Heat VAPO~ATION~HASECHANGE Section 507.8 Transfer FREEZING Page I Division CONTENTS, SYMBOLS August ! 996*
Page I. GENERAL " .-. I A. Contents...... 1 B. Symbols...... I C. References...... 2 D. Introduction, Applications, and Basic Physical Concepts to E. Predictive Methods...... II
II. PREDICTIVE EQUATIONS FOR FREEZING WITHOUT FLOW...... 12 A. One-Dimensional Freezing of Pure Materials without Density Change "...... 12 I. Solutions for Materials that are Initially at the Freezing Temperature... .. 12 2. Solutions for Materials that are Initially not at the Freezing Temperature. 13 B. One-Dimensional Freezing with Density Change...... 14 C. The Quasi-Static Approximation...... 15 I. Examples of the Quasi-Static Approximation for a Slab...... 15 2. Examples of the Quasi-Static Approximation for a Cylinder... J5 D. Estimation of Freezing Time...... 16 I. Freezing Time of Foodstuff...... 16 2. Other Approximations for Freezing Time...... 17
III. PREDICTrvE EQUATIONS FOR FREEZING WITH FLOW...... 18
TV. SOME METHODS FOR SIMPLIFYING SOLUTION...... 20 A. The Integral Method, with Sample Solution for Freezing of a Slab . ...... 20 B. The Enthalpy Method ,...... 22
V. APPLICATIONS BIBLIOGRAPHy...... 23 A. Casting, Molding, Sintering...... 23 B. Multi-component Systems, Freeze-separation, and Crystal-growth...... 24 C. Welding, Soldering, and Laser, EJectron-beam, and Electro-discharge Processing 31 D. Coating and Vapor and Spray Deposition...... 32 E. Frost Formation 33 F. Medical Applications and Food Preservation...... 33 G. Nuclear Power Safety...... 35 H. Thermal Energy Storage...... 35 1. Geophysical Phenomena and Freezing in Porous Media...... 36
B.SYMBOLS A area, m2 :y shape coefficient in Plank's equalion c specific heat, J!kg K q he,ll nux, J/m2s d diameter r radius, m hiS latent heat of fusion (melting or freezing), Jlkg K R radial position of the freeze-front, m II enthalpy, h R shape coefficient in Plank's equation h convective heat transfer coefficient, WIm 2s I time. s k thermal conductivity, W/m K T temperature, K. or OC L characteristic length II velocity of lhe freeze· from due to density change, m the time-variation conStant of the ambient mls 3 temperature Ta' KJs v specific volume, m /kg V volume of the body, m3 Biot number, == h Uk x coordinate Fourier number, o:tfL2 X position of the freezing front along the x-direclion, Stefan number, ::: c(To - Tf)/h ts m dimensionless time parameter y coordinale pressure, Pa z coordinate
GENIUM PUBLISHING Section 507.8 vAPORlZATIONIPHASE CHANGE Heat Page 2 FREEZING Transfer August 1996* SYMBOLS, REFERENCES Division
Greek Symbols Subscripts CI. lhennal diffusivity, m2/s o at the surface (x:::O) ~ lhennal "boundary layer" growth constant a ambient (fluid surrounding the freezing object) S lhennal "boundary layer" c coolant t:.vl.s (specific volume of the solid phase) - (specific f fusion (melting or freezing) volume of the liquid phase), m3/kg inner; or initial (at t=O) f. liquid fs phase-change from liquid to solid K ==t; o outer A. freezing rate parameter, dimensionless s solid I-l == pip, w at wall S dimensionless position of the freezing front, =XIL p density, kg/m3 e dimensionless temperature, Eq. (8-51) X. concentration, kglkg; or dimensionless distance x/L 00 shape factor [Eq. (8-38)]
C. REFERENCES Air-Conditioning Engineers). 1993. Cooling and freezing times of foods. Ch, 29 in Fundamentals. ASHRAE, Atlanta, GA. There has always been an intense interest in predicting 6. Bankoff, S. G. 1964. Heat conduction or diffusion with freezing phenomena as related to such applications as food change of phase. Adv. Chern. Eng.. vol. 5. p. 75. preservation, climate and Its control, navigation, and 7. Beckermann, C. and Wang, C. Y. 1994. lncorporaling materials processing. This interest expanded with lime into interfacial phenomena in solidification models. JaM. vol. new areas such as power generation and medicine. Increas 46. pp. 42-47. ingly rigorous quantitative predictions staned in the 19th 8. BelL G. E. 1979. Solidific.ation of a liquid about a century, and the number of published papers is now in .the cylindrical pipe. Int. 1. Heat Mass Transfer, vol. 22, pp. tens of thousands. The main books and reviews on the topic, 1681-1685. representative key general papers, as well as some of the 9. Boeninger, W.J. and Perepezko, J. H. 1985. Fu ndamentals references on appropriate thennophysical and transport of rapid solidificlltion. In Rapidly Solidified Cryslalline properties, are listed in references [I ]-[2731 below. A Alloys, S.K. Das, S.H. Kear. and C. M. Adam. cds., pp. 21-58. Metall. Soc. AIME, Warrendale, PA, further ralher extensive, yet not complete, list of references is given in Subsection