Update on The Science and Technology of Diesel Particulate Filters †

As diesel emission regulations become more stringent, diesel particulate filters (DPFs) have become possibly the most important and complex diesel emission control device. This paper provides an update on the science and technology of diesel particulate emission control, drawing from the results of many research projects carried out by the authors in collaboration with the automotive industry and its suppliers. Both fundamental as well as application-oriented approaches are presented to study the physicochemical characteristics of diesel soot particles and soot deposits formed in DPFs, which are viewed as multifunctional separators/reactors. Theroretical and experimental aspects of filtration efficiency, pressure drop, ash accumulation and soot reactivity (with emphasis on catalyst-assisted soot oxidation) are addressed, employing systems ranging from small-scale filter samples to full-scale devices installed in the exhaust system of diesel engines. Properly combined, the current state of knowledge, experimental methods and simulation approaches all provide a rational and systematic route for enhancing the design and reliability of future diesel particulate emission


Introduction
Advanced fuel injection technology in conjunction with an inherent high thermal efficiency have led to an increased market share of diesel-powered vehicles especially in Europe, and the trend is expected to continue assuming the diesel engine is able to meet ever more stringent emission limits in the future.This is dependent on the employment of cost-efficient, advanced emission control systems for particulate and NOx emissions.However, the adverse health effects of combustion-generated nanoparticles such as diesel soot generate a highly visible public debate where the role of aerosol scientists and technologists becomes very important both in terms of measurement and assessment technologies as well as technological solution developers/providers.
Particulate emission control entails measures to reduce both the solid (soot) particle emissions and the liquid particle (droplet) emissions formed due to condensation of the exhaust as it cools down.The latter can be easily reduced by regulating their precursors (reducing the fuel sulfur level and installing advanced diesel oxidation catalysts to oxidize the complete spectrum of hydrocarbon (HC) emissions).The solid soot emissions, however, remain intimately connected with the nature of the diesel combustion process, and their reduction can be achieved either by in-cylinder measures improving the combustion process itself, or by employing aftertreatment systems such as diesel particulate filters (DPFs).As diesel emission regulations become more stringent, the use of DPFs increases progressively in both lightduty/passenger car and heavy-duty/commercial vehicle applications.DPFs represent an important and highly complex type of multifunctional chemical reactor combining multiphase, separation, chemical reactions and material transformations over many disparate temporal and spatial scales (Fig. 1).Modern DPFs exhibit very high filtration efficien-cies, however, they need to be periodically cleaned (regenerated) in order to achieve efficient and safe operation of the vehicle.As typical diesel exhaust conditions are not hot enough to initiate and maintain particulate (soot) oxidation, active (engine) means are employed to raise the exhaust gas temperature up to the point that particulate oxidation can be selfsustained in the filter at fast enough rates (>650 C).To achieve the oxidation of soot particles at lower temperatures (250 -550 C), a number of direct and indirect catalytic measures can be employed ranging from fuel additives, generation of reactive species, catalytic combustion of post-injected fuel and filter coatings promoting soot oxidation.Modern trends 1,2,3,4) in passenger car emission control systems are now focusing on so-called "fitfor-life" solutions, thus overcoming the need for servicing (ash removal) of the DPF during the vehicle lifetime.Robustness and durability of the engine and emission control system is also clearly a first priority in heavy-duty surface transportation.These trends pose specific challenges and create new opportunities in the area of diesel emission control technology.State-of-the-art diesel emission control systems are complex assemblies of chemical reactors and separators, sometimes integrating different functionalities on the same monolithic support to achieve demanding requirements in space and cost, especially in passenger cars 5,6,7) .Such complex emission control systems require advanced simulation tools for their cost-effective design, development, system-level integration and optimization.In addition, requirements for robust on-board monitoring and control generate the need for efficient algorithms which are implementable in computationally limited engine control units (ECUs) and which will provide accurate knowl-edge of the state of the emission control system during vehicle operation, and which can be used in control loops for management of the integrated powertrain-emission control system.In the present paper, we provide an over view of our work in the area of diesel particulate emission control technologies, drawing from an over ten-year participation in several research projects in collaboration with the automotive industry and its suppliers.The structure of the paper is as follows: Initially, we discuss the physicochemical characteristics of diesel soot and then descibe the operation of a DPF as a separator, addressing its filtration efficiency and pressure drop behavior.We then discuss the operation of the DPF as a reactor and address the regeneration process (the oxidation of accumulated soot by various techniques, with emphasis on catalyst-assisted soot oxidation).Subsequently, we discuss the application of advanced simulation methods to DPF systems and provide the conclusions of this study.

