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KINETICS, CATALYSIS, AND REACTION ENGINEERING Back-Diffusion Modeling of NO2 in Catalyzed Diesel Particulate Filters Onoufrios A. Haralampous and Grigorios C. Koltsakis* Laboratory of Applied Thermodynamics, Aristotle University Thessaloniki, 541 24 Thessaloniki, Greece
The present study emphasizes the coupling between reaction and diffusion phenomena in catalyst-coated diesel particulate filters. The main emphasis is placed on the effect of NO2 backdiffusion, which is responsible for higher-than-expected reaction rates at low temperatures. Unlike traditional modeling approaches that neglect diffusion effects, the model presented in this study considers the coupled diffusion-reaction phenomena by proper discretization in the porous layer and wall. With the aid of the model, a theoretical analysis is performed on the importance of individual operating parameters on the regeneration performance of a catalystcoated filter. It is shown that the contribution of diffusion is responsible for about one-third of the total regeneration rate for the range of conditions typically found in medium-sized diesel engines. The results are sensitive to the values of the effective diffusivities of the species (in particular NO2) in the porous catalytic wall and the soot layer. For a deeper understanding of the involved processes, further work is needed to determine the effective diffusivities experimentally. Introduction Diesel particulate filter (DPF) technology is currently recognized as a technically feasible solution for the emission control of diesel engines.1 The wall-flow ceramic honeycomb particulate filter, introduced more than 20 years ago,2,3 is currently the most mature filter type. This type of filter exhibits an excellent filtration efficiency (higher than 95%). The main issue associated with the application of such filters is the accumulation of soot in the filter channels, which can increase the exhaust back-pressure to unacceptable levels. Filter regeneration via thermal soot oxidation is still the most challenging aspect of this technology. Because of the relatively low temperatures of the diesel exhaust gas, catalytic assistance in the form of fuel-borne catalysts4 or catalytic filter coatings5 is currently the state-of-theart for commercial systems. In any case, the regeneration system is largely supported by the diesel engine injection and control systems, which are responsible for increasing the exhaust gas temperature by “postinjection”, thereby initiating the regeneration process. The main oxidizing agent in uncatalyzed filters is oxygen, which is contained in appreciable quantities in the exhaust gas of the diesel engine. The same is true for regeneration systems supported with fuel-borne catalysts that are able to accelerate the reaction of oxygen with soot. Practical regenerations are then possible at temperatures on the order of 500 °C, rather than 600 °C in the case of noncatalytic systems. The situation is somewhat different in the case of catalystcoated filters. These filters are coated or impregnated * To whom correspondence should be addressed. Tel.: +30-2310-995870. Fax: +30-2310-996019. E-mail: greg@ antiopi.meng.auth.gr.
with Pt-based catalysts, which are very effective in promoting the oxidation reaction of NO, which is present in the raw diesel exhaust, to NO2. The latter is a strong oxidizing agent and is able to react with deposited soot at temperatures as low as 300 °C. However, because NO2 is formed on the catalytic wall, which is downstream of the soot layer deposit, the reaction with soot would not be possible unless NO2 were able to diffuse back to the soot layer, driven by the concentration gradient. Moreover, one has to take into account the fact that, in typical filter wall structures, the pores are partially filled with soot during filter loading. Therefore, a certain amount of soot will actually be downstream of the catalytic sites on which the NO2 is formed. To study the above phenomena involved in catalystcoated filters, mathematical modeling could be a helpful tool. This is especially true considering that detailed in situ measurements of species concentrations at the microscale is practically impossible. The purpose of this paper is to present a reaction-diffusion model for the phenomena occurring in the coated filter channels. According to the underlying assumptions, the various published models are categorized as zero-, one-, or twodimensional. A classical zero-dimensional model was presented by Bissett and Shadman.6 The main assumptions of the zero-dimensional approach can be summarized as follows: 1. All channels behave in an exactly identical way, which is true only when the heat losses are negligible, the flow at the filter face is radially uniform, and the soot is deposited uniformly in all channels. 2. The exhaust gas flows through two layers: the particle deposit, which shrinks uniformly with time during regeneration, and the porous ceramic channel wall. The soot deposition is assumed to be uniform along the monolith channels.
