Model-Based Analysis of TWC-Coated Filters Performance

Abstract

The wall flow filter technology is already an important element in gasoline engine emissions control. In addition to the filtration functions, the active catalytic coating can contribute to the reduction of gaseous emissions slipping from the upstream three-way catalytic converters (TWC). The substrate and coating technology require precise engineering to achieve the emission target without sacrificing fuel penalty associated with excessive backpressure and active regeneration. The present contribution addresses the main challenges in developing predictive models supporting the design and control of coated gasoline particulate filters (cGPFs), including filtration and pressure drop, soot oxidation, and effects of ash. The effect of soot and ash on TWC functions as well as the O2 competition between soot and the reactive surface are studied by analyzing the coupled transport–reaction processes at wall scale. The modeling study is supported by measurements with filters of different cell structures and washcoat amounts performed under steady-state and transient modes.

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Abbreviations

a 1 :

constant in channel pressure drop correlation, −

A :

pre-exponential factor, units depend on rate expression

b :

effective channel width, m

c :

concentration, mol/m3

C P :

specific heat capacity, J/(kg ∙ K)

cpsm:

channels per square meter, cells/m2

d :

hydraulic diameter of clean channel, m

d h :

hydraulic diameter of a channel, m

D knud :

Knudsen diffusivity, m2/s

D mol :

molecular diffusivity, m2/s

D part :

particle diffusion coefficient, m2/s

d pore :

mean pore size, m

D w :

effective diffusivity, m2/s

E :

activation energy, J/mol

f w :

geometric parameter, −

h :

heat transfer coefficient, W/(m2 ∙ K)

H :

heat source component, W/m3

k :

permeability, m2

k p :

apparent permeability of the particulate layer, m2

k 0 :

permeability of the clean filter, m2

l w :

substrate wall discretization length, m

M g :

gas molecular mass, kg/mol

M j :

molecular mass of each species, kg/mol

m p :

cake soot mass, kg

m w :

deposited particulate mass per unit collector, kg

n :

filtration efficiency, −

n clean, D :

diffusion mechanism correction factor, −

n clean, R :

interception mechanism correction factor, −

p j :

partial pressure, Pa

p sat, j :

saturation pressure, Pa

R :

universal gas constant, J/(mol ∙ K)

r :

radial coordinate, −

R k :

species reaction rate, mol/(m3 ∙ s)

Rk :

soot reaction rate, mol/(molC ∙ s)

S :

heat source term, W/m3

S F :

monolith specific surface area, m2/m3

T :

temperature, K

t :

time, s

u w :

velocity perpendicular to wall, m/s

v :

velocity, m/s

w :

dimension perpendicular to wall surface, −

W 0 :

total volume of micropores per reactor volume, m3/m3

w c :

washcoat layer thickness, m

w p :

soot layer thickness, m

w w :

wall thickness, m

x i :

wall thickness coordinate, m

y j :

molar fraction, −

z :

axial coordinate along monolith, m

a :

dependency from oxygen concentration, −

a 1 :

constant in channel pressure drop correlation, −

ΔH k :

reaction heat, J/mol

Δw :

elementary width in the wall direction, m

ε :

macroscopic void fraction, −

ε 0 :

porosity of the clean filter, −

λ :

thermal conductivity, W/(m ∙ K)

μ :

dynamic viscosity, Pa ∙ s

ρ :

density, kg/m3

ρ p :

soot packing density, kg/m3

ρ soot, w :

soot packing density in wall, kg/m3

g :

exhaust gas

j :

species index

k :

reaction index

s :

solid

w :

substrate wall

CC :

close-coupled

DPF :

Diesel particulate filter

FWC :

four-way catalyst

GDI :

gasoline direct injection

GPF :

gasoline particulate filter

PM :

particulate matter

RDE :

real driving emissions

TWC :

three-way catalyst

UF :

underfloor

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Funding

Part of the results presented here was obtained within the project UPGRADE. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 724036.

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Correspondence to G. Koltsakis.

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Appendix

Appendix

Wall Scale Pressure Drop Modeling

This section presents the main mathematical equations that are used to describe the wall scale pressure drop phenomena.