Diesel Soot
The characterization of soot particles emitted from modern diesel engines is a prerequisite for any particulate emission control approach, and has attracted enormous interest in the literature, with some representative recent studies being those in reference 8,9) .From a control technology point of view, we are interested in the state of the soot particles in the raw exhaust upstream of the DPF.Depending on the exhaust temperature, organic components from the lubrication oil and/or the fuel could condense on the soot particles and form the so-called soluble organic fraction (SOF) of diesel particulate.However, this is typically a phenomenon occurring upon dilution of Fig. 1 Example of integrated soot-NO x emission control system.DPF stands for diesel particulate filter.Sensory inputs of temperature (T), pressure (P), pressure drop (ΔP), soot and NO x concentration will be required along the exhaust pipe.
the exhaust, sometimes leading to the formation of additional particles, the so-called nucleation mode 9) .A high-SOF-content soot is sometimes referred to as "wet soot", and a low-SOF-content soot as "dry soot".It should be noted though, that reference to the SOF content of diesel particulate matter almost always implies particulate matter sampled on a Teflon-coated filter after a relatively long residence time and a dilution process in a dilution tunnel at a temperature of 52℃ , just as the legislated sampling protocols stipulate.
It must be borne in mind that sampling lines and flame ionization analyzers for hydrocarbon emissions measurements from the exhaust typically operate at 190℃ to avoid condensation-induced sampling artifacts.In studies of diesel particulates under raw exhaust conditions using gravimetric raw exhaust sampling techniques 10) , it was obser ved that at an exhaust temperature of 350℃ no particle-bound SOF was present.
Most modern DPF systems are preceded by a diesel oxidation catalyst (DOC), which is employed to oxidize exhaust hydrocarbons (injected in the engine cylinder or in the exhaust) and thus raise the exhaust temperature to the levels needed for DPF regeneration.The DOC almost eliminates any SOF that might be present on the soot particles, hence for the remainder of this paper, the term soot particle will imply the predominantly solid aggregates that exist in diesel exhaust (also known as the accumulation mode particles 9) .The available measurements indicate that the majority of emitted solid diesel aggregate particles have electrical mobility and aerodynamic diameters in the range of 10-300 nm.Primary particle diameters are found to lie in the range of 8 to 40 nm 8,9,11) .
The composition and morphology of diesel soot particles in the raw exhaust is ver y important, as this is the aerosol that challenges the DPF.The morphology of the soot particles affects the structure of deposits 12) that are formed in the DPF (hence the engine backpressure), while their composition (which is predominantly carbonaceous) affects their oxidation potential and hence the ease of DPF regeneration 13,14,15,16,17) .The substructure of soot primary particles obtained by different combustion sources and fuels has been studied by high-resolution transmission electron microscopy (TEM) 13,14,15,18,19) , and it is connected to the soot reactivity.
Harris and Maricq 20) have shown that soot particle size distributions from various types of diesel engines (model years from 1995-1998), running on different fuels under a wide range of operating conditions, can be approximated by a "signature distribution" when the number-based particle size distribution (PSD) is normalized with respect to the total particle number concentration and the particle size is scaled with respect to the mean particle size.We refer to this way of presentation as Harris-Maricq (HM) coordinates.More recently 21,22) , we have measured soot size distributions from the exhaust of 5 turbo-charged, directinjection diesel engines (model years 1997-2003) with displacements in the 1.9-2.4Lrange and advanced fuel-injection systems (3 common-rail, one pump unit injector, and one rotar y-pump-based fuel-injection system).Particles were measured employing a Scanning Mobility Particle Sizer (SMPS) and a Long Path Multi-wavelength Extinction (LPME) analyser as described in 11) , and the obtained size distributions are plotted in Fig. 2, in HM coordinates.The average value of σg for each engine (operating at several steady state points) is also given in Fig. 2. A remarkable constancy of σg is observed, σg = 1.89 ± 0.08.When the Harris and Maricq 20) data are included, all data are consistent to within 4% with a constant shape for the particle size distribution with an average σg = 1.84.The constant shape for the size distribution arises as the steady state solution of the soot aggregate population balance that accounts for coagulation and oxidative fragmentation processes 21) .
The oxidative fragmentation of soot par ticles has been observed in flame studies by 23) and limits further aggregate growth.The resulting soot fragments re-collide with the soot aggregates, leading to the establishment of a steady state at which point coagulation balances fragmentation and therefore the emergence of a size distribution with a "constant shape".Naturally the simultaneous occurrence of these two processes will lead to a distribution of soot aggregate morphologies.It is well known that diffusion limited cluster-cluster aggregation, (DLCCA), a much-studied growth mechanism 24,25,26) , leads to aggregate structures with a Df = 1.8, while diffusion limited aggregation (DLA) (a growth mechanism where monomers collide with clusters) leads to aggregate structures with a Df = 2.5.Employing a simultaneous measurement of electrical-mobility-based and aerodynamic-diameter-based distributions, we plot in Fig. 3 how the fractal dimension of diesel soot aggregates is distributed according to their electrical mobility diameter, as presented in 27) .
We observe a very robust pattern over the different engines and operation conditions tested, namely that small soot particles have a nearly spherical/ compact morphology (Df = 3) which rapidly evolves toward the value Df= 1.8 (the DLCCA limit) as the soot aggregate size increases up to about 100 nm.Then further increase of the aggregate size leads Fig. 2 Measured soot PSDs for five different engines at several operation points.N is the particle number concentration, D p the particle diameter, D g the geometric mean particle diameter and σ g the geometric standard deviation of the PSD.
g to an increase of Df to a range of around 2.4-2.5.As mentioned above, such a fractal dimension is consistent with a mechanism of addition of monomers to a cluster, and we rationalize it by considering the removal by oxidative fragmentation of smaller units from a large soot aggregate and their subsequent recollision with the same or other aggregates, through a type of DLA process.The data are consistent with an overall average Df = 2.4, and a similar average Df has been obtained by 28) , among others.
In 21) it is shown that in principle, σg does not have a universal value but rather depends on the soot aggregate morphology (as described by the soot aggregate fractal dimension, Df) and the mode of the oxidative fragmentation process (e.g.attrition, random fragmentation, equal size fragmentation, etc).An average fractal dimension Df= 2.4 for the diesel soot aggregates 28) and a physically realistic random fragmentation mode are shown to be consistent in 21) with the experimentally observed values of σg.An explicit connection between the combustion process, the aggregate morphology and the easily measured σg can therefore be made.

The DPF as a Separator: Filtration and Pressure Drop
Filters separate the diesel soot aggregates from the exhaust by means of a variety of transport mechanisms.The main deposition mechanisms are those of Brownian diffusion and direct interception, while thermophoresis can be important in the presence of temperature gradients 30) .Over the last 20 years, many DPF system concepts have appeared which incorporate different filter media and geometric configurations, regeneration technologies and control/ monitoring options.The interested reader should consult the literature 31,32,33) as well as on-line databases 34) .

1) Filter media and configurations
The filter media and geometric configuration is a key element in the DPF system, and the selection process calls for a careful balancing of different criteria including pressure drop performance, particulate collection ef ficiency, regeneration, durability and cost.Geometric configurations of filters are listed in Table 1 35,36) .
The wall-flow monolith honeycomb design originally introduced in 1981 37) still remains the most popular configuration, since it is a ver y compact arrangement, thus exhibiting a low pressure drop without having to make sacrifices in filtration rate or available space.
Representative material properties are given in 36) .
Since the majority of filter configurations use some sort of wall-flow arrangement, we will focus our investigation for the remaining of the paper on the wallflow geometry and honeycomb structures.It should be noted though that the developments/results can be extended direct to the other geometries 38) .In the following text, we examine the factors that affect the Fig. 3 Fractal dimension of soot aggregates with different mobility diameters obtained with different engines and operating conditions.Measurements from one Euro II and two Euro III diesel engines and a diesel generator 29) as presented in 27) . 1.20

2) Filtration theor y
The first applications and validations of filtration theor y to diesel particulate filters were made in 39) for fibrous structures and in 30) for wall-flow extruded filters.As flow in porous media represents a challenging area of fluid mechanics, initial approaches 39,40,41) have employed so-called unit-cell models where the porous filter wall is approximated as a collection of "cells", each hosting an object of simple geometry (a sphere for granular filters and a cylinder for fibrous filters).In these studies, it was shown that the classic filtration theory employing the concept of the "unit collector" (spheres for extruded filters and cylinders for fibrous and foamy structures) can give a good estimate of the size-specific collection efficiency of "clean" DPFs with respect to solid particles.Fig. 4 demonstrates that application of the unit-cell filtration theory can successfully describe the emitted size distribution of diesel aggregates at the DPF outlet for widely varying pore sizes of prototype SiCbased extruded wall-flow monoliths with a porosity of 42%.
To account for the ef fect of par ticle accumulation on the filtration process, the unit-cell-based filtration theory was extended to include a local recomputation of the evolving unit-cell geometry (Fig. 5) caused by deposition of particles 35) .The transient filtration model derived was tested with ver y good agreement against experimental data with ceramic, metallic and fibrous filters, see for example 42) 3) Flow resistance and pressure drop of filters Based on the fundamental principles of fluid mechanics and flow through porous media, Konstandopoulos and Johnson 30) published the first analytical solutions for the flow fields and pressure drop of wall-flow monoliths in terms of the filter media microstructure and geometric configuration that were validated experimentally for a particular extruded monolith design.The analytical model (extended for non-Darcian flow effects) was later shown to be in excellent agreement with 3-D computational fluid dynamics (CFD) simulations and was further validated against a larger variety of filter media 42,43,44) .The Konstandopoulos and Johnson 30) model has been extensively tested against a variety of filter samples, and it is reported to give excellent a-priori predictions of the pressure drop, opening up new development possibilities 41,45,46) .The approach was later extended to include the influence of the accumulated soot accounting explicitly for the soot layer microstructure and its dependence on the operating conditions of the DPF 12,47) .In the present section, we outline the dominant factors that are responsible for the pressure drop of clean and soot-loaded DPFs 48) .

