Characterization of secondary pores in washcoat layers and their effect on effective gas transport properties

Chemical Engineering Journal 324 (2017) 370–379

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Characterization of secondary pores in washcoat layers and their effect on effective gas transport properties Satoru Kato a,⇑, Satoshi Yamaguchi a, Takeshi Uyama a, Hiroshi Yamada b, Tomohiko Tagawa b, Yasutaka Nagai a, Toshitaka Tanabe a a b

Toyota Central R&D Labs., Inc., Nagakute City, Aichi Prefecture 480-1192, Japan Nagoya Univ. Dept. Chem. Eng., Chikusa Ku, Nagoya City, Aichi Prefecture 464-8603, Japan

h i g h l i g h t s

g r a p h i c a l a b s t r a c t


Pore morphology of washcoat layers

was analyzed with X-ray computed tomography.
Effective gas permeability of washcoat layers was measured.
Effect of pore properties on effective gas permeability was investigated.
New model for calculating gas permeability of washcoat layers was developed.

a r t i c l e

i n f o

Article history: Received 26 January 2017 Received in revised form 29 April 2017 Accepted 8 May 2017 Available online 10 May 2017 Keywords: Catalyst Transport Washcoat layer X-ray CT Pore

a b s t r a c t It is well known that the performance of monolithic catalysts is limited by gas transport resistance in the washcoat layer. Gases are transported via two types of pores within the washcoat layer: primary pores, which exist inside the particle of catalyst support material (e.g., Al2O3), and secondary pores, which are voids among particles of the catalyst support material. Primary pores play an important role in the effectiveness of catalytically active components such as Pt, Rh, and Pd, while secondary pores facilitate gas transport in the washcoat layer. This paper reports the characterization of secondary pores and their effect on gas transport. In order to evaluate the gas transport properties of a washcoat layer, effective gas permeability was measured. Four samples with different pore morphologies were prepared, and their secondary pore properties were characterized with scanning electron microscopy, mercury porosimetry and synchrotron X-ray computed tomography. The obtained pore properties were correlated with the effective gas permeability, and based on the obtained correlation, we formulated a model for the gas permeability of the pores. This new model was compared to the conventional Kozeny-Carman equation. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Monolithic catalysts are widely employed in the abatement of pollution from moving sources, such as cars, and from stationary ⇑ Corresponding author. E-mail address: [email protected] (S. Kato). http://dx.doi.org/10.1016/j.cej.2017.05.055 1385-8947/Ó 2017 Elsevier B.V. All rights reserved.

sources, such as factories [1]. Other applications of monolithic catalysts include partial oxidation of methane [2], dimethyl ether steam reforming [3], CO-PROX reaction [4] and combustion chambers for gas turbines used in power generation [5]. In many monolithic catalysts, the catalyst layer is applied on the wall of a honeycomb-type substrate. The catalyst layer is thin (10– 150 lm), porous, and commonly referred to as the washcoat layer.

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371

Nomenclature b di m n r K e;a K e;sub Ke,wc L Lpath;i PðrÞ

empirical factor that varies in the range 0.6 < b < 2 path size of percolation path i empirical cementation that varies in the range 1 < m < 4 number of percolation paths pore radius (m) apparent effective permeability (m2) effective permeability of the cordierite substrate (m2) effective gas permeability of the washcoat layer (m2) length over which the pressure drop is taking place (m) length of percolation path i probability density function of pore