To solve the velocity and pressure distribution in the inlet and outlet channels, it is necessary to calculate the pressure drop through the soot layer and the porous filter wall. A schematic of the front view of a filter channel is given in Fig. 13.

Fig. 13
figure13

Front view of a filter channel and schematic of the mass transfer limitations across a filter’s wall

With the definitions shown above, the channel width available to the flow is given by the following relationship:

$$ b(w)=d-2\left({w}_p-w\right) $$
(5)

The particulate layer thickness wp can be calculated from the following expression:

$$ {w}_p=\frac{1}{2}\left(d-\sqrt{d^2-\frac{2{m}_p^{\prime }}{\rho_p}}\right) $$
(6)

where \( {m}_p^{\prime } \) is the soot mass in a channel per unit length.

Starting from the elementary Darcy law, the differential pressure drop over dw will be as follows:

$$ \frac{dP}{dw}=\frac{\mu \bullet v(w)}{k_p} $$
(7)

Although mass flow rate is steady throughout the soot layer, the gas velocity varies due to changes in gas density and flow area, as shown by the continuity equation:

$$ v(w)=\frac{\rho_w\bullet {v}_w\bullet d}{\rho (w)\bullet b(w)} $$
(8)

The gas density across the soot layer is a function of the local pressure. With the ideal gas assumption, the density can be expressed as follows:

$$ \rho (w)=\frac{p(w)\bullet {M}_g}{RT} $$
(9)

The permeability of the particulate deposit depends on the gas mean free path due to slip phenomena, which may be described by the following expression [14]:

$$ {k}_p={k}_{p,0}\left(1+{C}_4\frac{p_0}{p}\mu \sqrt{\frac{T}{M_g}}\right) $$
(10)

The above formula expresses the dependency of soot permeability on local temperature and pressure for a constant soot density. The parameters kp,0 and C4 are estimated based on experimental data. Integration of Eq. 7 provides the pressure drop over the soot layer thickness:

$$ {\Delta P}_{\mathrm{soot}}=\frac{R\bullet T\bullet \mu \bullet d\bullet {\rho}_w\bullet {v}_w}{M_g\bullet \overline{p}\bullet 2\bullet {k}_p\left(\overline{p}\right)}\ln \left(\frac{d}{d-2{w}_p}\right) $$
(11)

where \( \overline{p} \) is the mean value of inlet and soot layer outlet pressures. The pressure drop through the porous filter wall, assuming constant flow velocity across the wall, can be described by the Darcy law:

$$ {\Delta P}_{\mathrm{wall}}=\frac{\mu \bullet {v}_w}{k_w}{w}_w $$
(12)

The dependence of wall permeability kw on temperature T and pressure p is taken into account, similarly to Eq. 10. Furthermore, an extra correction is included to account for the sudden decrease of wall permeability at the initial stage of soot loading, as soot particles block a portion of the wall pores:

$$ {k}_w=\frac{1}{\frac{1}{k_{w,0}}+{C}_1{\rho}_p+{C}_2{\rho}_p^2}\left(1+{C}_4\frac{p_0}{p}\mu \sqrt{\frac{T}{M_g}}\right) $$
(13)

The clean wall permeability kw,0 and parameters C1 and C2 define a trinomial function of ρp which is defined as total soot accumulated in the wall per wall volume. In order to calculate the extra parameters, the following values are estimated based on experimental data:

  • Clean wall permeability kw,0

  • Loaded wall permeability kw,1 at reference soot loading ρp,1

Based on the above analysis for the wall and deposit layer, the pressure drop between the inlet and the outlet channel can be evaluated as follows:

$$ {p}_1-{p}_2={\Delta P}_{\mathrm{soot}}+{\Delta P}_{\mathrm{wall}} $$
(14)

p1, p2, ΔPsoot, and ΔPwall are all calculated locally for each axial element. The mass and momentum balance equations can be solved to compute the velocity v1(z), v2(z), the pressure field p1(z), p2(z) in the inlet and outlet channels, as well as the velocity through the wall vw(z) with given inlet flow rate and the exit pressure. It should be noted that it is theoretically possible to have inverse wall flow in small channel region under extreme soot loading conditions.