V, "Applications Bibliography" 10. Bonacina. c., Comini. G., Fasano, A., and Prirnicerio, M. \974. On lhe eSlimalion of I hermophysical properties in below under the specific topics in which freezing plays an nonlinear heat conduction problems. Int. J. Heal Mass important role. In addition to the identification of past work Transfer, vol. 17, pp. 861-867. on specific topics, lhis extensive list of references also helps II. Budhia. H. and Kreilh, F. 1973. Hc.altransfer with melting identify various applications in which freezing plays a role, of freezing in a wedge. In!. J. Heat Mass TransJer. vol. 16. and the journals that typically cover the field. While pp. 195-211. encompassing sources from many countries, practically all \2. Chadam, J. and Rasmussen, H.. cds. 1992. Free Boundary of the references listed here were selected from the archival Problems: Theory and Applications. Longman, New York. refereed literature published in English. Many pertinent 13. Chalmers. B. 1964. Principles of Solidification. John publications on this topic also exist in other languages. Wiley, New York. 14. Chan, S. H., Cho, D. H. and KocamuSlafaoQullari. G. 1983. I. Alexiades. V., Solomon. A. D. and Wilson, D.G. 1988. The Melling and solidification wilh imemal radiative transfer formation of a wlid nucleus in a supercooled liquid, Pt. I. A Generalized Phase Change Model. Int. J. Heat Mass 1. Non-Equilibrium Thermodynamics. vol. 13. pp. 281 Transfer, vol. 26, pp. 62l-633. 300. 15. Charach. Ch. and Kahn, P.B. 1987. Solidificalion in finite 2. Alexiades. V. and Solomon. A. D. 1989. The formation of bodies with prescribed heat flux; bounds for the freeZing a solid nucleus in a supercooled liquid, PI. n. J. Non time and removed energy. Inl. J. Heal Mass Transfer, vol. Equilibrium Thermodynamics, vol. 14, pp. 99-109. 30. pp. 233-240. 3. Alexiades, Y. and Solomon, A. D. 1993. Mathematical 16. Cheng, K. C. and Seki, N. eds. I991. Freezing and Melli ng Modeling of Melting and Freezing Processes. Hemisphere, Heat Transfer in Engineering. Hemisphere, Washington, Washington, D.C. D.C. 4. ASHRAE (American Soc. of Heating, Refrigerating, and 17. Cheung. F.B. and Epstein, M. 1984. Solidification and Air-Conditioning Engineers). 1990. Refrigeration. Melting in Fluid Flow. In Adv. in Transport Processes ASHRAE, Atlanta, GA. Vol. J,el!. A.S. Majumdar. p.35-117. Wiley Eastern, New Delhi. 5. ASHRAE (American Soc. of Heating. Refrigerating. and
GENIUM PUBLISHING Heat VAPO~ATION~HASECHANGE Section 507.8 Transfer FREEZING Page 3 Division REFERENCES August 1996*
18. Cho. S. H. and Sunderland,). E. 1974. Phase change Mass Transfer in Materials Processing, ed.. 1. Tanasawa and problems with temperature dependent thermal properties. J. N. Liar. pp. 247-264. Hemisphere. New York. Heat Transfer, vol. 96, p. 214. 38. !ida, T. and Guthrie, R.I.L ] 988. The Physical Propenies 19. Chung, B. T. F. and Yeh. L. T. 1975. Solidification and of Liquid Metals. Clarendon Press, Oxford. melting of materials subject to convection and radialion. 1. 39. Jones, H. Prediction versus experimental fact in the Spacecraft, vol. 12, pp. 329-334. formation of rapidly soHdified microsrrucrure. ISU Int .. 20. Colucci·Mizenko, Lynn M.; Glicksman. Manin E.: Smith. vol. 35.1995, pp. 751-756 Richard N, 1994. Thermal recalescence and mushy wne 40. Knight, C. 1976. The Freezing of Supercooled Liquids_ coarsening io undercooled melts. 10M vol. 46. pp. 5 I-55. Van Nostrand, Princeton. 21. COnli. M. 1995. Planar sol idification of a fi nite slab: 41. Kurz.. W. and Fisher. D. 1984. Fundamentals of Solidifica· effects of pressure dependence on the freeling point. Int. 1. lion. Trans. Tech. Publications, Z.rich. Heat Mass Transfer. vol. 38. pp. 65-70. 42. Lamvik. M .. Zhau. 1.M. ExperimenLal study of thermal 22. Crank. 1. 1984. Free and Moving Boundary Problems. conductivily of solid and liquid phases at the phase Oxford University Press (Clarendon), London and New transitIOn. Tnt. 1. Thennophyslcs, vol. 16. 1995. pp. York. 567-576 23. Dean. N. F.; Mortensen, A.: Rcmings, M. C. 1994. 43. Ledemmn.J.M. and Boley, B.A. 1970. Axisymmelric Microsegregation in cellular solidification. Source: melting or sol idi ficalion of ci rcu Iar cyl inders. [nt. 1. Heat Metallurgical and Materials Trans. A. vol. 25A pp. 2295 Mass Transfer, vol. 13, pp. 413-427. 2301 44. Lee, L Moh, 1.H.. Hwang, K.Y. Effect of lhermal COnUK( 24. Dulikravich, G. S.; Ahuj GENIUM PUBLISHING Section 507.8 VAPOillZATION~HASECHANGE Heat Page 4 FREEZING Transfer August 1996* REFERENCES Division 58. Saraf, G.R. and Sharif, L.K.A. 1987, Inward freezing of 77. Zief, M. and Wilcox, W. R. 1967. Fractional Solidification, water in cylinders. Int. 1. Refrigeration, vol. 10, pp. 342 Marcel Dekker, New York. 348. 