Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-29T07:03:07.361Z Has data issue: false hasContentIssue false
  • English
  • Français

Theoretical modelling of non-equilibrium reaction–diffusion of rarefied gas on a wall with microscale roughness

Published online by Cambridge University Press:  20 June 2023

Shun-Liang Zhang Open the ORCID record for Shun-Liang Zhang [Opens in a new window]  and
Zhi-Hui Wang Open the ORCID record for Zhi-Hui Wang [Opens in a new window]
Shun-Liang Zhang
Affiliation:
University of Chinese Academy of Sciences, 100049 Beijing, PR China
Zhi-Hui Wang*
Affiliation:
University of Chinese Academy of Sciences, 100049 Beijing, PR China
*
†Email address for correspondence: wisdom@ucas.ac.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

Heat and mass transports through a rough surface are among the most fundamental and important phenomena in either natural or engineering problems. In this paper, theoretical modelling and direct simulation Monte Carlo method are employed to study the heterogeneous reaction–diffusion features induced by microscale roughness which is comparable to the molecular mean free path of the ambient gas. A quasi-one-dimensional homogeneous model is proposed, and it consists of an external diffusion region outside the roughness elements and an internal reaction–diffusion region which could be equivalent to a smooth surface with an effective chemical property. The external macroscopic diffusion can be characterized by a non-equilibrium criterion – the Damköhler number. The internal diffusion in micro-cavities must be analysed by considering the rarefied gas effects on the diffusivity, and another non-equilibrium criterion, the Thiele number, is introduced to evaluate the effective boundary condition imposed on the external region. Analytical formulae based on these criteria are derived to predict the equivalent surface reaction–diffusion performance, and the predictions compare well with the numerical results of different types of surface reaction, even on the three-dimensional roughness. This reveals that the roughness could either enhance or weaken the apparent reaction rate depending on the non-equilibrium degree. This study could enrich our understanding of the gas–surface interactions on a rough wall, such as the oxidation, catalysis and energy accommodation, and also preliminarily provides a practical method for evaluation of the aerothermochemical performance of coating materials of hypersonic vehicles.

JFM classification

Compressible Flows: Hypersonic Flow Micro-/Nano-fluid dynamics: Microscale transport
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 965 , 25 June 2023 , A21
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Achambath, A.D. & Schwartzentruber, T.E. 2018 Molecular simulation of boundary layer flow over thermal protection system microstructure. AIAA Paper 2018-0493.CrossRefGoogle Scholar
Achdou, Y., Pironneau, O. & Valentin, F. 1998 Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comput. Phys. 147 (1), 187–218.CrossRefGoogle Scholar
Alkandry, H., Boyd, I.D. & Martin, A. 2014 Comparison of transport properties models for flowfield simulations of ablative heat shields. J. Thermophys. Heat Transfer 28 (4), 569–582.CrossRefGoogle Scholar