Channel Scale Pressure Drop Modeling

Assuming a spatially uniform inlet flow and temperature field, the mathematical solution of the complete particulate filter can become equivalent to the solution of a representative pair of inlet and outlet channels (see Fig. 14).

Fig. 14
figure14

Side view of a filter, inlet (1), and outlet (2) channels

Filter-induced backpressure, attributed to pressure losses along the channels and through the substrate wall, was discussed in the previous section.

Mass–Momentum Balance

Mass conservation of channel gas: The mass balance equation for the gas flowing in the inlet and outlet channels is as follows:

$$ \frac{\partial }{\partial z}\left({d}_i^2{\rho}_i{v}_i\right)={\left(-1\right)}^i\bullet 4d{\rho}_{\mathrm{w}}{v}_{\mathrm{w}} $$
(15)

where the subscript i identifies regions 1 (inlet channel) and 2 (outlet channel).

Conservation of axial momentum of channel gas: Considering the mass gain/loss through the porous wall and the friction in the axial direction z, the momentum balance of exhaust gas can be written as follows:

$$ \frac{\partial {p}_i}{\mathrm{\partial z}}+\frac{\partial }{\mathrm{\partial z}}\left({\rho}_i{v}_i^2\right)=-{\alpha}_1\mu {v}_i/{d}_i^2 $$
(16)

Species Balance

The importance of species transfer to account for oxygen convection to the reacting soot layer was initially recognized for the case of uncontrolled regeneration modeling of DPF. In addition, the solution of the species transfer equation is essential to model the coated filters with respect to pollutant conversion and their interactions with the accumulated soot.

The governing advection–reaction–diffusion equation for the conservation of mass of any species within the soot layer and wall is as follows:

$$ {v}_{\mathrm{w}}\frac{\partial {y}_j}{\partial w}-{D}_{\mathrm{w},j}\frac{\partial }{\partial w}\left({f}_{\mathrm{w}}\frac{\partial {y}_j}{\partial w}\right)=\frac{f_{\mathrm{w}}}{c_{\mathrm{m}}}\sum \limits_{\mathrm{k}}{n}_{j,\mathrm{k}}{R}_{\mathrm{k}} $$
(17)

The calculation of the effective diffusivity Dw,j is based on the mean transport pore model which uses the following expression:

$$ \frac{1}{D_{\mathrm{w},\mathrm{j}}}=\frac{\tau }{\varepsilon_{\mathrm{pore}}}\left(\frac{1}{D_{\mathrm{mol},\mathrm{j}}}+\frac{1}{D_{\mathrm{knud},\mathrm{j}}}\right) $$
(18)

with the Knudsen diffusivity:

$$ {D}_{\mathrm{knud},\mathrm{j}}=\frac{d_{\mathrm{pore}}}{3}\sqrt{\frac{8 RT}{\pi {M}_{\mathrm{j}}}} $$
(19)

The porosity εpore and the mean pore size dpore can be extracted from the microstructural properties of the washcoat while tortuosity τ is an empirical parameter which can be tuned by dedicated experiments.

The boundary conditions couple the wall with the gas concentrations in the channels. The “film” approach is used to compute the convective mass transfer from the bulk gas to the wall surface. The mass transfer coefficients ki,j correspond to laminar flow for both inlet and outlet channels:

$$ \frac{\partial \left({v}_1{y}_{1,\mathrm{j}}\right)}{\mathrm{\partial z}}=-\frac{4}{d\bullet {f_{\mathrm{w}}}^2}{v}_{\mathrm{w}}{y}_{1,\mathrm{j}}+\frac{4}{d\bullet {f}_{\mathrm{w}}}{k}_{1,\mathrm{j}}\left({y}_{1\mathrm{s},j}-{y}_{1,\mathrm{j}}\right) $$
(20)
$$ \frac{\partial \left({v}_2{y}_{2,\mathrm{j}}\right)}{\partial z}=\frac{4}{d\bullet {f_{\mathrm{w}}}^2}{v}_{\mathrm{w}}{y}_{2\mathrm{s},\mathrm{j}}+\frac{4}{d\bullet {f}_{\mathrm{w}}}{k}_{2,\mathrm{j}}\left({y}_{2\mathrm{s},\mathrm{j}}-{y}_{2,\mathrm{j}}\right) $$
(21)

