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

Björn Augner; Dieter Bothe

doi:10.3934/dcdss.2020406

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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 70^th 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, L_p-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 and Continuous Dynamical Systems - S, 2021, 14 (2) : 533-574. doi: 10.3934/dcdss.2020406

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.
[2]	H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math., 45 (1983), 225-254. doi: 10.1007/BF02774019.
[3]	H. Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math., 360 (1985), 47-83. doi: 10.1515/crll.1985.360.47.
[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.
[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.
[6]	H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence, Math. Z., 202 (1989), 219-250. doi: 10.1007/BF01215256.
[7]	H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.
[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.
[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.
[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.
[11]	B. Augner and D. Bothe, A thermodynamically consistent model for bulk-surface systems with sorption and surface chemistry, in preparation.
[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.
[13]	E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math., 112 (1994), 13-49.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[23]	P. Clément, B. de Pagter, F. A. Sukochev and H. Witvliet, Schauder decomposition and multiplier theorems, Studia Math., 138 (2000), 135-163.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[34]	R. Haase, Thermodynamik Irreversibler Prozesse, Fortschritte der physikalischen Chemie, 8, Steinkopff, Darmstadt, 1963.
[35]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840, Springer-Verlag, 1981.
[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.
[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.
[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.
[39]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995.
[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.
[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.
[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.
[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.
[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.
[45]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall Inc., Englewood Cliffs, N. J., 1967.
[46]	F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, Springer, Berlin, 1984. doi: 10.1007/BFb0099278.
[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.
[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.
[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.

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.
[2]	H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math., 45 (1983), 225-254. doi: 10.1007/BF02774019.
[3]	H. Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math., 360 (1985), 47-83. doi: 10.1515/crll.1985.360.47.
[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.
[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.
[6]	H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence, Math. Z., 202 (1989), 219-250. doi: 10.1007/BF01215256.
[7]	H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.
[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.
[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.
[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.
[11]	B. Augner and D. Bothe, A thermodynamically consistent model for bulk-surface systems with sorption and surface chemistry, in preparation.
[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.
[13]	E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math., 112 (1994), 13-49.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[23]	P. Clément, B. de Pagter, F. A. Sukochev and H. Witvliet, Schauder decomposition and multiplier theorems, Studia Math., 138 (2000), 135-163.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[34]	R. Haase, Thermodynamik Irreversibler Prozesse, Fortschritte der physikalischen Chemie, 8, Steinkopff, Darmstadt, 1963.
[35]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840, Springer-Verlag, 1981.
[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.
[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.
[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.
[39]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995.
[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.
[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.
[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.
[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.
[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.
[45]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall Inc., Englewood Cliffs, N. J., 1967.
[46]	F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, Springer, Berlin, 1984. doi: 10.1007/BFb0099278.
[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.
[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.
[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.

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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2021 Impact Factor: 1.865

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2021 Impact Factor: 1.865

American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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