Amirat, Y. & Bodart, O. 2001 Boundary layer correctors for the solution of laplace equation in a domain with oscillating boundary. Z. Anal. Anwend. 20 (4), 929–940.CrossRefGoogle Scholar
Amirat, Y., Bodart, O., de Maio, U. & Gaudiello, A. 2004 Asymptotic approximation of the solution of the laplace equation in a domain with highly oscillating boundary. SIAM J. Math. Anal. 35 (6), 1598–1616.CrossRefGoogle Scholar
Amirat, Y., Chechkin, G.A. & Gadyl'shin, R.R. 2006 Asymptotics of simple eigenvalues and eigenfunctions for the laplace operator in a domain with an oscillating boundary. Comput. Maths Math. Phys. 46 (1), 97–110.CrossRefGoogle Scholar
Amirat, Y., Chechkin, G.A. & Gadyl'shin, R.R. 2007 Asymptotics for eigenelements of laplacian in domain with oscillating boundary: multiple eigenvalues. Appl. Anal. 86 (7), 873–897.CrossRefGoogle Scholar
Arya, G., Chang, H.-C. & Maginn, E.J. 2003 Knudsen diffusivity of a hard sphere in a rough slit pore. Phys. Rev. Lett. 91 (2), 026102.CrossRefGoogle Scholar
Bhagat, A., Gijare, H. & Dongari, N. 2019 Modeling of Knudsen layer effects in the micro-scale backward-facing step in the slip flow regime. Micromachines 10 (2), 118.CrossRefGoogle ScholarPubMed
Bird, G.A. 1994 Molecular Gas Dynamics and the Direct Simulation of Gas Flows, 2nd edn. Oxford University Press.Google Scholar
Bird, G.A. 2005 The DS2V/3 V program suite for DSMC calculations. In Rarefied Gas Dynamics: 24th International Symposium (ed. M. Capitelli), vol. 762, pp. 541–546. AIP Conference proceedings.CrossRefGoogle Scholar
Bird, R.B., Stewart, W.E. & Lightfoot, E.N. 2002 Transport Phenomena, 2nd edn. J. Wiley.Google Scholar
Blyth, M.G. & Pozrikidis, C. 2003 Heat conduction across irregular and fractal-like surfaces. Intl J. Heat Mass Transfer 46 (8), 1329–1339.CrossRefGoogle Scholar
Bottaro, A. 2019 Flow over natural or engineered surfaces: an adjoint homogenization perspective. J. Fluid Mech. 877, 211.CrossRefGoogle Scholar
Chapman, S. & Cowling, T.G. 1990 The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction, and Diffusion in Gases, 3rd edn. Cambridge University Press.Google Scholar
Chechkin, G.A., Friedman, A. & Piatnitski, A.L. 1999 The boundary-value problem in domains with very rapidly oscillating boundary. J. Math. Anal. Appl. 231 (1), 213–234.CrossRefGoogle Scholar
Chung, D., Hutchins, N., Schultz, M.P. & Flack, K.A. 2021 Predicting the drag of rough surfaces. Annu. Rev. Fluid Mech. 53 (1), 439–471.CrossRefGoogle Scholar
Clausing, P. 1932 Über die strömung sehr verdünnter gase durch röhren von beliebiger länge. Ann. Phys. 404 (8), 961–989.CrossRefGoogle Scholar
Colson, F. & Barlow, D.A. 2019 Statistical method for modeling knudsen diffusion in nanopores. Phys. Rev. E 100 (6–1), 062125.CrossRefGoogle ScholarPubMed
Coppens, M.-O. & Froment, G.F. 1995 Knudsen diffusion in porous catalysts with a fractal internal surface. Fractals 03 (04), 807–820.CrossRefGoogle Scholar
Corral-Casas, C., Gibelli, L., Borg, M.K., Li, J., Al-Afnan, S.F.K. & Zhang, Y. 2021 Self-diffusivity of dense confined fluids. Phys. Fluids 33 (8), 082009.CrossRefGoogle Scholar
Cussler, E.L. 2009 Diffusion: Mass Transfer in Fluid Systems, 3rd edn. Cambridge University Press.CrossRefGoogle Scholar
