Tomography based simulation of reactive flow at the micro-scale: Particulate filters with wall integrated catalyst

Chemical Engineering Journal 378 (2019) 121919

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Tomography based simulation of reactive flow at the micro-scale: Particulate filters with wall integrated catalyst Robert Greinera,b, Torben Prillc, Oleg Ilievc, Barry A.A.L. van Settenb, Martin Votsmeiera,b,

T ⁎

a

Technische Universität Darmstadt, Ernst-Berl-Institut, Alarich-Weiss-Straße 8, 64287 Darmstadt, Germany Umicore AG & Co. KG, Rodenbacher Chaussee 4, 63457 Hanau, Germany c Fraunhofer ITWM, Fraunhofer-Platz 1, 67663 Kaiserlautern, Germany b

HIGHLIGHTS

GRAPHICAL ABSTRACT

based pore-scale reactive • Tomography flow simulation of gasoline particulate filter.

shows diffusion limitations • Simulation in the catalyst domains. effective diffusion length was de• An termined from the pore-scale solu-

• •

tions. Conversion well described by effectiveness factor based on generalized Thiele modulus. Pore-scale simulations were coupled with channel-scale model.

ARTICLE INFO

ABSTRACT

Keywords: X-ray microtomography Particulate filter Pore-scale Reactive flow Simulation Mass transfer

Three-way catalyst material was deposited inside the pores of a ceramic particulate filter and the pore geometry as well as the distribution of the catalyst in the pores was determined by X-ray microtomography. On the resulting 3D geometry, the flow field through the pores was computed and the convection-diffusion-reaction equation in the open pores and the catalyst particles was solved assuming a first order model reaction taking place in the catalyst. The conversion in the filter wall was compared to a homogeneous model with the same dimensions and catalyst content and it was found that the conversion in the pore network is lower than predicted by the homogeneous model, indicating the presence of some kind of in-pore micro-scale transport limitation. Analysis of the flow field showed channeling of the flow through a few large pores and blocking of many other pores by the catalyst that leads to a broadening of the residence time distribution. While the broadened residence time distribution has some negative effect on the conversion, diffusion limitation in the catalyst particles was identified as the main reason for the reduced conversion compared to the homogeneous model. When diffusion in the washcoat was described by a standard effectiveness factor based on a generalized Thiele modulus with an effective diffusion length fitted to the results of the full pore-scale simulation, a very good agreement between the pore-scale simulation and the homogeneous model was found. The effective diffusion length obtained by this fit is surprisingly large, compared to the apparent size of the washcoat particles. This can be explained by the limited accessibility of the washcoat due to the confinement in the pore structure. Finally, the 3D micro-scale model was coupled to a channel-scale model representing one pair of inlet and outlet channels. It is shown that micro-scale diffusion limitations have a higher impact on the overall reactor performance than the channel-scale mass transport. This suggests that for filter design and for reactor modeling more emphasis should be placed on micro-scale transport effects.

⁎

Corresponding author at: Technische Universität Darmstadt, Ernst-Berl-Institut, Alarich-Weiss-Straße 8, 64287 Darmstadt, Germany. E-mail address: [email protected] (M. Votsmeier).

https://doi.org/10.1016/j.cej.2019.121919 Received 1 April 2019; Received in revised form 5 June 2019; Accepted 7 June 2019 Available online 08 June 2019 1385-8947/ © 2019 Elsevier B.V. All rights reserved.

Chemical Engineering Journal 378 (2019) 121919

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List of abbreviations c cgas cin cout cs,in, cs,out c wall c wc cgas, slice c wc, slice

Deff Dgas DH d wall Dwc

Jx k1D, k1D k wc Leff n

p rsphere S u u wall v V x Xeffective Xmax z in , out

concentration (mol·m−3) concentration in the gas phase (mol·m−3) inlet channel concentration (mol·m−3) outlet channel concentration (mol·m−3) channel/washcoat interface concentration (mol·m−3) averaged wall concentration perpendicular to x (mol·m−3) concentration in the washcoat (mol·m−3) gas concentration averaged perpendicular to x (mol·m−3) washcoat concentration averaged perpendicular to x (mol·m−3) effective diffusion coefficient of the filter wall (m2·s−1) bulk diffusion coefficient (m2·s−1) hydraulic diameter (m) wall thickness (m) effective pore-scale washcoat diffusion coefficient (m2·s−1) molar flux along radial coordinate effective volume averaged reaction constant (s−1) volume averaged reaction constant (s−1) pore-scale washcoat reaction constant (s−1) effective diffusion length (m) normal vector pore/washcoat interface (−)

3D, slice

filter

,sphere

µ

wc

1. Introduction

pressure (Pa) sphere radius (m) pore-scale washcoat surface to gas phase (m2) fluid velocity vector (m·s−1) wall (radial) velocity (m·s−1) channel (axial) velocity (m·s−1) pore-scale washcoat volume (m3) radial coordinate (m) actual conversion (−) best case conversion (−) axial coordinate (m) channel/wall mass transfer coefficient (m·s−1) wall porosity (−) pore-scale effectiveness factor for a single slice along x (−) overall particulate filter efficiency (−) Thiele modulus effectiveness factor for spherical particle (−) dynamic viscosity (kg·m−1·s−1) fluid density (kg·m−3) wall tortuosity (−) Thiele modulus (−) washcoat volume share in wall (−)