The thinness of the washcoat layer has advantages for reducing gas transport resistance. Nevertheless, it is well known that catalytic performance is limited by gas transport resistance, which takes place within the fluid-washcoat interface (external, interface) and in the washcoat layer (internal, intraface). An internal transport limitation has been reported by many researchers. For example, it was reported that gas transport resistance in the washcoat layer can play an important role in determining the light-off behavior of automotive catalysts [6]. Santos and Costa reported that automotive catalysts operate in a mixed regime, in which both external and internal gas transport resistance play a role, and even at high temperature, the purely external gas transport controlled regime is difficult to obtain [7]. In order to meet emissions regulations, it is necessary to reduce gas transport resistance in the washcoat layer. This needs deep understanding about gas transport phenomena in the washcoat layer. Measurement method of effective gas transport in the washcoat layer is well studied [5,8,9–13]. But the guideline of optimizing the pore properties of the washcoat layer has not been established. This is because the influence of pore properties on the gas transport phenomena, and even the pore properties themselves are not well studied. Since washcoat layers usually consist of porous metal oxide particles (e.g., Al2O3, ZrO2, CeO2), the pores can be classified into two types according to size: smaller pores, on the order of nm, are voids inside the porous metal oxide particles, and are called primary pores, while larger sub-lm or lm pores are voids among porous metal oxide particles, and are called secondary pores. Primary and secondary pores can be observed as two or three peaks in a pore size distribution graph obtained from mercury porosimetry. For example, Haya et al. reported pore sizes of 5 nm and 250 nm as primary and secondary pores, respectively [5]. We reported two peaks of 5 nm and 200 nm for an experimentally simulated washcoat layer [8]. Stary´a et al. reported three peaks of 200 nm, 1 lm and 5 lm [9]. Both primary and secondary pores play important roles in catalytic performance. Fig. 1 shows a simple diagram of gas transport in a washcoat layer. The reactants first diffuse through the secondary pores (among secondary particles), and then diffuse through the primary pores (inside a secondary particle), and finally react on active sites (e.g., Pt, Rh, Pd)

NO, CO

N2, CO2

Secondary particle Secondary pore

DP U V total Z sub Z wc Dp

e s l

k

pressure drop (Pa) superficial flow velocity (m s1) volume of domain for calculating percolation path thickness of substrate (m) thickness of washcoat layer (m) diameter of particles of the porous material (m) porosity (–) tortuosity (–) viscosity (Pa s) mean free path (m)

at the wall of a primary pore. To analyze these gas transport phenomena, conventional mercury porosimetry does not seem to be appropriate. For example, mercury porosimetry has been used to predict effective gas diffusivity with the random pore model (RPM) [14]. Hayes et al. reported that RPM predicts an effective diffusivity of the washcoat layer 8 times higher than the experimentally measured value [5]. The reason for this discrepancy may be due to the fundamental difference between mercury intrusion and gas transport phenomena. The pores of the washcoat layer may be too complicated to analyze with mercury porosimetry. Imaging the 3D structure using a technique such as X-ray CT should therefore give us new perspectives on the pore properties of washcoat layers. Recently, secondary pores of the washcoat layer of a commercial Rh-cAl2O3 monolithic catalyst has been studied by Karakaya et al. [15]. They reconstructed the 3D structures from a series of sliced images obtained with focused ion beam – scanning electron microscopy (FIB-SEM). A volume of 26.7  13.8  37.4 lm3 was imaged with a spatial resolution of approximately 0.1 lm. From the 3D structure, 3D images of secondary pores are reconstructed. Their morphologies varied significantly, ranging from crevice-like features to tortuous capillary-like features. They developed new model and the model was successfully applied to investigate reaction-diffusion process within the reconstructed 3D secondary pore volume. With this result, Blasi et al. develop macroscale models that can be applied at larger length scale [16]. These studies contributed new quantitative insight about the influence of gas diffusion on the catalytic performance. But the pore properties of primary and secondary pores, and their effect on the gas transport phenomena remains to be researched. As a first step to reveal all of the pores of washcoat layers and to investigate gas transport phenomena in washcoat layers, in this research, we imaged secondary pores of washcoat layers with synchrotron X-ray CT. The obtained pore properties were compared with those obtained by scanning electron microscopy (SEM) and mercury porosimetry. The gas transport properties were evaluated based on the gas permeability. It is appropriate to investigate the effect of secondary pores only, because primary pores can be neglected in the total permeability, as reported by Salejova et al. [17] The correlation between gas permeability and pore properties was investigated to establish a new model that can link secondary pore properties to gas transport properties.

2. Experimental Primary pore Substrate

Primary particle

Fig. 1. Diagram of gas transport phenomena in washcoat layer.

2.1. Sample preparation Four types of monoliths were prepared with washcoating. In the washcoating process, a slurry that included ZrO2 powder and

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zirconium nitrate, or Al2O3 powder and aluminum nitrate, was coated on a cordierite honeycomb substrate with a diameter of 30 mm, length of 25 mm, cell density of 400 cells per square inch, and hexagonal channel shape. After washcoating, the monolith was dried at 393 K for 12 h and then calcined at 773 K for 1 h. Table 1 summarizes the loading amount of the washcoat layer and the properties of the material, such as the specific surface area.

