Skip to main content

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

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Abbreviations

\( {a}_{\mathrm{n}}^{(l)} \) :

active site density of reaction n in layer l (mol − site/m3)

\( {A}_{\mathrm{k}}^{(l)} \) :

active site density for coverage k in layer l (mol − site/m3)

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 (m2/s)

\( {D}_{\mathrm{i},\mathrm{eff}}^{(l)} \) :

effective diffusivity of species i in washcoat layer l (m2/s)

\( {D}_{\mathrm{i},\mathrm{kn}}^{(l)} \) :

Knudsen diffusivity of species i in washcoat layer l (m2/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/(m2 s K))

h x :

external heat transfer coefficient (J/(m2 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)

MWi :

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))

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

nth reaction rate in layer l (mol/(mol − site s))

\( {R}_{\mathrm{i}}^l \) :

ith species rate in layer l (kg/(m3 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 nth 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))

Λs :

effective thermal conductivity of reactor (J/(m s K))

ρ g :

density of bulk gas (kg/m3)

ρ s :

density of gas within washcoat (kg/m3)

ρ (l) :

density of washcoat layer l (kg/m3)

ρ sb :

density of substrate (kg/m3)

σ k,n :

stoichiometric coefficient for coverage k in reaction n

ψs :

effective heat capacity of reactor (J/(m3 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

References

  1. Heck, R.M., Farrauto, R.J., Gulati, S.T.: Catalytic Air Pollution Control: Commercial Technology. Wiely publication (2016)

  2. Joshi, S.Y., Harold, M.P., Balakotaiah, V.: Overall mass transfer coefficients and controlling regimes in catalytic monoliths. Chem. Eng. Sci. 65, 1729–1747 (2010)

    Article  Google Scholar 

  3. Metkar, P.S., Harold, M.P., Balakotaiah, V.: Experimental and kinetic modeling study of NH3-SCR of NOx on Fe-ZSM-5, Cu-chabazite and combined Fe- and Cu-zeolite monolith catalysts. Chem. Eng. Sci. 87, 51–66 (2013)

    Article  Google Scholar 

  4. Vaclavik, M., Koci, P., Novak, V., Thompsett, D.: NOx conversion and selectivity in multi-layer and sequential DOC-LNT automotive exhaust catalysts: influence of internal transport. Chem. Eng. J. 329, 128–134 (2017)

    Article  Google Scholar 

  5. Gundlapally, S.R., Papadimitriou, I., Wahiduzzaman, S., Gu, T.: Development of ECU capable models from detailed models—application to a SCR reactor. Emission Control Science and Technology. 2, 124–136 (2016)

    Article  Google Scholar 

  6. Aris, R.: The Mathematical Theory of Diffusion And Reaction in Permeable Catalysts, Volume I, the Theory of the Steady State. Oxford University Press, London (1975)

    Google Scholar 

  7. Balakotaiah, V.: On the relationship between Aris and Sherwood numbers and friction and effectiveness factors. Chem. Eng. Sci. 63, 5802–5812 (2008)

    Article  Google Scholar 

  8. Bissett, E.J.: An asymptotic solution for washcoat pore diffusion in catalytic monoliths. Emission Control Science and Technology. 1(1), 3–16 (2015a)

    Article  Google Scholar 

  9. Bissett, E.J.: Small Washcoat Diffusion Resistance, Further Developments. CLEERS Workshop, Dearborn (2015b)

    Google Scholar 

  10. Mozaffari, B., Tischer, S., Votsmeier, M., Deutschmann, O.: A one-dimensional modeling approach for dual-layer monolith catalysts. Chem. Eng. Sci. 139, 196–210 (2016)

    Article  Google Scholar 

  11. Rink, J., Mozaffari, B., Tischer, S., Deutschmann, O., Votsmeier, M.: Real-time simulation of dual-layer converters based on the internal mass transfer coefficient approach. Top. Catal. 60, 225–229 (2017)

    Article  Google Scholar 

  12. Groppi, G., Belloli, A., Tronconi, E., Forzatti, P.: A comparison of lumped and distributed models of monolithic catalytic combustors. Chem. Eng. Sci. 50, 2705–2715 (1995)

    Article  Google Scholar 

  13. Gundlapally, S. R. (2011) Effect of non-uniform activity and conductivity on the steady-state and transient performance of catalytic reactors. PhD thesis, University of Houston

  14. Gundlapally, S.R., Balakotaiah, V.: Heat and mass transfer correlations and bifurcation analysis of catalytic monoliths with developing flows. Chem. Eng. Sci. 66(9), 1879–1892 (2011)

    Article  Google Scholar 

  15. Fuller, E.N., Ensley, K., Giddings, J.C.: Diffusion of halogenated hydrocarbons in helium. The effect of structure on collision cross sections. J. Phys. Chem. 73(11), 3679–3685 (1969)

    Article  Google Scholar 

  16. Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena, p. 258. John Wiley and Sons, New York (1960)

    Google Scholar 

  17. Scheuer, A., Hauptmann, W., Drochner, A., Gieshoff, J., Vogel, H., Votsmeier, M.: Dual layer automotive ammonia oxidation cataysts: experiments and computer simulation. Appl. Catal. B Environ. 111–112, 445–455 (2012)

    Article  Google Scholar 

  18. Nova, I., Colombo, M., Tronconi, E., Schmeisser, V., Bandle-Konrad, B., Zimmermann, L.: Experimental and modelling study of a dual-layer NH3 slip monolith catalyst for automotive SCR aftertreatment systems. Top. Catal. 56, 227–231 (2013)

    Article  Google Scholar 

  19. Oh, S.H., Bissett, E.J., Battiston, P.A.: Mathematical modeling of electrically heated monolith converters: model formulation, numerical method, and experimental verification. I&EC research. 32, 1560–1567 (1993)

    Google Scholar 

  20. Zhang, F., Hayes, R.E., Kolaczkowski, S.T.: A new technique to measure the effective diffusivity in a catalytic monolith washcoat. Chem Eng Research and Design. 82, 481–489 (2004)

    Article  Google Scholar 

  21. Holder, R., Bollig, M., Anderson, D.R., Hochmuth, J.K.: A discussion on transport phenomena and three-way kinetics of monolithic converters. Chem. Eng. Sci. 61(24), 8010–8027 (2006)

    Article  Google Scholar 

  22. Sampara, C.S., Bissett, E.J., Chmielewski, M.: Global kinetics for a commercial diesel oxidation catalyst with two exhaust hydrocarbons. Ind. Eng. Chem. Res. 47, 311–322 (2008)

    Article  Google Scholar 

  23. Chatterjee, D., Burkhardt, T., Weibel, M., Nova, I., Grossale, A., Tronconi, E.: Numerical simulation of zeolite- and v-based SCR catalytic converters. SAE 2007–01-1136 (2007)

  24. Pant, A., Schmieg, S.J.: Kinetic model of NOx SCR using urea on commercial Cu-zeolite catalyst. Ind. Eng. Chem. Res. 50, 5490–5498 (2011)

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Santhosh R. Gundlapally.

Ethics declarations

Conflict of Interest

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gundlapally, S.R., Dudgeon, R. & Wahiduzzaman, S. Efficient Solution of Washcoat Diffusion-Reaction Problem for Real-Time Simulations. Emiss. Control Sci. Technol. 4, 90–102 (2018). https://doi.org/10.1007/s40825-018-0083-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40825-018-0083-9

Keywords

  • Washcoat
  • Diffusion
  • Asymptotic
  • Dual-layer
  • Real time