Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-19T03:48:39.122Z Has data issue: false hasContentIssue false
  • English
  • Français

Melting and dripping of a heated material with temperature-dependent viscosity in a thin vertical tube

Published online by Cambridge University Press:  26 October 2020

Benjamin M. Sloman ,
Colin P. Please  and
Robert A. Van Gorder Open the ORCID record for Robert A. Van Gorder [Opens in a new window]
Benjamin M. Sloman
Affiliation:
Elkem ASA, Technology, Fiskaaveien 100, 4621Kristiansand, Norway
Colin P. Please
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, OxfordOX2 6GG, UK
Robert A. Van Gorder*
Affiliation:
Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin9054, New Zealand
*
†Email address for correspondence: rvangorder@maths.otago.ac.nz
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the flow of a thermoviscous fluid within a vertical tube which is heated from below, modelling a scenario where a fluid melts, flows and eventually drips due to a temperature-dependent viscosity. To do so, we develop a two-dimensional axisymmetric model comprising three regions, a solid granular upper region (modelled as a region of hydrostatic pressure), a middle highly viscous ‘crust’ region which flows and a lower cavity region within which the material can drip. New material is continuously added to the top, yet the highly viscous middle region can slow mass transfer from the top region to the cavity if it becomes too thick or does not drip fast enough. In the limit of a tall, thin geometry, akin to what is often seen in industrial applications, the resulting model comprises a moving boundary problem governed by an energy equation with a Stefan condition, both subject to a non-local radially averaged convective term. We carry out numerical simulations and an asymptotic analysis of the model in this tall, thin limit, for a variety of physically relevant parameter regimes. Our results reveal a variety of qualitatively different behaviours, and enable us to explore how various parameter regimes influence the salient features of the flow, including the ‘crust’ thickness and the flux of material through the lower moving boundary.

JFM classification

Phase change: Solidification/melting
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 905 , 25 December 2020 , A16
Check for updates
Copyright
© The Author(s), 2020. 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