2.2. Characterization A cross-sectional micrograph of the washcoat layer was obtained by SEM. For the measurement, the monolith was embedded in an epoxy resin. The fixed sample was cut and ground down to create a smooth surface to measure. The pore size distribution was measured by mercury porosimetry. For the measurement, the monolith was cut into a small piece with a diameter of 10 mm and a length of 25 mm. The 3D structure was reconstructed from synchrotron X-ray CT imaging. For the synchrotron X-ray CT experiments, the wall of the monolith was cut into a small piece. The X-ray CT imaging experiments were carried out at BL33XU of Spring-8, Japan Synchrotron Radiation Research Institute (JASRI) [18,19]. A volume of 650  650  650 lm3 was imaged with 1800 projections and photon energy of 29 keV. Tomographic reconstruction was achieved using software published by JASRI (http://www-bl20.spring8.or. jp/xct/). This software uses a convolution back projection method (CBP) [20]. And with this software, gray values in the images were normalized with 50 cm1 of the liner adsorption coefficient. The voxel size of the resulting 3D image was 0.325 lm and the spatial resolution was approximately 0.65 lm. This spatial resolution was verified by measuring a phantom (wood chip which have tracheal diameter of 0.65 lm). Therefore pore of 1 voxel was neglected in this research. The pore morphology was extracted from 3D images. Segmentation parameter for extracting pores was decided based on the analysis of the brightness histogram. The brightness histogram of 3D images was bimodal distribution. The one component was attributed to pores and the other component represented material. These 2 components were decomposed by the normal distribution curve. The point of intersection between these curves was used to decide segmentation parameter. This procedure was done by the software packages ExFact VRÒ (Nihon Visual Science). Extracted 3D pore image was denoised by kriging method [21] with ExFact VRÒ Analysis Particle/Pore (Nihon Visual Science). The software package GeoDictÒ (Math2MarketÒ GmbH) was used to calculate the porosity, pore size distribution, and pore connectivity. The pore size distribution was determined by granulometry by fitting virtual spheres into the pore, and the pore size was determined by the particle size. The pore connectivity was evaluated by percolation path analysis, and the percolation paths were calculated by moving virtual spheres in the pores. The path size was defined as the diameter of the largest virtual sphere that could move through the path. The path length was determined by the trajectory of the moving virtual sphere, and was calculated via the shortest route. In other words, the shortest paths of the largest

virtual particles were calculated for each path. The ratio of the path length to percolation length represents the geometric tortuosity of the path.

2.3. Evaluating gas transport property The effective permeability of helium through the washcoat was evaluated. A permeation cell was made by cutting a 25-mm-long block of seven channels from the monolith. As shown in Fig. 2a, the cell consists of a central channel and six surrounding channels. The six surrounding channels at the end of the monolith were plugged with epoxy resin. At the opposite end of the monolith, the central channel was plugged with epoxy resin. The outside surface of the monolith was treated with epoxy resin. The permeation cell was connected to the flow system via a connecter pipe. Epoxy resin was applied to the connections to seal the joints between the monolith and the connecter. During the measurement, helium was fed through the connector in the central channels, and was then allowed to permeate throughout the washcoat layer and substrate. Finally, helium was exhausted from the six surrounding channels at the opposite side of the monolith. The experimental setup is shown in Fig. 2b. The gas flow rate was measured with a digital flowmeter. The pressure drop across the washcoat layer and substrate was measured. In each case, three samples were used and averaged. The flow rate

Fig. 2a. Perspective view of the seven channels comprising the permeation cell.

Fig. 2b. Simple diagram of experimental set up.

Table 1 Properties of washcoat layer and materials. Sample ID

A B C D 1)

Specific surface area.

Loading amount (g/L)

200 200 300 100

Material of washcoat Component

SSA1) (m2/g)

Particle size (lm)

ZrO2 ZrO2 ZrO2 Al2O3

100 24 5 200

4 2 2 8

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was in the range of 8–20 cc/min. The pressure difference was in the range of 50–5000 Pa. The effective permeability was calculated by Darcy’s law [22]

U¼

K e;a DP l L

ð1Þ

where U is the superficial flow velocity (ms1), K e;a is the apparent effective permeability and represents the overall permeation coefficients of helium through the composite wall, which consists of the washcoat and the substrate, l is the viscosity (Pa s), DP is the pressure drop and L is the length over which the pressure drop takes place (m). The effective permeability of the washcoat is calculated by using the standard resistance in series concept established by Hayes et al. [5] Assuming that two washcoat layers have the same thickness, Eq. (2) is obtained.

