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Close-contact melting on an isothermal surface with the inclusion of non-Newtonian effects

  • Y. Kozak (a1), Yi Zeng (a2), Rabih M. Al Ghossein (a2), J. M. Khodadadi (a2) and G. Ziskind (a1)...


The present study deals with a theoretical investigation of a close-contact melting (CCM) process involving a vertical cylinder on a horizontal isothermal surface, where the liquid phase is a non-Newtonian viscoplastic fluid that behaves according to the Bingham model. Accordingly, a new approach is formulated based on the thin layer approximation and different quasi-steady process assumptions. By analytical derivation, an algebraic equation that relates the molten layer thickness and the solid bulk height is developed. The problem is then solved numerically, coupled with another equation for the melting rate. The new model shows that as the yield stress increases the melting rate decreases and the molten layer thickness increases. It is found that under certain conditions, the model can be reduced to a form that allows an analytical solution. The approximate model predicts an exponential dependence of both the melt fraction and the molten layer thickness. Comparison between the numerical and analytical solutions shows that the analytical approximation provides an excellent estimation for sufficiently large values of the yield stress. Dimensional analysis, which is supported by the analytical model results, reveals the dimensionless groups that govern the problem. For the general case, the melt fraction is a function of two dimensionless groups. For the analytical approximation, it is shown that the melt fraction is governed by a single dimensionless group and that the molten layer thickness is governed by two dimensionless groups.


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Afoakwa, E. O., Paterson, A., Fowler, M. & Vieira, J. 2008 Relationship between rheological, textural and melting properties of dark chocolate as influenced by particle size distribution and composition. Eur. Food Res. Technol. 227 (4), 12151223.10.1007/s00217-008-0839-5
Bahrami, P. A. & Wang, T. G. 1987 Analysis of gravity and conduction-driven melting in a sphere. Trans. ASME J. Heat Transfer 109, 806809.10.1115/1.3248166
Bareiss, M. & Beer, H. 1984 An analytical solution of the heat transfer process during melting of an unfixed solid phase change material inside a horizontal tube. Intl J. Heat Mass Transfer 27, 739746.10.1016/0017-9310(84)90143-1
Bejan, A. 1992 Single correlation for theoretical contact melting results in various geometries. Intl Commun. Heat Mass Transfer 19 (4), 473483.10.1016/0735-1933(92)90003-Z
Betzel, T. & Beer, H. 1988 Solidification and melting heat transfer to an unfixed phase change material (PCM) encapsulated in a horizontal concentric annulus. Wärme- Stoffiibertrag. 22, 335344.10.1007/BF01387889
Bingham, E. C. 1916 An investigation of the laws of plastic flows. Bull. U.S. Bur. Stand. 13, 309353.10.6028/bulletin.304
Bingham, E. C. 1922 Fluidity and Plasticity. McGraw-Hill.
Bird, R. B., Dai, G. C. & Yarusso, B. J. 1983 The rheology and flow of viscoplastic materials. Rev. Chem. Engng 1 (1), 170.10.1515/revce-1983-0102
Cantor, B. & O’Reilly, K. 2016 Solidification and Casting. CRC Press.