78. Zubair, S.M. and Chaudhry, MA 1994. Exact solutions of 59. Schaefer, R.l and Glicksman, M.E. 1969. Fully tlme solid-liquid phase-change heat transfer when subjected to dependent theory for the growth of spherical crystal nuclei. convective boundary conditions. Waerme- und 1. Crystal Growth, vol. 5, pp. 44-58. Swffuebenragung. vol. 30, pp. 77-81. 60. Schulze, T. P. and Davis, S. H. 1994, Influence of osci lla 79. Aboutalebi, M. R., Hasan, M., Guthrie, R.I.L. Coupled lOry and steady shears on interfacial stability during turbulent flow, heat, and solute transport in continuous directional solidification. 1. Crystal Growth, vol. 143, pp. casting processes. Metallurgical and Materials Trans. B, 317-333. vol. 26, 1995, pp. 731·744 61. Sekerka, R.F. 1984. Morphological and hydrodynamic 80. Aboutalebi, M. R., Hasan, M., Guthrie. R.LL 1995. instabilities during phase transformation. Physica D. vol. Numerical study of coupled turbulent flow and solidifica 12D, pp. 212-214. tion for steel slab casters. Num. Heat Transfer A, voL 28, 62. Stefan, J. 1891. Ueber die theorie der eisbi ldung. pp.279-297 insbesondere ueber die eisbildung im polarmcere. Ann. 81. Brewster. R. A. and Gebhart. S. 1994. The effects of Phys. Chem., N.s. vol. 42. pp. 269-286. supercooling and freezing on natural convection in 63. Sulfredge, C.D.. Tagavi. K.A. and Chow, L.c. 1994. Void seawater. lnL 1. Heat Mass Transfer. vol. 37. pp. 543-552. proliles in unidirectional solidification: The role of 82. Chen. c.F. Experimental study of convection in a mushy capillary forces and gravity, J, Thermophysics Heat layer during directional sol idi fication. J. Fluid Meeh.. vol. Transfer, vol. 8 pp. 608-61 5. 293. 1995, pp. 81-98 64. Tanasawa, '- and Liar, N. cd. 1992. Heat and Mass 83. Chen. Y., 1m. Y.-T., Lee. 1.. 1995. Finitt: element simula Transfer in Materials Processing. Hemisphere. New York. tion of solidification with momentum. heat and specIes N.Y. transport. J. Materials Processing Tcchno!.. vol. 48, pp. 65. Trivedi, R. and Kurz, W. 1994, Dendritic growth., Int. 571-579 Materials Rev., voL 39, pp. 49-74, 84, Choudhary, S.K., Mazumdar, D. 1995. Mathematical 66. Vanfleet. R.R. and Mochel. 1.M. 1995.111ermodynamics mode II ing of fluid flow. heat transfer and ,,) Iidi lication of melting and frcezing in small panicles. Surf. Sci., voL phenomena in cont inuous easti ng of steel. Steel Res.. vol, 341. pp. 40-50, 66. pp. 199-205 67. Viswanath, R., Jalurin, Y. Numerical slUdy of conjugate 85. Cheng, K. C. and Wong, S. L. 1977. Liquid solidification transient solidification in an enclosed region. Num. Heat ina conveetively-cooled parallei·pi ate channel. Can. J. Transfer A. vol. 27, 1995. PI', 519·536 Chem. Eng.. vol. 55, pp. 149-155, 68. Wang, Ch.-H., Fuh, K.H .. and Chien. L.-Ch. 1994. liqUid 86. Chiarcli. A,O.P.; Huppert, H. E.: Worster. M. G. 1994. solidification under microgravily condition. Microgravity Segregation and flow during the solidification of alloys, J. Sci. Technol.. vol. 7. pp. 142-147 Crystal Growth. vol. 139. pp. I34-146 69. Weaver, 1. A. and Viskanta. R. 1986. Freezing of liquid. 87. Chiesa, F. M. and Guthric. R. l. L. 1974. Natural convec saturated porous media, 1. Heat Transfer, vol. 108, pp. 654 tion heat transfer rates during the solidification and melting 659. of mdals and alloy systems. J. Heat Transier, vol, 96. pp. 377-384, 70. Wolfgardl. M., Basehnagel. 1.. Binder, K. On the equation of state for thermal polymer solutions and mehs. 1. Chem. 88. Cho. C. and Ozisik, M. N. 1979. Transient frcezi ng of Phys., vol. 103, pp. 7166-7179 liquids in turbulent flow inside tubes. J. Heat Transfer, \'01 101 . pp. 465-468. 71. Woodruff. D.P. 1973. The Solid-Liquid Interface. Cambridge Univ. Prcss.. Oxford. 89. Dietsche, C. and Muller, U. 1985. Influence of Bernard convection on solid-liquid interfaces. 1. FlUld Mech" \'01. 72. Yao, L. S. and Prusa, 1. 1989. Melting and Freezi ng, If} 161. p. 249. Advances in Heat Transfer Vol. 19. ed, J. P, Hartnen and T. F. Irvine, p. 1-95. Academic Press, San Diego. CA. 90. Epstein, M, 1976. The growth and decay of GENIUM PUBLISHING Heat VAPORIZAllONIPHASE CHANGE Seclion 507.8 Transfer FREEZING Page 5 Division REFERENCES August 1996* 95. Fukusako, S., Yamada, M" Horibe, A., Kawai, H, In. Kroeger, P, G. and Ostrach, S. 1974. The solution of a twO Solidification of aqueous binary solution on a venical dimensional freezing problem including convection effecLS cooled plate with main flow. Waerme- und in the liquid region. In!. J. Heat Mass Transrer. vol. 17, pp. Stoffuebertragung, Yol. 30, 1995, pp. 127-134 1191-1207. 