Microsoft Word - 476hernandez.docx CHEMICAL ENGINEERING TRANSACTIONS VOL. 43, 2015 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Chief Editors: Sauro Pierucci, Jiří J. Klemeš Copyright © 2015, AIDIC Servizi S.r.l., ISBN 978-88-95608-34-1; ISSN 2283-9216 Analysis of the Gas Diffusivity in the Simulated Washcoat Layer Based on Mean Transport Pore Model and the Mean Molecular Speed Satoru Kato*a, Hironobu Ozekib, Hiroshi Yamadab, Tomohiko Tagawab, Naoki Takahashia, Hirofumi Shinjoha a Toyota Central R & D Labs., Inc., Nagakute City, Aichi prefecture, 480-1192 Japan b Nagoya Univ. Dept. Chem. Eng. Chikusa Ku, Nagoya City, Aichi prefecture, 464-8603 Japan e1325@mosk.tytlabs.co.jp The pore structure of a simulated washcoat layer was evaluated using the Mean Transport Pore Model (MTPM). The MTPM provides the mean diffusive pore radius (rm) of a simulated washcoat layer from the experimentally measured effective diffusion coefficient (De). However, data previously published in the literature indicated that the value of rm depended on the choice of diffusive gas for the measurement of De. Therefore, a new, more efficient method was developed to estimate rm. To obtain base data, De was measured in a binary gas diffusion system using a modified Wicke-Kallenbach diffusion cell at room temperature, 473 K, and 673 K. The diffusive gases tested were H2, He, CH4, Ne, N2, O2, C3H6, CO2, and C3H8, and a simulated washcoat layer was used. The results confirmed that the value of rm depended on the type of diffusive gas used. However, many measurements of De are required to avoid problems. In the catalytic R&D process, the measurement of many De values for every experiment is impractical. To balance validity against efficiency for estimating rm, a combination of experimentally measured De and predicted De was used. This method is effective for predicting De using the mean molecular speed instead of a porous structure, and is more efficient than the conventional method of using MTPM. This method also can contribute to investigations of automotive catalysts. 1. Introduction A washcoated catalyst on a honeycomb substrate is used widely for automotive catalysts to improve fuel economy while reducing exhaust back pressure. Modeling the reaction and gas diffusion in the washcoat layer helps design an effective catalytic converter. Corbetta et al. (2013) proposed a reactive computational fluid dynamics based modeling approach, and reported the kinetics of the automotive catalyst converter. To evaluate gas diffusivity, the pore radius of the washcoat layer was an important factor. The mean diffusive pore radius (rm) was estimated from a set of experimentally measured effective diffusion coefficient (De) values using the mean transport pore model (MTPM) (Pazdernik and Schneider, 1982). The rm value is a parameter unique to the MTPM that represents the average pore size of the gas diffusion route. The rm value for a simulated washcoat layer has been estimated (Kato et al., 2013a). But since evaluating rm requires a pair of experimentally measured De values (Pazdernik and Schneider, 1982), data previously reported (Pazdernik and Schneider, 1982, Kato et al., 2013a, Schneider and Gelbin, 1985) indicated that the value of rm depended on the choice of diffusive gas used to measure De. This issue is investigated in the present report along with an effective method for estimating rm from a minimum number of De values. DOI: 10.3303/CET1543266 Please cite this article as: Kato S., Ozeki H., Yamada H., Tagawa T., Takahashi N., Shinjoh H., 2015, Analysis of the gas diffusivity in the simulated washcoat layer based on mean transport pore model and mean molecular speed, Chemical Engineering Transactions, 43, 1591- 1596 DOI: 10.3303/CET1543266 1591 2. Experimental A simulated washcoat layer was made using a procedure reported previously (Kato et al., 2013a). The simulated washcoat layer was applied to a metal mesh by dip coating. The used slurry contained ZrO2 powder and zirconium nitrate. After dip coating, the sample was dried at 393 K for 12 h and then calcined at 773 K for 1 h. Using a metal mesh as the substrate instead of a cordierite honeycomb substrate to measure the gas diffusivity in the washcoat layer was advantageous because the metal mesh had no diffusion resistance. The gas diffusivity was measured in a binary gas diffusion system using a modified Wicke-Kallenbach diffusion cell (Kato et al., 2015b) at room temperature, 473 K, and 673 K. The tested diffusive gases included H2, He, CH4, Ne, N2, O2, C3H6, CO2, and C3H8. For a pair of tested diffusive gases, Ar was used. The concentration of the diffusive gases tested was 100 %. 