Filter Scale Pressure Drop Modeling

Additional pressure drop losses, due to flow entrance/exit effects, are considered by using the following equations [15]:

$$ {\Delta P}_{\mathrm{contraction}}=\left[1.1-0.4\frac{{\left(d-2{w}_p\right)}^2}{2{\left(d+{w}_w\right)}^2}\right]\frac{{\left.{\rho}_1{v}_1^2\right|}_{z=0}}{2} $$
(22)
$$ {\Delta P}_{\mathrm{expansion}}={\left[1-\frac{d^2}{2{\left(d+{w}_w\right)}^2}\right]}^2\frac{{\left.{\rho}_2{v}_2^2\right|}_{z=L}}{2} $$
(23)

Energy Balance

Gas phase: The conservation of energy of the channel gas for the inlet and the outlet channels is given as follows:

$$ {\left.{C}_{\mathrm{p},\mathrm{g}}{\rho}_1{v}_1\right|}_{\mathrm{z}}\frac{\partial {T}_1}{\mathrm{\partial z}}={h}_1\frac{4}{d_1}\left({T}_{\mathrm{s}}-{T}_1\right) $$
(24)
$$ {\left.{C}_{\mathrm{p},\mathrm{g}}{\rho}_2{v}_2\right|}_{\mathrm{z}}\frac{\partial {T}_2}{\mathrm{\partial z}}=\left({h}_2+{C}_{\mathrm{p},\mathrm{g}}{\rho}_{\mathrm{w}}{v}_{\mathrm{w}}\right)\frac{4}{d}\left({T}_{\mathrm{s}}-{T}_2\right) $$
(25)

Solid phase: The one-dimensional temperature field in the filter is described by the transient heat conduction equation with heat sources:

$$ {\rho}_{\mathrm{s}}\bullet {C}_{\mathrm{p},\mathrm{s}}\frac{\partial {T}_{\mathrm{s}}}{\partial t}={\lambda}_{\mathrm{s},\mathrm{z}}\frac{\partial^2{T}_{\mathrm{s}}}{\partial {\mathrm{z}}^2}+S $$
(26)

If the non-uniformities in the inlet temperature and flow distribution need to be considered, a 2D or a 3D model is required. In this case, the transient energy balance equation is extended to two or three dimensions. The heat conduction equation is formulated in polar coordinates for 2D simulations and in Cartesian coordinates for 3D simulations.

$$ {\rho}_{\mathrm{s}}{C}_{\mathrm{p},\mathrm{s}}\frac{\partial {T}_{\mathrm{s}}}{\partial t}={\lambda}_{\mathrm{s},z}\frac{\partial^2{T}_{\mathrm{s}}}{\partial {z}^2}+{\lambda}_{\mathrm{s},r}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial {T}_{\mathrm{s}}}{\partial r}\right)+S $$
(27)
$$ {\rho}_{\mathrm{s}}{C}_{\mathrm{p},\mathrm{s}}\frac{\partial {T}_{\mathrm{s}}}{\partial t}={\lambda}_{\mathrm{s},x}\frac{\partial^2{T}_{\mathrm{s}}}{\partial {x}^2}+{\lambda}_{\mathrm{s},y}\frac{\partial^2{T}_{\mathrm{s}}}{\partial {y}^2}+{\lambda}_{\mathrm{s},z}\frac{\partial^2{T}_{\mathrm{s}}}{\partial {z}^2}+S $$
(28)

The source term S includes the contribution of the convective heat transfer of the gas flow in the channel Hconv and through the wall Hwall, as well as the exothermic heat release Hreact.

