Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-21T11:33:52.894Z Has data issue: false hasContentIssue false
  • English
  • Français

Modelling and analysis of an endothermic reacting counter-current flow

Published online by Cambridge University Press:  29 September 2022

Ellen K. Luckins Open the ORCID record for Ellen K. Luckins [Opens in a new window] ,
James M. Oliver ,
Colin P. Please ,
Benjamin M. Sloman  and
Robert A. Van Gorder Open the ORCID record for Robert A. Van Gorder [Opens in a new window]
Ellen K. Luckins*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
James M. Oliver
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
Colin P. Please
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
Benjamin M. Sloman
Affiliation:
Elkem ASA, Technology, Fiskaaveien 100, Kristiansand 4621, Norway
Robert A. Van Gorder
Affiliation:
Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand
*
†Email address for correspondence: luckins@maths.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the endothermic reaction and flow of a granular solid reactant, where energy for the reaction is provided by a counter-current flow of hot gases through the porous reactant bed. Research into reacting flows typically focusses on exothermic combustion processes. However, endothermic processes are common in the metallurgy industry, including the production of cement, silicon and rutile titanium dioxide. Several common features are observed in experimental and numerical studies of these processes, including critical temperatures of the reactant at which the chemical reaction begins, and regions of the reactor with uniform reactant temperature. Motivated specifically by the processes in a silicon furnace, we analyse a model of endothermic, reacting counter-current flow using the method of matched asymptotic expansions. Assuming the Péclet number in the solid is large, we explore the full range of values for the dimensionless inter-phase heat-transfer rate, finding six distinguished limits. In all limits, we find a diffusive boundary layer in which there is a fast chemical reaction rate due to the high temperatures, analogous to exothermic flame fronts. Outside this region, the counter-current flow is crucial to the chemical processes. For intermediate values of the heat-transfer rate, we find the same qualitative properties as those observed across the metallurgy industry, and we quantify the dependence of these properties on the flow rate and heat-transfer rate. In the limit of large heat-transfer coefficient, we derive the single-temperature limit, in which the solution structure is dependent on the direction of net heat flux through the domain.

JFM classification

Multiphase and Particle-laden Flows: Reacting multiphase flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 949 , 25 October 2022 , A21
Check for updates
Copyright
© The Author(s), 2022. 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