96. Fukusako, 5., Sugawara, S. and Seki, N. 1977. Onset of 114. Kuiken, H. K. 1977. Sol idi fication of a Iiquid on a movi ng free convection and heat transfer in a melt water layer. 1 sheet. Tnt. J. Heat Mass Transfer, vol. 20, pp. 309-314. JSME, YOI. SO, pp. 445-450. 115. Kuzay, T. M. and Epstein. M. 1979. Penetration of a 97. Gau, C. and Viskanta, R. Effect of natura! convection on freezing liquid into an annular channeL AIChE Symp. Ser. solidification from above and melting from below. Int. 1. vol. 75. pp. 95-102. Heal Mass Transfer, vol. 28, pp. 573, 116. Lee, S.L and Hwang. GJ. 1989. Liquid solidification in 98. Gilpin, R.R. 1976. The influence of narural convection on low Peclel number pipe flows. Canad. J. Chem Engng. vol. dendritic ice growth. J. Crystal Growth, vol. 36, pp. 101 67. pp. 569-578. lOS. 117. Lein, H. and Tankin, R.S. 1992. Natural convection in 99. Gi Ipi n. R. R. 1977, The effects of cooling rate on the porous media. II. Freezing. lnt. J, Heat Mass Transfer. vol. formation of dendritic ice in a pipe with no main flow. J. 36, pp. 187-194, Heat Transfer, vol. 99, pp. 419-424. 118. Li, B.O.. Anyalebechi. P.N. Micro/macro model for fluid 100. G iIpi n, R. R. 1977. The effects of dendritic Ice formation flow eVOlution and microstructure [ormation in solidifica· in water pipes. Inc J. Heat Mass Transfer, vol. 20, pp. 693 lion processes. Int. J. Heat Mass Transfer, vol. 38. 1995. 699. pp. 2367-2381 101. Gilpin, R. R. 1981. Ice fonnation in a pipe containing 119. Madejski. J. 1976. Solidification in flow through channels flows in the transition and turbulent regime.,. J. Heal and into cavities. lnt. J. Heal Mass Transfer. vol. 19. pp. Transfer vol. 103, pp. 363-368. 1351-1356 102. Gilpin, R. R. 1981. Modes of ice formation and flow 120 McDaniel. D. J. and Zabaras. Nicholas. 1994. Leasl blockage thai occur while filling a cold pipe. Cold Regions squares front-tracking finite element melhod analysis of Science and Technology. vol. 5. pp. 163-17 I. phase-change wilh natural conveclion. 1m. J. Num. Meth. 103. Guemguall. R. and Poots. G. 1985. Effects of natural Engng, vol. 37. pp, 2755-2777 convection on the inward solidification of spheres and 121. Nishi mura. T.. ImalO. T. and MiY3shita. H.. 1994. cylinders. lnt. J. Heal Mass Transfer. vol. 28, pp. 1229. Occurrence and development of double-diffusive convec 104. Hirala, T., Gilpin, R. R, Cheng. K.C. and Gales. E.M. tion during solidificalion of a binary system. In\. J. Heat 1979. The steady Mate ice layer pronIe on a constant Mass Transfer, vol. 37, pp. 1455-1464. temperarure plate in a forced convection now: l. Lamioflr 122.. Ou. J.W. and Cheng, K.C. 1975. Buoyancy effects on heat regime. Int. J. Heat Mass Transfer. vol. 22, pp. 1425·1433. transfer and liquid solidit1cation-frce zone in a 105. Hirata, T., Gilpin, R. R.. and Cheng. C. 1979. The steady convectively-cooled horizontal square channel. App!. Sci. state ice layer profile on a constant temperature plate in a Res.. vol. 30, pp, 355·366, forced convection /low: II. The transition and Turbulent 123. Petrie. D. J .. Linehan, J. H., Epstein. M.. Lambert. G. A.. Regimes. Int. J. Heat Mass Transfer, vol. 22. pp. 1435 Stachyra, L. 1. 1980. Solidificalion in two-phase flow. J. 1443. Heal Transfer 102:784-86. 106. Hirata, T. and Hanaoka, C. 1990. Laminar-now heal 124, Prescoll. P.J.. Incropera. F.P. and Bennon. W.O. Modeling transfer in a horizontal tube with internal freeZing (effects of dendrilic solidification systems: Reassessment of lhe of f10w acceleration and llarural convection). Heat Transfer continuum momentum equation. Int. J, Hea.\ Mass -Japanese Res.. vol. 19, pp, 376-390. Transfer. vol. ? 107. Hirata, T. and Ishihara, M. 1985. Freeze-off conditions of a 125. Ramachandran, N.. Gupta. J. P.. and lalul'i GENIUM PUBLISHING Section 507.8 VAPORlZATIONfPHASE CHANGE Heat Page 6 FREEZING Transfer August 1996* REFERENCES Division rnt. J. Heat Mass Transfer, vol. 25, pp. 119. 148. Wilson, N. W., and Vyas, B. D. 1979. Velocity profiles 131. Savino, J. M. and Siegel, R. 1969. An analytical solution near a vertical ice surface melting into fresh water. 1. Heat for solidification of a moving warm liquid onto an Transfer, vol. 101, pp. 313-317. isothermal coJd wall. Int. 1. Heat Mass Transfer, vol. 12, 149. Yao, L. S. 1984. Natural convection effects in the pp.803-809. continuous casting of horizontal cyl inder. Jnl. J. Heal Mass 132. Savi no, J. M. and Siegel, R. 1971. A discussion of the Transfer, voL 27, pp. 697. paper 'On the solidification of a warm liquid flowing over 150. Yeah, G.H., Betmia, M., Davis, G. D. and Leonardi, E. a cold plate. Int. J. Heal Mass Transfer, vol. 14, p. J225. 