faster heat up during vehicle cold start. Catalytic coatings also can help to reduce the temperature required for filter regeneration by soot oxidation. In Diesel vehicles, mostly the Diesel oxidation catalyst functionality or the selective catalytic reduction (SCR) functionality have been integrated in the particulate filter [8–12]. For gasoline vehicles, particulate filters with integrated three-way catalytic functionality have been developed [13–15]. Since these devices simultaneously remove CO, hydrocarbons, NO and soot, they are also called four-way catalysts. Wall-flow filters are also interesting from a reactor engineering point of view. While in a conventional open monolith, diffusion is required to transport the reactants into the catalyst layer, in a wall flow filter the gas is forced through the porous catalyst by convection. Due to this improved mass transfer, the monolith filter reactor also has been proposed for other industrial catalytic processes beyond automotive catalysis, especially for reactions where mass transfer limitations in the porous catalyst can lead to reduced product selectivity [16,17]. There are several reactor engineering studies that investigate the role of mass transfer in the monolith wall-flow filter reactor. These studies focus on the transport effects at the channel-scale and describe the catalyst containing porous wall by a homogeneous volume averaged model. Such homogeneous models compute the flow- and concentration profiles in the wall in terms of volume averaged velocities, concentrations, diffusion coefficients and permeabilities. Knoth et al. and Votsmeier et al. outlined the different mass transfer mechanisms in conventional open monoliths and the wall flow filter [18,19]. Dardiotis et al. have compared the efficiency of Diesel oxidation catalyst coated on an open monolith and on a filter [20] and later extended this analysis towards zoned designs [12]. Similar studies for the SCR catalyst have been described by Park and Ruthland [21] and Karamitros and Koltsakis [22]. Opitz et al. compared the cold start behavior of gasoline particulate filters and conventional three-way catalysts [23]. Recently, we presented a systematic investigation of the efficiency of the catalytically active filter reactor [24]. An overall efficiency factor was defined relative to the ideal plug flow reactor and it was shown that the effectiveness of the filter reactor is reduced by channel-scale mass transfer limitations, resulting in effectiveness factors below one. However, using a solution of the mass balance equations in terms of dimensionless numbers, it was demonstrated that mass transfer limitations in the filter reactor are generally small, compared to the mass

Due to the need for high contact area, catalytic systems generally exhibit a micro-scale structure. The standard simulation approach today does not explicitly treat this micro-structure but uses approximate homogeneous models that represent the micro-scale structure in terms of volume averaged quantities. In this homogeneous approach, microscale effects are generally included in effective volume averaged diffusion coefficients and reaction rates. While these homogeneous descriptions of the micro-scale frequently are successful in describing reactor behavior at the larger scale, model-based design of the catalyst micro-structure requires simulation models that explicitly resolve the pore-scale. In recent years, X-ray tomography with sub-µm resolution has become available with benchtop devices [1–3] and more sophisticated synchrotron based tomography methods achieve resolutions down to ~25 nm [4], which opens up the possibility of nano- and micro-scale reactive flow simulations on real measured catalyst geometries. In this contribution we apply tomography based micro-scale reactive flow simulation towards automotive particulate filters with a catalyst material distributed in the pores of the filter wall. Today, particulate filters are a standard component of the exhaust purification system of Diesel vehicles and are currently also introduced for gasoline vehicles. Gasoline engines emit much less soot mass, but especially fuel efficient direct injection gasoline engines emit a large number of small nanometer sized soot particles that are believed to be more dangerous than the larger diesel soot particles [5,6]. Therefore, recent legislation limits particle numbers, which means that in the future also gasoline vehicles will require particulate filters. Current filter designs applied in automotive catalysis use honeycomb monolith structures with a large number of parallel channels that are alternatingly plugged at the inlet and outlet so that the gas is forced through the walls and the soot is deposited inside or on top of the wall [7]. Besides the particulate filter, current exhaust systems contain several different catalysts for the removal of NOx, CO and unburned hydrocarbons. There is a strong interest to integrate at least some of this catalytic functionality into the particulate filter, as such multifunctional catalytic filters allow to reduce the number of components of the exhaust system. The resulting more compact exhaust systems are advantageous in terms of weight, space requirements and cost, and allow 2

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transfer limitations observed in conventional monoliths and other reactors that require diffusive transport into the catalyst volume. It was found that over the full range of possible dimensionless operating parameters, efficiency factors of the filter reactor are above 85%. It was also shown, that only in a small range of operating conditions the filter reactor is more efficient that the conventional monolith and that the efficiency of the open monolith never exceeds that of the filter by more than a few percent. This result seems to be in disagreement with practical experience, where it is frequently found that it is difficult to achieve comparable performance in a catalytic filter compared to the conventional monolith, even if similar catalyst compositions are applied. A potential explanation for this apparent contradiction is that besides the channel-scale mass transfer limitations studied in the earlier investigations, the efficiency of the filter reactor is further decreased by additional mass transfer limitations at the pore-scale that were not considered in the earlier analysis. In the current paper we make a first attempt to extend the analysis of mass transfer effects in catalytically coated wall flow filters towards the pore-scale. The distribution of the catalyst within the filter pores has an impact on the backpressure, filtration efficiency and catalytic activity of a catalytically coated filter. The optimization of the washcoat distribution is a challenging task because the distribution can only be controlled indirectly by changes in the coating process, and because it is difficult to obtain good backpressure, filtration efficiency and catalytic activity at the same time. From the beginning of commercial filter development, pore-scale simulation has been used to support filter design. Initially this relied on reconstruction of the 3D pore geometry for example from 2D electron microscopy data [25–27]. With increasing capability of microtomography, also tomography based simulation on real pore geometries became possible [1,2,28,29]. To date, the application of pore-scale simulation mostly focused on the computation of the flow field (back pressure) and the filtration efficiency [25,26,30] and the oxidation of soot [2,27,28,31]. There is limited simulation work that covers the catalytic reactions in the pore-network at the micro-scale. Several authors performed pore-scale simulations of NO oxidation in the filter wall and the effect of NO2 back diffusion on soot oxidation [2,28,32]. In those, the catalyst was not resolved and the catalytic reaction was assumed to take place uniformly on the pore walls. Recently, Kočí et al. presented tomography based pore-scale simulation of the concentration profiles in catalytically coated filters, with a focused on a comparison of on-wall and in-wall coatings [33]. In this work we apply tomography based micro-scale simulations of the reactive flow in catalytically coated gasoline particulate filters, using a single model reaction with first order kinetics, to investigate the effect of pore-scale mass transfer on the overall catalytic performance of the catalytic filters. The results of the micro-scale simulations are compared to the volume averaged homogeneous model applied in earlier reactor engineering studies of catalytic particulate filters and it