Fyrillas, M.M. & Pozrikidis, C. 2001 Conductive heat transport across rough surfaces and interfaces between two conforming media. Intl J. Heat Mass Transfer 44 (9), 1789–1801.CrossRefGoogle Scholar
Gnoffo, P.A. 1999 Planetary-entry gas dynamics. Annu. Rev. Fluid Mech. 31, 459–494.CrossRefGoogle Scholar
Gupta, R.N., Scott, C.D. & Moss, J.N. 1985 Slip-boundary equations for multicomponent nonequilibrium airflow. NASA Technical Paper 2452.Google Scholar
Hermann Schlichting, K.G. 2017 Boundary-Layer Theory, 9th edn. Springer.CrossRefGoogle Scholar
Hewitt, G.F. & Sharratt, E.W. 1963 Gaseous diffusion in porous media with particular reference to graphite. Nature 198 (4884), 952–957.CrossRefGoogle Scholar
Hoogschagen, J. 1955 Diffusion in porous catalysts and adsorbents. Ind. Engng Chem. 47 (5), 906–912.CrossRefGoogle Scholar
Inger, G.R. 2001 Scaling nonequilibrium-reacting flows: the legacy of Gerhard Damköhler. J. Spacecr. Rockets 38 (2), 185–190.CrossRefGoogle Scholar
Jochen, M. 1997 Experimental determination of oxygen and nitrogen recombination coefficients at elevated temperatures using laser-induced fluorescence. AIAA Paper 1997-3879.Google Scholar
Kavokine, N., Netz, R.R. & Bocquet, L. 2021 Fluids at the nanoscale: from continuum to subcontinuum transport. Annu. Rev. Fluid Mech. 53 (1), 377–410.CrossRefGoogle Scholar
Keerthi, A., et al. 2018 Ballistic molecular transport through two-dimensional channels. Nature 558 (7710), 420–424.CrossRefGoogle ScholarPubMed
Kim, Y.C. & Boudart, M. 1991 Recombination of O, N, and H atoms on silica: kinetics and mechanism. Langmuir 7 (12), 2999–3005.CrossRefGoogle Scholar
Kim, I., Lee, S., Kim, J.G. & Park, G. 2020 a Analysis of nitrogen recombination activity on silicon dioxide with stagnation heat-transfer. Acta Astronaut. 177, 386–397.CrossRefGoogle Scholar
Kim, I., Yang, Y. & Park, G. 2020 b Effect of titanium surface roughness on oxygen catalytic recombination in a shock tube. Acta Astronaut. 166, 260–269.CrossRefGoogle Scholar
Knudsen, M. 1909 Die gesetze der molekularströmung und der inneren reibungsströmung der gase durch röhren. Ann. Phys. 333 (1), 75–130.CrossRefGoogle Scholar
Kraus, A.D., Aziz, A. & Welty, J.R. 2001 Extended Surface Heat Transfer. Wiley.Google Scholar
Lachaud, J., Bertrand, N., Vignoles, G.L., Bourget, G., Rebillat, F. & Weisbecker, P. 2007 A theoretical/experimental approach to the intrinsic oxidation reactivities of C/C composites and of their components. Carbon 45 (14), 2768–2776.CrossRefGoogle Scholar
Lachaud, J., Cozmuta, I. & Mansour, N.N. 2010 Multiscale approach to ablation modeling of phenolic impregnated carbon ablators. J. Spacecr. Rockets 47 (6), 910–921.CrossRefGoogle Scholar
Le, V.T., Ha, N.S. & Goo, N.S. 2021 Advanced sandwich structures for thermal protection systems in hypersonic vehicles: a review. Compos. B 226 (5), 109301.CrossRefGoogle Scholar
Levdanskii, V.V., Roldugin, V.I., Zhdanov, V.M. & Zdimal, V. 2014 Free-molecular gas flow in a narrow (nanosize) channel. J. Engg Phys. Thermophys. 87 (4), 802–814.CrossRefGoogle Scholar
Levet, C., Helber, B., Couzi, J., Mathiaud, J., Gouriet, J.-B., Chazot, O. & Vignoles, G.L. 2017 Microstructure and gas-surface interaction studies of a 3D carbon/carbon composite in atmospheric entry plasma. Carbon 114 (3–4), 84–97.CrossRefGoogle Scholar