$$ S={H}_{\mathrm{conv}}+{H}_{\mathrm{wall}}+{H}_{\mathrm{react}} $$
(29)
$$ {H}_{\mathrm{conv}}={h}_1\bullet {S}_{\mathrm{F}}\bullet \left({T}_1-{T}_{\mathrm{s}}\right)+{h}_2\bullet {S}_{\mathrm{F}}\bullet \left({T}_2-{T}_{\mathrm{s}}\right) $$
(30)
$$ {H}_{\mathrm{w}\mathrm{all}}={\rho}_{\mathrm{w}}\bullet {v}_{\mathrm{w}}\bullet {S}_{\mathrm{F}}\bullet {C}_{\mathrm{p},\mathrm{g}}\bullet \left({T}_1-{T}_{\mathrm{s}}\right) $$
(31)
$$ {H}_{\mathrm{react}}={S}_{\mathrm{F}}\sum \limits_k\left(\underset{-{w}_{\mathrm{p}}}{\overset{w_{\mathrm{w}}}{\int }}{f}_{\mathrm{w}}{R}_{\mathrm{k}} dw\right)\bullet {\Delta H}_{\mathrm{k}} $$
(32)

The definition of the geometrical parameter fw is as follows:

$$ {f}_{\mathrm{w}}=\frac{b(w)}{d} $$
(33)

TWC Reactions

The main catalytic reactions that take place in a TWC/GPF are the same as for the TWC [16]:

Oxidation reactions

$$ \mathrm{CO}+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathrm{O}}_2\to {\mathrm{CO}}_2 $$
(reac. 1)
$$ {\mathrm{H}}_2+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathrm{O}}_2\to {\mathrm{H}}_2\mathrm{O} $$
(reac. 2)
$$ {\mathrm{C}}_x{\mathrm{H}}_y+\left(x+\frac{y}{4}\right){\mathrm{O}}_2\to x{\mathrm{C}\mathrm{O}}_2+\frac{y}{2}{\mathrm{H}}_2\mathrm{O} $$
(reac. 3)

NO reduction

$$ \mathrm{CO}+\mathrm{NO}\to {\mathrm{CO}}_2+\frac{1}{2}{\mathrm{N}}_2 $$
(reac. 4)
$$ {\mathrm{C}}_x{\mathrm{H}}_y+\left(2x+\frac{y}{2}\right)\mathrm{NO}\to x{\mathrm{C}\mathrm{O}}_2+\frac{y}{2}{\mathrm{H}}_2\mathrm{O}+\left(x+\frac{y}{4}\right){\mathrm{N}}_2 $$
(reac. 5)
$$ {\mathrm{H}}_2+\mathrm{NO}\to {\mathrm{H}}_2\mathrm{O}+\frac{1}{2}{\mathrm{N}}_2 $$
(reac. 6)
$$ \mathrm{CO}+2\mathrm{NO}\to {\mathrm{CO}}_2+{\mathrm{N}}_2\mathrm{O} $$
(reac. 7)
$$ {\mathrm{H}}_2+2\mathrm{NO}\to {\mathrm{H}}_2\mathrm{O}+{\mathrm{N}}_2\mathrm{O} $$
(reac. 8)
$$ \mathrm{NO}+\frac{5}{2}{\mathrm{H}}_2\to {\mathrm{NH}}_3+{\mathrm{H}}_2\mathrm{O} $$
(reac. 9)
$$ \mathrm{N}{\mathrm{H}}_3+\frac{3}{4}{\mathrm{O}}_2\to \frac{1}{2}{\mathrm{N}}_2+\frac{3}{2}{\mathrm{H}}_2\mathrm{O} $$
(reac. 10)

Steam reforming

$$ {\mathrm{C}}_x{\mathrm{H}}_y+x{\mathrm{H}}_2\mathrm{O}\to x\mathrm{CO}+\left(x+\frac{y}{2}\right){\mathrm{H}}_2 $$
(reac. 11)

Water–gas shift

$$ \mathrm{CO}+{\mathrm{H}}_2\mathrm{O}\leftrightarrow {\mathrm{CO}}_2+{\mathrm{H}}_2 $$
(reac. 12)