Chen, W., Cheng, S., Luo, Z. & Gu, W. 1993 Study of contact melting inside isothermally heated vertical cylindrical capsules. J. Therm. Sci. 2, 190195.10.1007/BF02650856
Chen, H., Ding, Y. & Tan, C. 2007 Rheological behaviour of nanofluids. New J. Phys. 9, 367.10.1088/1367-2630/9/10/367
Covey, G. H. & Stanmore, B. R. 1981 Use of the parallel-plate plastometer for the characterisation of viscous fluids with a yield stress. J. Non-Newtonian Fluid Mech. 8 (3–4), 249260.10.1016/0377-0257(81)80024-9
Dai, G. & Bird, R. B. 1981 Radial flow of a Bingham fluid between two fixed circular disks. J. Non-Newtonian Fluid Mech. 8 (3–4), 349355.10.1016/0377-0257(81)80031-6
Dhaidan, N. S. & Khodadadi, J. M. 2015 Melting and convection of phase change materials in different shape containers: a review. Renew. Sustain. Energy Rev. 43, 449477.10.1016/j.rser.2014.11.017
Dhaidan, N. S. & Khodadadi, J. M. 2017 Improved performance of latent heat energy storage systems utilizing high thermal conductivity fins: a review. J. Renew. Sustain. Energy 9 (3), 034103.10.1063/1.4989738
Fomin, S. A., Saitoh, T. S. & Chugunov, V. A. 1997 Contact melting materials with non-linear properties. Heat Mass Transfer 33 (3), 185192.10.1007/s002310050177
Gadgil, A. & Gobin, D. 1984 Analysis of two-dimensional melting in rectangular enclosures in presence of convection. Trans. ASME J. Heat Transfer 106 (1), 2026.10.1115/1.3246636
Gartling, D. K. & Phan-Thien, N. 1984 A numerical simulation of a plastic fluid in a parallel-plate plastometer. J. Non-Newtonian Fluid Mech. 14, 347360.10.1016/0377-0257(84)80053-1
Groulx, D. & Lacroix, M. 2003 Effects of convection and inertia on close contact melting. Intl J. Therm. Sci. 42 (12), 10731080.10.1016/S1290-0729(03)00096-6
Hale, N. W. Jr. & Viskanta, R. 1980 Solid–liquid phase-change heat transfer and interface motion in materials cooled or heated from above or below. Intl J. Heat Mass Transfer 23 (3), 283292.10.1016/0017-9310(80)90116-7
Hirata, T., Makino, Y. & Kaneko, Y. 1991 Analysis of close-contact melting for octadecane and ice inside isothermally heated horizontal rectangular capsule. Intl J. Heat Mass Transfer 34, 30973106.10.1016/0017-9310(91)90079-T
Huang, D. C., Liu, B. C. & Jiang, T. Q. 1987 An analytical solution of radial flow of a Bingham fluid between two fixed circular disks. J. Non-Newtonian Fluid Mech. 26 (1), 143148.10.1016/0377-0257(87)85052-8
Khodadadi, J. M., Fan, L. & Babaei, H. 2013 Thermal conductivity enhancement of nanostructure-based colloidal suspensions utilized as phase change materials for thermal energy storage: a review. Renew. Sustain. Energy Rev. 24, 418444.10.1016/j.rser.2013.03.031
Khodadadi, J. M. & Hosseinizadeh, S. F. 2007 Nanoparticle-enhanced phase change materials (NEPCM) with great potential for improved thermal energy storage. Intl Commun. Heat Mass Transfer 34 (5), 534543.10.1016/j.icheatmasstransfer.2007.02.005
Khodadadi, J. M. & Zhang, Y. 2001 Effects of buoyancy-driven convection on melting within spherical containers. Intl J. Heat Mass Transfer 44 (8), 16051618.10.1016/S0017-9310(00)00192-7
Kole, M. & Dey, T. K. 2011 Effect of aggregation on the viscosity of copper oxide–gear oil nanofluids. Intl J. Therm. Sci. 50 (9), 17411747.10.1016/j.ijthermalsci.2011.03.027
Kozak, Y., Rozenfeld, T. & Ziskind, G. 2014 Close-contact melting in vertical annular enclosures with a non-isothermal base: theoretical modeling and application to thermal storage. Intl J. Heat Mass Transfer 72, 114127.10.1016/j.ijheatmasstransfer.2013.12.058