Clean filters
Based on previous work in the area 30,35,40,43,49,50) , the flow resistance of a clean DPF as expressed by its pressure drop can be very accurately described by totalling the individual pressure drop contributions, shown in Fig. 6.Each contribution to the pressure drop requires the specification of one or more parameters, which collectively define the required set of what we term "flow resistance descriptors".Flow resistance descriptors of DPF walls, experimental protocols for their determination and useful correlations have already been presented in 48) .The thin porous wall contributes a pressure drop that is described by the sum of a Darcy term and a so-called Forchheimer term 30,35,40,47,49)    Forchheimer coefficient are intensive properties of the porous medium and for a homogeneous medium, will not depend on the size of the sample measured.
In our past work 30,35,47) , we have introduced and validated a generalized representation for the permeability of DPFs, which can be stated as: Eq. 2 where the Stokes-Cunnigham factor 51) accounting for slipflow effects that are dependent on the local value of the Knudsen number of the flow through the porous filter wall, Kn = 2λ /d c (based on the prevailing mean free path λ and the typical grain diameter dc of the filter media), and the hydrodynamic interaction function fw(ε) for sintered granular media was derived in several interchangeable forms 48) .
Eq. 3 Slip-flow phenomena are not usually observed in bare (uncoated) DPFs since the pore/grain size of these filters is sufficiently greater than the gas mean free path.For some catalyst-coated filters, however, temperature-dependent slip-flow effects may cause the pressure drop to become smaller than that predicted from continuum flow.In this case, and as shown in 35) , one can employ a slip-flow-corrected permeability of the coating, kcoating.that depends on the local value of the Knudsen number of the flow through the coating, Kn = 2λ /d grain where dgrain is a characteristic length scale of the coating (which usually has a granular character).
grain × SCF The Forchheimer coefficient has dimensions of inverse length.Both k and β can then be related to a "pore size" (which is operationally defined by the employed measurement technique) and porosity of the porous medium.In 48) it was shown that the Forchheimer coefficient β and the Darcy permeability k were interrelated according to: The value of the constant in Eq. ( 4) will depend on the actual morphology of the elements making up the porous medium.Based on the known structural properties of commercially available wall-flow DPFs (porosity in the order of 50% and pore size in the 10-30 µm range) and ranges of DPF operating flow rates, it can be assumed that the Forchheimer contribution to the pressure drop will be negligible, unless filters are operated at relatively high filtration velocities.This can occur for some newer filter designs based on sheets of sintered metal, fibrous materials or foam structures.From experiments with granular sintered metal filter media 38) the value of the constant was found to be equal to 0.34, while from experiments with high-porosity fibrous filter media, the constant was evaluated to be 16.6 47) .Additional pressure drop occurs in a wall-flow filter due to frictional losses of the flow along the square channels of the filter.This pressure loss has a linear dependence on channel velocity for laminar flow, in a square channel of inlet opening a and length L, according to: where cf Re, for fully developed laminar flow in a  square cross-sectional channel, has a constant value of 14.227, and in our notation we designate:

F=2cfRe
Eq. 6 The assumption of fully developed laminar flow is usually satisfied in a wall-flow filter since the Re number based on the channel opening is typically less than 1000 for a DPF measuring 5.66 inches ×6 inches and with 200 cells/in 2 (cpsi).The factor ξ is a correction accounting for the effect of strong suction or injection on cf Re.For typical wall-flow filter applications, ξ can be taken as equal to 1 since local wallflow velocities uw are small enough to keep the wall Re number (uwaρ/µ) less than 2 30) .Finally, there are inertial losses due to contraction and expansion of the flow as it enters and leaves the filter channels.This component increases with the axial channel inlet velocity U according to: where ζ is the so-called contraction/expansion inertial loss coefficient, which in general depends on the filter fractional open cross sectional flow area and on the Reynolds number 47) .All these contributions are included in the following equation that gives the total pressure drop of a clean wall-flow filter: Eq. 8 Eq. 8 has been shown to be a predictive and validated macroscopic model for the pressure drop of wall-flow filters 30,35,40,43,49,50) , and requires as input the flow rate Q, the filter geometrical characteristics (VDPF, a, ws.L), the wall permeability k and the filter inertial loss coefficient, ζ.Eq. 8 can then be employed in conjunction with experimental data to determine the relevant flow resistance coefficients k and ζ, provided that all other parameters entered in Eq. 8 are known or can be independently measured.The estimation of k can also be attempted from the filter microstructure, employing any of a number of idealized geometrical descriptions of the porous medium structure, an approach that was used in the past 48) .In that case, the required input data are the "porosity" and "pore size", parameters that are themselves operationally defined based on the measurement technique.Such estimations, however, based as they are on idealized geometrical representations of the filter microstructure, provide only order-ofmagnitude estimates of the wall permeability.Trueto-the-geometry representations based on statistical computer reconstruction of the porous filter wall from microscope pictures and subsequent computational analysis of the flow through the wall have been introduced in 47,48,52) and provide more accurate estimates.Experiments for direct measurements of wall permeability can be performed using either filter disks of the same material formulation 35) or pieces from the filter wall, after the orthogonal walls have been removed carefully 53) .Such experiments are quite timeconsuming and are not easy to employ for the routine analysis of samples.An easier, non-destructive alternative for determination of the permeability of a wallflow filter sample is to rely on careful experimental measurements made on complete honeycombs and the application of predictive, continuum-level mathematical models, such as Eq. 8.This is now standard industrial practice 54) .For the determination from first principles of the inertial losses coefficient, ζ, the use of threedimensional CFD simulation has already proved to be promising 47) .However, the computational effort required for an extensive parametric study of a multichannel configuration 47) makes experimental testing a more flexible alternative.

Soot-loaded filters
Depending on their microstructure, filters can exhibit more a so-called deep bed filtration mode or a so-called cake (or surface) filtration mode 55) .With reference to Fig. 7, the deep-bed filtration mode occurs initially (i.e.all filter structures will exhibit it) and is characterized by a non-linear increase of the filter pressure drop as a function of the accumulated soot mass in the filter.This is due to the initial deposition of soot particles inside the porous structure of the filter wall, which block the flow paths locally.Depending on the microstructure of the porous filter, even a small amount of deposited soot may have a huge effect on the pressure drop, since it may block a dispropor tionately large par t of the pore structure, hence the non-linear character of the pressure drop evolution.As the porous wall becomes more and more blocked by the deposited soot, there is a smooth transition to the cake filtration mode, where a macroscopic soot layer grows on top of the filter wall, characterized by a linear dependence of the pressure drop on the accumulated particulate mass in the filter.
It is obvious that more porous wall structures will exhibit a more pronounced deep-bed mode of filtration, however, in all filter structures that are currently commercially employed for DPFs, a cake mode of filtration eventually sets in, since the amount of soot one might want to be stored in the DPF before its regeneration is initiated is higher by far than the amount that the porous wall can accommodate.Typically, the porous wall will store between 0.5-2 g/m 2 of soot mass per filtration area (depending on the filter structure), while the total amount of soot stored before regeneration is on the order of 10 g/m 2 (depending on the employed DPF material and configuration).For predominantly cake-mode-type DPFs, a closedform expression has been derived for the pressure drop as a function of soot loading 35) .The soot-loaded DPF pressure drop is expressed in terms of a uniform, effective soot deposit thickness, w on the filter wall 35) , which with reference to Fig. 8 is computed as follows: Eq. 9 Eq. 10 The analytical expression for the DPF pressure drop has been extensively validated (Fig. 9) against 3-D CFD and experimental results 35) .Fig. 10 depicts the experimental transient loading behavior of different DPFs and their simulation using the transient filtration model of Konstandopoulos et al. 35) .Whenever the DPF pressure drop is significant when compared to the atmospheric pressure, compressibility effects need to be taken into account.The DPF pressure drop in the case of compressible flow has been analyzed in 48) as explained in the Appendix.