10.1021/ie034187p CCC: $27.50 © 2004 American Chemical Society Published on Web 01/27/2004
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3. The weakest assumption is probably that of an equal exhaust gas temperature entering the deposit layer over the channel length. This simplifying model construction employs a single spatial variable x along the gas flow through the layers, whereas all variations in the direction perpendicular to x are neglected. 4. The soot and wall temperature along the x direction is assumed to be uniform. The one-dimensional approach initially presented by Bissett7 eliminated assumption 3 (uniform exhaust gas conditions flowing through the soot layer and wall) by including the flow and heat-transfer phenomena along the channel. This approach allows inherently the cancellation of assumption 2, as the soot layer need not be uniform and can be computed as function of the distance from filter inlet. This one-dimensional model is thus capable of computing the flow distribution through the wall as well as the axial temperature gradients in the gas and solid phases along a representative filter channel.8 Assumption 1 (identical behavior in all channels) is also lifted in more comprehensive two-dimensional models, which take into account heat losses and flow maldistribution.9-12 In the case of nonaxisymmetric filter geometries (i.e., commercial segmented filters), more complex three-dimensional models are necessary for more accurate predictions of the heat transfer and flow distribution.13 Assumption 4 was also reexamined by Haralampous and Koltsakis,14 who studied the intralayer temperature gradients expected in the soot and wall layer, in light of recent findings regarding the microstructural particulate deposit properties. In all previously published models, the molecular diffusion of species was not taken into account. In the case of NO2, which is the species of interest in this study, its concentration in the inlet channel was considered to be constant and equal to that of the inflowing gas. However, NO2 diffusion can occur within the filter as a result of the concentration gradients developed between NO2-rich regions of the flowing gas and NO2-poor regions in the reacting soot layer during regeneration. This paper considers the importance of these diffusion phenomena during regeneration. With the aid of mathematical modeling, a theoretical analysis is conducted on the importance of individual operating parameters on the regeneration performance of a catalyst-coated filter. For simplification and clearer illustration of the diffusion effects, the modeling study is based on a single, isothermal channel approach. As mentioned above, the single-channel approach is valid when all channels behave in the same way. This implies uniform gas inlet conditions, negligible heat losses, and identical soot loading distributions in all channels. Although these assumptions are not realistic under high-temperature uncontrolled regeneration conditions,12 they can be safely acceptable for the purposes of the present paper dealing with low- to moderatetemperature steady-state regenerations. Because we confine our study to steady-state inlet gas conditions, the assumption of isothermal filter behavior is also acceptable because, in the temperature range studied here, the exothermicity of the reactions is almost negligible. The reason for introducing these assumptions is to avoid modeling complexities that are not directly relevant to the back-diffusion effects. In any case,
Figure 1. Schematic of channel model: (a) front view, (b) side view.
extension of the single-channel approach to nonisothermal, multichannel full filter modeling is straightforward.12 Single Isothermal Channel Model Gas-Phase Flow. The calculation of the flow field in the single-channel model is based on the solution of the mass and momentum balance equations. A schematic of the side and front views of a channel model is presented in Figure 1. The governing equations are as follows
Conservation of the mass of the channel gas ∂ 2 (d F v ) ) (- 1)i4dFwvw ∂z i i i
(1)
Conservation of z component of momentum of channel gas Rµvi ∂pi ∂ + (Fivi2) ) - 2 ∂z ∂z d
(2)
i
Pressure drop across the deposit layer and ceramic wall The pressure difference across the inlet and exit channel at fixed z is the sum of the pressure losses due to the flows through the soot layer and the wall
p1 - p2 ) ∆psoot + ∆pwall
(3)
According to the analysis presented in ref 15, the
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pressure loss through the soot layer is given by