Levet, C., Lachaud, J., Ducamp, V., Memes, R., Couzi, J., Mathiaud, J., Gillard, A.P., Weisbecker, P. & Vignoles, G.L. 2021 High-flux sublimation of a 3D carbon/carbon composite: surface roughness patterns. Carbon 173 (3–4), 817–831.CrossRefGoogle Scholar
Li, S.T. & Dong, M. 2021 Verification of local scattering theory as is applied to transition prediction in hypersonic boundary layers. Adv. Mech. 51 (2), 364–375.Google Scholar
Lowell, S. & Shields, J.E. 1991 Powder Surface Area and Porosity. Springer.Google Scholar
Luo, J. & Wang, Z. 2020 Analogy between vibrational and chemical nonequilibrium effects on stagnation flows. AIAA J. 58 (5), 2156–2164.CrossRefGoogle Scholar
Mark Brady, C.P. 1993 Diffusive transport across irregular and fractal walls. Proc. R. Soc. Lond. A 442 (1916), 571–583.Google Scholar
Massuti-Ballester, B. & Herdrich, G. 2017 Experimental methodology to assess atomic recombination on high-temperature materials. J. Thermophys. Heat Transfer 32 (2), 353–368.CrossRefGoogle Scholar
Massuti-Ballester, B. & Herdrich, G. 2021 Heterogeneous catalysis models of high-temperature materials in high-enthalpy flows. J. Thermophys. Heat Transfer 35 (3), 459–476.CrossRefGoogle Scholar
Miró Miró, F. & Pinna, F. 2021 Decoupling ablation effects on boundary-layer stability and transition. J. Fluid Mech. 907 (14), 1552.Google Scholar
Murray, V.J., Recio, P., Caracciolo, A., Miossec, C., Balucani, N., Casavecchia, P. & Minton, T.K. 2020 Oxidation and nitridation of vitreous carbon at high temperatures. Carbon 167 (15), 388–402.CrossRefGoogle Scholar
Nevard, J. & Keller, J.B. 1997 Homogenization of rough boundaries and interfaces. SIAM J. Appl. Maths 57 (6), 1660–1686.CrossRefGoogle Scholar
Panerai, F., Cochell, T., Martin, A. & White, J.D. 2019 Experimental measurements of the high-temperature oxidation of carbon fibers. Intl J. Heat Mass Transfer 136 (3), 972–986.CrossRefGoogle Scholar
Pirrone, S.R.M., Agabiti, C., Pagan, A.S. & Herdrich, G. 2022 A fast thermal 1D model to study aerospace material response behaviors in uncontrolled atmospheric entries. Materials 15 (4), 1505.CrossRefGoogle ScholarPubMed
Poovathingal, S.J. & Schwartzentruber, T.E. 2014 Effect of microstructure on carbon-based surface ablators using DSMC. AIAA Paper 2014-1210.CrossRefGoogle Scholar
Poovathingal, S., Schwartzentruber, T.E., Srinivasan, S.G. & van Duin, A.C.T. 2013 Large scale computational chemistry modeling of the oxidation of highly oriented pyrolytic graphite. J. Phys. Chem. A 117 (13), 2692–2703.CrossRefGoogle ScholarPubMed
Present, R.D. 1958 Kinetic theory of gases. Am. J. Phys. 29, 649–650.CrossRefGoogle Scholar
Richardson, S. 1971 A model for the boundary condition of a porous material. Part 2. J. Fluid Mech. 49 (2), 327–336.CrossRefGoogle Scholar
Richardson, S. 1973 On the no-slip boundary condition. J. Fluid Mech. 59 (4), 707–719.CrossRefGoogle Scholar
Ringhofer, C.A. & Gobbert, M.K. 1998 An asymptotic analysis for a model of chemical vapor deposition on a microstructured surface. SIAM J. Appl. Maths 58 (3), 737–752.CrossRefGoogle Scholar
Rosner, D.E. & Papadopoulos, D.H. 1996 Jump, slip, and creep boundary conditions at nonequilibrium gas/solid interfaces. Ind. Engng Chem. Res. 35 (9), 3210–3222.CrossRefGoogle Scholar