Abd, A.S., Elhafyan, E., Siddiqui, A.R., Alnoush, W., Blunt, M.J. & Alyafei, N. 2019 A review of the phenomenon of counter-current spontaneous imbibition: analysis and data interpretation. J. Petrol. Sci. Engng 180, 456–470.CrossRefGoogle Scholar
Agrawal, A. & Ghoshdastidar, P.S. 2017 Numerical simulation of heat transfer during production of rutile titanium dioxide in a rotary kiln. Intl J. Heat Mass Transfer 106, 263–279.CrossRefGoogle Scholar
Andresen, B. 1995 Process model for carbothermic production of silicon metal. Master's thesis, NTH, Norway.Google Scholar
Baer, M.R. & Nunziato, J.W. 1986 A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Intl J. Multiphase Flow 12 (6), 861–889.CrossRefGoogle Scholar
Beals, R. 1981 Partial-range completeness and existence of solutions to two-way diffusion equations. J. Math. Phys. 22 (5), 954–960.CrossRefGoogle Scholar
Bilger, R.W. 1989 Turbulent diffusion flames. Annu. Rev. Fluid Mech. 21 (1), 101–135.CrossRefGoogle Scholar
Booty, M.R. & Matkowsky, B.J. 1991 Modes of burning in filtration combustion. Eur. J. Appl. Maths 2 (1), 17–41.CrossRefGoogle Scholar
Brennen, C.E. 2005 Fundamentals of Multiphase Flow. Cambridge University Press.CrossRefGoogle Scholar
Brunner, G. 2009 Counter-current separations. J. Supercrit. Fluids 47 (3), 574–582.CrossRefGoogle Scholar
Buckmaster, J.D. & Ludford, G.S.S. 1983 Lectures on Mathematical Combustion. SIAM.CrossRefGoogle Scholar
Byrne, H. & Norbury, J. 1994 Stable solutions for a catalytic converter. SIAM J. Appl. Maths 54 (3), 789–813.CrossRefGoogle Scholar
Byrne, H. & Norbury, J. 1997 The effect of solid conversion on travelling combustion waves in porous media. J. Engng Maths 32 (4), 321–342.CrossRefGoogle Scholar
Chapiro, G. & de Souza, A.J. 2016 Asymptotic approximation for counterflow combustion in porous media. Appl. Anal. 95 (1), 63–77.CrossRefGoogle Scholar
Chapiro, G. & Senos, L. 2018 Riemann solutions for counterflow combustion in light porous foam. Comput. Appl. Maths 37 (2), 1721–1736.CrossRefGoogle Scholar
Deendarlianto, H.T., Lucas, D. & Vierow, K. 2012 Gas–liquid countercurrent two-phase flow in a PWR hot leg: a comprehensive research review. Nucl. Engng Des. 243, 214–233.CrossRefGoogle Scholar
Egerton, A., Gugan, K. & Weinberg, F.J. 1963 The mechanism of smouldering in cigarettes. Combust. Flame 7, 63–78.CrossRefGoogle Scholar
Fitt, V., Ockendon, J.R. & Shillor, M. 1985 Counter-current mass transfer. Intl J. Heat Mass Transfer 28 (4), 753–759.CrossRefGoogle Scholar
Hagan, P.S. & Ockendon, J.R. 1991 Half-range analysis of a counter-current separator. J. Math. Anal. Appl. 160 (2), 358–378.CrossRefGoogle Scholar
Hinch, E.J. 1991 Perturbation Methods. Cambridge University Press.CrossRefGoogle Scholar
Ignatova, S., Hewitson, P., Mathews, B. & Sutherland, I. 2011 Evaluation of dual flow counter-current chromatography and intermittent counter-current extraction. J. Chromatogr. A 1218 (36), 6102–6106.CrossRefGoogle ScholarPubMed
Johansen, S.T., Tveit, H., Grådahl, S., Valderhaug, A.M. & Byberg, J. 1998 Environmental aspects of ferro-silicon furnace operations – an investigation of waste gas dynamics. In The 8th International Ferroalloys Congress, Pekin. Mintek.Google Scholar
Kaviany, M. 2012 Principles of Heat Transfer in Porous Media. Springer Science & Business Media.Google Scholar
Kierzenka, J. & Shampine, L.F. 2001 A BVP solver based on residual control and the MATLAB PSE. ACM Trans. Math. Softw. 27 (3), 299–316.CrossRefGoogle Scholar
Koopmans, R.J., Shrimpton, J.S., Roberts, G.T. & Musker, A.J. 2013 A one-dimensional multicomponent two-fluid model of a reacting packed bed including mass, momentum and energy interphase transfer. Intl J. Multiphase Flow 57, 10–28.CrossRefGoogle Scholar