is shown that pore-scale mass transfer can significantly limit the overall efficiency of the filter reactor. It will be furthermore shown that the micro-scale mass transfer can be efficiently represented by an effectiveness factor approach based on a generalized Thiele modulus and an effective diffusion length determined from the micro-scale simulations. This leads to a modified homogeneous model that well describes the overall efficiency of the catalytic filter reactor. Finally, we couple the wall-scale simulation to a channel-scale model and investigate the relative importance of micro-scale versus channel-scale mass transfer limitations. The investigated scales, as well as a particulate filter schematic are depicted in Fig. 1. 2. Methods 2.1. Gasoline particulate filter A model gasoline particulate filter was prepared based on a cordierite substrate with a length of 6 in., a diameter of 4.66 in., a cell density of 300 cells per inch2, a wall thickness of 12 mil (305 µm) and a nominal porosity of the filter wall of 65%. A three-way active washcoat was prepared containing the support oxides alumina and Ce/ZrO2 in a ratio of roughly 1/1. The washcoat was coated on the filter substrate using a simple laboratory hand coating procedure. The total washcoat loading on the final filter was ~125 g·L−1, the precious metal loading was 20 g·ft−3 with a ratio of Pd/Rh = 1/9. Out of this filter, a single channel including the neighboring intersections was mechanically extracted and analyzed by X-ray tomography. 2.2. X-ray tomography X-ray computed tomography was used to determine the three-dimensional shape of the pore structure in a channel segment. For the analysis, a phoenix nanotom m from GE was used. Reconstruction of the X-ray adsorption data for our sample yielded a 3D description of the filter geometry with a voxel (3D pixel) size of 1.4 µm. A cross-sectional view through the reconstructed wall is shown on the left side of Fig. 2. The relative intensity (brightness) is a measure for the X-ray absorption of the materials, where low-density material is shown by dark pixels and high-density material by bright ones. Since the absorption roughly corresponds to the atomic numbers of the elements and since the catalyst domains contain the heavier elements Zr and especially Ce while the cordierite of the wall contains only the light elements Mg, Al, Si and O, the catalyst domains and the wall can be distinguished in the reconstructed tomography data by their different absorption. Due to the very low density, the open pores can distinguished as well. In order to simulate the three dimensional filter wall, the grey-scale image resulting from the reconstruction of the tomographic images has to be segmented, i.e. each voxel has to be assigned as either pore, washcoat or substrate. This segmentation was achieved using the MATLAB Image

Fig. 1. Schematic with the different scales covered by the particulate filter simulation. 3

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Fig. 2. Cross section of the filter wall. Left: grey-scale image corresponding to the measured intensities, right: segmented (black: pores, light grey: cordierite, white: washcoat).

Processing Toolbox™. An exemplary result is shown on the right side of Fig. 2. Fig. 3 shows a 3D render view of the resulting simulated domain. In the blue circle, a zoom onto the voxel level of the geometry is shown.

(( u · ) u )

·(µ u ) =

p ,

·u = 0

(1)

In these equations, u is the fluid velocity vector, p is the pressure, = 0.617 kg·m−3 the density of the fluid and µ = 2.93·10−5 kg·m−1·s−1 its viscosity, where the physical parameters were chosen corresponding to air at 300 °C. A constant velocity is imposed at the inlet boundary (2) and constant pressure at the outlet boundary (3). At the tangential domain boundary, a symmetry boundary condition is applied (4) and at the solid-fluid boundary condition, a no-slip boundary condition is used (5).

2.3. Tomography based micro-scale simulation of the filter wall 2.3.1. Computation of the flow-field In a first step, the flow field in the open pore structure was computed. The segmentation of the tomography data results in a cubic mesh where each volume element is assigned as either open pore, catalyst or support. The simulation is carried out directly on the cubic mesh resulting from the reconstruction of the tomography data. We assume that the flow in the catalyst and in the substrate can be neglected due to their low permeability so that the Navier-Stokes Equation (1) is only solved in the open pore domain with a no-slip boundary condition applied at the pore-catalyst and at the pore-substrate interfaces.

u =

p=0

u in mm 0 , u in = 79 s 0

(2) (3)

Fig. 3. 3D render view of the simulated domain (456 × 779 × 801 voxels). General flow direction along x-axis. Blue circle: zoomed region, where single voxels are visible. 4

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(4)

p · n = 0, u· n = 0

u = 0,

coefficient in the catalyst domains in the filter wall since integrating the catalyst in the filter pores generally requires smaller particles sizes and a more compact pore structure so that the effective diffusion coefficient should be lower than the effective diffusion coefficient of on-wall coatings in conventional monoliths. The effective diffusion coefficient of the substrate is assumed to be low compared to the washcoat, so that the convection-diffusion-reaction equation Eqs. (1)–(11) is only solved in the open pore domain and in the catalyst domain, with a zero flux boundary condition applied at the substrate interface.