3. Result and Discussion 3-1. Experimentally measured De The experimentally measured De values are shown in Figure 1. The driving force for the gas diffusion phenomena was the random motion of the gas molecules. The mean molecular speed is given by Eq(1), in which R, T, and M are the gas constant, temperature, and molecular weight, respectively. An increase in De with increasing temperature or decreasing molecular weight is due to an increase in the mean molecular speed. 2/1 8      = M RT vm π (1) Figure 1. experimentally measured De. 3-2. Problem involved in conventional usage of MTPM The MTPM describes gas diffusivity in a porous material according to Eq(2). Here, De,ijb, Ψ, Dijb, De,ik, rm, and Ki are the effective bulk diffusion coefficient, a geometric factor, the bulk diffusion coefficient, the effective Knudsen diffusion coefficient, the mean diffusive pore radius, and the Knudsen constant, respectively. b ij b ije DD ψ=, (2a) im k ie KrD ψ=, (2b) Eq(3) can be obtained by substituting Eq(2) into the modified Stefan-Maxwell equation (e.g., Schneider and Gelbin, 1985), because bulk and Knudsen diffusion occur simultaneously in the simulated washcoat layer (Kato et al., 2013a and 2015b). In Eq(3), yi represents the molar fraction in the diffusion cell: ( ) bijiiimiie DyKrD ψαψ /)1(/11 , −+= (3) 2/1)(1 jii MM−=α (3a) For the MTPM, rm and Ψ are estimated by substituting experimental values of De into Eq(3). The rm value is the main focus of this research. To obtain rm, Eq(3) is converted to Eq(4), and then De, yi, Db and K are substituted into Eq(4): 1592 ( ) ( ) }/)1){(/1(/1, bijiiiijeffi DKyrDK αψψ −+= (4) Results obtained at room temperature, which produced mostly linear relations, are shown in Figure 2. The slope and intercept provided an rm value of 267 nm. Figure 2. Plot of Ki (1-αyi) /Dijb vs Ki /De,i In Figure 2, the correlation coefficient for the linear equation was 0.89, indicating that rm was dependent on the diffusive gas selected. In principle, evaluating rm requires at least a pair of experimentally measured De values (Pazdernik and Schneider, 1982). But if the diffusivity of only CH4 and C3H8 was measured, the rm value would be approximately -31354 nm. This value is unrealistic and far from the value of 267 nm, estimated from multiple De values. This situation is problematic for the use of the MTPM to evaluate rm for the simulated washcoat layer. Figure 3(a) demonstrates the dependence of rm on the selected number of De value at room temperature. When the number of De values was 2, 3, 4, 5, 6, 7, or 8, the estimated rm value was 36, 84, 126, 126, 84, 36, or 9, respectively. Figure 3. Dependence of number of De values on (a) value of rm, (b) variation range of rm As shown in Figure 3(a), a negative value of rm often occurred when the number of De values was less than 6. When the number of De values was greater than 7, rm was never negative. When 8 values of De were used, the minimum and maximum values of rm were 646 and 170 nm, respectively. These values are both close to 267 nm. Figure 3(b) shows that the range of variation in rm decreases as the number of De values increases. This was also found to be the case for De measured at 473 and 673 K. Thus, the value of rm depended on the number of experimentally measured De values. This was related to the inherent conventional use of the MTPM. Although Figure 3(b) implies that more than 8 De values may be required to obtain reasonable value of rm, in catalytic R&D processes measuring many De values is impractical. Thus, a new method is proposed to estimate reasonable rm values with minimal experimental effort. 