∆psoot )
RT µdFwvw d ln Mgp j 2kp(p d - 2w j)
(
)
(4)
The soot permeability is a function of the local mean free path length and can be expressed as16
(
x )
p0 kp ) k0 1 + C4 µ p
T Mg
(5)
The pressure loss through the porous filter wall can be described by the Darcy law
∆pwall )
µvw w ks s
(6)
The parameters characterizing the permeability of the soot layer, k0 and C4, can be calibrated by engine experiments at various soot loadings, flow rates, and temperatures as described in a previous publication.15 A set of boundary conditions is necessary to solve the system of differential eqs 1-5. The mass flow rate and the gas temperature are given at the entrance of the inlet channel, and additionally, the pressure is given at the outlet channel exit. The solution of the system yields the channel flow field (vw, v1, v2) and the pressure field (p1, p2). Diffusion. NO2 is consumed in the soot layer and produced in the catalyzed wall. Consequently, NO is produced in the soot layer and consumed in the wall, and at the same time, O2 reacts in the soot layer due to thermal oxidation and in the wall due to NO oxidation. As a whole, these reactions induce concentration gradients of NO, NO2, and O2 between the bulk gas flow in the channels, the soot layer, and the wall, which are the driving forces of the intralayer diffusion. The governing equation17 for mass conservation of any species in the soot layer and wall is
vw
∂yj ∂x
- Dj
∂ ∂x
( ) fx
∂yj
)
fx
∑k cj,kRk
cm
∂x
(7)
where the geometrical parameter fx is defined as
fx )
b(x) d
(8)
{
d + 2x, x < 0 d, x g 0
(9)
The effective diffusivity is calculated according to the mixed diffusion model
(
τ 1 1 1 ) + Dj p Dmol,j Dknud,j
)
∂ 1 1 (v y ) ) vwy1,j + k (y - y1,j) ∂z 1 1,j df-w 1,j 1s,j df-w2 ∂ 1 1 (v y ) ) v y + k2,j(y2s,j - y2,j) 2 w 2s,j ∂z 2 2,j df df ws
x
8RT πMj
(13)
The mass-transfer coefficient for each channel and species is
ki,j )
ShDj di
(14)
The molecular flow at the surface of the deposit layer can also be expressed as
vwy1s,j - Djf-w
∂yj ∂ | ) - df-w2 (v1y1,j) ∂x 1s ∂z
(15)
The molecular flow at the wall surface in the outlet channel is given by
vwy2s,j - Djfws
∂yj ∂ | ) - dfws2 (v2y2,j) ∂x 2s ∂z
(16)
Combining eqs 12 and 15 yields the boundary condition for the inlet channel
vwy1s,j - Djf-w
∂yj | ) vwy1,j - k1,jf-w(y1s,j - y1,j) (17) ∂x 1s
Similarly eqs 16 and 13 yield the boundary condition for the outlet channel
∂yj | ) k1,j(y2s,j - y2,j) ∂x 2s
(18)
Equations 17 and 18 are equivalent to the Danckwerts boundary conditions extended with an additional term for diffusive mass transfer with the bulk gas flowing parallel to the wall. The following global reactions are considered to take place in the soot layer and the wall. They include soot oxidation with O2 and NO2, as well as catalytic conversion of species on the active sites of the wall
(
C + R1O2 f 2 R1 -
(10) R1 )
with the Knudsen diffusivity given by
dp Dknud,j ) 3
(12)
ws
-Djfws
The width available for flow b(x) varies in the particulate layer and remains constant in the wall
b(x) )
size dp are based on the microstructural properties of the soot layer and the filter wall. The boundary conditions should “couple” the phenomena in the wall with the gas conditions in the inlet and outlet channels. At these boundaries, one should consider the convective mass transfer from the bulk gas to the wall surface, which can be computed as usual according to the “film” approach with mass-transfer coefficients ki,j corresponding to laminar flow of both the inlet and outlet channels
1 CO2 + 2(1 - R1)CO 2
)
sFpA1e-E1/RTpO2 Mc
(19)
C + R2NO2 f R2NO + (2 - R2)CO + (R2 - 1)CO2 (11)
The values of the porosity p, tortuosity τ, and mean pore
R2 )
sFpA2e-E2/RTpNO2 Mc
(20)
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{
NO +
1 O T NO2 2 2
A3Fe-E3F/RTpNOpO2
(
G
1-
)
K′ Kp
K′ < Kp R3 ) A3Be-E3B/RTpNO2 Kp 1G K′ K′ > Kp
(
)
(21)
The parameter R is used in reactions 19 and 20 as an index of the completeness of carbon oxidation. Although these stoichiometry parameters might depend on the operating conditions, it is realistic to employ constant values for the present low-temperature investigations, as shown in a previous publication.18 Because of the high convective mass-transfer rates through the porous layer and wall, the species concentrations in the gas phase and at the gas-solid interface can be assumed to be equal. This justifies the use of uniform values for species concentrations in both the rate expressions and the mass balance (eq 7). The bidirectional oxidation reaction of NO to NO2 depends on the thermodynamic equilibrium function Kp(T). The reaction rates are modified to account for the equilibrium limitation as shown in the rate expression (eq 21), using the parameter K′, defined as