REFERENCES

Ali, A., Underwood, A., Lee, Y.-R. & Wilson, D. I. 2016 Self-drainage of viscous liquids in vertical and inclined pipes. Food Bioprod. Process. 99, 38–50.CrossRefGoogle Scholar
Bechiri, M. & Mansouri, K. 2019 Study of heat and fluid flow during melting of pcm inside vertical cylindrical tube. Intl J. Therm. Sci. 135, 235–246.CrossRefGoogle Scholar
Becker, L. E. & McKinley, G. H. 2000 The stability of viscoelastic creeping plane shear flows with viscous heating. J. Non-Newtonian Fluid Mech. 92 (2–3), 109–133.CrossRefGoogle Scholar
Benham, G. P., Hildal, K., Please, C. P. & Van Gorder, R. A. 2016 a Penetration of molten silicon into a bed of fines. Intl Commun. Heat Mass Transfer 75, 323–327.CrossRefGoogle Scholar
Benham, G. P., Hildal, K., Please, C. P. & Van Gorder, R. A. 2016 b Solidification of silicon in a one-dimensional slab and a two-dimensional wedge. Intl J. Heat Mass Transfer 98, 530–540.CrossRefGoogle Scholar
Bergstrøm, T., Cowley, S., Fowler, A. C. & Seward, P. E. 1989 Segregation of carbon paste in a smelting electrode. IMA J. Appl. Maths 43 (1), 83–99.CrossRefGoogle Scholar
Brinkman, H. C. 1952 The viscosity of concentrated suspensions and solutions. J. Chem. Phys. 20 (4), 571–571.CrossRefGoogle Scholar
Brosa Planella, F., Please, C. P. & Van Gorder, R. A. 2019 Extended stefan problem for solidification of binary alloys in a finite planar domain. SIAM J. Appl. Maths 79 (3), 876–913.CrossRefGoogle Scholar
Ceseri, M. & Stockie, J. M. 2014 A three-phase free boundary problem with melting ice and dissolving gas. Eur. J. Appl. Maths 25 (4), 449–480.CrossRefGoogle Scholar
Chang, C. & Powell, R. L. 1994 Effect of particle size distributions on the rheology of concentrated bimodal suspensions. J. Rheol. 38 (1), 85–98.CrossRefGoogle Scholar
Chong, J. S., Christiansen, E. B. & Baer, A. D. 1971 Rheology of concentrated suspensions. J. Appl. Polym. Sci. 15 (8), 2007–2021.CrossRefGoogle Scholar
Clanet, C. & Lasheras, J. C. 1999 Transition from dripping to jetting. J. Fluid Mech. 383, 307–326.CrossRefGoogle Scholar
Costa, A. & Macedonio, G. 2005 Viscous heating effects in fluids with temperature-dependent viscosity: triggering of secondary flows. J. Fluid Mech. 540, 21–38.CrossRefGoogle Scholar
Das, N., Takata, Y., Kohno, M. & Harish, S. 2016 Melting of graphene based phase change nanocomposites in vertical latent heat thermal energy storage unit. Appl. Therm. Engng 107, 101–113.CrossRefGoogle Scholar
Dhaidan, N. S. & Khodadadi, J. M. 2015 Melting and convection of phase change materials in different shape containers: a review. Renew. Sust. Energ. Rev. 43, 449–477.CrossRefGoogle Scholar
Dowden, J., Davis, M. & Kapadia, P. 1983 Some aspects of the fluid dynamics of laser welding. J. Fluid Mech. 126, 123–146.CrossRefGoogle Scholar
Fitt, A. D. & Aitchison, J. M. 1993 Determining the effective viscosity of a carbon paste used for continuous electrode smelting. Fluid Dyn. Res. 11 (1–2), 37–59.CrossRefGoogle Scholar
Fitt, A. D. & Howell, P. D. 1998 The manufacture of continuous smelting electrodes from carbon-paste briquettes. J. Engng Maths 33 (4), 353–376.CrossRefGoogle Scholar
Fowler, A. C. 1985 Fast thermoviscous convection. Stud. Appl. Maths 72 (3), 189–219.CrossRefGoogle Scholar
Fowler, A. C. 1992 Modelling ice sheet dynamics. Geophys. Astrophys. Fluid Dyn. 63 (1–4), 29–65.CrossRefGoogle Scholar
Griffiths, R. W. 2000 The dynamics of lava flows. Annu. Rev. Fluid Mech. 32 (1), 477–518.CrossRefGoogle Scholar
Griffiths, I. M. & Howell, P. D. 2008 Mathematical modelling of non-axisymmetric capillary tube drawing. J. Fluid Mech. 605, 181–206.CrossRefGoogle Scholar
Halvorsen, S. A., Schei, A. & Downing, J. H. 1992 A unidimensional dynamic model for the (ferro)silicon process. In Electric Furnace Conference Proceedings, vol. 50, pp. 45–59. Iron and Steel Society, Warrendale, USA.Google Scholar
Hassanein, A. M., Kulcinski, G. L. & Wolfer, W. G. 1984 Surface melting and evaporation during disruptions in magnetic fusion reactors. Nucl. Engng Des. Fusion 1 (3), 307–324.CrossRefGoogle Scholar
Hejazi, S. H. & Azaiez, J. 2012 Stability of reactive interfaces in saturated porous media under gravity in the presence of transverse flows. J. Fluid Mech. 695, 439–466.CrossRefGoogle Scholar