(5)

p·n = 0

The Navier-Stokes equation was solved using the SimpleFFT-Solver implemented in GeoDict [34]. The code works directly on image data and uses a SIMPLE projection scheme. To accelerate convergence the code uses the Fast Fourier Transform for solving the pressure equation. Since the discretization of the equation results in a staggered grid, the velocity values have to be interpolated to the cell centers for the subsequent reactive transport simulation.

2.4. Channel model

2.3.2. Computation of the concentration-field In a second step, the flow field obtained as described in the previous section was used to compute concentration profiles in the open pores and in the washcoat domain. In this study we assume a simple model reaction A → B with first order kinetics. The mass balance equation was solved in the open pore space and in the catalyst domain. In the open pore space the species mass conservation equation

·(cgas u ) = Dgas

2c

gas

To study the contribution of micro-scale mass transfer effects on the overall catalytic conversion in the filter, a channel-scale model was set up. The model describes the flow in an inlet channel, an outlet channel and the wall connecting the two channels. In the 1D-Model, the flow in the inlet and outlet channel is described in terms of velocity averaged concentrations. Mass transfer between the channels and the wall is described by mass transfer coefficients using the correlation of Bissett et al. and Kostoglou et al. [38,39] that takes into account the effect of the wall flow on the gas phase mass transfer. The mass balance equations for the channels read:

(6)

was solved, where cgas is the concentration of a pollutant, Dgas is the molecular diffusion coefficient and u is the velocity in the open pores computed as described in Section 2.3.1. In the catalyst domain, the equation

k wc c wc = Dwc

2c

wc

(7)

was solved where c wc is the interstitial concentration in the porous catalyst, k wc is the effective first order reaction rate in the catalyst domain and Dwc is the effective diffusion coefficient in the washcoat domains. As an inlet boundary condition a constant concentration has been chosen (8). On the outlet side, two different boundary conditions were used throughout this paper: either constant concentration, or no diffusion (9). Perfect interface conditions were assumed at the open pore – washcoat interface, i.e. continuity of the concentration and the flux (10) and (11), where n is the normal vector of the open pore/washcoat interface. At the interface between substrate and the other materials zero diffusive flux has been enforced. In combination with the velocity being zero at those boundaries, no flux enters the substrate.

c = cin

(9)

cgas = c wc

(10)

Dgas cgas·n = Dwc c wc· n

vin·cin z

0=

vout·cout z

in ·

4 (cin DH

out ·

cs,in )

4 (cout DH

u wall·

4 ·cs,in DH

cs,out ) + u wall ·

4 ·cs,out DH

(12) (13)

where the indices ‘in ’ and ‘out ’ stand for the inlet and outlet channel, respectively. v is the velocity in axial direction, the mass transfer coefficient, DH the hydraulic diameter of the open channel cross section, u wall the gas velocity through the wall and cs the concentration at the channel-wall interface, which can be defined as:

cs,in = c wall |x= dwall,0 cs,out = c wall |x= dwall,end

(14)

Mass balances of the channels are coupled to either the full 3D tomography based micro-scale model described in Section 2.3 or to the 1D homogeneous model described in Section 3.3. In each case the coupling between the wall model and the channel models is achieved by matching the inlet and outlet fluxes of the wall to the flux from the channel-wall mass transfer:

(8)

c·n = 0

0=

Deff ·

(11)

Deff ·

Again, the equations were solved using the PoreChem simulation tool on the grid corresponding to the voxel grid of the µCT image, using a cell centered finite volume discretization. The resulting linear systems were solved using the BiCGStab algorithm with an incomplete LU factorization (ILU(0)) as preconditioner [35]. As a diffusion coefficient for the gas/pore phase Dgas a value of 65 mm2·s−1 is used, which represents the diffusion coefficient of nitrogen at 300 °C obtained from [36]. Diffusion in the washcoat domain is described by an effective diffusion coefficient Dwc . To our knowledge, there is no experimental data on the effective diffusion coefficients for the pore-integrated catalyst domains in a catalytic particulate filter. We therefore performed our simulations with two different effective diffusion coefficients Dwc , which were chosen so that they cover the range of possible diffusion coefficients that can be expected in this situation. The lower value of 1.0 mm2·s−1 is slightly higher than pure Knudsen diffusion in the nanopores which certainly represents the lower limit of the expected mass transfer coefficients. The higher value of 6.5 mm2·s−1 is taken from Zhang et al. [37] who measured the effective diffusion coefficient for a washcoat coated on the wall of a standard open monolith. This should be an upper limit for the effective diffusion

dc wall dx

x = dwall,0

dc wall dx

x = dwall,end

=

=

in ·(cs,in

out ·(cout

cin )

cs,out )

(15) (16)

To reduce the number of pore-scale simulations, we assume that the average velocity entering the porous wall is constant along the length of the channel. Two solutions of the 3D micro-scale model are then computed using two different sets of boundary conditions. Making the further assumption that concentrations and fluxes are uniform across the inlet and outlet boundaries, the wall-scale solutions for arbitrary boundary conditions can then be represented by a linear combination of the two computed solutions. To validate the assumptions allowing the representation of arbitrary solutions as a linear combination of two computed solutions, Fig. S1 in the Supplementary material compares the numerical 3D solution obtained with a fixed inlet concentration and a Neumann zero gradient outlet boundary conditions to the respective solution obtained from the linear combination of two different solutions (Fixed concentrations of c wall = 1 mol·m−3 and c wall = 0 mol·m−3 at the inlet and outlet respectively, and vice versa). Fig. S2 in the Supplementary material compares the outlet conversions of the full numerical 3D solution to the conversion obtained by the linear combination reproducing the 5