K′ )
pNO2 pNOpO21/2
(22)
According to the Langmuir-Hinshelwood assumptions, the heterogeneous reaction rate can be expressed accounting for the so-called inhibition term G in the denominator. According to the pioneering work of Voltz et al.19 on Pt catalysts, the inhibition by NO is given by the expression
G ) T(1 + AGe-EG/RTyNO0.7)
∑cC,kRk
is filtered in the wall pores, thus substantially reducing the wall’s permeability. This part of the accumulated soot is nearby or even downstream of the NO2 produced on the catalytic sites and is therefore subject to more favorable reaction conditions than the “cake” deposit. The distribution of soot as a function of distance in the cross-flow direction is more difficult to assess. It is possible to rely on filtration models, which take into account the filter pore structure, as well as the flow conditions and the particle size distribution of the exhaust gas.20,21 Although these models are based on the simplification of equivalent “fiberlike” filtration elements, they can be adjusted to produce macroscopically equivalent behavior compared to the respective experiments. For the conditions typically found in diesel particulate filters with over 95% efficiency, the resulting soot mass distribution decreases exponentially from its maximum value near wall entrance to near-zero values near the filter exit. For the purposes of the present modeling study, we assume that the soot mass in the filter wall follows this exponential behavior, which can be mathematically formulated as
(23)
Because the concentrations of other inhibiting species (CO, hydrocarbons) are very low in diesel exhaust, their inhibition contributions can be neglected. Soot Mass Balance. The rate of soot mass change due to reaction in the soot layer and inside the wall is
1 dmp Mc ) mp dt Fp
Figure 2. Qualitative representation of a typical pressure drop measurement as a function of filter soot loading. After a certain soot loading, the pressure drop increases linearly, indicating cake filtration.
(24)
Soot “Trapped” within the Filter Wall. During the initial phase of filter loading, starting from a clean filter, the pores of the wall are partially filled with soot. For the present intralayer model, it is important to introduce the amount of soot in the wall separately to the soot cake deposit, as it is expected to significantly affect the reaction-diffusion phenomena. Although it is difficult to measure the exact distribution of soot in the wall, it is possible to estimate the mass of this trapped soot indirectly by examining the pressure drop behavior of the filter during loading. As shown in Figure 2, the pressure drop increase becomes linear only after a certain amount of filtered soot mass. The linear behavior is the indication of cake deposition, whereas the initial “asymptotic” form of the curve indicates that the soot
g(x) ) gmax e-βx
(25)
Moreover, we assume that this soot loading profile remains constant during regeneration. This implies that the soot mass consumed by the reaction in the wall is instantaneously “replaced” by the same amount of soot arriving from the remaining soot upstream. Although this assumption cannot be experimentally supported, it is used because of the lack of in-depth knowledge of such microscale soot migration phenomena in the wall. Nevertheless, this assumption is not expected to affect significantly the validity of the modeling study. Solution Procedure. Given the wall flow rate and the concentrations in the inlet and outlet channel, the intralayer concentrations can be calculated using eq 7 and its boundary conditions in eqs 17 and 18. The control volume including the soot layer and wall is divided into 20 elements. As shown in Figure 1, the soot layer has a trapezoidal shape when only one quadrant is taken into account because of symmetry. A numerical solution is necessary because of the nonlinearities imposed by the trapezoidal shape (denoted by the term fx) and the pressure variation inside the soot layer, which affects the reaction rates. Because of the deposit depletion, the soot layer is redivided at every time step
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Figure 3. Calculated soot mass in the filter with and without diffusion taken into account. Table 1. Geometry and Thermophysical Properties of the DPF Used in the Simulations filter material cell density wall thickness diameter length plug length wall density wall permeability porosity mean pore size