3-3. Estimating reasonable rm values with minimal experimental effort If De can be predicted accurately, combining experimentally measured De and predicted De values to provide a reasonable rm value. Although many models are available for predicting De for a porous structure, the De 1593 values obtained using these models often have large errors due to the difficulty in factoring in the porous structure. For example, Hayes et al. (2000), measured the diffusivity of CH4 in a washcoat layer and compared the experimentally measured De value with the predicted De value. The De value was predicted using the pore size distribution of the washcoat layer based on the random pore model (Wakao and Smith, 1962). The pore size distribution was measured using mercury porosimetry. The results indicated that the predicted De was approximately seven times greater than the experimentally determined De value. This difference was too large to use for estimating a valid rm value. The present study focused on the mean molecular speed (vm) to predict De because gas diffusivity depends on the vm value, the vm value is independent of the pore structure, and vm can be estimated from gas kinetics. If De can be expressed as a function of vm, other unmeasured De values could be predicted. The relation between experimentally measured De and vm values is shown in Figure 4. De values measured at the same temperature can be described using a linear function of vm. Although a rigorous theoretical explanation for this linear relationship is difficult to provide, the theoretical diffusion equation can explain part of the linear relation. Bulk diffusion is described by the Chapman-Enskog equation (Smith, 1981) as: ( ) πσ /8 /1 0018583.0 2 2/1 , RP MMTv D ijij jiimb ij Ω + = (5) where T and Mj are independent of vm,I, indicating that the linear relation between vm and De requires De to be measured at the same temperature and with the same gas. Figure 4 shows the relation between vm and De and the temperature dependence of the slope. The relation between vm and De can be verified with data previously reported by Pazdernik and Schneider (1982), Valuš and Schneider (1985). They determined De in porous material with the chromatographic method. Pazdernik and Schneider (1982) tested commercial porous pellet. Valuš and Schneider (1985) examined samples of bidisperse alumina with changing volume and size of pores. Figures 5(a) and 5(b) show the relation between De and vm estimated from literature values. The relation between vm and De is linear when measured with same gas pair. Figure 4. Relation between De and vm Figure 5. relation between De (diffusive gas / pair gas) and vm estimated from values reported by (a) Pazdernik and Schneider (1982) and (b) Valuš and Schneider (1985). 1594 The linear relation between De and vm is reasonable based on the data presented and allows the reliable prediction of any De value from one pair of experimentally measured De values. To estimate rm from experimentally measured De values and predicted De values: 1. Measure two De values, such as for H2 and C3H8. These two values can be used to create a line. 2. The new De value, not obtained from measurements, can be predicted from this line using the vm of the specific molecule. 3. Thus, the rm value can be estimated from the measured and predicted De values. This method was verified using nine De values measured at room temperature. For example, the De values using H2 and C3H8 were selected to provide a linear equation in Step 1. The linear relation between these values was used to predict De for He, CH4, Ne, N2, O2, C3H6, and CO2. Lastly, a combination of selected De values (H2, C3H8) and predicted De values (He, CH4, Ne, N2, O2, C3H6, and CO2) were used to estimate rm in Step 3. Because 36 pairs of De values were provided from nine sets of De values, 36 rm values were estimated in Step 3. In comparison, conventional use of the MTPM results in 36 pair of De values for estimating rm without the use of predicted De values. Figure 6 shows the relation between rm and the ratio of the molecular weight of the two molecules. This new method is possible only when yi in Eq(4) can be approximated by zero because the experimental conditions used cannot approximate yi. Hence, measured yi values were used for verification. The conventional use of the MTPM estimated rm within the range of -32000 to 600 nm, and approximately 42 % of the values were negative. In contrast, the new method did not produce a negative rm value when the molecular-weight