American Institute of Mathematical Sciences

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February  2021, 14(2): 533-574. doi: 10.3934/dcdss.2020406

The fast-sorption and fast-surface-reaction limit of a heterogeneous catalysis model

Björn Augner and  Dieter Bothe ,

Fachbereich Mathematik, Technische Universität Darmstadt, Alarich-Weiss-Straße 10, 64287 Darmstadt, Germany

* Corresponding author: Dieter Bothe, bothe@mma.tu-darmstadt.de

Dedicated to Michel Pierre on the occasion of his 70th anniversary

Received  November 2019 Revised  June 2020 Published  February 2021 Early access  July 2020

Within this paper, we consider a heterogeneous catalysis system consisting of a bulk phase $ \Omega $ (chemical reactor) and an active surface $ \Sigma = \partial \Omega $ (catalytic surface), between which chemical substances are exchanged via adsorption (transport of mass from the bulk boundary layer adjacent to the surface, leading to surface-accumulation by a transformation into an adsorbed form) and desorption (the reverse process). Quite typically, as is the purpose of catalysis, chemical reactions on the surface occur several orders of magnitude faster than, say, chemical reactions within the bulk phase, and sorption processes are often quite fast as well. Starting from the non-dimensional version, different limit models, especially for fast surface chemistry and fast sorption at the surface, are considered. For a particular model problem, questions of local-in-time existence of strong and classical solutions and positivity of solutions are addressed.

Keywords: Heterogeneous catalysis, dimension analysis, reaction diffusion systems, surface chemistry, surface diffusion, sorption, quasilinear PDE, Lp-maximal regularity, positivity, blow-up.
Mathematics Subject Classification: Primary 35K57; Secondary 35K51, 80A30, 92E20.
Citation: Björn Augner, Dieter Bothe. The fast-sorption and fast-surface-reaction limit of a heterogeneous catalysis model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 533-574. doi: 10.3934/dcdss.2020406
References:
[1]

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H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.  doi: 10.1016/0022-0396(88)90156-8.  Google Scholar

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H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence, Math. Z., 202 (1989), 219-250.  doi: 10.1007/BF01215256.  Google Scholar

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H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.   Google Scholar

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H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), Teubner-Texte Math. Teubner, Stuttgart, 133 (1993), 9–126. doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

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B. Augner and D. Bothe, Analysis of some heterogeneous catalysis models with fast sorption and fast surface chemistry, submitted (2020), https://arXiv.org/abs/2006.12098. Google Scholar

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D. Bothe and W. Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures, Acta Mech., 226 (2015), 1757-1805.  doi: 10.1007/s00707-014-1275-1.  Google Scholar

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D. Bothe, M. Köhne, S. Maier and J. Saal, Global strong solutions for a class of heterogeneous catalysis models, J. Math. Anal. Appl., 445 (2017), 677-709.  doi: 10.1016/j.jmaa.2016.08.016.  Google Scholar

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D. Bothe and M. Pierre, Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate, J. Math. Anal. Appl., 368 (2010), 120-132.  doi: 10.1016/j.jmaa.2010.02.044.  Google Scholar

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D. Bothe and M. Pierre, The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 49-59.  doi: 10.3934/dcdss.2012.5.49.  Google Scholar

[20]

D. Bothe and G. Rolland, Global existence for a class of reaction-diffusion systems with mass action kinetics and concentration-dependent diffusivities, Acta Appl. Math., 139 (2015), 25-57.  doi: 10.1007/s10440-014-9968-y.  Google Scholar

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H. Brenner, Is the tracer velocity of a fluid continuum equal to its mass velocity?, Phys. Rev. E, 70 (2004), 061201. doi: 10.1103/PhysRevE.70.061201.  Google Scholar

[22]

J. A. Cañizo, L. Desvillettes and K. Fellner, Improved duality estimates and applications to reaction-diffusion equations, Comm. Partial Differential Equations, 39 (2014), 1185-1204.  doi: 10.1080/03605302.2013.829500.  Google Scholar

[23]

P. Clément, B. de Pagter, F. A. Sukochev and H. Witvliet, Schauder decomposition and multiplier theorems, Studia Math., 138 (2000), 135-163.   Google Scholar

[24]

R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114 pp. doi: 10.1090/memo/0788.  Google Scholar

[25]

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[26]

R. Denk and M. Kaip, General Parabolic Mixed Order Systems in $ \mathrm{L}_p$ and Applications, Operator Theory: Advances and Applications, 239. Birkhäuser/Springer, Cham, 2013. doi: 10.1007/978-3-319-02000-6.  Google Scholar