SiC 300 cells/in.2 0.356 mm 144 mm 152 mm 10 mm 1800 kg/m3 1.2 × 10-12 m2 50% 25 µm
into elements of equal mass. With an initial assumption about the intralayer concentrations, the reaction rates are calculated in each element. Equation 7 is then solved with an implicit finite difference method, resulting in a tridiagonal algebraic system for each species. The solution of the tridiagonal system yields the intralayer concentrations, which are then used to recalculate the reaction rates. This procedure is repeated until the concentrations converge to steady values. The concentrations of the first and the last elements are used to calculate the molecular flows due to diffusion from the inlet and outlet channels. These molecular flows are then used to calculate the downstream channel concentrations, which are used as input for the intralayer calculation of the next axial node. This procedure is repeated until the channel exit. Regeneration Analysis The geometrical and thermophysical properties of interest for the catalyzed filter considered are listed in Table 1. For the regeneration analysis, the inlet gas properties are considered to be constant with time and are given in Table 2. These gas and filter properties are characteristic for a medium-size passenger car diesel engine, operating at moderate-to-high speed and load. It should also be noticed that the temperature of 350 °C is favorable for the production of NO2 on the catalyst and is sufficiently high to initiate the reaction of NO2 with soot. Figure 3 shows the computed soot mass in the filter during the steady-state test, with and without diffusion effects being taken into account. In the former case, the reaction with NO2 is based on three mechanisms: (a) reaction of soot with incoming NO2; (b) production of NO2 on catalytic sites and reaction locally or downstream with soot trapped in the wall; and (c) production of NO2 on catalytic sites, back-diffusion, and reaction with soot upstream of the catalytic sites. In the com-
Figure 4. Computed profiles of NO2 concentration in the filter (a) inlet and (b) outlet channels with and without diffusion taken into account. Table 2. Exhaust Gas Properties Used for the Regeneration Analysis property
value
mass flow rate temperature O2 concentration NO concentration NO2 concentration
0.05 kg/s 350 °C 10% 1000 ppm (vol) 100 ppm (vol)
prehensive reaction-diffusion model, the predicted reaction rate is overall higher, indicating the importance of diffusion phenomena. For a better understanding of the NO2 diffusion phenomena occurring during regeneration, Figure 4 shows the computed axial profiles of NO2 in the inlet and outlet channels at a time point corresponding to 0 s. As expected, for the reaction-only model, the NO2 concentration in the inlet channel remains constant, whereas the reaction-diffusion model predicts a slow exponential decrease across the inlet channel, as a result of diffusion induced by the lower NO2 concentration at the interface between the bulk gas and the soot layer. The concentration in the outlet channel remains practically constant for both models, which is attributed to chemical equilibrium control of the reaction rates in the region. The quantitative difference is discussed further below. The concentration profile of NO2 in the soot layer and the wall is shown in Figure 5 at two different axial points in the channel at time ) 0 s. Focusing on Figure 5a (near filter entrance) and the reaction model, the NO2 concentration initially decreases as the flow enters the soot layer as a result of the reaction with soot, until it the soot is fully consumed. After entering the wall region, the concentration rises steeply in the beginning and more slowly further below, approaching chemical equilibrium at the wall exit. In the diffusion-reaction model, the NO2 produced in the wall back-diffuses” in the soot layer, and a slow increase in NO2 concentration is evident from the soot layer region. Meanwhile, a
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Figure 5. Computed profiles of NO2 concentration in the soot layer and the filter wall with and without without diffusion taken into account: (a) near filter entrance, (b) near filter exit.
Figure 7. (a) Mean reaction rate as a function of temperature for two different initial soot loadings. (b) Percent contribution of back-diffusion to overall reaction rate.