Z sub þ 2Z wc Z sub 2Z wc ¼ þ K e;a K e;sub K e;wc

ð2Þ

Here, K e;sub is the effective permeability of the cordierite substrate with a thickness of Z sub , and K e;wc is the effective permeability of the washcoat layer with a thickness of Z wc . Measuring Z sub , Z wc , K e;a and K e;sub enables the calculation of K e;wc from Eq. (2). 3. Results 3.1. Characterization Fig. 3 shows SEM images of cross-sectional views of monoliths A, B, C and D. The gray and black areas represent material and secondary pores, respectively. The secondary pore sizes are different in each sample: pore sizes of samples A and D are larger than those

of samples B and C, and pore sizes of sample D are larger than those of sample A. These differences can be calculated with image analysis. Fig. 4 shows the pore size distribution for washcoat layers of monoliths A, B, C and D. The average pore sizes (50% value in cumulative curve) are 0.58 lm, 0.14 lm, 0.24 lm and 1.08 lm for washcoat layers of monoliths A, B, C and D, respectively, and the calculated porosities for washcoat layers of monoliths A, B, C and D are 43%, 34%, 54%, and 40%, respectively. Pore size distributions obtained by mercury porosimetry are shown in Fig. 5. In Fig. 5, the substrate has 1 peak, while monoliths have 3 peaks. Considering the particle size for the material of the washcoat layer, the smallest peak under 0.05 lm represents the primary pore size. The medium peak is in the region of 0.1– 1 lm, and the larger peak is over 1 lm. These two peaks represent secondary pore sizes. The largest peak for the washcoat layer of every monolith has similar pore size and pore volume. This means that the washcoat layer for every monolith has pores that connect the surface with the substrate. During the measurement, the mercury first intrudes into the secondary pores of the washcoat layer. If the primary pore size of the washcoat layer is smaller than the pores of the substrate, such as shown in Fig. 5, the mercury intrudes into the substrate before the mercury intrudes into the primary pore. Hence, the porosity of the washcoat layer could be calculated by subtracting the pore volume of the substrate from that of the monolith. The porosities and peak values are summarized in Table 2. A series of sliced images for the washcoat layer were obtained from X-ray CT and are shown in Fig. 6. The gray and black areas represent material and secondary pores, respectively. These sliced images are similar to the SEM images shown in Fig. 3. Fig. 7 shows a reconstructed 3D structure of the washcoat layer of monolith A. The black areas represent secondary pores, while the material is

B

A

20 μm

20 μm

D

C

20 μm Fig. 3. SEM images of cross sectional views of washcoat layers.

20 μm

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100

Porosity (%)

80 60

B

C

A

D

40 20 0 0.01

0.10

1.00

10.00

coat B, whereas washcoats C and D do not contain micro cracks. The path size distribution and tortuosity distribution are calculated from the percolation path. The path size distribution is shown in Fig. 10. In the region of path size 0.65–1 lm, the path densities for monoliths A and D are higher than those for monoliths B and C. For path sizes over 1.5 lm, the washcoat layers of monoliths A and B continue, whilst monoliths C and D do not have paths in that region. It should be noted that minimum value of path size is 0.65 lm due to the spatial resolution of the X-ray CT (0.65 lm/ vox) in this research. Fig. 11 shows the tortuosity factor distribution of the washcoat layers. The tortuosity factors for the washcoat layer of monoliths A and B are similar, while the washcoat layers of monoliths C and D have relatively larger tortuosity.

Pore diameter, dp (μm) Fig. 4. Pore size distribution for washcoat layers calculated by SEM image analysis.