Soot deposit microstructure
Konstandopoulos et al. 12) demonstrated for the first time that during filter loading, the microstructure of the soot cake is determined by the relative strength of convective vs. diffusive transport of the soot aggregates towards the deposit Fig. 11.The soot cake packing density (ρsoot) and permeability (ksoot) were therefore shown to be not just "static" intrinsic physical properties of the soot cake, as it was commonly assumed in the literature.They were instead shown to be "dynamic" material properties of the soot porous cake that depend on the deposit growth mechanism and its histor y, and that they are determined by the prevailing value of the dimensionless mass transfer Peclet number, Pe = uw dpr/Dp, where dpr is the primary particle size of the soot aggregates and Dp is the soot aggregate diffusion coefficient.
The soot cake packing density (ρsoot) and permeability (ksoot) can be related to the porosity ε and primary particle size (dpr) of the soot aggregates as follows 12) : Eq. 11 where SCF is the Stokes-Cunningham factor calculated with a Knudsen number (Kn) based on the primary soot particle size (dpr) from: and f(ε) is the Kuwabara hydrodynamic function which depends on the porosity ε, through 30,55) : Further insight into soot cake properties has been obtained by model experiments with soot aggregates generated by a Combustion Aerosol Standard (CAST) burner (Matter Engineering, Switzerland).The CAST is a quenched diffusion flame gas (propane) burner that allows the stable and controlled generation of soot aggregates over a much larger size range than that found in diesel exhaust.Fig. 12 depicts the hydrodynamic resistance factor (ρ ×k)soot of CAST sootdeposited cakes on flat disk-shaped glass-fiber filters as a function of the Peclet number and aggregate mobility diameter dag.The values of the aggregate mobility diameters shown in Fig. 12 are values measured by the Scanning Mobility Particle Sizer (SMPS).The data are consistent with a scaling relation of the form: where the porosity of the deposits follows a power law in terms of the Peclet number 12,56) Eq. 16 generalizes earlier porosity-Peclet number power-law correlations 12) obtained at Pe > 0.3 down to the diffusion-limited deposition limit.Pe0 is a characteristic cross-over Pe number defining the scale beyond which the convective mechanism will take over the diffusive mechanism of deposition, and ε∝ the large Peclet number asymptote of the porosity.Using Eq. 16, the experimental data of Fig. 12 can be collapsed on a single curve as shown in Fig. 13.
At sufficiently high values of the pressure drop, soot deposit compaction starts to set in.Konstandopoulos et al. 47) accounted for deposit compaction in the pressure drop model, treating the soot deposit as a Bingham-type of material that remains undeformed below a yield pressure and that deforms with a power law in the post-yield region: Eq. 17 with φ denoting the solid fraction (1-ε) and φ the solid fraction of the deposit in the uncompacted stage.∆ Pcr is the critical or yield pressure drop for the onset of deposit compaction, and ∆P* is a scaling constant to make the equation dimensionally correct.
The extended pressure drop model that accounts for deposit compaction was applied in 47) to obtain for the first time the evolution of the soot deposit microstructure under compaction.As shown in Fig. 14, critical pressure drops for the onset of compaction of soot The continuous lines are plotted using the scaling relation form Eq. 15, Eq.16 47) .
Fig. 13 Dependence of the soot cake porosity for different soot aggregate sizes at the prevailing Peclet number.
Fig. 11 Soot deposit growth mechanism: competition between diffusionand convection-dominated growth, as determined by the Peclet number, leads to more porous deposits in the case of diffusion-limited deposition 12) .
cakes created by CAST-generated soot aggregates of different size are in the vicinity of 200 mbar and cause a gradual increase of the solid fraction (1-ε) of the compacted soot deposit, which can reach 25-30% for some aggregate sizes at high enough values of the pressure drop (> 400 mbar).These high pressure drops are non-typical for appropriately sized DPFs and under regular DPF operation, but it is possible to quantitatively take into account such phenomena if the need arises with the simple deposit compaction model mentioned above.

Effect of ash accumulation
The DPF pressure drop model has been extended to account for the presence of ashes in the channels of the DPF in 27,47) .Ash deposit growth dynamics was described with a mechanistic model that exhibits different ash deposition profiles: deposition along the filter channel walls as well as deposition at the end of the filter channel.A comparison to experimental data available in the literature (Fig. 15) showed good quantitative agreement, and the model can be used to describe the dynamic ash transport and deposition phenomena inside the DPF.
Considering two idealized modes of ash accumulation, namely (i) ash only on the wall (forming a layer in series with the porous wall and the soot cake) and (ii) ash only at the end of the DPF channel (forming a plug that reduces the DPF length), it was possible to derive in 57) analytic approximations for the optimum cell density of wall-flow filters that minimizes the DPF pressure drop, under the combined constraints of a prescribed filter volume, exhaust flow, temperature, as well as different soot and ash loadings inside the filter, and thus to facilitate the task of selection and reliable employment of diesel particulate filters over the vehicle life-cycle.

The DPF as a Reactor
As already mentioned, modern DPFs need to be periodically cleaned (regenerated) in order to achieve efficient and safe operation of the vehicle.As typical diesel exhaust conditions are not hot enough to initiate and maintain particulate (soot) oxidation, active (engine) means are employed to raise the exhaust gas temperature up to the point that particulate oxidation can be self-sustained in the filter, and therefore the filter acts as a soot oxidation reactor.DPF regeneration can be achieved by employing a number of different approaches including direct and indirect catalytic measures ranging from fuel-borne catalysts (also known as fuel additives), generation of reactive species, catalytic combustion of post-injected fuel and filter coatings promoting soot oxidation, as discussed in the subsequent sections.

1) Regeneration measures
It is convenient to classify regeneration measures 31, 32, 33, 34) as active (employing external or engine means, see Table 2) or passive (usually employing catalytic means, although catalytic means can also be applied in active systems, see Table 3).Combinations of measures are also common.

2) Soot oxidation rate during regeneration
During regeneration tests with an increasing temperature, the pressure drop of the soot-loaded filters decreases and at the same time CO and CO2 gases are emitted as the soot collected inside the filters oxidizes.In order to evaluate the regeneration behavior of different DPF technologies, the soot oxidation rate has to be calculated as a function of temperature.The normalized soot oxidation rate (s -1 ) is defined as: where m0 is the initial amount of soot mass collected.The soot consumption rate dm/dt is typically computed by adding the CO and CO2 produced during the oxidation in a synthetic exhaust gas stream which does not contain CO or CO2 in order to detect the soot-derived CO/CO2.The evolution of the normalized soot oxidation rate as a function of temperature, an example of which is given in Fig. 16, provides a means to compare and evaluate different DPF technologies with respect to their soot oxidation activity.
In addition, the CO selectivity which is defined as: Fig. 14 Pressure-drop-induced compaction of soot cake deposits as a function of soot aggregate size.
Eq. 19 can be also calculated.The CO selectivity is an im-portant parameter because it affects the total heat release during regeneration.