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R. Denk, J. Prüss and R. Zacher, Maximal $ \mathrm{L}_p$-regularity of parabolic problems with boundary dynamics of relaxiation type, J. Funct. Anal., 255 (2008), 3149-3187.  doi: 10.1016/j.jfa.2008.07.012.  Google Scholar

[28]

L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176.  doi: 10.1016/j.jmaa.2005.07.003.  Google Scholar

[29]

L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, Global existence for quadratic systems of reaction-diffusion, Adv. Nonlinear Stud., 7 (2007), 491-511.  doi: 10.1515/ans-2007-0309.  Google Scholar

[30]

L. Desvillettes and K. Fellner, Entropy methods for reaction-diffusion equations: Slowly growing a-priori bounds, Rev. Mat. Iberoam., 24 (2008), 407-431.  doi: 10.4171/RMI/541.  Google Scholar

[31]

L. Desvillettes, K. Fellner and B. Q. Tang, Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks, SIAM J. Math. Anal., 49 (2017), 2666-2709.  doi: 10.1137/16M1073935.  Google Scholar

[32]

P.-E. Druet and A. Jüngel, Analysis of cross-diffusion systems for fluid mixtures driven by a pressure gradient, SIAM J. Math. Anal., 52 (2020), 2179-2197.  doi: 10.1137/19M1301473.  Google Scholar

[33]

J. B. Duncan and H. L. Toor, An experimental study of three component gas diffusion, A. I. Ch. E. Journal, 8 (1962), 38-41.  doi: 10.1002/aic.690080112.  Google Scholar

[34]

R. Haase, Thermodynamik Irreversibler Prozesse, Fortschritte der physikalischen Chemie, 8, Steinkopff, Darmstadt, 1963. Google Scholar

[35]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840, Springer-Verlag, 1981.  Google Scholar

[36]

M. Herberg, M. Meyries, J. Prüss and M. Wilke, Reaction-diffusion systems of Maxwell-Stefan type with reversible mass-action kinetics, Nonlinear Anal., 159 (2017), 264-284.  doi: 10.1016/j.na.2016.07.010.  Google Scholar

[37]

N. J. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators, Math. Ann., 321 (2001), 319-345.  doi: 10.1007/s002080100231.  Google Scholar

[38]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968.  Google Scholar

[39]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995.  Google Scholar

[40]

A. Lunardi, Interpolation Theory, $3^\text{rd}$ Edition, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), 16, Edizioni della Normale, Pisa, 2018. doi: 10.1007/978-88-7642-638-4.  Google Scholar

[41]

R. H. Martin and M. Pierre, Nonlinear reaction-diffusion systems, in Nonlinear Equations in the Applied Sciences, (eds. W.F. Ames and C. Rogers), Math. Sci. Engrg., Academic Press, Boston, MA, 185 (1992), 363–398. doi: 10.1016/S0076-5392(08)62804-0.  Google Scholar

[42]

M. Pierre, Weak solutions and supersolutions in $L^1$ for reaction-diffusion systems, J. Evol. Equ., 3 (2003), 153-168.  doi: 10.1007/s000280300007.  Google Scholar

[43]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan J. Math., 78 (2010), 417-455.  doi: 10.1007/s00032-010-0133-4.  Google Scholar

[44]

M. Pierre and D. Schmitt, Blow up in reaction-diffusion systems with dissipation of mass, SIAM J. Math. Anal., 28 (1997), 259-269.  doi: 10.1137/S0036141095295437.  Google Scholar

[45]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall Inc., Englewood Cliffs, N. J., 1967.  Google Scholar

[46]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, Springer, Berlin, 1984. doi: 10.1007/BFb0099278.  Google Scholar

[47]

R. Schnaubelt, Stable and unstable manifolds for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions, Advances in Differential Equations, 22 (2017), 541-592.   Google Scholar

[48]

O. Souček, V. Orava, J. Málek and D. Bothe, A continuum model of heterogeneous catalysis: Thermodynamic framework for multicomponent bulk and surface phenomena coupled by sorption, Int. J. Eng. Sci., 138 (2019), 82-117.  doi: 10.1016/j.ijengsci.2019.01.001.  Google Scholar

[49]

B. Terreni, Hölder regularity results for nonhomogeneous parabolic initial-boundary value linear problems, in Semigroup Theory and Applications (Trieste, 1987), Lecture Notes in Pure and Appl. Math. Dekker, New York, 116 (1989), 387–401.  Google Scholar

show all references

References:
[1]