Sarkar, K. & Prosperetti, A. 1996 Effective boundary conditions for stokes flow over a rough surface. J. Fluid Mech. 316, 223–240.CrossRefGoogle Scholar
Scott, C.D. 1992 Wall Catalytic Recombination and Boundary Conditions in Nonequilibrium Hypersonic Flows—With Applications, vol. 2. Springer.Google Scholar
Shindo, Y., Hakuta, T., Yoshitome, H. & Inoue, H. 1983 Gas diffusion in microporous media in Knudsen's regime. J. Chem. Engng Japan 16 (2), 120–126.CrossRefGoogle Scholar
Sigal, A. & Danberg, J.E. 1990 New correlation of roughness density effect on the turbulent boundary layer. AIAA J. 28 (3), 554–556.CrossRefGoogle Scholar
Sing, K.S.W. 1985 Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity (recommendations 1984). Pure Appl. Chem. 57 (4), 603–619.CrossRefGoogle Scholar
Smoluchowski, M.V. 1910 Zur kinetischen theorie der transpiration und diffusion verdünnter gase. Ann. Phys. 338 (16), 1559–1570.CrossRefGoogle Scholar
Song, S., Yang, X., Xin, F. & Lu, T.J. 2018 Modeling of surface roughness effects on stokes flow in circular pipes. Phys. Fluids 30 (2), 023604.CrossRefGoogle Scholar
Swaminathan-Gopalan, K. & Stephani, K.A. 2016 Recommended direct simulation monte carlo collision model parameters for modeling ionized air transport processes. Phys. Fluids 28 (2), 027101.CrossRefGoogle Scholar
Taylor, G.I. 1971 A model for the boundary condition of a porous material. Part 1. J. Fluid Mech. 49 (2), 319–326.CrossRefGoogle Scholar
Thoemel, J. & Chazot, O. 2009 Surface catalysis of rough surfaces. AIAA Paper 2009-3931.CrossRefGoogle Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics, annotated edn. Parabolic Press.Google Scholar
Vérant, J.L., Perron, N., Pichelin, M.B., Chazot, O., Kolesnikov, A., Sakharov, V., Gerasimova, O. & Omaly, P. 2012 Microscopic and macroscopic analysis for TPS SiC material under earth and mars re-entry conditions. Intl J. Aerodyn. 2 (2–4), 152.CrossRefGoogle Scholar
Wang, C.Y. 2003 Flow over a surface with parallel grooves. Phys. Fluids 15 (5), 1114–1121.CrossRefGoogle Scholar
Wang, Z. 2014 Theoretical Modelling of Aeroheating on Sharpened Noses Under Rarefied Gas Effects and Nonequilibrium Real Gas Effects. Springer.Google Scholar
Xu, B. & Ju, Y.G. 2005 Concentration slip and its impact on heterogeneous combustion in a micro scale chemical reactor. Chem. Engng Sci. 60 (13), 3561–3572.CrossRefGoogle Scholar
Zade, A.Q., Renksizbulut, M. & Friedman, J. 2008 Slip/jump boundary conditions for rarefied reacting/non-reacting multi- component gaseous flows. Intl J. Heat Mass Transfer 51 (21–22), 5063–5071.CrossRefGoogle Scholar
Zalc, J.M., Reyes, S.C. & Iglesia, E. 2004 The effects of diffusion mechanism and void structure on transport rates and tortuosity factors in complex porous structures. Chem. Engng Sci. 59 (14), 2947–2960.CrossRefGoogle Scholar
Zhang, S. & Wang, Z. 2022 Effects of chemical energy accommodation on nonequilibrium flow and heat transfer to a catalytic wall. Chin. J. Aeronaut. 35 (10), 165–175.CrossRefGoogle Scholar
Zhou, H., Khayat, R.E., Martinuzzi, R.J. & Straatman, A.G. 2002 On the validity of the perturbation approach for the flow inside weakly modulated channels. Intl J. Numer. Meth. Fluids 39 (12), 1139–1159.CrossRefGoogle Scholar