3) Catalytic DPFs
The development of catalytic DPFs (CDPFs) aims at achieving: (i) some soot oxidation activity under moderate exhaust temperature to prolong as much as possible the intervals between fixed regenerations, exploiting direct (i.e. through oxygen transfer) as well as indirect (through NO2 generation) soot oxidation; (ii) reduced soot ignition temperatures compared to uncatalyzed filters to allow for energy savings during regeneration; (iii) tolerance to ash accumulation.To achieve these goals, it is important to understand the different soot oxidation mechanisms and the significance of the soot-catalyst geometric proximity.

Catalytic oxidation mechanisms
Catalytic soot oxidation has a long history 63) , while some recent reviews appear in 64,65) .Catalyst chemistry is an important factor affecting the performance of a CDPF.Currently available CDPFs in the market feature predominantly catalytic coatings based on Ptgroup metal (PGM) formulations aimed at oxidizing NO into NO2 in order to achieve soot combustion at a lower temperature, motivated by the success of the CRT TM NO2-assisted system 62) .The influence of NO2 on soot oxidation, in conjunction with a highly selective NO to NO2 oxidation promoting catalytic coating on a DPF, was studied experimentally and theoretically for the first time in 35) .At that time, NO2-regenerative technologies were practiced with uncatalyzed filters 60,62) , and it was suggested that combinations of NO2-regenerative technologies with catalytic filters could lower the dependence of NO2-regenerative technologies on high engine-out NOx concentrations, as NOx emission standards become tighter.This has now become standard practice 60) .PGM-based CDPFs do not have a strong direct soot oxidation activity 11) and their operation depends on the balance between NO2 and soot in the exhaust.It has nevertheless become clear from these developments that a noble metal such as Pt could have a beneficial role if used in conjunction with a direct soot oxidation catalyst, since Pt can also deliver "active oxygen" to the soot-catalyst interface in addition to oxidizing NO into NO2.Soot oxidation catalysts based on base metal oxides are thought to act through two mechanisms 64,65) .(i) Redox mechanism: carbon oxidation is caused by lattice oxygen from the catalyst (reduction step) and re-oxidation of the catalyst by oxygen from the gas phase (oxidation step) (ii) Spill-over mechanism: dissociation of adsorbed gas-phase oxygen over the catalyst surface occurs, followed by surface diffusion to the soot surface where carbon oxidation occurs.The above mechanisms are not mutually exclusive and a soot oxidation catalyst can exhibit both.Direct catalytic soot oxidation requires soot-catalyst proximity, a fact that has been known for almost half a century 66) .The problem of soot-catalyst contact in diesel emission control systems was recognized in the 1980s 63) as a barrier for active catalytic filter development, and it has become popularized in more recent laboratory studies of powdered carbon blackcatalyst mixtures, with the introduction of so-called "loose" and "tight" contact 64,65) samples.Direct demonstration of soot-catalyst contact effects on diesel soot oxidation in filters has been published in 67) .A mathematical description of the incomplete sootcatalyst contact, the so-called "Two-Layer Model" was introduced a decade ago, and it has been since incorporated into state-of-the-art DPF simulators 68,69,70,71) .This forms the basic analysis tool that we employ in analysing soot oxidation rates in CDPFs as discussed in the next section.

The two-layer model of CDPF
Soot particle-catalyst contact is determined by the details of catalyst distribution in the filter (a type of "frozen" randomness) and the details of soot particle deposition and resulting deposit microstructure as well as soot deposit restructuring (a type of evolving randomness), it is therefore important to study it under realistic conditions, i.e. depositing fractal diesel soot aggregates from an engine under similar conditions of Peclet number, Pe 12) .The effect of Pe on the  soot deposit microstructure, soot-catalyst contact and reactivity is addressed in 72) .CDPF development has to address two key areas 67) : Chemistr y and geometr y and their interaction through the catalyst deposition process.The works in the literature dealing with chemistry at the powder synthesis level are too numerous to be cited here, while quantitative analyses of geometric effects have not been published as extensively.In the present work, in the geometric aspects of soot oxidation, we emphasize the kinetics in CDPFs by applying the two-layer formalism to analyze well-controlled direct catalytic soot oxidation experiments.Fig. 17 depicts a schematic of the two-layer model 68) .Soot and catalyst coexist in a region (Layer I), on top of which a soot-only layer exists (Layer II).

Layer I: Catalyst-affected layer
As the catalyst coating can interpenetrate or overlap partially with the top part of the wall, we can define Layer I to be the region over which a spatial "field of catalyst activity" exists.Particles that oxidize when found within the "field of activity" are considered in contact with the catalyst.The fraction β ∈ [0-1] of the soot surface in contact with the catalyst generally depends on the filter and coating structure and the deposition mode of soot in the filter.In general, β can vary dynamically due to reaction (oxidative fragmentation) and/or restructuring of soot microstructure.This layer can "store" a certain amount of soot (which depends on coating structure and filtration velocity) until it is filled up.

Layer II: Soot cake layer
This layer is formed by soot particles which form a "queue" on top of the filled-up catalyst-affected Layer I.These particles can locally migrate into the catalystaffected layer depending on an interaction parameter ξ∈ [0-1].This also depends on the filter and coating structure and can vary dynamically for the same reasons as β.
The interested reader can consult the original references for the mathematical formulation of the twolayer model 68) .A simple way, however, to understand the two-layer model dynamics is to consider the following system of kinetics in Eq. 20 and Eq. 21.In Eq. 20, m1 is the mass of soot in Layer I, a fraction β of which reacts with a catalytic rate and the rest of it reacts non-catalytically (thermal oxidation).In Eq. ( 2), m2 is the mass of soot in Layer II which is consumed by thermal oxidation, but which also migrates in proportion to the parameter ξ into any "open" space that becomes available in Layer I after some of the m1 soot reacts.
In the simplest applications of the two-layer model, the parameters β and ξ are taken to be constant.
The reaction constants are assumed to follow global modified Arrhenius forms 68) , and for well-controlled experiments with negligible oxygen variation across the filter, they can be assumed to incorporate into their pre-exponentials the oxygen concentration as well as the surface area of the soot.The two-layer formulation can also describe a fuel-borne catalyst system considering only Layer I and setting m2 and ξ equal to zero.
A number of filter structures have been catalyzed in-house with base-metal catalysts with a variety of coating technologies, aimed at direct catalytic soot oxidation.All filter structures had a disk form of 60 mm in diameter and they were loaded with soot under identical exhaust conditions in side-stream reactors, already described in the past 33,73) , and they were subsequently subjected to a slow temperature ramp (3℃/min) under a 10% O2-in-N2 atmosphere in order to study soot oxidation in them by following the evolution of CO and CO2 and effecting overall carbon balance.
For all the catalyzed materials there was a similar soot oxidation pattern.In Fig. 18, a typical soot oxidation rate cur ve for the aforementioned materials is depicted, where two regions of soot oxidation are observed.The first peak (in the region of 400-500℃ ), at lower temperature, originates mainly from the soot oxidized due to the soot-catalyst contact in Layer I, and the second peak, at a higher temperature, originates from the rest of the soot that is oxidized in Layer II.The first peak depends on the level of the contact that exists between the soot and the catalyst, whereas the second one occurs at the typical temperature range of thermal soot oxidation.Fig. 19 demonstrates how the two modes of soot oxidation (catalytic and non-catalytic) add up to give the total soot oxidation behavior according to the two-layer model.
It is clear that the first peak results from the soot that already exists in contact with the catalyst in Layer I as well as from contributions of soot which migrates from Layer II into Layer I.The double peak is therefore a direct manifestation of the importance that catalyst layer geometry plays in establishing different modes of contact between the deposited soot particles and the catalytic coating.It is not possible to explain this double peak structure of the soot oxidation curves with a model based only on chemistry, since uncatalyzed soot oxidation exhibits a single peak and the same is true for mixtures of soot-catalyst powders 67) .
An example of application of the two-layer model to analyze the soot conversion on a catalytic filter is shown in Fig. 20, while Fig. 21 demonstrates the ability of the two-layer model to describe the soot oxidation of several catalytic filter samples to within +/-5% 74) .
The reduced activation energy (E/R) for the thermal oxidation for these samples is in the range of 21000 -22000 (K) 74) , and it is within the values usually reported in the literature, e.g. 63,64,65,66,67) . CCatalytic oxidation activation energies are almost the same or up to 15% lower, and depend on the type of catalyst and coating technology employed 74) .Depending on the catalyst coating technology, the state of soot-catalyst contact expressed by β can span a large range (values of β from 0.0006 up to 0.214 have been measured in 74) ), while migration of soot from Layer II into the catalyst-affected Layer I can occur with a varying degree of intensity as measured by the ξ parameter, which is measured to range from 0 (no migration) up to 0.642 74) .These findings demonstrate the important effect that the filter/catalyst micro-structural environment can have on the global oxidation kinetics.Essentially, it is possible to oxidize a small amount of soot in the catalyst-affected Layer I and still obtain a sufficiently high macroscopic reaction rate provided that the migration of soot from the top Layer II can proceed at an appropriate rate 74) .