1990. Numerical study of three~dimensionalnarural 133. Schaefer, K J. and Coriell, S. R. 1984. Convection-induced convection during freezing of water. Int. J. Num. Meth. distortion of a solld-liquid interface. Metallurgical Engng., vol. 30, pp. 899-914. Transactions A, vol. 15A, pp. 2109-2115. 151. Zabaras, Nicholas, Nguyen, T. Hung, 1995. Control of the 134. Schneider, W. 1980. Transient solidification of a flowing freeZing interface morphology in solidification proccsses in liquid at a heat conducting wall. Jnt. J. Heat Mass Transfer, the presence of narural con vcclion. 1m. J. Num. Mel hods in vol. 23, pp. )377-1383. Engng, vol. 38, pp. 1555-1578 135. Seki. N.. Fukusako S. and Younan. G. W. 1984. Ice 152. Zerkle, R. D., Sunderland, J. E. 1968. The effect of liquid formation phenomena for water flow between two cooled solidification in a rube upon laminar-flow heat lransfer and parallel plates. J. Heal Transfer. vol. 106. pp. 498-505. pressure drop. J. Hcal Transfer, vol. 90. pp. 183-90. 136, Shibani. A. A. and Ozisik, M. N. 1977. A solUlion of 153. Adomalo. P. M. and Brown, R. A. 1987. Pelrov-Galerkin freezing of liquids of low Prandll Number in turbulent flow melhods for nalural convection in directional solidification between parallel plales. 1. Heat Transfer. vol. 99, pp. 20 of binary alloys. [n/. J. Num. Meth. Auids. vol. 7, p. 5. 24. 154. Bell, G.E. 1982. On the per[onnance of Ihe enlhalpy method. Int. J. Heal Mass Transfer, vol. 25. pp. 587-589. 137. Shibani, A. A. and Ozisik, M. N. 1977. Freezing of liquIds in turbulent flow inside tubes. Can. J. Chern. Eng., vol. 55. 155. Bell, G. E. 1985. Inverse formulalions as a method for pp.672-677. solving certain mehing and freeling problems. Numerical Methods in Heat Transfer, Vol. Ill. pp. 59-78. Wiley. New 138. Siegel, R. and Savino, J. M. 1966. An analysis of the York. transient solidification of a Oowing wann liquid on a convectiveJy cooled wall. Proc. 3rd Int. Heal Transfer J56. Binder, K. 1994. Monte Carlo simulation of thc glass Canf.. vol 4. pp. 141- 151, Elsevier. Amsterdam. transition of polymer meJts. Progr. Colloid & Polymer Sci" vol. 96, pp. 7-J5. 139. Sparrow. E.M. and Ohkubo, Y. 1986, Numerical analysis of two-di mensional transient frcczi ng including sol id phase 157, BiOI. M. A. 1959. Further developmems of new methods in and lube-wall conduction and liqUId-phase natural heal now analysis. J. Aero/Space Sci. Vol. 26. pp. 367 convection. Num. Heat Transfer. vol. 9. pp. 59-77. 381. 140. Sparrow. E. M., Ramsey, J. W" and Kemink. R. G, 1979. 158. Boley, B. A. 1964. Upper and lower bounds in problems of Freezing controlled by natural convection. J. Heat melting and solidifying slabs. Quart. Appl. Math., vol. 17. Transfer, vol. 10 J. p. 578. pp. 253-269. 141. Stephan, K. J969, !nOuenee of heatlransfer on melting nnd 159. Boley, B.A, and Yagoda. H.P. 1969. The starting solution solidification in forced flow. Inl. J. Heat H. Mass Trnnsfer, for lwo-dimensional heal conduclion problems with change vol. 12. pp. 199-214. of phase. Quart. Appl. Math., vol. 27. pp. 223-246. 142. Szekely, J. and Stanek, V. 1970. NaturaJ convection 160. Bonacina, C.. Comini, G., Fasano, A.. and Primicerio, M. transients and their effecls in unidirectional solidification. 1973. Numerical solution of phase-change problems. Int. J. Metallurgical Trans.. vol. I. pp. 2243-2251. Heat Mass Transfer. vol. 16. pp. J825- 1832. 143. Tankin. R.S. and Farhadieh. R. 1971. EfCects of thermal 16 I. Brown. S.G.R., Bruce, N.B. Three-dimensional cellular conveclion currenls on formation of Ice. Int J. Heal Mass aUlOmalOn models of mlcroslruClural evohllion during Transfer. Vol. 14, pp, 953-961. solidification. J. Malerials Sci.. vol. 30. 1995. pp. 1144- J 150 144. Thomason, S. B.. Mulligan, J. C. and Everharl. J. 1978. The effecl of internal solidification on turbulem flow heal 162. Brown. S.G.R.. Bruce, N.B. 3-Dimensional cellular lraosfer and pressure drop in horilOnlai lube. J. Heal aUlOmaton model of 'frcc' dendlitic growth. ScriPlll Transfer, vol. 100, pp. 387-394. tvIelallurgica et Materialia. vol. 32, 1905. pp. 241-246 145. Udaykumar, H.S.. Shyy, W. 1995. SImulation of interfacial 163. Caldwell, J. and Chiu, C.-K, 1994. Numcril.:at solution of instabilities during solidificalion ' L Conduction and melling/solidification problems in cylindl'ical geometry. capillarily effects. Int. J. Heal Mass Transfer. vol. 38, pp. Engrg. Compulations, vol. II. pp. 357-363. 2057-2073 164. ChawaJa, T. c., Pederson, D. R.. Leaf. G .. Minkowycz, W. 146. Viskanta. R. 1985. NafuraJ convection in melting and 1.. and Shouman. A. R. 1984, Adaptive collocmion melhod soJidification. Natural Convection: Fundamentals and for simullaneous heat and mass diffusion witli phase Applications, pp. 845. (S. Kakac. W. Aung. and R. change. J, Heal Transfer, vol. 106, pp. 491. Viskanta. cds.), Hemisphere, Washington. D. C. 165. Chen, Long-Qing, Novel computer simulalion lechnique J47. Weigand. B., Ncumann. 