Hejazi, S. H., Trevelyan, P. M. J., Azaiez, J. & De Wit, A. 2010 Viscous fingering of a miscible reactive a + b $\rightarrow$ c interface: a linear stability analysis. J. Fluid Mech. 652, 501–528.CrossRefGoogle Scholar
Huang, H., Wylie, J. J., Miura, R. M. & Howell, P. D. 2007 On the formation of glass microelectrodes. SIAM J. Appl. Maths 67 (3), 630–666.CrossRefGoogle Scholar
Huppert, H. E. 1989 Phase changes following the initiation of a hot turbulent flow over a cold solid surface. J. Fluid Mech. 198, 293–319.CrossRefGoogle Scholar
Jones, B. J., Sun, D., Krishnan, S. & Garimella, S. V. 2006 Experimental and numerical study of melting in a cylinder. Intl J. Heat Mass Transfer 49 (15–16), 2724–2738.CrossRefGoogle Scholar
Joseph, D. D. 1964 Variable viscosity effects on the flow and stability of flow in channels and pipes. Phys. Fluids 7 (11), 1761–1771.CrossRefGoogle Scholar
Kadkhodabeigi, M., Tveit, H. & Johansen, S. T. 2011 Modelling the tapping process in submerged arc furnaces used in high silicon alloys production. ISIJ Intl 51 (2), 193–202.CrossRefGoogle Scholar
Kerschbaum, S. & Rinke, G. 2004 Measurement of the temperature dependent viscosity of biodiesel fuels. Fuel 83 (3), 287–291.CrossRefGoogle Scholar
King, J. R. & Riley, D. S. 2000 Asymptotic solutions to the stefan problem with a constant heat source at the moving boundary. Proc. R. Soc. Lond. A 456 (1997), 1163–1174.CrossRefGoogle Scholar
Kiradjiev, K. B., Halvorsen, S. A., Van Gorder, R. A. & Howison, S. D. 2019 Maxwell-type models for the effective thermal conductivity of a porous material with radiative transfer in the voids. Intl J. Therm. Sci. 145, 106009.CrossRefGoogle Scholar
Ksiazek, M., Tangstad, M. & Ringdalen, E. 2016 Five furnaces five different stories. In Silicon for the Chemical and Solar Industry XIII, vol. 2016, pp. 33–42.Google Scholar
Lister, J. R. & Dellar, P. J. 1996 Solidification of pressure-driven flow in a finite rigid channel with application to volcanic eruptions. J. Fluid Mech. 323, 267–283.CrossRefGoogle Scholar
Liu, Z., Yao, Y. & Wu, H. 2013 Numerical modeling for solid–liquid phase change phenomena in porous media: shell-and-tube type latent heat thermal energy storage. Appl. Energy 112, 1222–1232.CrossRefGoogle Scholar
Mader, H. M., Llewellin, E. W. & Mueller, S. P. 2013 The rheology of two-phase magmas: a review and analysis. J. Volcanol. Geotherm. Res. 257, 135–158.CrossRefGoogle Scholar
Manickam, O. & Homsy, G. M. 1994 Simulation of viscous fingering in miscible displacements with nonmonotonic viscosity profiles. Phys. Fluids 6 (1), 95–107.CrossRefGoogle Scholar
Mavroyiakoumou, C., Griffiths, I. M. & Howell, P. D. 2019 Mathematical modelling of a viscida network. J. Fluid Mech. 872, 147–176.CrossRefGoogle Scholar
Mueller, S., Llewellin, E. W. & Mader, H. M. 2009 The rheology of suspensions of solid particles. Proc. R. Soc. Lond. A 466 (2116), 1201–1228.Google Scholar
Myers, T. G., Charpin, J. P. F. & Tshehla, M. S. 2006 The flow of a variable viscosity fluid between parallel plates with shear heating. Appl. Math. Model. 30 (9), 799–815.CrossRefGoogle Scholar
Myrhaug, E. H., Tuset, J. & Tveit, H. 2004 Reaction mechanisms of charcoal and coke in the silicon process. In Proceedings: Tenth International Ferroalloys Congress, vol. 1, pp. 108–121.Google Scholar
Ockendon, H. 1979 Channel flow with temperature-dependent viscosity and internal viscous dissipation. J. Fluid Mech. 93 (4), 737–746.CrossRefGoogle Scholar
Ockendon, J., Howison, S., Lacey, A. & Movchan, A. 2003 Applied Partial Differential Equations. Oxford University Press.Google Scholar
Ockendon, H. & Ockendon, J. R. 1977 Variable-viscosity flows in heated and cooled channels. J. Fluid Mech. 83 (1), 177–190.CrossRefGoogle Scholar
Ockendon, J. R., Tayler, A. B., Emerman, S. H. & Turcotte, D. L. 1985 Geodynamic thermal runaway with melting. J. Fluid Mech. 152, 301–314.CrossRefGoogle Scholar
Pabst, W. & Gregorová, E. V. A. 2013 Elastic properties of silica polymorphs–a review. Ceram.-Silikaty 57 (3), 167–184.Google Scholar
Pearson, J. R. A. 1977 Variable-viscosity flows in channels with high heat generation. J. Fluid Mech. 83 (1), 191–206.CrossRefGoogle Scholar