ΔVp / Δ(ln dp)

Monolith D Monolith C Monolith B Monolith A substrate 0.001

0.01

0.1

1

10

100

Pore diameter, dp (μm) Fig. 5. Pore size distribution for substrate and monoliths measured by mercury porosimetry.

not visualized. Pore properties such as pore size cannot be estimated visually due to the complexity of the pore morphology. The calculated pore size distribution for secondary pores of the washcoat layer is shown in Fig. 8. The washcoat layer of monolith D has larger pores relative to monoliths A, B and C. This is consistent with the SEM images shown in Fig. 3. Calculated porosities for the washcoat layers with X-ray CT are 16%, 12%, 19% and 29% for monoliths A, B, C and D, respectively. The pore connectivity in the reconstructed 3D structure was analyzed by percolation path analysis. The percolation path for the washcoat layer that connects the substrate to the surface of the washcoat layer was then calculated. Fig. 9 visualizes part of the percolation path in the region of 180 lm  180 lm  18 lm for the washcoat layer. In order to distinguish differences among the samples, the short depth of 18 lm is shown in Fig. 9. For the percolation path of washcoat B, almost all the path is arranged in a curved line. This line represents a micro crack, which is a type of pore that can be considered to be an aggregation of connected pores. Micro cracks of washcoat A are smaller than those of wash-

3.2. Effective gas permeability Fig. 12 shows the effective gas permeability for the washcoats of monoliths A, B, C and D and of the substrate. The substrate exhibited the largest effective permeability. The effective gas permeabilities of samples A and D are similar, with values larger than those of samples B and C. 4. Discussion 4.1. Pore characterization In this research, the properties of secondary pores were evaluated by SEM, mercury porosimetry and synchrotron X-ray CT. Each measurement evaluates similar or different pore properties. Comparison of these properties is expected to improve our understanding of secondary pores in washcoat layers. Comparison of the porosity is shown in Fig. 13, and pore size is compared Fig. 14. As shown in Fig. 13, mercury porosimetry and SEM yield similar porosities. This means that the influence of the substrate can be subtracted successfully in mercury porosimetry. Mercury porosimetry classifies secondary pores as the medium and largest peaks, as shown in Fig. 5. The largest peak corresponds to the connective pores between the substrate and the surface of the washcoat. This connective pore is visualized as the percolation path shown in Fig. 9. The path size of the percolation path is in the range of 0.65–3 lm, which is roughly consistent with the range of the largest peak (1–5 lm) in the pore size distribution obtained from mercury porosimetry. In Fig. 5, not only the largest peak but also the medium peak is corresponded to secondary pores. The pores represented as the medium peak may be partly imaged with Xray CT because the spatial resolution of 0.65 lm of X-ray CT is close to or less than the medium peak value, as shown in Table 2. This explains why X-ray CT tends to measure porosities that are lower than those measured using SEM, as shown in Fig. 13. In detail, porosity of monolith D obtained from X-ray CT is relatively close to the value obtained from SEM. This is due to that the secondary pore size of monolith D is larger than monolith A, B and C. As the measurement for characterizing pore size, there are fundamental differences between mercury porosimetry and SEM. The

Table 2 Properties of washcoat layers determined by mercury porosimetry. Monolith

A B C D 1)

Pore size (lm)

Porosity (%)1) Primary pore

Secondary pore

Total

Primary pore

Secondary pore

29 25 13 42

53 42 55 45

83 67 68 87

0.014 0.030 0.009 0.010

0.228 0.155 0.312 1.420

Under 0.05 lm is considered to be primary pore; secondary pore is considered as over 0.05 lm.

4 3.78 3.15 4.14

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A

B

20 μm

20 μm

D

C

20 μm

20 μm

Fig. 6. Sliced images of washcoat layers obtained from X-ray CT.

Fig. 7. 3D structure for washcoat layer of monolith A obtained from X-ray CT.

Fig. 8. Pore size distribution for washcoat layers obtained from X-ray CT.

pore size derived from mercury porosimetry is affected by the bottleneck size of the pore route for mercury intrusion. On the other hand, the bottleneck size does not affect results of SEM image analysis, because the granulometric method is used in SEM image analysis. However, Fig. 14 shows that the pore size of medium peak derived from mercury porosimetry correlates with the value obtained from SEM. This indicates that the ratio of pore bottleneck size and larger space is similar in each sample.

effective gas permeability and pore size or porosity was not found, suggesting that pore size and porosity independently affect the gas permeability. The contribution of each of these pore properties should be evaluated to determine the influence of secondary pores on effective gas transport. A large number of formulas to calculate effective permeability by using pore properties have been proposed. Eq. (3) [23] is based on the Kozeny-Carman approach [24,25], which is one of the most popular methods.