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Neumann outlet condition. Again, very good agreement is observed, indicating the validity of the assumptions underlying the linear combination approach. Due to the assumption of a constant wall velocity, the concentration profiles in the entire wall can be obtained by just two numerical solutions of the 3D wall-scale model. Constant wall velocity is a good assumption whenever the flow resistance in the wall dominates the flow resistance along the open channels, i.e. at higher soot loading or for less porous filter substrates. Modern highly porous filters show deviations from a constant wall flow velocity. The effect of non-uniform wall flow profiles has been studied in our previous study [24] and it was shown there that the non-uniform flow profiles do not qualitatively change the reactor behavior. 3. Results & discussion

many pores are blocked by the washcoat and that through this blocking, dead-end zones are formed with nearly stagnant flow and flow velocities below 20% of the inlet (again shown in green color). The blocking of many pores and the formation of dead-end zones further contributes to the acceleration of the flow in the remaining open pores. Fig. 5 suggests that due to the inhomogeneous distribution of the flow, many washcoat domains are not well accessible by convection, which might lead to diffusion resistances in those washcoat domains. This aspect will be discussed in more detail in the following sections of this paper. The inhomogeneous distribution of the flow also suggests a broadening of the residence time in the wall. Fig. 6 shows a residence time distribution computed based on the flow field. A substantial broadening of the distribution is observed as a consequence of the flow channeling through the few large and connected pores.

3.1. Flow field and residence time distribution

3.2. Concentration fields in the open pores and in the washcoat

In a first step, the flow field in the porous wall was solved. A wall inlet velocity of 79 mm·s−1 was chosen. This velocity corresponds to a space velocity of 120 000 h−1 for the whole particulate filter, which is in the upper range of the conditions encountered in the real world application. Fig. 4 shows velocity profiles through a number of planes perpendicular to the flow direction. It is observed that only a small number of larger pores with connectivity from the inlet to the outlet are responsible for a large part of the flow. In the pores with the main flow, flow velocities above 1 m·s−1 are reached, a significant acceleration compared to the inlet wall velocity before the wall of 0.079 m·s−1. On the other hand, a large fraction of the pores does not contribute significantly to the overall convective transport and shows low flow velocities, frequently below 20% of the inlet velocity (shown in green in Fig. 4 and Fig. 5a). Fig. 5a shows the effective flow velocity in a cut parallel to the flow direction. The cut has been chosen so that it shows a large pore connecting the inlet and outlet side of the wall. It is observed that at the inlet of the pore the flow accelerates from 0.079 m·s−1 to over 1 m·s−1. The figure also shows that the flow distribution in the filter wall is very inhomogeneous with only few pores accounting for nearly all of the flow and many other pores seeing very low flow. One can observe that

Using the flow field from the previous section, in a next step concentration fields in the open pores and in the washcoat domains were calculated. Fig. 5b shows a concentration profile for the same plane as shown in Fig. 5a. It is observed that in the large pore with the high flow velocity the conversion at the outlet is lower than in other pores. It is also observed that the concentration profiles in the washcoat domains show large concentration gradients with low reactant concentrations in the inner washcoat so that these washcoat inner segments contribute little to the overall conversion. Both, the observation of a broadened residence time distribution, as well as the observation of diffusion limitations in the washcoat domain suggest that the efficiency of the wall reactor might be reduced by micro-scale transport effects. Therefore, in the following section we compare the overall conversion of the wall reactor to the conversion in an ideal homogeneous reactor that does not show any transport limitations or broadening of the residence time distribution. 3.3. Comparison of 3D pore-scale model with a homogeneous 1D model Today, device level simulations of a particulate filter are generally carried out with homogeneous models that do not resolve the micro-

Fig. 4. Effective velocity profile in a series of slices in the micro-scale simulation. Gray: substrate; brown: washcoat. The velocity axis was cut off at 1500 mm·s−1. The inlet velocity was 79 mm·s−1. The green color means that the flow velocity is below 20% of the inlet velocity. 6

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Fig. 5. Slice along the flow direction (gray: substrate; brown: washcoat). The general flow direction is from top to bottom. a: effective velocity in the gas phase; b: concentration of the non-reacted reactant in the gas phase and washcoat with k wc : 18675 s−1 and D wc : 1.0 mm2·s−1.

required for the definition of the 1D model can either be performed based on the total volume or based on the open gas volume. The standard approach in the literature on particulate filter simulation is to average concentration based on the open pore volume and velocity based on the total volume including the catalyst and substrate domains. In this way, continuity in both concentration and velocity is obtained at the interface of the porous medium and the gas phase. The resulting mass balance equation for the homogeneous model is:

0=

u wall ·

dc wall + Deff · dx

2c

wall

x2

k1D·c wall

(17)

In Eq. (17), u wall is the volume averaged velocity which is equal to the inlet velocity of the 3D micro-scale model, c wall is the average concentration in the pore domain. The volume averaged rate constant k1D is obtained from the rate constant in the catalyst domains k wc and the volume fraction of the catalyst in the filter wall wc , which is 17%, as:

k1D =

wc · k wc

(18)

Deff is the volume averaged diffusion coefficient that is determined from a simulation of the micro-scale model without flow and chemical reaction. To determine Deff , fixed concentration boundary conditions are imposed on both sides of the wall and the diffusive flux through the wall is computed in the micro-scale model. An average flux (based on the open pore volume) is computed from the 3D solution and Deff is obtained from this average flux via:

Fig. 6. Cumulative residence time distribution of the gas in the simulated domain (including the inlet- and outlet zone) computed from the flow field, i.e. not considering diffusion.

structure and solve the balance equations in terms of volume averaged concentrations and velocities using volume averaged diffusion coefficients and rate constants. To investigate in how far the neglect of microscale transport effects in the 1D models is justified, in the following we compare the results of our 3D micro-scale simulation with the corresponding 1D homogeneous model. A simple 1D model is set up that reflects the geometry of our 3D domain, i.e. an inlet gas phase zone of length of 166 µm, a porous wall of length 316 µm and a gas phase outlet zone of length 156 µm. As in the 3D case, a concentration is imposed as a boundary condition at the inlet and zero diffusive flux is imposed as the outlet boundary condition. In principal, the volume averaging of concentrations and velocities