Parametric Study
Figure 6. Computed evolution of soot layer thickness along the channel for different time points with and without without diffusion taken into account.
In this section, the effect of NO2 diffusion is studied parametrically as a function of the operating parameters. Because the diesel engine operates over a wide range of conditions regarding the exhaust gas flow rate, temperature, and NO2 concentration, the purpose is to identify the conditions under which the effects of diffusion are especially important. For quantification purposes, we express the results in terms of a mean reaction rate (MRR) over a time period ∆t, defined as
MRR ) reduction of the concentration in the wall is observed, due to the same phenomenon, also resulting in smaller concentrations in the outlet channel. An additional NO2 flow is induced as a result of the concentration gradient between the inlet channel and the soot layer. A similar behavior is observed in Figure 5b (near the filter exit). Because the inlet NO2 concentration is very low (as also shown in Figure 4), back-diffusion is responsible for supplying the soot layer with NO2. The resulting soot layer profiles across the channel are displayed in Figure 6, which shows a preferential consumption of soot at the front rather than the rear part of the filter when accounting for diffusion. In contrast, uniform soot consumption is observed for the model without diffusion.
∆msoot ∆t
(26)
The time period ∆t selected for the evaluation of MRR was equal to 200 s, which is a value lower than that needed for complete regeneration in all cases. The model parameters in the parametric study are essentially the same as those used in the previous section. In all cases, it is assumed that the concentration of NO2 is equal to 10% of the NO concentration. The results are generated by changing one parameter at a time. Figure 7 presents the regeneration rate as a function of the temperature for two different initial soot loadings. The reaction rate increases with temperature, as expected, although not exponentially. This is because the overall soot consumption rate is controlled not only by the rate of soot oxidation with NO2 but also by the rate
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Figure 9. Contribution of diffusion as a function of tortuosity for two different temperatures.
Figure 8. (a) Mean reaction rate as a function of flow rate for two different NOx concentrations. (b) Percent contribution of backdiffusion to overall reaction rate. Figure 10. Percent contribution of back-diffusion to overall reaction rate for three different oxidation catalyst activity factors.
of NO2 production in the wall, which is limited by thermodynamic equilibrium at higher temperatures. Interestingly, there seems to be no dependence of the reaction rate on the initial soot loading, at least for temperatures less than 400 °C. This is not the case in noncatalyzed filters, in which the reaction rate is almost directly proportional to the soot mass. In catalyzed filters, the governing reaction of soot with NO2 takes place inside the catalytic layer and in the soot layer near the wall via back-diffusion. Therefore, the additional soot mass in the case of 10 g/L is not accessible to the NO2 produced in the catalytic layer. At temperatures higher than 400 °C, the reaction with oxygen starts to contribute, and therefore, the higher soot loading results in a slightly increased reaction rate. Figure 7b presents the percent contribution of backdiffusion to the overall reaction rate, which shows that the reaction-only model predicts reaction rates that are 25-35% lower than those predicted by the reactiondiffusion model. The discrepancy is higher at lower temperatures. The effect of the flow rate and NOx concentration in terms of the mean reaction rate is shown in Figure 8a. The temperature in these simulations is considered to be constant at 350 °C. The range of flow rate shown in the graph is representative of the conditions likely to be found in a diesel engine equipped with a filter of the size studied here. Considering the percent contribution of diffusion, it seems that the effect is higher at lower flow rates. This is expected because back-diffusion in the soot layer is dependent on the Peclet number of the flow through the porous layer and wall. The dependence