0., Slrohmayer. T.. Beer. H. 1995. for modeling grain growth. Scripta Metallurgica el Combined free and forced convection flow in a cooled Materialia 32, 1995, pp. 115-120 veniclli duct with internal soJidlficalion, Waerme- und 166. Churchill. S. W. and Gupla, J. P. 1977. Approximations for Stoffuebertragung, vol. 30, pp. 349-359 conduclion with freezing or melting, lnt. J. Heat Mass GENIUM PUBLISHING Heal VAPORIZATIONIPHASE CHANGE Section 507.8 Transfer FREEZING Page 7 Division REFERENCES August 1996" Transfer. vol. 20, pp. 1251-1952. boundary condition. Int. J. Heat Mass Transfer. VoL 24, 167. Cleland. D.1., Cleland. A.C. and Earle, R.L 1987. pp.25]-259. Prediction of freezing and thawing times for multi 185. Gupta. R. S. and Kumar. A. 1985. Treatment of multi dimensional shapes by simple fonnulae. Pan I: regular dimensional mOVing boundary problems by coordinate shapes: Pan 2: irregular shapes. Int. J. Refrig. 10: pp. 156 transformalion. Jnt. J. Heat Mass Transfer, voL 28. pp. 166: 234-240. 1355-1366. 168. Comilli. G"> del Guidice. S., Lewis, R. W. and Zienkiewicz, 186. Hajj-Sheikh, A. and Sparrow, E. M. 1967. The solution of O. C. 1976. Finite element solution of nonlinear heal heat conduction problems by probabiljty methods. Trans.. conduclion problems with special reference to phase 1. Heat Transfer, voL 89, pp. 121-131. change. Int. 1. Num. Melhods Engng, vol. 8, pp. 613-624. 187. Hsiao. J.S. and Chung, B.T.F. 1986. Efficient algorithm for 169. Crowley. A, B. 1978. Numerical solution of Slefan finile element solution to lwo-dimensional heat transfer problems. Int. J. Heat Mass Transfer. vol. 21, pp. 215-219. wilh melting and freezing. J. Heat Transfer. vol. 108. pp. 170. Dilley, J.E and Liar, N. 1986. The evalualion of simple 462-464. analytical solutions for the prediction of freeze-up time, 188. Hsu, C. F. and Sparrow. E. M. 1981. A closed-form freezing. and melling. Proc. 8th Intemational Heat Transfer analytical Solulion for freezing adjacent 10 a plane wall ConL.4: 1727-1732, San Francisco. CA. cooled by forced convection. J. Heal Transfer, vol. 103. pp. 17!. Oliley. J.F. and. Lior. N. 1986. A mixed implicil-explicil 596-598. variable grid scheme for a transient environmental ice 189. Hsu. C. E, Sparrow. E. M. and Patankar. V. 1981. model. Num. Heat Transfer. vol. 9. pp. 381-402. Numerical Solulion of moving boundary problems by [72, EI-Genk. MS and Cronenberg, A. W. 1979. Slefan-like boundary immObilization and a control. volume based, probl ems in fl nile geometry. AIChE Symp. Scr.. No. 189. finile differ~nce sclleme. lnl. J. Heat Mass Transfer. vol. vol. 75, pp.69-80. 24, pp. 1335-1343. 173.Ellioll. C M, 1987. Error analysis of the enthalpy melhod for 190. Hu. Hongfa. Argyropoulos, A.1995. Modeling of Stefan the Stefan problem, IMA 1. Num. Anal., vol. 7. pp. 61-71. problems in complex configurations involving two different metals using lhe enthalpy method, Modeling and 174. Elliol. C. M. and Ockendon, J. R. 1982. Weak and Simulation in Materials Sri. and Engng, . vol. 3, pp. 53-64 Varialional Melhods for Moving Boundary Problems. Research nOles in Mathematics 59. Pilman Advanced 19l.Huang. C. L. and Shih. Y. P. 1975. Perturbation solution for Publishing Program. Boston. MA. planar solidification of a saturated liquid with convection al lhe wall. Inl. J. Heal Mass Transfer. vol. 18. pp. 1481. 175. Fox, L. 1974. What are the best numerical methods? In Moving Boundary Problems in Heat Flow and Diffusion. J. 192. Hyman. J.M. 1984. Numerical methods for tracking R. Ockendon and W.R. Hodgkins. cds, . pp. 210. Oxford illlerfaces. Physica D. vol. 12D. pp. 396-407. Universily Press, London. 193. Kang. S" Zabaras, N. 1995. Conlrol of the freezing 176. Frederick. D. and Greif, R. 1985. A method for the solution inlerface mOlion in lwo-dimensional SOlidification of heal transfer problems with a change of phase. J. Heat processes using the adjoint method. Inl. J. Num. Methods Transfer. vol. 107. pp. 520. in Eogng. vol. 38. pp. 63-80. 177. Friedman. A. 1968. The Slefan problem in several space 194. Kern, J. 1977. A simple and apparently safe solution to the variables. Trans. Am. Math. Soc.. vol. 133. pp. 51-87; generalized Stefan problem. IIll. J. Heal Mass Transfer. Correclion 1969. vol. 142. pp. 557. vol. 20. pp. 467-474. 178. Friedman. A. 1982. Variational Principles and Free 195. Kern. 1. and We\ Is, G. 1977. Simple analysis and worki ng Boundary Problems. Wiley. New York. equations for {he solidification of cylinders and spheres. Mel. Trans.. Vol. Sb. pp, 99·105. ! 