4.2. Influence of secondary pores on effective gas permeability based on the Kozeny-Carman equation In this section, the influence of secondary pores on the effective gas permeability is discussed. The pore size and porosity are evaluated with SEM, mercury porosimetry and synchrotron X-ray CT. These pore properties are expected to be effective for explaining gas transport properties. However, a simple correlation between

Ke ¼

D2p e3 72sð1  eÞ2

ð3Þ

Here, Dp is the diameter of the particles of the porous material, e is the porosity, and s is the tortuosity, which is related to the porosity by the empirically based Archie relationship [26]:

s ¼ be1m

ð4Þ

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Fig. 9. Visualized percolation paths of washcoat layers obtained from X-ray CT.

Fig. 10. Path distribution for washcoat layers obtained from X-ray CT.

Fig. 12. Experimentally measured effective permeability of washcoat layers.

Ke ¼

Fig. 11. Tortuosity distribution for washcoat layers obtained from X-ray CT.

D2p emþ2 72bð1  eÞ2

ð5Þ

In this research, the particle size of the material shown in Table 1 can be applied for Dp , and the secondary pore porosity obtained from mercury porosimetry can be applied for e. We can then calculate the effective permeability, which depends on empirical factors b and m. Fig. 15 shows a comparison between the calculated effective permeability and the experimentally measured values. In Fig. 15, the calculated value is expressed as a range belonging to (b, m) = (2, 1) and (0.6, 4), where the former and latter represent the minimum and maximum values, respectively. As shown in Fig. 15, the range of calculated values is too wide to discuss the influence of secondary pore properties on effective gas permeability. Therefore, a new model that does not contain an empirical factor seems to be necessary. 4.3. Developing a new equation

where b is an empirical factor that varies in the range 0.6 < b < 2 and m is an empirical factor that varies in the range 1 < m < 4. Considering relationship (4) in Eq. (3), we have:

As shown in the previous section, the conventional KozenyCarman equation cannot be used to calculate a valid value of the

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Mercury porosimetry

Porosity of secondary pore (%) obtained from X-ray CT or mercury porosimetry

60

C A D

B 40

X-ray CT D C

20 B

A

0 0

10

20 30 40 Porosity of secandary pore (%) obtained from SEM

50

60

Fig. 13. Comparison of porosity between SEM, X-ray CT and mercury porosimetry.

effective permeability due to the wide range of empirical factors. Miyakoshi et al. [27] proposed a theoretical model that does not have an empirical factor:

Pore size obtained from mercury porosimetry

2

D

K e;a ¼ 1

C B

A

0 0

1 Average pore size (μm) obtained from SEM

2

Fig. 14. Comparison of pore sizes between SEM and mercury porosimetry (Value of Y axis are those of medium peak which is summarized in Table 2).

e s

Z

1

PðrÞðr 2 =8Þð1 þ 4k=rÞdr

ð6Þ

0

where e is the porosity, s is the tortuosity, r is the pore radius, PðrÞ is the probability density function of pore, and k is the mean free path. Eq. (6) is developed on the assumption that: 1) The pore shape is tube-like. 2) The pores are connected in the direction of the gas flow. 3) The pores have tortuosity but not branches. 4) Pore radius, r, is constant in the direction of the gas flow. These assumptions are not consistent with real pore morphology, and in fact, this model is rarely used. To the best of our knowledge, studies that employ Miyakoshi’s model have not

Fig. 15. Comparison between experimentally measured effective permeability and values calculated with Kozeny-Carman equation.

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been reported. However, we noticed that tube-like pores in Eq. (6) can be regarded as connected pores that are visualized as the percolation path in Fig. 9. Therefore, the correlation between the percolation path parameter and the effective gas permeability was investigated. As a result, a correlation between effective permeability and path density of 0.92 lm for path sizes was found, as shown in Fig. 16. Since path size represents size for the bottleneck of the path, as explained in the experimental section, Fig. 16 indicates that the gas flow in the washcoat layer is controlled by the bottleneck of connected pores. With this result, Eq. (6) can be modified by redefining the pore radius as the path size:

Ke ¼

e s

Pn

2 i¼0 ðdi =32Þð1

n

þ 8k=di Þ

ð7Þ

Here, n is the number of percolation path and di is the path size of path i. These parameters can be obtained from percolation path analysis. The average value of tortuosity for the percolation path can be used for s. For porosity e, two types of porosities can be applied: 1) secondary pore (total) porosity, which is obtained from X-ray CT, or 2) effective porosity. Effective porosity, eeff , is calculated using the following equation:

Fig. 16. Relationship between path density and experimentally measured effective permeability.

eeff ¼

Pn

i¼0 ð

pd2i Lpath;i =4Þ V total

ð8Þ

where Lpath;i is the length of percolation path i, and V total is the volume of the 3D structure for calculating the percolation path. The effective porosity is obtained by extracting specific pores that can contribute to the gas flow. For example, gas cannot flow through dead-end pores and closed pores. These types of pores can be omitted by using the concept of effective porosity. The comparison between the calculated effective permeabilities with Eq. (7) and the experimentally measured values is shown in Fig. 17. In the case of using the secondary pore (total) porosity, the calculated value was higher than the experimental value. This is reasonable because the secondary pore porosity includes the contribution of some pores that do not contribute to the gas flow. In the case of using the effective porosity, the calculated value was lower than the experimental value. This is also reasonable because effective porosity neglects the expansion of the pores other than the bottleneck in the path. The effective permeation coefficient calculated by Eq. (7) should be considered to fall in a certain range whose upper limit is calculated using the secondary pore porosity, and whose lower limit is calculated using the effective porosity. The range is smaller than the case of using the Kozeny-Carman equation as shown in Fig. 15. Regardless of whether porosity or effective porosity is used, Eq. (7) calculates effective permeation coefficients relatively close to the experimental values. With these results, the potential of Eq. (7) is successfully demonstrated; however, its validity should be verified with more samples, which will be the subject of one of our future works. Finally, the contribution of the path size and the path density on the effective permeability is evaluated. Since Eq. (7) summarizes the contribution of each path to the total effective permeability, the effect of the path size and path density on the gas permeability can be calculated. The result is shown in Fig. 18. The smaller path had a dominant effect on the gas permeability in every sample. This is due to the high density of the smaller path, as shown in Fig. 10. The contribution of the smallest pore is different in each sample due to the difference in the larger path. In the case of having a larger path, such as with samples A and D, the influence of the larger path cannot be neglected, as shown in Fig. 18. Even if the

Fig. 17. Comparison between experimentally measured effective permeability and values calculated with new model.

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Reference

Fig. 18. Contribution of each path on effective permeability.

path density of the larger path is small, as shown in Fig. 10, since the path size effect is squared as described in Eq. (7), the effect of the larger path cannot be neglected. Eq. (7) is expected to be applied to research on gas transport phenomena in various porous materials. 6. Conclusion Four samples of washcoat layers were prepared to analyze the influence of secondary pore properties on effective gas transport. Secondary pore properties of washcoat layers were characterized with SEM, mercury porosimetry and synchrotron X-ray CT. The porosity and pore size distribution obtained by each method were compared. The porosities as measured by SEM and mercury porosimetry were similar in the range of 35–55%. On the other hand, X-ray CT gave lower porosities in the range of 10–30%. This is due to the resolution of the X-ray CT (0.325 lm/vox), which was lower than the value of SEM (0.05 lm/pix). Mercury porosimetry gave two peaks for secondary pores in the pore size distribution graph. The larger peak that connects the substrate to the surface of the washcoat layer was visualized and characterized with percolation path analysis of the 3D structure obtained from Xray CT. The smaller peak was consistent with the average pore size calculated by SEM image analysis. The effective gas permeability was measured in the range 1  1015–6  1015 m2 for washcoat layers. The effect of the pore parameters on the effective gas permeability was discussed using a modified version of Miyakoshi’s theoretical model. The key to the analysis was the use of the pore property obtained from percolation path analysis. Acknowledgement We thank Ms. Yamamura of Toyota Central R&D Labs for carrying out the mercury porosimetry measurements, and thank Dr. Ikuta of Toyota Central R&D Labs for helpful comment about the model. For using GeoDictÒ, we thank Mr. Fukushige of SCSK Corporation for careful technical supports, and also thank Dr. Wiegmann, Dr. Becker and Dr. Glatt of Math2MarketÒ GmbH for fruitful discussion about analyzing pores. The synchrotron radiation experiments were performed at the BL33XU and BL16B2 of SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2014B5370, 2015A5370, 2015A7012, 2015B7112 and 2016A7112).

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