on the value of the NO concentration in the range of 500-1000 ppm is weak. An important parameter of the diffusion model is the effective diffusivity of the species in the porous wall and the soot layer. The experimental determination of these properties is not straightforward, especially in porous media with nonuniform pore distributions. Experimental techniques addressing this problem are discussed in the recent work of Kolaczkowski.22 This work presents measurements of effective diffusivities in catalyst-coated monoliths whose tortuosity factors can be determined. In the case of catalytic coatings in flow-through catalysts, typical values for the tortuosity factor range from 1 to 10.23,24 In Figure 9, the effect of tortuosity on the percent contribution of diffusion is shown for tortuosity values ranging from 1 to 10. The simulation parameters are the standard ones described in the previous section. The sensitivity of the results to tortuosity is very important, especially for values below 3. For the lowest tortuosity value of 1, the contribution of diffusion can be as high as 50% of the total reaction rate. On the other hand, even for high tortuosity values, the effect of diffusion is not negligible. The above results emphasize the importance of accurate determinations of the species effective diffusivities in the porous wall and the soot layer. Finally, the role of the oxidation catalyst activity is illustrated in Figure 10. The activity of the catalyst is modified by changing the preexponential factor in the rate expressions by a factor of 2 or 0.5. The results show
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that the contribution of diffusion phenomena is higher with more active oxidation catalysts. Conclusions The main conclusions of this paper can be summarized as follows: The regeneration of catalyzed diesel particulate filters systems is associated with diffusion phenomena coupled with the reaction processes in the soot layer and the catalytic wall. This paper presented a new model able to account for the combined reaction and diffusion phenomena. In low-temperature, controlled regenerations of Ptbased catalytic filters, NO2 plays a significant role. Apart from the small quantities of incoming NO2, NO2 is produced on the catalyst sites and can diffuse back to the soot layer to enhance the regeneration rate. With the aid of the model, the NO2 concentration profiles at the inlet and outlet channels as well as within the soot layer were computed and revealed an interesting NO2 diffusion behavior. The contribution of diffusion was evaluated computationally and shown to be responsible for about onethird of the total regeneration rate for the range of conditions typically found in medium-sized diesel engines. To validate the quantitative contribution of such phenomena, real-world experiments are necessary.25 The results are sensitive to the values of the effective diffusivities of the species (in particular NO2) in the porous catalytic wall and the soot layer. Further work is needed to determine these values experimentally. Acknowledgment We gratefully acknowledge the financial support of NGK Insulators, Inc. Notation Variables A ) reaction-rate frequency factor b ) geometric parameter defined in Figure 1, m C4 ) slip correction factor, m s/(kg mol K)1/2 cj,k ) stoichiometric coefficient of species j in reaction k cm ) molecular density, mil/m3 cpsi ) cell density, cells/in2 D ) mass diffusivity, m2/s d ) hydraulic diameter of clean channel, m di ) hydraulic diameter of channel i, m dp ) mean pore size, m E ) activation energy, J/mol fx ) b/d G ) inhibition term g ) capacity of soot mass in wall, kg/m3 K′ ) pNO2/(pNOpO21/2) ki,j ) mass-transfer coefficient of species j in channel i, m/s ko ) permeability of the particulate layer, m2 Kp ) chemical equilibrium constant kp ) apparent permeability of the particulate layer, m2 ks ) permeability of the ceramic substrate, m2 L ) filter length, m M ) molecular weight, kg/mol m ) mass, kg p ) pressure, Pa p j ) mean pressure of the inlet and outlet channels, Pa p0 ) reference pressure for apparent permeability calculation, Pa R ) universal gas constant, J/(mol K)
R ) reaction rate, mol/(m3 s) s ) specific area of deposit layer, m-1 Sh ) Sherwood number T ) temperature, K t ) time, s v ) velocity, m/s w ) particulate layer thickness, m ws ) channel wall thickness, m x ) space variable perpendicular to the wall surface, m yj ) mole fraction of species j z ) axial distance, m Greek Letters R1, R2 ) completeness indexes of reactions 1 and 2, respectively R ) constant in channel pressure drop correlation β ) constant in eq 25, m-1 ∆p ) back-pressure, Pa p ) porosity µ ) exhaust gas viscosity, kg/(m s) F ) density, kg/m3 τ ) tortuosity Subscripts 1s ) inlet channel-soot surface interface (x ) -w) 2s ) outlet channel-wall surface interface (x ) ws) g ) exhaust gas i ) channel index (1 ) inlet channel, 2 ) outlet channel) j ) species index k ) reaction index p ) particulate layer s ) solid, ceramic substrate w ) wall-outlet channel interface z ) axial direction
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Received for review October 15, 2003 Revised manuscript received December 12, 2003 Accepted December 18, 2003 IE034187P