Jx =

Deff =

Deff ·

dc wall dx

· Dgas

(19) (20)

The obtained effective diffusion coefficients for the two different diffusivities in the catalyst domain are summarized in Table 1. The effective diffusion coefficient for the lower Dwc corresponds to the value obtained with a parallel pore model Eq. (20) and a tortuosity factor of 2.7 (porosity of 46%). This is in qualitative agreement with the experimental effective diffusion coefficients inside the wall of a coated 7

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filter determined in [40]. For the case of higher Deff , diffusion through the catalyst domains significantly contributes to the overall effective diffusion through the wall, which leads to an increased Deff , compared to the low Deff case, where most of the diffusive transport occurs through the open pores. Fig. 7 compares the concentration profile obtained by the simple 1D model to the concentration profile obtained by the 3D pore-scale model for a given rate constant. Obviously, the homogeneous model predicts a significantly lower concentration at the outlet than the 3D pore-scale model, indicating that the volume averaged model overestimates the efficiency of the wall integrated catalyst. A major objective of the remaining part of this paper will be to analyze the physical origin of the reduced efficiency in the micro-scale model and to provide a quantitative description of the reduced efficiency that allows to set up an improved homogeneous model.

concentration profiles in the open pores and in the washcoat. According to the concept of the generalized Thiele modulus, any of the standard textbook equations for the effectiveness factor in different washcoat geometries can be applied in this fit, since for a given generalized diffusion length Leff they all give very similar values for the effectiveness factor. We have used the equation for a spherical washcoat shape: ,sphere

c wc, slice cgas, slice

(21) −1

For a rate coefficient of k1D : 3162 s , the obtained effectiveness factors as a function of the x-position are plotted in Fig. 8 for the two different washcoat diffusion coefficients Dwc . Obviously, independently of the axial position, effectiveness factors below 100% are obtained. As expected, lower effectiveness factors are obtained for the lower washcoat diffusion coefficient. The effect of diffusion resistance in the washcoat naturally increases with increasing rate coefficient of the first order reaction. For each rate constant k , average effectiveness factors were obtained by averaging the effectiveness factors along the flow direction in the area marked in gray in Fig. 8, neglecting the transition regions from wall to open gas space and vice versa. Fig. 9 plots the so obtained average effectiveness factors as a function of the rate coefficient k1D . For each effective diffusion coefficient, a decrease in the effectiveness factor is observed as the rate constants increase. As expected, stronger diffusion limitations are observed for the smaller diffusion coefficient. We further observe that the functional form of the effectiveness factor as a function of the rate coefficient well resembles the functional form discussed in textbooks for simple washcoat geometries such as infinite slab, infinite cylinder or sphere. 3.4.2. Description of pore-scale diffusion limitations by an effective diffusion length A further quantification of the diffusion limitations in the intra-pore washcoat can be obtained by the concept of the generalized Thiele modulus proposed by Aris [41]. This concept is based on the observation that for arbitrary catalyst shapes the effectiveness factor as a function of a generalized Thiele modulus is described by a unified functional relationship, if the generalized Thiele modulus is defined in terms of a generalized diffusion length Leff , computed as the ratio of the catalysts outer surface S and the catalyst volume V :

Leff =

V ; S

= Leff

k wc D wc

·

1 tanh(3· )

1 3·

(23) (24)

The fit was performed here for the simulation results with the lower effective washcoat diffusion coefficient of 1.0 mm2·s−1. Fig. 9 shows that the effectiveness factors computed from the 3D concentration profiles are very well described by the fit with an effective diffusion length Leff of 22 µm which corresponds to sphere with a radius of rsphere : 67 µm. This effective length and the size of the corresponding sphere seems large compared to the apparent size of the washcoat segments in the pores, see Fig. 10. One should keep in mind though, that the washcoat is to a large extend confined in the pore structure, so that only a small fraction of the washcoat interface is accessible to the gas phase. Due to the confinement, effective diffusion paths in the washcoat are longer than suggested by the relatively small size of the washcoat domains. We also attempted to compute an effective diffusion length based on Eq. (22) directly from the reconstructed 3D washcoat geometry, i.e. as a ratio of the volume and surface area of the washcoat domains. The computation of the volume to surface ratio based on the 3D geometry is not straightforward since noise in the reconstructed geometry leads to an overestimation of the surface area and hence to an underestimation of the effective diffusion lengths. Furthermore, discretization error in the computed volume to surface ratio may cause a further underestimation of the effective diffusion length. From the evaluation of the surface area, an effective diffusion length of 11 µm is obtained (corresponding to a sphere with a radius of 33 µm), compared to an effective length of 22 µm (corresponding to a radius of 67 µm) obtained from the fit of the effectiveness factor model. The factor of two deviation between the different methods for determining the effective length seems too large to be completely explained by the uncertainty in the determination of the geometric surface area. An additional explanation might be provided by the presence of additional transport limitations in the open pores. In Fig. 5a it was shown that a significant fraction of the pore structure forms dead zones with flow velocities below 20% of the inlet velocity. We speculate that these dead zones function as additional diffusion barriers that lead to the increased apparent diffusion length obtained by our fit. A potential better description of the diffusion resistance in the dead zones might be by an additional external mass transfer coefficient. Fig. 9 shows that also the effectiveness factors for the higher effective washcoat diffusion coefficient are reasonably well described by the effective length obtained from the fit to the effectiveness factors for the lower effective diffusion coefficient. The fact that diffusion resistances are slightly overpredicted when using the effective length determined for the lower diffusion coefficients might be in line with our hypothesis of an additional diffusion resistance in dead zones, which would be present independent of the diffusion coefficient in the washcoat.