79, Fukusako. S. and Seki. N. 1987. FundarneOlal Aspecls of Analytical and Numerical Melhods on Freezing and 196. Ki m. CJ. and Kaviani. M. 1992. A numerical melhod for Melling Heat-Transfer Problems. Ch. 7 in Annual Rev. of phase-change problems wilh conveclion and diffusion. Inl. Numerical Fluid Mechanics and Heat Transfer Vol. I. cd. J, Heal and Mass Tran~fer. Vol. 36, pro 457-467. T. C. Chawla. p. 351-402. Hemisphere, Washington, D.C. 197. Kim. CJ.. Ro. S.T. and Lee, J.S. i ')!)3. All efficient 180. Furzeiand. R. M. 1980. A comparative sludy of numencal compulalional technique 10 solve the moving boundary melhods for moving boundary problems. J. Insl. Math. problems in the axisymmelric geomelries. [nl. J HC- GENIUM PUBLISHING Section 507.8 VAPORIZATIONIPHASE CHANGE Heal Page 8 FREEZING Transfer AuguSl 1996* REFERENCES Division 202. Lee, S.L., Tzong. R.Y. 1995. Latent heal method for temperaNre range. 1. Heal Transfer, vol. 101, pp. 331-334. solidification process of a binary alloy system. Int. J. Heal 222. Pedraza, A.J.• Harnage, S., and Fainstein-Pedraza, D. Mass Transfer. vol. 38, pp. 1237-1247 1980. On the use of analytic approximations for describing 203. Li, D. 1991. A 3-dimensionall1yperbolic Stefan problem the macroscopic heat flow during solidification. Metall. with discontinuous temperature. Quart. Appl. Math., vol. Trans. ASM & MS-AIME, vol. liB, pp. 321-330. 49, pp. 577-589. 223. Pedroso. R. I. and Domoto, A. 1973. Inward spherical 204. Little, T.O. and Showalter, R.E. 1995. Super-Stefan solidification-solution by the method of strained coordi problem. Int. J. Engng Sci., VO!. 33, pp. 67-75. nates. InL J. Heat Mass Transfer, vol. 16. pp. 1037-1043. 205. Liu, J., Liu, Z., Wu, Z. Monte-Carlo approach to ceUular 224. Pedroso, R. I. and Domoto, G. A. 1973. Exact Solulion by pattern evolution for binary alloys during directional perturbation method for planar solidification of a saturated solidification. J. Materials Sci. and Techno!., vol. II. 1995, liqUId Wilh convection at ll1e wall. Int. J. Heat Mass pp. 127-132 Transfer, vol. 16. pp. 1816-1819. 206. Lobe, 8.; Baschnagel, J.; Binder, K.1994. Monte Carlo 225. Pham. Q. T. 1985. A fast unconditionally stable fi ni IC simulation of the glass tmnsition in two - and three difference scheme for the heal conduction wilh phase dimensional polymer melts. J. Non-Crystalline Solids, vol. change. InC J. Heat Mass Transfer, vol. 28, p. 2079. l72-7, pp. 384-390 226. Pham, Q. T. 1986. The use of lumped capacilance in the 207. Majchrzak, E .• Mochnacki. B. 1995. ADpl ication of the tinite-element SolUlion of hcal conduction problems with BEM in the thennaltheory of foundry. Engng Analysis phase change. Int. 1 Heat Mass Transfer. vol. 29. p. 285. with Boundary Elements. vol. 16, 1995, pp. 99-121 227. Rabin. Y. and Korin. E. 1993. EfficieOl numencal solution 208. Majchrzak. E., Mendakiewicz, J. 1995. Numerical analysis for lhe multidimensional solidificalion (or melting) of cast iron solidi tication process. J. Materials Proccssi ng problem using a microcomputer. lnt. J, Heal Mass Techno!., vol. 53, pp, 285-292 Transfer. vol. 36, pp. 673-683. 209. Meirmanov, A.M. 1992. The Stefan Problem. Walter de 228. Ramos, M.. Sanzo P.O., Aguirre-Puente, J. and Posado. R. Gruyler, Berlin-New York. 1994. Use of the equivalenl volumetric enlhalpy variation 210. Mennig, J. and 021Sik, M. N. 1985. Coupled integral in non-linear phase-change processes: freezing-wne progression and thawing time. [nt. J. Refrig.. vol. 17. pp. equation approach for solving melting or solidification. [01. J. Heat Mass Transfer. vol. 28, p. 1481. 374-380. 211. Meyer, G, H. 1973, Multidimensional Stefan problems. 229. Rao, P. R. and Sastri. V. M. K. 1984, Efficienl numerical SIAM J. Num. Anal .. Vol. 10. pp. 522-538. melhod for tlVO dimensional phase change problems. Int. J. Heat Mass Transfer. vol. 27. pp. 2077·2084. 212. Meyer, G.H. 1977. An application of the method of lines to multidi mensional free boundary problems. J. Inst. Math 230. Raw. M. J. and Schneider, G.E. 1985. A new implicit AppL vol. 20, pp. 317-329. solution procedure for multidimensional finile-di fference modeling of the Stefan problem. Numerical Heat Transfer, 213. Meyer. G.H. 1978. Direct and iterative one-dimensional vol. 8. pp. 559-571. fronl lracking melhods for lhe lwo-dimensional Stefan problem. Num. Heal Transfer, vol. 1. pp. 351-364. 231. Ruan, Y. 1994. Lagrangian-Eulerian finite-element fonnulalion for steady-stale solidification problems. Num. 214. Meyer. G.H. 1981. A numerical method for (he solidifica Heat Transfer B, vol. 26, pp. 335-351. tion of a binary alloy. [n(. 1. Heal Mass Transfer. vol. 24. pp.778-781. 