ratio was greater than 15. The range obtained was 121 to 400 nm, which contains the value of 267 nm estimated from 9 sets of experimentally measured De values. Figure 6. comparison of methods for estimating rm values ×rm estimated from pairs of experimentally measured De values ○rm estimated from a combination of experimentally measured De and predicted De values --- indicates the value estimated from 9 data points for experimentally measured De values 4. Conclusions The validity of estimating rm based on the MTPM was verified with experimentally measured De values. The diffusive gases tested were H2, He, CH4, Ne, N2, O2, C3H6, CO2, and C3H8. Their diffusivity was measured in a binary gas diffusion system using a modified Wicke-Kallenbach diffusion cell at room temperature, 473 K, and 673 K. With the conventional MTPM, rm depends on the number of De data sets, and many De values are required in order to obtain a reasonable rm value. This is a problem with the conventional use of the MTPM. To obtain a reasonable rm value with minimal experimental effort, a method combining measured De and predicted De values was proposed. Although many models exist to predict De for a porous structure, the De values predicted from these models often contain a large margin of error due to the difficulty in determining the pore structure. Hence, the relation between De and vm was focused on to predict De because vm is independent of the pore structure and can be estimated from kinetic gas theory. The results showed that De could be described as a linear expression of vm, thus allowing the prediction of reliable De values from one pair of measured De values. The results also confirmed that combining measured De and predicted De values -31800 -31200 -1800 -1500 -1200 -900 -600 -300 0 300 600 0 5 10 15 20 25 Ratio of molecular weight of gas molecule For which De was experimentally measured rm 1595 provided a reasonable value for rm when the ratio of the molecular weights was greater than 15. This new method combining measured De and predicted De values, estimated from vm for the diffusive gases, is more efficient than the conventional use of the MTPM and can contribute to investigations involving automotive catalysts. Nomenclature De,i effective diffusion coefficient for component i, m2/s De,ik effective Knudsen diffusion coefficients for component i, m/s De,ijb effective binary bulk diffusion coefficient for pair i - j, m2/s Dijｂ binary bulk diffusion coefficient for pair i - j, m2/s Ki Knudsen diffusion constant of component i, m/s M molecular weight, kg R gas constant, J/(K mol) T temperature, K rm mean diffusive pore radius, m vm mean molecular speed, m/s yi molar fraction of component i Ψ ratio of transport pore porosity and tortuosity σAB collision integral ΩAB Lennard-Jones potential parameter References Corbetta. M, Manenti. F, Visconti. C.G., 2013, Development of a kinetic model of Lean-NOx-Trap and validation through a reactive CFD approach, Chemical Engineering Transactions 32, 643-648 Hayes. R.E., Kolaczkowski. S.T, Li. P.K.C, Awdry. S, 2000, Evaluating the effective diffusivity of methane in the washcoat of a honeycomb monolith, Applied Catalysis B: Environmental 25, 93-104 Kato. S, Ozeki. H, Yamada. H, Tagawa. T, Takahashi. N, Shinjoh. H, 2013a, Direct measurements of gas diffusivity in a washcoat layer under steady state conditions at ambient temperature, Journal of Industrial and Engineering Chemistry 19, 835 Kato. S, Ozeki. H, Yamada. H, Tagawa. T, Takahashi. N, Shinjoh. H, 2015b, Chemical Engineering Journal, in press, DOI: 10.1016/j.cej.2015.02.033 Pazdernik. O, Schneider. P, 1982, Chromatographic determination of transport parameters of porous solids, Applied Catalysis 4, 321-331 Schneider. P, Gelbin. D, 1985, Direct transport parameters measurement versus their estimation from mercury penetration in porous solids, Chemcal Engineering Science. 40, 1093-1099 Smith. J.M, in: Chemical Engineering Kinetics, 3rd ed., McGraw Hill Book Co, New York, 1981, Ch. 11. Valuš. J, Schneider. P, 1985, Transport Parameters of porous catalysts via chromatography with a single- pellet string column, Chemical Engineering Science, 40, 1457-1462 Wakao. W, Smith. J.M, 1962, Diffusion in catalyst pellets, Chemical Engineering Science 17, 825 1596