3.4.1. Catalyst effectiveness determined from concentration profiles in the pores and in the washcoat To quantify the diffusion limitations in the intra-pore washcoat, average reactant concentrations in the gas phase and in the washcoat were determined from the 3D concentration profiles for each plane perpendicular to the flow direction. Since we study a first order reaction, an effectiveness factor of the plane can be computed as:

=

1

Leff = 1/3· rsphere

3.4. Quantification of diffusion resistance in intra-pore catalyst

3D, slice

=

Table 1 Pore, washcoat and effective diffusion coefficients for the different diffusion cases that are applied in the different wall-scale models.

(22)

In a first step, we determine an effective diffusion length Leff by using Leff as a fit parameter and fitting this parameter to the effectiveness factors of Fig. 9, which were computed from the 3D 8

Diffusion case

Dgas (3D)/mm2·s−1

D wc (3D)/mm2·s−1

Deff (1D)/mm2·s−1

Low High

65 65

1.0 6.5

11 16

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Fig. 9. Average effectiveness factor obtained from the 3D concentration profiles as a function of the first order rate constant k1D . Effectiveness factors were first computed by Eq. (21) for each plane perpendicular to the flow direction. The effectiveness factors for the different x-positions are then averaged to obtain the average effectiveness factors plotted in the figure. Average effectiveness factors were computed for two different effective washcoat diffusion coefficients: 6.5 mm2·s−1 (red) and 1.0 mm2·s−1 (blue). The figure also shows the effectiveness factors computed by a fit of Eq. (23) to the average effectiveness factors obtained from the 3D concentration profiles for the lower effective diffusion coefficient. An average effective length of Leff : 22 µm was obtained from this fit.

Fig. 7. Concentration profiles along the flow direction for the 3D micro-scale model with D wc : 1.0 mm·s−1 and for the volume averaged homogeneous 1D model at k1D : 3162 s−1. For the pore-scale model, at each axial position the concentration was averaged in the open pore space over the plane perpendicular to the flow.

Fig. 8. Effectiveness factors for each slice along the flow direction x, where washcoat is present for different diffusion coefficients at k1D : 3162 s−1. Each data point in this figure has been obtained by averaging over the corresponding plane perpendicular to the flow direction and computing an average effectiveness factor by Eq. (21). The gray area marks the range considered for the computation of the wall-averaged effectiveness factors plotted in Fig. 9. Fig. 10. Spheres with the effective diameters determined by the fit of the effective diffusion length to the effectiveness factors in Fig. 9 (pink, Leff : 22 µm, rsphere : 67 µm) and determined directly from the segmented geometry as the ratio of catalyst volume and surface area according to Eq. (22) (blue, Leff : 11 µm, rsphere : 33 µm). For reference, the background shows the filter wall structure.

An equally good fit as obtained in Fig. 9 for the lower diffusion coefficient can also be obtained for the higher diffusion coefficient see Fig. S3 in the Supplementary material. In this case an effective diffusion length of 29 µm (rsphere : 87 µm) is obtained. From a practical model reduction perspective, the fact that the fitted diffusion length slightly varies with washcoat diffusion coefficient might be of minor relevance, since, in general, model reduction will be performed for a specific washcoat and hence a specified effective diffusion coefficient.

3.4.3. Implementation of the effectiveness factor in the 1D model The good description of the observed effectiveness factors through Eq. (23) suggests that the concept of fitting an effective diffusion coefficient to the 3D solution might provide a convenient way to obtain 9

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a simplified homogeneous 1D model. To test this approach, the effectiveness factor computed by Eq. (23) with the effective diffusion length, obtained by the individual fit to the 3D solutions, was implemented in the 1D model. This means that in the 1D model the rate constant k1D , previously calculated by Eq. (18) is replaced by:

k1D, =

,sphere ·k1D

The reactor-scale effectiveness factors with and without the contribution of the micro-sale transport limitations are compared in Fig. 12b. The figure shows that channel-scale transport effects only account for a decrease in the reactor efficiency of 12%. This is in line with our previous study [24] where it was shown that with a homogeneous wall model, reactor-scale efficiencies larger than 85% are obtained for the full range of possible operating conditions. According to Fig. 12b, micro-scale mass transfer accounts for the main part of the overall mass transfer limitation. Based on this result, it is clear that meaningful filter models of catalyzed filters should include micro-scale transport effects. We therefor conclude that in future simulation and catalyst development work, more emphasis should be placed on pore-scale transport effects. Fig. 12a also shows the conversion computed using the channelscale model with the wall represented by a homogeneous model incorporating the micro-scale mass transfer limitations by a local effectiveness factor as discussed in Section 3.4, i.e. based on an effective diffusion length obtained from the detailed 3D pore-scale simulation. Very good agreement is obtained between the homogeneous model incorporating our effectiveness factor approach and the model using the full 3D solution of the micro-scale mass transport. This result underlines the promise of our effective diffusion length based approach for the implementation of micro-scale mass transfer in reactor-scale simulations of catalytically coated particulate filters.