Catalyst formulations
Following the results of basic research on catalysts, e.g. 64,65,75,76,77,78,79,80) , we hve embarked on a program of synthesis and screening soot oxidation catalysts with the goal of depositing them on filters.Numerous catalyst formulations with different characteristics have been studied 67) in order to arrive at a family of mixed oxide catalysts with high intrinsic soot oxidation activity, designated as E37.Although these catalyst formulations are proprietary and will not be listed here, the basic ingredient is cerium oxide doped with different rare earth and transition metals.

Catalyst deposition techniques
Catalyst deposition on the porous filter substrates can be performed either with a pre-formed catalyst powder slurry filtration procedure or by dipping the substrates into catalyst precursor solutions and subsequent firing.The coated filters are then thermally treated in a furnace.Permeability changes on the order of 15% can be noticed for filter samples coated by wet chemistry techniques 67) .
We have introduced 81,82) the aerosol spray pyrolysis technique (ASP), which combines catalyst synthesis and deposition on a porous substrate in one step, as an alternative to wet chemistry multi-step techniques (which include powder synthesis, slurry deposition and firing steps).The ASP procedure also has the advantages of precise control of the catalyst particle composition and of the quantity deposited, as well as its spatial deposition profile along the porous filter wall.An added benefit is the fact that the catalyst is deposited with the same mechanism as the soot particles, a feature which is expected to maximize the contact of the soot particles with the pre-deposited catalyst sites during filter operation.
In the aerosol spray pyrolysis technique, catalyst nanoparticles of controlled size are synthesized from the thermal decomposition/evaporation of precursor solution droplets introduced in a hot-tube aerosol reactor.Subsequently, these are deposited in-situ on porous substrates introduced in the reactor.Factors affecting the particle formation are the type of solution precursor, the temperature and the residence time in the reactor prior to deposition on the filter wall.Aerosol/vapor-phase techniques lead to nanostructured catalysts with controlled deposition profiles on the porous filter wall 67) .

4) DPF Assessment in the exhaust
The assessment of CDPF technologies is ultimately performed by exposing full-size CDPFs on the engine exhaust and evaluating their regeneration behavior according to a certain methodology.In the present section, we provide an example of such an assessment, employing prototype in-house catalyzed full-scale DPFs, as well as a description of our testing methodology.

Engine and DPF characteristics
Full-scale DPFs were tested in the exhaust of a Euro III passenger car diesel engine (displacement 1.9 L, rated power 60 kW) with common-rail fuel injection, coupled to a servo-controlled dynamometer.
The engine was operated under steady-state conditions as well as under transient conditions.Transient operation was achieved by simulating the speed and torque profiles of the New European Driving Cycle (NEDC) for the specific engine in the dynamometer, in collaboration with the engine manufacturer.The characteristics of the DPFs (SiC wall-flow filters, 5.66 inches diameter×6 inches long, 200 cells/in 2 ) in terms of their catalyst loading are shown in Table 4, and are denoted as DPF A-D.Overall, 4 catalyzed DPFs (at different loadings of the E37 catalyst in combination with a noble metal) were tested.
An uncatalyzed DPF was also used as a reference.In Table 4, x is the amount of catalyst load (referred to as E37) and y is the amount of noble metal (NM) added in the filter.The noble metal is from the platinum group.

Regeneration method and control
Modifications of the engine tuning in conjunction with fuel post injection through the common rail system, EGR rate variation, boost pressure variation or throttling offers great opportunities to increase the exhaust temperature without adversely affecting fuel penalty and driveability conditions.Specific strategies that focus on a combination of boost pressure, EGR rate increase, and fuel injection modification have therefore been implemented by engine manufacturers 4,11) .
In some cases 83) , the exhaust temperature is (in addition to engine measures) raised by exhaustpor t fuel injection in front of a diesel oxidation catalyst (DOC).The same technique is also applied for some retrofit applications 84) .The applicability of the technique relies on having the DOC above the hydrocarbon light-off temperature (about 150 C), a condition,which is met with our test engine running over the NEDC, consisting of four city cycles and one extra urban (EU) cycle.Fig. 22 shows the exhaust gas temperature of our engine over the cycle.As our Euro III test diesel engine has a "closed ECU" (i.e. a production ECU), we have selected exhaust port injection in front of a DOC as the means to raise the exhaust gas temperature, and in turn to effect regeneration of the DPF.
The exhaust port fuel injection set-up was constructed from standard automotive spare parts and consists of an air-assisted nozzle and auxiliary fuel and air pumps.Fuel flow to the nozzle is controlled by a custom electronic circuit defining the nozzle opening time and the period between injections.The fuel injection system has been optimized with the aid of a phase-Doppler analyzer in terms of droplet size.The fuel injection location was selected with the aid of CFD calculations so as to allow the fuel droplets to evaporate and the resulting diesel vapor to reach the DOC inlet with an even distribution.Fuel dosing and calibration of the system has been performed at representative engine operating conditions, and its effect on the increase of exhaust temperature has been studied providing an operational map.
Proprietary improvements in the virtual soot sensor include a detailed account of the effects of soot deposit microstr ucture under reactive and nonreactive conditions 12,47) and emergency procedures in the case of virtual sensor failure (caused either by a hardware malfunction or by deviations from pre-programmed specific criteria and error bounds).When this occurs, regeneration would be initiated based on estimating the soot mass load from stored emission data.

