# Efficient Solution of Washcoat Diffusion-Reaction Problem for Real-Time Simulations

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## Abstract

As countries around the globe adapt more stringent emissions standards set by Real Driving Emissions (RDE) legislation, mathematical models are becoming ever more widely used as plant models for devising vehicle control strategies. It is important for the model to run on Hardware-in-Loop (HIL) and engine control unit (ECU) systems which have significantly less computational power and memory than modern personal computers. Washcoat diffusion limitations play a very important role in the efficient design of a catalytic converter. Numerical solution of aftertreatment models that include diffusion-reaction equations in the washcoat are computationally demanding. There are several simplified approaches proposed in the literature for the solution of diffusion-reaction equations in the washcoat to avoid the computational demand of the full numerical solution. In this paper, we use the recently proposed asymptotic solution and compare the results with that of the full numerical solution for the following aftertreatment reactor models with both single- and dual-layer washcoat configurations for the practical range of operating conditions; three-way catalyst (TWC), diesel oxidation catalyst (DOC), Selective Catalytic Reduction (SCR), and ammonia slip catalyst (ASC). These reactor models are constructed using published kinetic mechanisms and represent the global kinetics mechanisms (including non-linear reaction orders and inhibition functions) commonly used in the aftertreatment modeling community. We also discuss the importance of adaptive mesh, quasi-steady state assumption, and occurrence of concentration jumps in the simulation of aftertreatment reactors.

## Keywords

Washcoat Diffusion Asymptotic Dual-layer Real time## Abbreviations

- \( {a}_{\mathrm{n}}^{(l)} \)
active site density of reaction n in layer l (mol − site/m

^{3})- \( {A}_{\mathrm{k}}^{(l)} \)
active site density for coverage k in layer l (mol − site/m

^{3})*C*_{pg}heat capacity of bulk gas (J/(Kg K))

- \( {C}_{\mathrm{p}}^l \)
heat capacity of washcoat layer l (J/(Kg K))

*C*_{p,sb}heat capacity of substrate (J/(Kg K))

*D*_{h}hydraulic diameter of channel (m)

*D*_{i,m}diffusivity of species i in the mixture (m

^{2}/s)- \( {D}_{\mathrm{i},\mathrm{eff}}^{(l)} \)
effective diffusivity of species i in washcoat layer l (m

^{2}/s)- \( {D}_{\mathrm{i},\mathrm{kn}}^{(l)} \)
Knudsen diffusivity of species i in washcoat layer l (m

^{2}/s)- \( {\boldsymbol{D}}_{\mathbf{inv}}^{\left(\boldsymbol{l}\right)} \)
diagonal matrix of dimensionless effective diffusion resistances in washcoat layer

*l*.- \( {d}_{\mathrm{p}}^{(l)} \)
pore diameter of washcoat layer

*l*(m)*f*^{(l)}volume fraction of layer

*l**f*_{sb}volume fraction of substrate

*f*_{vd}void fraction of reactor

*h*heat transfer coefficient (J/(m

^{2}s K))*h*_{x}external heat transfer coefficient (J/(m

^{2}s K))*k*_{i}mass transfer coefficient of species i (m/s)

**K**diagonal matrix of dimensionless mass transfer coefficients

*L*length of reactor (m)

- MW
_{i} molecular weight of species i (kg/mol)

*n*_{rxns}total number of reactions

*n*_{sp}total number of species

*R*_{g}gas constant (J/(mol K))

- \( {r}_{\mathrm{n}}^{(l)} \)
n

^{th}reaction rate in layer*l*(mol/(mol − site s))- \( {R}_{\mathrm{i}}^l \)
i

^{th}species rate in layer*l*(kg/(m^{3}s))**R**^{(l)}vector of species rates in layer

*l**s*_{i,n}stoichiometric coefficient of species i in reaction n

*S*specific area per reactor volume (m

^{−1})*S*_{x}external surface area per reactor volume (m

^{−1})*t*time (s)

*T*_{g}gas temperature (K)

*T*_{s}solid temperature (K)

*T*_{x}external temperature (K)

*u*average gas velocity (m/s)

*x*position through the washcoat thickness (m)

*z*axial position (m)

*δ*^{l}effective thickness of washcoat layer

*l*(m)*ε*^{(l)}porosity of layer

*l*- \( {\Delta H}_{\mathrm{n}}^{(l)} \)
heat of n

^{th}reaction in layer*l*(J/mol)- \( {\theta}_{\mathrm{k}}^{(l)} \)
surface coverages of species k in layer

*l**λ*^{(l)}thermal conductivity of layer

*l*(J/(m s K))*λ*_{sb}thermal conductivity of substrate (J/(m s K))

- Λ
_{s} effective thermal conductivity of reactor (J/(m s K))

*ρ*_{g}density of bulk gas (kg/m

^{3})*ρ*_{s}density of gas within washcoat (kg/m

^{3})*ρ*^{(l)}density of washcoat layer

*l*(kg/m^{3})*ρ*_{sb}density of substrate (kg/m

^{3})*σ*_{k,n}stoichiometric coefficient for coverage k in reaction n

- ψ
_{s} effective heat capacity of reactor (J/(m

^{3}K))*ω*_{g,i}mass fraction of species i in the bulk gas

**ω**_{g}vector of species mass fractions in the bulk gas

*ω*_{i}mass fraction of species i in the washcoat

**ω**vector of species fractions in the washcoat

*ω*_{s,i}mass fraction of species i at the washcoat/channel surface

**ω**_{s}vector of species fractions at the washcoat/channel surface

## Notes

### Acknowledgments

We are grateful to Dr. Ed Bissett who spent significant amount of time discussing the asymptotic solution, adaptive meshing, and inner workings of ODE solvers. We also thank Jonathan Brown for setting up some of the models used in this work.

### Compliance with Ethical Standards

### Conflict of Interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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