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As shown in Fig. 11, if the effectiveness factor is taken into account, the 1D model well describes the results of the full 3D simulation. This suggests that indeed the reduced efficiency of the in-wall washcoat is to a large extend explained by diffusion limitations in the wall integrated washcoat particles and that the broadening of the wall residence time due to flow channeling discussed in Section 3.1 only has a minor contribution to the overall loss in catalyst effectiveness. Our results also suggest that despite the very irregular shape of the washcoat particles, the diffusion limitations in the filter wall can be well described by the concept of a generalized Thiele modulus, i.e. by assuming an effective diffusion length in combination with the well-known textbook equations for effectiveness factors in regular washcoat shapes. This opens the possibility for an efficient implementation of porescale diffusion effects into volume averaged reactor-scale filter models that are today routinely used for exhaust system design. In a first step, an effective length characteristic of the respective in-wall coating is determined from a single micro-scale simulation with a simple first order reaction, based on tomography of a representative volume of the filter wall. Then, in the filter scale simulations, the effect of washcoat diffusion limitation can be considered by a simple effectiveness factor based on the effective diffusion length, obtained from the detailed wallscale simulation. The implementation of the wall-scale diffusion limitations into the channel-scale models for arbitrary complex kinetics could be further simplified by applying the concept of internal mass transfer coefficients towards the intra-pore diffusion problem [42]. The idea here would be to compute internal mass transfer coefficients as a function of the first order rate constant from a set of detailed wall-scale simulations with different k. The so obtained internal mass transfer coefficient could then be applied in the channel-scale simulation with arbitrary kinetics. Work in this direction is in progress in our group.

4. Conclusion We have demonstrated the tomography based micro-scale simulation of the reactive flow in the pores of a particulate filter with wall integrated catalyst. The conversion obtained by the micro-scale simulation was compared to the conversion obtained by a conventional homogeneous model of the filter wall that does not resolve the pore structure in the wall. It is found that the volume averaged model significantly overestimates the conversion, indicating the presence of some kind of micro-scale transport limitation that is not resolved by the volume averaged model. Diffusion limitation within the catalyst domains was identified as the main effect behind the observed reduced conversion in the micro-scale model and the effect was well described by a standard effectiveness factor model based on a generalized Thiele modulus with the effective diffusion length fitted to the pore-scale

3.5. Pore-scale versus channel-scale: The effect of pore-scale transport limitations on overall reactor efficiency So far, we have focussed our discussion on the transport processes in the wall and we have shown that the micro-scale transport limitations can have a significant impact on conversion. Finally, it is interesting to see the contribution of the micro-scale transport limitations to the overall performance of the filter reactor and to assess their importance relative to the channel-scale transport effects. To this end, we have set up a channel-scale filter model that represents one pair of inlet/outlet channels and the porous wall connecting the two channels. The mass balance equations for the gas phase in the inlet/outlet channels are coupled either with our 3D micro-scale model or with a simple 1D homogeneous model of the reactive flow through the filter. A more detailed description of the reactor-scale model and the underlying assumptions and simplifications is provided in Section 2.4. Fig. 12a compares the conversion of two channel-scale models where in one case the reactive flow in the wall is computed by our 3D micro-scale model and in the other case the wall is described by a simple 1D model. The figure also shows the expected conversion of an ideal plug flow reactor with identical catalyst loading and residence time. Following our approach in [24], we define a reactor-scale effectiveness factor filter which relates the observed conversion of the channel-scale models Xeffective to the expected conversion of an ideal plug flow reactor Xmax : filter

=

X effective Xmax

Fig. 11. Comparison of the overall conversion of the pore-scale model with consideration of the fitted effectiveness factor for the 1D rate constant.

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Fig. 12. a) The computed conversion for the low diffusion case as a function of the first order rate constant k1D , obtained by the channel-scale model with the wall represented by: the 3D micro-scale model (blue circles), a conventional homogeneous model not taking into account micro-scale diffusion limitations (green line) and a homogeneous model with the rate constant reduced by an effectiveness factor obtained from a solution of the 3D micro-scale model (blue line). Also shown: The conversion of an ideal plug flow reactor with a residence time identical to the wall residence time and with the same catalyst loading (black line). b) The reactor-scale efficiency factor according to Eq. (26) for the channel-scale model.

for this approach due to the relatively large pore sizes that can be well resolved by current tomography equipment, in the near future higher resolution tomography will become available, which will make this approach applicable to many more catalytic systems.

effectiveness factors. Analysis of the flow field in the pores showed channeling of the flow through a few larger pores. Due to the channeling, significant parts of the pore space see negligible flow, mainly because access is blocked by the catalyst deposited in the pores. The non-uniform flow distribution leads to a broadened residence time distribution that reduces the efficiency of the catalyst. In our case, this residence time effect is minor, compared to the effect of diffusion limitation in the catalyst domains. To analyze the relative importance of micro-scale intra pore mass transport limitations relative to channel-scale transport effects, the micro-scale model was implemented in a channel-scale model representing one pair of inlet/outlet channels. It was shown that microscale transport limitations can be significantly larger than channel-scale effects. In conclusion, this paper demonstrates the importance of microscale mass transport effects in particulate filters so that in the future more emphasis should be placed on these micro-scale effects, both in simulation and in catalyst development. Simulation so far focussed on homogeneous models and channel-scale effects. The generalized Thiele modulus based effectiveness factor approach with an effective diffusion length fitted to a micro-scale simulations might be a promising route to include micro-scale effects in device-scale simulations, although our results indicate that also external diffusion limitations gas phase, especially in the dead zones with little flow velocity, might contribute to the overall catalyst effectiveness. Taking into account these additional diffusion limitations might lead to further improvements in the model quality. In terms of filter development, the results of this work show that the catalyst distribution in the pore network is an important parameter and that significant improvements in catalytic performance can be expected by optimizing the catalyst distribution. In part, these improvements have been realized already in filter development work progressing in parallel to the here presented simulation study so that state of the art commercial particulate filters can be expected to show higher catalyst efficiency than the lab samples investigated in this study that were obtained by a relatively simple coating process. In this paper we demonstrated the usefulness of a tomography based simulation approach using the application example of the automotive particulate filters. While the particulate filter is particularly well suited

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