Cerium (reduced state)

$$ {\mathrm{Ce}}_2{\mathrm{O}}_3+\frac{1}{2}{\mathrm{O}}_2\to 2\mathrm{Ce}{\mathrm{O}}_2 $$
(reac. 13)
$$ {\mathrm{Ce}}_2{\mathrm{O}}_3+\mathrm{NO}\to 2\mathrm{Ce}{\mathrm{O}}_2+\frac{1}{2}{\mathrm{N}}_2 $$
(reac. 14)
$$ {\mathrm{Ce}}_2{\mathrm{O}}_3+{\mathrm{H}}_2\mathrm{O}\to 2\mathrm{Ce}{\mathrm{O}}_2+{\mathrm{H}}_2 $$
(reac. 15)
$$ {\mathrm{Ce}}_2{\mathrm{O}}_3+{\mathrm{CO}}_2\to 2\mathrm{Ce}{\mathrm{O}}_2+\mathrm{CO} $$
(reac. 16)

Cerium (oxidized state)

$$ 2\mathrm{Ce}{\mathrm{O}}_2+\mathrm{CO}\to {\mathrm{Ce}}_2{\mathrm{O}}_3+{\mathrm{CO}}_2 $$
(reac. 17)
$$ 2\mathrm{Ce}{\mathrm{O}}_2+{\mathrm{H}}_2\to {\mathrm{Ce}}_2{\mathrm{O}}_3+{\mathrm{H}}_2\mathrm{O} $$
(reac. 18)
$$ {\mathrm{C}}_x{\mathrm{H}}_y+2\left(x+\frac{y}{2}\right)\mathrm{Ce}{\mathrm{O}}_2\to \left(x+\frac{y}{2}\right){\mathrm{C}\mathrm{e}}_2{\mathrm{O}}_3+x\mathrm{CO}+\frac{y}{2}{\mathrm{H}}_2\mathrm{O} $$
(reac. 19)

Soot Reactions

More than 90% of the engine emitted particles consist of soot, i.e., elemental carbon and small hydrocarbon fractions derived by lube oil and fuel. A GPF thermal regeneration involves the oxidation of solid particles to gaseous products. This process can be described by the following reactions:

$$ \mathrm{C}+{\mathrm{O}}_2\to {\mathrm{CO}}_2 $$
(reac. 20)
$$ \mathrm{C}+\frac{1}{2}{\mathrm{O}}_2\to \mathrm{C}\mathrm{O} $$
(reac. 21)

The respective rate expressions of the above reactions can be formulated as follows:

$$ {R}_{20}^{\prime }={A}_{20}\bullet {\exp}^{\frac{-{E}_{20}}{RT}}\bullet {P}_{O_2}^{\alpha_{20}} $$
(34)
$$ {R}_{21}^{\prime }={A}_{21}\bullet {\exp}^{\frac{-{E}_{21}}{RT}}\bullet {P}_{O_2}^{\alpha_{21}} $$
(35)

where the pre-exponential factor A, the activation energy E, and the reaction order α are tunable parameters which are identified based on dedicated experiments.

Soot oxidation may occur even at the absence of O2 via the reaction.

$$ \mathrm{C}+\mathrm{C}{\mathrm{O}}_2\to 2\mathrm{CO} $$
(reac. 22)

The respective rate expression can be formulated as follows:

$$ {R}_{22}^{\prime }={A}_{22}\bullet {\exp}^{\frac{-{E}_{22}}{RT}}\bullet {P}_{CO_2}\bullet E{q}_f $$
(36)
$$ E{q}_f=1-\frac{P_{CO}^2}{P_{C{O}_2}\bullet {K}_p} $$
(37)
$$ {K}_{\mathrm{p}}={\exp}^{\frac{-\Delta G}{RT}} $$
(38)

This reaction is thermodynamically limited at low temperatures [17].

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Mitsouridis, M.A., Karamitros, D. & Koltsakis, G. Model-Based Analysis of TWC-Coated Filters Performance. Emiss. Control Sci. Technol. 5, 238–252 (2019). https://doi.org/10.1007/s40825-019-00124-3

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Keywords

  • Gasoline
  • GPF
  • Pressure drop
  • Filtration
  • Soot