P. Acquistapace and B. Terreni, Fully nonlinear parabolic systems, Recent Advances in Nonlinear Elliptic and Parabolic Problems (Nancy, 1988), Pitman Res. Notes Math. Ser. Longman Sci. Tech., Harlow, 208 (1989), 97–111.  Google Scholar

[2]

H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math., 45 (1983), 225-254.  doi: 10.1007/BF02774019.  Google Scholar

[3]

H. Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math., 360 (1985), 47-83.  doi: 10.1515/crll.1985.360.47.  Google Scholar

[4]

H. Amann, Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations, Nonlinear Anal., 12 (1988), 895-919.  doi: 10.1016/0362-546X(88)90073-9.  Google Scholar

[5]

H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.  doi: 10.1016/0022-0396(88)90156-8.  Google Scholar

[6]

H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence, Math. Z., 202 (1989), 219-250.  doi: 10.1007/BF01215256.  Google Scholar

[7]

H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.   Google Scholar

[8]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), Teubner-Texte Math. Teubner, Stuttgart, 133 (1993), 9–126. doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[9]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[10]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. II. Function Spaces, Monographs in Mathematics, 106. Birkhäuser/Springer, Cham, 2019. doi: 10.1007/978-3-030-11763-4.  Google Scholar

[11]

B. Augner and D. Bothe, A thermodynamically consistent model for bulk-surface systems with sorption and surface chemistry, in preparation. Google Scholar

[12]

B. Augner and D. Bothe, Analysis of some heterogeneous catalysis models with fast sorption and fast surface chemistry, submitted (2020), https://arXiv.org/abs/2006.12098. Google Scholar

[13]

E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math., 112 (1994), 13-49.   Google Scholar

[14]

D. Bothe, The instantaneous limit of a reaction-diffusion system, in Evolation Equations and their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Appl. Math., Dekker, New York, 215 (2001), 215–224.  Google Scholar

[15]

D. Bothe, On the Maxwell-Stefan approach to multicomponent diffusion, in Parabolic problems, Progr. Nonlinear Differential Equations Appl., Birkhüser/Springer Basel AG, Basel, 80 (2011), 81–93. doi: 10.1007/978-3-0348-0075-4_5.  Google Scholar

[16]

D. Bothe and W. Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures, Acta Mech., 226 (2015), 1757-1805.  doi: 10.1007/s00707-014-1275-1.  Google Scholar

[17]

D. Bothe, M. Köhne, S. Maier and J. Saal, Global strong solutions for a class of heterogeneous catalysis models, J. Math. Anal. Appl., 445 (2017), 677-709.  doi: 10.1016/j.jmaa.2016.08.016.  Google Scholar

[18]

D. Bothe and M. Pierre, Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate, J. Math. Anal. Appl., 368 (2010), 120-132.  doi: 10.1016/j.jmaa.2010.02.044.  Google Scholar

[19]

D. Bothe and M. Pierre, The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 49-59.  doi: 10.3934/dcdss.2012.5.49.  Google Scholar

[20]

D. Bothe and G. Rolland, Global existence for a class of reaction-diffusion systems with mass action kinetics and concentration-dependent diffusivities, Acta Appl. Math., 139 (2015), 25-57.  doi: 10.1007/s10440-014-9968-y.  Google Scholar

[21]

H. Brenner, Is the tracer velocity of a fluid continuum equal to its mass velocity?, Phys. Rev. E, 70 (2004), 061201. doi: 10.1103/PhysRevE.70.061201.  Google Scholar

[22]

J. A. Cañizo, L. Desvillettes and K. Fellner, Improved duality estimates and applications to reaction-diffusion equations, Comm. Partial Differential Equations, 39 (2014), 1185-1204.  doi: 10.1080/03605302.2013.829500.  Google Scholar

[23]

P. Clément, B. de Pagter, F. A. Sukochev and H. Witvliet, Schauder decomposition and multiplier theorems, Studia Math., 138 (2000), 135-163.   Google Scholar

[24]

R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114 pp. doi: 10.1090/memo/0788.  Google Scholar

[25]

R. Denk, M. Hieber and J. Prüss, Optimal $ \mathrm{L}_p$-$ \mathrm{L}_q$ estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.  doi: 10.1007/s00209-007-0120-9.  Google Scholar

[26]

R. Denk and M. Kaip, General Parabolic Mixed Order Systems in $ \mathrm{L}_p$ and Applications, Operator Theory: Advances and Applications, 239. Birkhäuser/Springer, Cham, 2013. doi: 10.1007/978-3-319-02000-6.  Google Scholar

[27]