Richet, P., Bottinga, Y., Denielou, L., Petitet, J. P. & Tequi, C. 1982 Thermodynamic properties of quartz, cristobalite and amorphous sio2: drop calorimetry measurements between 1000 and 1800 K and a review from 0 to 2000 K. Geochim. Cosmochim. Acta 46 (12), 2639–2658.CrossRefGoogle Scholar
Schei, A., Tuset, J. K. & Tveit, H. 1998 Production of High Silicon Alloys. Tapir.Google Scholar
Shmueli, H., Ziskind, G. & Letan, R. 2010 Melting in a vertical cylindrical tube: numerical investigation and comparison with experiments. Intl J. Heat Mass Transfer 53 (19–20), 4082–4091.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., Valderhaug, A. M., Birkeland, R. G. & Wegge, H. 2017 A heat and mass transfer model of a silicon pilot furnace. Metall. Mater. Trans. B 48 (5), 2664–2676.CrossRefGoogle Scholar
Solomatov, V. S. 1995 Scaling of temperature-and stress-dependent viscosity convection. Phys. Fluids 7 (2), 266–274.CrossRefGoogle Scholar
Sparrow, E. M. & Broadbent, J. A. 1982 Inward melting in a vertical tube which allows free expansion of the phase-change medium. Trans. ASME J. Heat Transfer 104, 309–315.CrossRefGoogle Scholar
Sparrow, E. M., Gurtcheff, G. A. & Myrum, T. A. 1986 Correlation of melting results for both pure substances and impure substances. Trans. ASME J. Heat Transfer 108, 649–683.CrossRefGoogle Scholar
Stickel, J. J. & Powell, R. L. 2005 Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech. 37, 129–149.CrossRefGoogle Scholar
Stokes, Y. M. 1998 Very viscous flows driven by gravity with particular application to slumping of molten glass. PhD thesis, University of Adelaide.CrossRefGoogle Scholar
Stokes, Y. M. & Tuck, E. O. 2004 The role of inertia in extensional fall of a viscous drop. J. Fluid Mech. 498, 205–225.CrossRefGoogle Scholar
Stokes, Y. M., Tuck, E. O. & Schwartz, L. W. 2000 Extensional fall of a very viscous fluid drop. Q. J. Mech. Appl. Maths 53 (4), 565–582.CrossRefGoogle Scholar
Tan, C. T. & Homsy, G. M. 1986 Stability of miscible displacements in porous media: rectilinear flow. Phys. Fluids 29 (11), 3549–3556.CrossRefGoogle Scholar
Tronnolone, H., Stokes, Y. M., Foo, H. T. C. & Ebendorff-Heidepriem, H. 2016 Gravitational extension of a fluid cylinder with internal structure. J. Fluid Mech. 790, 308–338.CrossRefGoogle Scholar
Tuck, E. O., Stokes, Y. M. & Schwartz, L. W. 1997 Slow slumping of a very viscous liquid bridge. J. Engng Maths 32 (1), 27–40.CrossRefGoogle Scholar
Valderhaug, A. M. 1992 Modelling and control of submerged-arc ferrosilicon furnaces. PhD thesis, The Norwegian Institute of Technology, Trondheim.Google Scholar
Valderhaug, A. M. & Sletfjerding, P. 1991 A non-interacting electrode current controller for submerged-arc furnaces. In Electric Furnace Conference Proceedings, vol. 49, pp. 311–320.Google Scholar
White, D. E., Moggridge, G. D. & Wilson, D. I. 2008 Solid–liquid transitions in the rheology of a structured yeast extract paste, marmite. J. Food Engng 88 (3), 353–363.CrossRefGoogle Scholar
Wilson, S. D. R. 1988 The slow dripping of a viscous fluid. J. Fluid Mech. 190, 561–570.CrossRefGoogle Scholar
Woods, A. W. 1999 Liquid and vapor flow in superheated rock. Annu. Rev. Fluid Mech. 31 (1), 171–199.CrossRefGoogle Scholar
Woods, A. W. & Fitzgerald, S. D. 1993 The vaporization of a liquid front moving through a hot porous rock. J. Fluid Mech. 251, 563–579.CrossRefGoogle Scholar
Wu, Y. K. & Lacroix, M. 1995 Melting of a PCM inside a vertical cylindrical capsule. Intl J. Numer. Meth. Fluids 20 (6), 559–572.CrossRefGoogle Scholar
Wylie, J. J. & Huang, H. 2007 Extensional flows with viscous heating. J. Fluid Mech. 571, 359–370.CrossRefGoogle Scholar
Wylie, J. J. & Lister, J. R. 1995 The effects of temperature-dependent viscosity on flow in a cooled channel with application to basaltic fissure eruptions. J. Fluid Mech. 305, 239–261.CrossRefGoogle Scholar
Zhang, D. F. & Stone, H. A. 1997 Drop formation in viscous flows at a vertical capillary tube. Phys. Fluids 9 (8), 2234–2242.CrossRefGoogle Scholar
Zippelius, A., Halperin, B. I. & Nelson, D. R. 1980 Dynamics of two-dimensional melting. Phys. Rev. B 22 (5), 2514.CrossRefGoogle Scholar