Luckins, E.K., Oliver, J.M., Please, C.P., Sloman, B.M. & Van Gorder, R.A. 2022 Homogenised model for the electrical current distribution within a submerged arc furnace for silicon production. Eur. J. Appl. Maths 33 (5), 828–863.CrossRefGoogle Scholar
Marias, F., Roustan, H. & Pichat, A. 2005 Modelling of a rotary kiln for the pyrolysis of aluminium waste. Chem. Engng Sci. 60 (16), 4609–4622.CrossRefGoogle Scholar
MATLAB 2021 version 9.10.0 (R2021a). The MathWorks Inc.Google Scholar
Merzhanov, A.G. & Khaikin, B.I. 1988 Theory of combustion waves in homogeneous media. Prog. Energy Combust. Sci. 14 (1), 1–98.CrossRefGoogle Scholar
Mitchell, J.W. & Myers, G.E. 1968 An analytical model of the counter-current heat exchange phenomena. Biophys. J. 8 (8), 897–911.CrossRefGoogle ScholarPubMed
Mujumdar, K.S. & Ranade, V.V. 2006 Simulation of rotary cement kilns using a one-dimensional model. Chem. Engng Res. Des. 84 (3), 165–177.CrossRefGoogle Scholar
Ni, J. & Beckermann, C. 1991 A volume-averaged two-phase model for transport phenomena during solidification. Metall. Trans. B 22 (3), 349–361.CrossRefGoogle Scholar
Norbury, J. & Stuart, A.M. 1988 Travelling combustion waves in a porous medium. Part I–existence. SIAM J. Appl. Maths 48 (1), 155–169.CrossRefGoogle Scholar
Nunge, R.J. & Gill, W.N. 1965 Analysis of heat or mass transfer in some countercurrent flows. Intl J. Heat Mass Transfer 8 (6), 873–886.CrossRefGoogle Scholar
Please, C.P., Liu, F. & McElwain, D.L.S. 2003 Combustion waves with exothermic/endothermic reactions. Combust. Theor. Model. 7, 129–143.CrossRefGoogle Scholar
Ravikrishna, R.V. & Sahu, A.B. 2018 Advances in understanding combustion phenomena using non-premixed and partially premixed counterflow flames: a review. Intl J. Spray Combust. Dyn. 10 (1), 38–71.CrossRefGoogle Scholar
Schei, A., Tuset, J.K. & Tveit, H. 1998 Production of High Silicon Alloys. Tapir.Google Scholar
Schmidt-Nielsen, K., Hainsworth, F.R. & Murrish, D.E. 1970 Counter-current heat exchange in the respiratory passages: effect on water and heat balance. Respir. Physiol. 9 (2), 263–276.CrossRefGoogle ScholarPubMed
Schult, D.A., Bayliss, A. & Matkowsky, B.J. 1998 Traveling waves in natural counterflow filtration combustion and their stability. SIAM J. Appl. Maths 58 (3), 806–852.Google Scholar
Sirignano, W.A. 2021 Mixing and combustion in a laminar shear layer with imposed counterflow. J. Fluid Mech. 908, A35.CrossRefGoogle Scholar
Skalicka-Woźniak, K. & Garrard, I. 2014 Counter-current chromatography for the separation of terpenoids: a comprehensive review with respect to the solvent systems employed. Phytochem. Rev. 13 (2), 547–572.CrossRefGoogle Scholar
Sloman, B.M., Please, C.P. & Van Gorder, R.A. 2018 Asymptotic analysis of a silicon furnace model. SIAM J. Appl. Maths 78 (2), 1174–1205.CrossRefGoogle Scholar
Sloman, B.M., Please, C.P. & Van Gorder, R.A. 2020 Melting and dripping of a heated material with temperature-dependent viscosity in a thin vertical tube. J. Fluid Mech. 905, A16.CrossRefGoogle Scholar
Spang III, H.A. 1972 A dynamic model of a cement kiln. Automatica 8 (3), 309–323.CrossRefGoogle Scholar
Stadler, K.S., Poland, J. & Gallestey, E. 2011 Model predictive control of a rotary cement kiln. Control Engng Pract. 19 (1), 1–9.CrossRefGoogle Scholar
Sutherland, I.A. 2007 Recent progress on the industrial scale-up of counter-current chromatography. J. Chromatogr. A 1151 (1–2), 6–13.CrossRefGoogle ScholarPubMed
Tieszen, S.R. 2001 On the fluid mechanics of fires. Annu. Rev. Fluid Mech. 33 (1), 67–92.CrossRefGoogle Scholar
Winters, R.W. & Davies, R.E. 1961 The role of countercurrent mechanisms in urine concentration: a review. Ann. Intern. Med. 54 (4), 810–826.Google ScholarPubMed