Screening of DPF systems
The DPFs were tested in series with a DOC (Cordierite, 5.66 inches×4 inches, 400 cells/in 2 ) in an exhaust set-up in order to study the influence of NO2 in conjunction with the catalytic coating on soot oxidation.
Initial tests employed the DPFs A-D and an engine operating point where direct NO2-soot oxidation was very low (due to a NOx/soot ratio of 4) in order to test the ability of the catalytic coatings to deliver NO2 to the soot collected inside the filter.The results of the DPF loadings at 2400 rpm and 6 bar brake mean effective pressure are shown in Fig. 23, while the soot mass load obtained from the virtual sensor is shown in Fig. 24.
As seen in Fig. 24, the presence of a DOC (hence NO2 in the DPF inlet) causes an acceleration of soot oxidation with respect to the uncoated filter.Interestingly enough, DPF B -despite its zero noble metal content -exhibits continuous regeneration behavior very quickly, and this is maintained up to a challenge mass load of 8 g/m 2 .This is attributed to the higher catalyst load (2.7× ) it had with respect to the other filters.The thick porous coating creates an effective filter medium with soot and catalyst in close proximity, and at the same time provides the extra residence time for increased NO2 turnover.DPFs C and D exhibit similar loading behavior with a progressively lower rate than the uncatalyzed DPF A.
The soot loading curves shown in Fig. 23 indicate that the catalyzed DPFs B, C and D would reach a continuous regeneration state if they were exposed to the exhaust gas for sufficiently long periods, as a simple first-order reaction-deposition dynamic model would indicate 38) .These results demonstrate that the soot oxidation catalyst content as well as the noble  The same DPF samples were also tested under a stepped temperature increase procedure to study their catalytic activity at higher temperatures.To this end, increasing quantities of engine fuel were injected upstream of the DOC in order to increase the exhaust gas temperature at the entrance of the DPFs, in steps of about 50 C.A representative result for DPF B is shown in Fig. 25 in terms of DPF pressure drop and in Fig. 26 in terms of soot mass load estimated by the virtual sensor.
The comparative assessment of the DPFs A-D in terms of the soot mass oxidation rate is shown in Fig. 27.It is evident that all DPFs exhibit significant and similar high temperature catalytic activity with respect to the uncatalyzed DPF A.
Based on these results, DPF C was chosen for subsequent study as an optimum compromise between catalyst load (directly affecting the cost and base system pressure drop), direct and indirect reactiv-ity with NO2 and a good catalytic reactivity at higher temperature with soot.

Transient Testing
The DOC-DPF C system was tested over a number of new European driving cycles in order to evaluate its behavior under transient conditions.With reference to Fig. 28, it is evident that the DPF exhibits    Fig. 29 depicts two successive fuel injections (each one with a duration of 2 min) performed at different soot mass load levels in the DPF, leading to partial regenerations.Fig. 30 depicts a complete regeneration achieved by extending the duration of fuel injection to 33 min.Prolonged operation of the system would result in an increase of the "clean" DPF mass load due to the ash accumulation.The virtual sensor is programmed to take this into account by containing information on the flow resistance of ashloaded filter materials 38) .
Using such tests in conjunction with a state-of-theart real-time DPF simulator 70,71) , it is possible to calibrate and to fully integrate model-based regeneration strategies with the soot virtual sensor module (see 86) for an early description of such strategies).Our current implementation relies on maintaining a target soot mass load in the DPF by injecting a quantity of fuel upstream of the DOC at appropriate instants.Since an exponential trade-off between the soot mass load in the DPF and fuel injection duration exists as shown in Fig. 31, it is evident that maintaining a target (constant) soot load in the DPF permits a priori optimization of the DPF size and cell density as well as significant fuel penalty savings.
As an example and based on our data with DPF C, we estimate that over the EUDC cycle, it is possible to maintain a target soot load of 5 g/m 2 with a fuel penalty of about 0.23% (a model-based controlled regeneration to bring the soot load whenever it reaches 6 g/m 2 down to 4 g/m 2 ).In comparison, a regeneration strategy that aims at a periodically completely   cleaned DPF whenever the soot load reaches 10g/m 2 would result in a fuel penalty of 0.52%.
Even though state-of-the ar t actively managed DPF systems 4) employing various aspects of the early virtual sensor models 50,85) achieve impressively minimum amounts of fuel penalty (about 0.7%), it is clear that significant gains could be made by bringing novel and computationally fast simulation algorithms (also known as vir tual sensors) into the developments of future ECUs.

Simulation Approaches to DPFs
Based on a traditional design of experiments approach, diesel particulate filter design, system integration and control becomes ver y time-consuming and costly due to the high number of tests required.This provides a privileged window of opportunity for the application of simulation.A recent review of advances in DPF simulation technology is given in 47,87) .While the interested reader is encouraged to consult these references and their cited literature for more detailed information on the underlying assumptions regarding the treatment of the various physicochemical phenomena (including soot particle transport, deposition and oxidation), in the present paragraph we provide an overview of the mode of use of the different simulation models in practice.
The multiscale nature of the problem lends itself to a hierarchical organization of the different models.The models at each spatial scale are classified according to their complexity and detail in the representation of the actual situation.Three sub-models corresponding to the three size scales (wall, channel, entire filter) must be combined to give an overall simulation model of the DPF.Ideally, we would like to employ the most detailed treatment from each scale.Such an approach would thus entail a true-to-the-geometry description of the porous filter wall, along with a 3-D CFD simulation of each channel, coupled through heat transfer to the other numerous (on the order of a few thousand) channels of the DPF.As this is for obvious reasons impossible even with computing resources of the foreseeable future, the employed strategy is to use the most detailed models at each spatial scale in order to validate and extract parameters for simpler (lower-degrees-offreedom) models applicable to the next scale.These relatively simple lower-order models are then connected with the hierarchically superior models in the next spatial scale, and the procedure is repeated over the entire DPF scale.
At the wall-scale algorithmic as well as processbased reconstruction techniques are employed to generate 3-D "digital materials" that are realistic representations of DPF microstructures.This is especially important for the development of new filter materials, the optimization of catalyst deposition inside the porous wall and for the design of gradient-functional filter microstructures where multiple functionalities in terms of particle separation and catalyst distribution (for combined gas and particle emission control) can be exploited.We refer to this approach as microflow simulation.Examples of computer-reconstructed DPF porous media are given in Fig. 32 and encompass all currently available filtration media: extruded ceramic filters (including reaction-formed media as cordierite and grain-sintered media as SiC), fibrous filters, foams and sintered metal powder/wire mesh.Using the reconstructed porous material and Lattice Boltzmann-based methods (see e.g. 88)), the flow field inside the filter porous wall can be obtained.The computation of soot deposition in the filter wall can then be carried out based on iterations between convective-diffusion soot deposition and flow field updates (as the porous wall structure is being progressively blocked by the deposited soot) 47,52) .An example of soot deposition in a granular-structured DPF "digital material" is shown in Fig. 33.
In practice, the detailed microflow simulation is employed when one is interested in developing new filter wall structures that need to meet specific requirements of flow resistance, filtration efficiency and catalyst coating accommodation.The models at this scale are therefore of the highest interest to the manufacturers of filter media.In this case, the majority of the experimental development revolves around small-scale filter samples, frequently in disk form, that are quite convenient for use in laboratory-scale experiments 35) .Depending on the outcome of this de- Fuel injection duration (min) Normalized soot mass load velopment, interesting filter structures can be scaledup into monolithic honeycomb samples.The use of a unit-cell structural description of the filter wall can be very advantageous at this stage, since with a minimal number of physically relevant parameters 35,42) and well-characterized experiments, a description of the filtration and pressure drop behavior of the DPF can be achieved in a very computationally efficient manner.However, this information cannot be embedded directly into a filter channel simulation without significant computational resources, since the number of state variables needed for this is an order of magnitude higher.The already presented two-layer model in this case offers a compact description that can be embedded into a channel-scale simulation.Channel-scale simulation is appropriate for the initial design and sizing of the DPF for an application.In this case, 3-D CFD is typically employed to assess the applicability of the classic perimeteraveraged approach 89) , e.g. for filter channels of other shapes 44) .Otherwise, the single-channel perimeteraveraged formalism 30,44,89) represents a fast and accurate approach which can be employed in conjunction with well-defined experimental campaigns to obtain needed physicochemical parameters of the DPF in consideration.Single-channel simulation accounts for the majority of the simulation activities of the industry (DPF suppliers, catalyst coaters, emission control system integrators, engine manufacturers), and can provide useful estimates of axial soot loading distributions, pressure drop behavior and magnitude and location of temperature exotherms and thermal gradients during regeneration.After the initial design phase, a DPF becomes part of an exhaust emission control system where its behavior also depends on the other components of the system.DPF models coupled with simulations of other emission control devices are then the appropriate tools to employ for system optimization and control.An example of a coupled simulation of a diesel oxidation catalyst (DOC) and a DPF in series is shown in  In addition, incomplete regenerations of the DPF need to be described.The continuum multichannel approach 69,70,71) represents a computationally tractable and accurate tool to address the previously mentioned issues.The development of highly integrated simulators of multi-functional exhaust emission control systems requires the interfacing of multichannel models of DPFs as well as other honeycomb-type con-  While computing limitations still remain the barrier for the routine employment of detailed simulations of coupled emission control components over the entire exhaust system scale, we anticipate that in the near future such simulations will be widely employed by the industry, exploiting grid-computing environments.This means that from a research point of view, DPF simulation will focus on providing a deeper understanding and more detailed description of the coupled transport, structural and reaction phenomena occurring at the wall and pore scales, to permit materialization of the vision for an a-priori design of advanced microstructures, hosting multifunctional catalysts for the highly compact and efficient emission control devices of the future.

