rs rc
ð12Þ
where es is the porosity of the sample, rc is the density of carbon and rs is the density of the sample. The addition of Teflon changes the total density of the sample. Using Eq. (12), the corresponding porosity for each Teflon treatment percentage is calculated and given in Table 4. The dependency of the diffusibility on the porosity of the carbon paper GDL is shown in Fig. 8. In this figure, a comparison between the determined and the model predicted diffusibility is also given. It is evident that the models significantly overpredict the value of the diffusibility. Thus, the use of these theoretical models can result in inaccurately simulating the mass transport in the GDL of PEM fuel cells and their use should be cautioned, especially for the high current density conditions. To further investigate the effect of porosity on the overall diffusibility, the measured data is fitted to an expression, which is a derivative of the Bruggeman Approximation and is given as
Table 4 Estimated porosity values for TORAY TPGH-120 with different PTFE content by weight. PTFE content by weight (%)
Porosity
0 10 20 30 40
0.755 0.731 0.706 0.682 0.657
Fig. 9. Determined and fitted diffusibility values.
follows Q¼
Deff i Dbulk i
¼ AeB
ð13Þ
where A and B are fitting parameters. Using Fig. 9, A and B are found to be 1.63 and 5.65, respectively. The corresponding coefficient of determination R2 is found to be 0.981, which is defined as P ðSSerr Þ ð14Þ R2 ¼ 1 Pi i ðSStot Þ P P where i SSerr ¼ i ðyi fi Þ2 is the total sum of squared errors and P P 2 i SStot ¼ i ðyi yÞ is the total sum of squares. The determined data are represented by yi, fi denotes the diffusibility predicted by the expression in Eq. (13) and y is the average value of the determined data. The results of the curve fit, as shown in Fig. 9 and the values of A and B (1.63 and 5.65, respectively) in Eq. (13) when compared to the well-known Bruggeman (1935) correlation (A= 1 and B = 1.5) as shown in Table 1, suggest that carbon paper is more tortuous than the geometries used to derive the existing effective diffusion coefficient models summarized in Table 1. From this curve fit, it could also be concluded that the tortuosity factor of the GDL is 1.63. Thus, the validity of using these existing models in literature to predict the effective diffusion coefficient in the carbon paper GDL of PEM fuel cells should be cautioned as well. The significant difference between the predicted and the measured values of the effective diffusion coefficient raises concerns over many modeling/simulation studies for PEM fuel cell performance in literature where the existing effective diffusion coefficient models like the Bruggeman correlation has been routinely used, and is the motivation for more ongoing work on the effective diffusion coefficient in gas diffusion layers of PEM fuel cells.
4. Conclusions
Fig. 8. Effect of porosity on the diffusibility of oxygen–water vapor in nitrogen– water vapor—determined and model predicted values.
In this study, a Loschmidt cell was used to measure the effective diffusion coefficient in the carbon paper GDL of PEM fuel cells. The effects of temperature, Teflon treatment and porosity on the overall diffusibility are investigated. It was found that the temperature does
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not affect the overall diffusibility of the gas in carbon paper. The addition of Teflon to carbon paper results in decreasing the overall porosity of the GDL; hence, decreasing the overall diffusibility of gases in the layer. A comparison between the determined data and that predicted by the existing effective diffusion coefficient models available in literature indicated that these models overpredict the effective diffusion coefficient significantly. This overprediction can lead to underestimating the mass transport limitations in the gas diffusion layer of PEM fuel cells, especially at high current densities and low porosity values. Further work on the effective diffusion coefficient in carbon paper is under way.
Acknowledgments The financial support by the Natural Sciences and Engineering Research Council (NSERC) of Canada, Auto21 NCE and NRCInstitute for Fuel Cell Innovation is greatly appreciated. References Astrath, N.G.C., Shen, J., Song, D., Rohling, J.H., Astrath, F.B.G., Zhou, J., Navessin, T., Liu, Z.S., Gu, C.E., Zhao, X., 2009. The effect of relative humidity on the binary gas diffusion. Journal of Physical Chemistry A 113, 8369–8374. Baker, D., Wieser, C., Neyerlin, K.C., Murphy, M.W., 2006. The use of limiting current to determine transport resistance in PEM fuel cells. ECS Transactions 3, 989–999. Bruggeman, D.A.G., 1935. Calculation of various physics constants in heterogenous substances I: dielectricity constants and conductivity of mixed bodies from isotropic substances. Annalen der Physik (Leipzig) 24, 636–664.
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Crank, J., 1975. The Mathematics of Diffusion, second ed Oxford University Press, New York. Das, P.K., Li, X., Liu, Z.S., 2009. Estimation of effective transport properties for the PEM fuel cells. Applied Energy, in press, doi: 10.1016/j.apenergy.2009.05.006. ¨ ¨ Fluckiger, R., Freunberger, S.A., Kramer, D., Wokaun, A., Scherer, G.G., Buchi, F.N., 2008. Anistropic, effective diffusivity of porous gas diffusion layer materials for PEFC. Electrochimica Acta 54, 551–559. Fuller, E.N., Schettler, P.D., Giddings, J.C., 1966. New method for prediction of binary gas phase diffusion coefficients. Industrial and Engineering Chemistry 58 19–27 ¨ ¨ Kramer, D., Freunberger, S.A., Fluckiger, R., Schneider, I.A., Wokaun, A., Buchi, F.N., Scherer, G.G., 2008. Electrochemical diffusimetry of fuel cell gas diffusion layers. Journal of Electroanalytical Chemistry 612, 63–77. Nam, J.H., Kaviany, M., 2003. Effective diffusivity and water-saturation distribution in single and two-layer PEMFC diffusion medium. International Journal of Heat and Mass Transfer 46, 4595–4611. Neale, G.H., Nader, W.K., 1973. Prediction of transport processes within porous media: diffusive flow processes within a homogeneous swarm of spherical particles. AIChE Journal 19, 112–119. Rohling, J.H., Shen, J., Zhou, J., Gu, C.E., 2006. Application of the diffraction theory for photothermal deflection to the measurement of the temperature coefficient of the refractive index of a binary gas mixture. Journal of Applied Physics 99, 103–107. Rohling, J.H., Shen, J., Wang, C., Zhou, J., Gu, C.E., 2007. Determination of binary diffusion coefficients of gases using photothermal deflection technique. Journal of Applied Physics B 87, 355–362. Stumper, J., Haas, H., Granados, A., 2005. In situ determination of MEA resistance and electrode diffusivity of a fuel cell. Journal of the Electrochemical Society 152, A837–A844. Tomadakis, M.M., Sotirchos, S.V., 1993. Ordinary and transition regime diffusion in random fiber structures. AIChE Journal 39, 397–412. Ye, X., Wang, C.-Y., 2007. Measurement of water transport properties through membrane electrode assemblies. Journal of the Electrochemical Society 154, B683–B686.