Kozak, Y. & Ziskind, G. 2017 Novel enthalpy method for modeling of PCM melting accompanied by sinking of the solid phase. Intl J. Heat Mass Transfer 112, 568586.10.1016/j.ijheatmasstransfer.2017.04.088
Krieger, I. M. & Dougherty, T. J. 1959 A mechanism for non-Newtonian flow in suspensions of rigid spheres. Trans. Soc. Rheol. 3 (1), 137152.10.1122/1.548848
Lipscomb, G. G. & Denn, M. M. 1984 Flow of Bingham fluids in complex geometries. J. Non-Newtonian Fluid Mech. 14, 337346.10.1016/0377-0257(84)80052-X
Michaelides, E. E. 2014 Nanofluidics. Springer.10.1007/978-3-319-05621-0
Moallemi, M. K., Webb, B. W. & Viskanta, R. 1986 An experimental and analytical study of close-contact melting. Trans. ASME J. Heat Transfer 108, 894899.10.1115/1.3247030
Motahar, S., Nikkam, N., Alemrajabi, A. A., Khodabandeh, R., Toprak, M. S. & Muhammed, M. 2014 Experimental investigation on thermal and rheological properties of n-octadecane with dispersed TiO2 nanoparticles. Intl Commun. Heat Mass Transfer 59, 6874.10.1016/j.icheatmasstransfer.2014.10.016
Myers, T. G. & Mitchell, S. L. 2008 Unsteady contact melting of a rectangular cross-section material on a flat plate. Phys. Fluids 20 (10), 103101.10.1063/1.2990751
Ockendon, H. & Ockendon, J. R. 1995 Viscous Flow. Cambridge University Press.10.1017/CBO9781139174206
Roy, S. K. & Sengupta, S. 1987 The melting process within spherical enclosures. Trans. ASME J. Heat Transfer 109, 460462.10.1115/1.3248104
Rozenfeld, A., Kozak, Y., Rozenfeld, R. & Ziskind, G. 2017 Experimental demonstration, modeling and analysis of a novel latent-heat thermal energy storage unit with a helical fin. Intl J. Heat Mass Transfer 110, 692709.10.1016/j.ijheatmasstransfer.2017.03.020
Rozenfeld, T., Kozak, Y., Hayat, R. & Ziskind, G. 2015 Close-contact melting in a horizontal cylindrical enclosure with longitudinal plate fins: demonstration, modeling and application to thermal storage. Intl J. Heat Mass Transfer 86, 465477.10.1016/j.ijheatmasstransfer.2015.02.064
Sahoo, B. C., Vajjha, R. S., Ganguli, R., Chukwu, G. A. & Das, D. K. 2009 Determination of rheological behavior of aluminum oxide nanofluid and development of new viscosity correlations. Petrol. Sci. Technol. 27 (15), 17571770.10.1080/10916460802640241
Saito, A., Utaka, Y., Akiyoshi, M. & Katayama, K. 1985a On the contact heat transfer with melting: 1st report: experimental study. Bull. JSME 28 (240), 11421149.10.1299/jsme1958.28.1142
Saito, A., Utaka, Y., Akiyoshi, M. & Katayama, K. 1985b On the contact heat transfer with melting: 2nd report: analytical study. Bull. JSME 28 (242), 17031709.10.1299/jsme1958.28.1703
Saito, A., Utaka, Y., Shinoda, K. & Katayama, K. 1986 Basic research on the latent heat thermal energy storage utilizing the contact melting phenomena. Bull. JSME 29 (255), 29462952.10.1299/jsme1958.29.2946
Sparrow, E. M. & Geiger, G. T. 1986 Melting in a horizontal tube with the solid either constrained or free to fall under gravity. Intl J. Heat Mass Transfer 29, 10071019.10.1016/0017-9310(86)90200-0
Wilson, S. D. R. 1993 Squeezing flow of a Bingham material. J. Non-Newtonian Fluid Mech. 47, 211219.10.1016/0377-0257(93)80051-C
Yoo, H., Hong, H. & Kim, C. J. 1998 Effects of transverse convection and solid–liquid density difference on the steady close-contact melting. Intl J. Heat Fluid Flow 19 (4), 368373.10.1016/S0142-727X(98)10011-5
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