R. Denk, J. Prüss and R. Zacher, Maximal $ \mathrm{L}_p$-regularity of parabolic problems with boundary dynamics of relaxiation type, J. Funct. Anal., 255 (2008), 3149-3187.  doi: 10.1016/j.jfa.2008.07.012.  Google Scholar

[28]

L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176.  doi: 10.1016/j.jmaa.2005.07.003.  Google Scholar

[29]

L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, Global existence for quadratic systems of reaction-diffusion, Adv. Nonlinear Stud., 7 (2007), 491-511.  doi: 10.1515/ans-2007-0309.  Google Scholar

[30]

L. Desvillettes and K. Fellner, Entropy methods for reaction-diffusion equations: Slowly growing a-priori bounds, Rev. Mat. Iberoam., 24 (2008), 407-431.  doi: 10.4171/RMI/541.  Google Scholar

[31]

L. Desvillettes, K. Fellner and B. Q. Tang, Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks, SIAM J. Math. Anal., 49 (2017), 2666-2709.  doi: 10.1137/16M1073935.  Google Scholar

[32]

P.-E. Druet and A. Jüngel, Analysis of cross-diffusion systems for fluid mixtures driven by a pressure gradient, SIAM J. Math. Anal., 52 (2020), 2179-2197.  doi: 10.1137/19M1301473.  Google Scholar

[33]

J. B. Duncan and H. L. Toor, An experimental study of three component gas diffusion, A. I. Ch. E. Journal, 8 (1962), 38-41.  doi: 10.1002/aic.690080112.  Google Scholar

[34]

R. Haase, Thermodynamik Irreversibler Prozesse, Fortschritte der physikalischen Chemie, 8, Steinkopff, Darmstadt, 1963. Google Scholar

[35]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840, Springer-Verlag, 1981.  Google Scholar

[36]

M. Herberg, M. Meyries, J. Prüss and M. Wilke, Reaction-diffusion systems of Maxwell-Stefan type with reversible mass-action kinetics, Nonlinear Anal., 159 (2017), 264-284.  doi: 10.1016/j.na.2016.07.010.  Google Scholar

[37]

N. J. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators, Math. Ann., 321 (2001), 319-345.  doi: 10.1007/s002080100231.  Google Scholar

[38]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968.  Google Scholar

[39]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995.  Google Scholar

[40]

A. Lunardi, Interpolation Theory, $3^\text{rd}$ Edition, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), 16, Edizioni della Normale, Pisa, 2018. doi: 10.1007/978-88-7642-638-4.  Google Scholar

[41]

R. H. Martin and M. Pierre, Nonlinear reaction-diffusion systems, in Nonlinear Equations in the Applied Sciences, (eds. W.F. Ames and C. Rogers), Math. Sci. Engrg., Academic Press, Boston, MA, 185 (1992), 363–398. doi: 10.1016/S0076-5392(08)62804-0.  Google Scholar

[42]

M. Pierre, Weak solutions and supersolutions in $L^1$ for reaction-diffusion systems, J. Evol. Equ., 3 (2003), 153-168.  doi: 10.1007/s000280300007.  Google Scholar

[43]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan J. Math., 78 (2010), 417-455.  doi: 10.1007/s00032-010-0133-4.  Google Scholar

[44]

M. Pierre and D. Schmitt, Blow up in reaction-diffusion systems with dissipation of mass, SIAM J. Math. Anal., 28 (1997), 259-269.  doi: 10.1137/S0036141095295437.  Google Scholar

[45]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall Inc., Englewood Cliffs, N. J., 1967.  Google Scholar

[46]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, Springer, Berlin, 1984. doi: 10.1007/BFb0099278.  Google Scholar

[47]

R. Schnaubelt, Stable and unstable manifolds for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions, Advances in Differential Equations, 22 (2017), 541-592.   Google Scholar

[48]

O. Souček, V. Orava, J. Málek and D. Bothe, A continuum model of heterogeneous catalysis: Thermodynamic framework for multicomponent bulk and surface phenomena coupled by sorption, Int. J. Eng. Sci., 138 (2019), 82-117.  doi: 10.1016/j.ijengsci.2019.01.001.  Google Scholar

[49]

B. Terreni, Hölder regularity results for nonhomogeneous parabolic initial-boundary value linear problems, in Semigroup Theory and Applications (Trieste, 1987), Lecture Notes in Pure and Appl. Math. Dekker, New York, 116 (1989), 387–401.  Google Scholar

Figure 1.  Physical and chemical mechanisms in a bulk-surface reaction diffusion system
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Figure 2.  Free sites on the surface are interpreted as an additional species $ A_0^\Sigma $
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