Conclusions
The diesel particulate filter (DPF) has evolved into the most complex emission control device, due to its multiscale and multitemporal operational nature in combination with the different functionalities (particle separation/gas and particle catalytic reactions) embedded in it.Despite the challenges offered by this state of affairs, it has been possible to develop a systematic understanding of the science and technology of DPFs.Our starting point has been fundamental studies of diesel soot particle size, composition and morphology, and the mechanisms that determine the microstructural properties of soot deposits in DPFs.Filtration efficiency, pressure drop, ash accumulation and soot reactivity (with emphasis on catalyst-assisted soot oxidation) were then addressed, employing experimental methodologies ranging from small-scale filter set-ups to full-scale devices installed in the exhaust of diesel engines.The experimental methods are complemented by computational approaches that range from true-to-geometry representations of porous DPF microstructures up to entire emission control system simulations.Properly combined, the current state of knowledge, experimental methods and simulation approaches provide a rational and systematic route for enhancing the design  and reliability of future diesel particulate emission control systems.

Fig. 4
Fig.4 Effect of pore size on the size distribution of emitted soot aggregates.The DPFs tested are prototype SiC-based extruded wall-flow monoliths with a porosity of 42%.The theoretical lines are the predictions of the unit-cell-based filtration theory30, 35)

Fig. 5
Fig. 5 Unit-cell filtration model.The collector size dc and the empty envelope b are matched to the macroscopic porosity of the filter.The unit-cell blocks when the size of the collector becomes a fraction ψ of b 35,42)

Fig. 6
Fig. 6 Schematic of channels adjacent to filter inlet and outlet depicting local pressure values for the derivation of the effects of compressibility on pressure drop.

Fig. 8
Fig. 8 Schematic and nomenclature of soot cake accumulation inside the cross-section of a channel (left) and image of soot accumulated in a DPF channel (right).

Fig. 9
Fig.9 Validation of an analytical pressure drop model with 3-D CFD (left) from35)  , and experiments (right) with 10 filters in 31 experimental runs with light-and heavy-duty diesel engines from43) .

Fig. 10 Fig. 12
Fig.10 Transient loading of a SiC (top), cordierite (middle) and fibrous metal (bottom) filter in the exhaust of a diesel engine.Experimental data and simulation results42) .

Fig. 15
Fig. 15 Left: Simulated ash profile along the normalized filter length (z/L) vs. experimental data of Bardasz et al. with ash of different qualities, generated by high-sulfur oil doped into fuel (top), low-sulfur oil doped into fuel (middle), high-sulfur oil, regular use undoped into fuel (bottom).Right: Effect of ash accumulation on the DPF pressure drop.Experimental data courtesy of Ibiden Co. Ltd. and simulation results 47) .

Fig. 16
Fig.16 Soot oxidation rate as a function of temperature.

Fig. 17
Fig.17 Schematic illustrating the DPF wall, catalytic-coating-influenced Layer I and the top Layer II that is not in the sphere of influence of the catalyst.

Fig. 22
Fig.22 Exhaust gas temperature and engine speed over the NEDC cycle.

Fig. 24 Fig. 26
Fig. 24 Soot mass load of DPFs A, C and D at steady state.DPF B soot mass load is estimated to be close to zero.DPF inlet temperature: 370 C, exhaust mass flow rate: 50 g/s, NO x /soot ratio: 4.

Fig. 27
Fig. 27 Comparison of DPFs A-D over stepped temperature soot oxidation.

Fig. 25
Fig. 25 Pressure drop and injection profile at stepped temperature soot oxidation for DPF B.

Fig. 23
Fig. 23 Pressure drop of DPFs A, B, C and D as a function of challenge soot mass at steady state.DPF inlet temperature: 370 C, exhaust mass flow rate: 50 g/s, NO x /soot ratio: 4.

Fig. 30
Fig.30 Complete regeneration over the NEDC by exhaust-port fuel injection initiated at a soot loading of 9 g/m 2 .

Fig. 31
Fig.31 Normalized soot mass load in the DPF vs. fuel injection duration.

Fig. 34 .
We observe how a hydrocarbon pulse injection upstream of the DOC raises the exhaust temperature and causes regeneration of the DPF.Such simulation tools are very useful for the development and optimization of post-injection strategies for DPF regeneration.Important issues for the DPF performance at this

Fig. 32
Fig. 32 Computer reconstruction of various porous filters.

Fig. 33
Fig. 33 Visualisation of soot deposition at different surface mass loads in an extruded ceramic (granular) filter wall.(a) Development of soot deposits (black)and soot mass fraction in the wall (solid material is gray) to the onset of cake formation.Soot mass fraction scale is from 0 (violet) to the inflow value (red).In (b), the velocity on a section through the filter wall is shown, with overlay of the soot deposit shapes.

Fig. 35 3
Fig. 35 3-D -DPF simulation.Example of of temperature field evolution in a 11.25 in×12 in DPF during regeneration.

Table 1
Diesel Particulate Filter Materials and Configurations are the Darcy permeability k and the Forchheimer coef ficient β.The permeability has dimensions of length squared, and k 1/2 represents a pore level length scale, characteristic of the porous medium.It should be emphasized that the permeability and the

Table 2
Active Regeneration Measures

Table 3
Passive Regeneration Measures

Table 4
Characteristics of the DPFs tested