232. Ruan. Y., Li, B. Q.. Liu, J. C. 1995. Finite element method for steady-slale conduction-adveclion phase change 215. Miller. M. L. and Jiji, L. M. J970, Application of point problems. Finite Elements in Analysis and Design, vol 19. malching techniques 10 tlVo-dimensional solidificalion of pp.153-168 viscous flow over a scmi-infinile flat plate. J. Appl. Mech.. vol. 37, p. 508. 233. Rubinshtcin, L. 1.,1967. The Stefan Problem. Translations of Mathematical Monogmphs. Am. Math. Soc.. Vol. 27. 216. Mingle, J. 1979. Slefan problem sensilivity and uncer Providence, R.l. 02904. tainty. Num. Heal Transfer. vol. 2, pp. 387-393. 234. Rubinsky. B. and ShiLZer. A. 1978. Allalylical solul1ons of 217. Morgan, K., Lewis, R. W .. and Zienkiewic2, O.c. 1978. An lhe heal equations involving a moving boum!:lry wilh improved algorithm for heat conduction problems lVith applicalion to the change of phase problem (The inverse phase change. rn(. J. Num. Methods Engng. vol. 12. pp. Stefan problem), J. Heal Transfer, vol. 100. p..,00·304, 1191-1195. 235. Rubiolo de Reinick. A.C. 1992. ESlimmion 01' awragc 218. Mori, A. And Araki. K. /976. Methods for analysis of the freezing and thawing lemperalure vs. lime wilh infinite moving boundary-surface problems, Int. Chem, Engng.. slab correlation. Computers In Industry. vol. 19. pp. 297 vol. 14, No.4, pp. 734·743, 303. 219. Ockendon, J. R. and Hodgkins, W. R.. cds. 1975. Moving 236. Sailoh. T. 1978. Numencal melhod for multidimensional Boundary Problems in Heat Flow and Diffusion. Oxford freezing problems in arbitrary domains. J. Heal Tnul:>tcr. University Press (Clarendon), London. vol. 100, pp. 284-299. 220. Ouyang, T. and Tamma, K. K.1994. Finite-element 237. Saitoh. T .. Nakamura. M. and Goml. T. 1994, Time.space developmcnts for two-dimensional. multiple-intcrfalX method (or multidimensional melting and freeZing phasc-change problems. Num. Heat Transfer B. vol. 26. problems. Int. 1. Num. Melh, Engng. vol. 37. pp. 1793· pp. 257-271. 1805. 221. +zisik. M.N. and U7.2ell, J.c. 1979, Exact solution for 238. SailOh. T. 1980. Recent dcvelopments of solution methods freezing in cylindrical symmetry with extcnded freezing for the freel.ing problem. Refrigeration. Vol. 55. No. 636. GENIUM PUBLISHING Heat VAPORUATION~HASECHANGE Section 507.8 Transfer FREEZING Page 9 Division REFERENCES August 1996* pp. 875-83 and pp. 1005-1015. PaIl B Fundamentals, vol. 27. 1995, pp. 127-153 239. Saunders, B.V. Boundary conforming grid generation 257. Verdi, C and Visintin, A. 1988. Error estimates for a semi system for interface tracking. Computers Math AppJ., vol. explicit numerical scheme for Stefan-type problems. 29, 1995, pp. 1-17 Numerische Mathematik. vol. 52, pp. 165-185. 240. Shamsundar, N. and Sparrow, E. M. 1975. Analysis of 258. Vick, B. and Nelson. D 1993. Boundary element method multidimensional conduction phase change via the applied to freezing and melting problems. Num. Heat enthalpy model, Trans. J. Heat Transfer, vol. 97. pp. 333 Transfer B, vol. 24, pp. 263-277 340. 259. Voller, V. R.. and Cross. M. 1981. Accurate solutions of 241. Shamsundar, N. and Srinivasan R. 198!. A new similarity moving boundary problems using the enthalpy method. InL method for analysis of muilidimensional solidification, J. 1. Heat Mass Transfer, vol. 24. pp. 545-556. Heat Transfer, vol. 103, pp. 173-175. 260. Voller, V. R., Cross. M.. and Walton. P. 1979. Assessment 242. Shishkaev, S.M. 1988. Mathematical thermal model of lake of weak solution techniques for solving Stefan problems, freeZing and its testing. Water Resources, vol. 15, PD. 18 (ed.). Numerical Methods in Thermal Problems. p. 172. 24. Pineridge Press. 243. Shyy, Wei, Rao, M.M .. Udaykumar. H.S. Scaling 261, Voller. V.R. and Prakash, C. 1987. A fixed grid numerical procedure and finite volume computations of phase-change modeling methodology for convection-di ffusion mushy problems with convection. Engng AnalYSIS with Boundary region phase-change problems. lnl. J. Heat Mass Transfer, Elements, vol. 16, 1995. pp. 123·147 vol. 30. pp. 1709-1719. 244. Siegel, R. 1986. Boundary perrurbation methods for free 262. Weinbaum. 5. and Jiji. L. M. 1977. Singular perturbation boundary problem in convectively cooled continuous theory for melting or freezing in finile domains not at the casting. J. Heat Transfer vol. 108. pp. 230. fusion temperature. J. Appl. Mech. vol. 44. pp. 25-30. 245. Siegel, R., Goldstein. M.E and Savino, J.M. 1970. 263. Westphal, K. O. 1967. Series solutions of freeZing Conformal mapping procedure for transient and steady problems with the ftxed surface radiating into a medium of state two-dimensional solidificalion. Proc. 4th lnt. Heal Tr. arbilrary varying temperalure, In\. J. Heat Mass Transfer. Conf., vol. I, paper Cu 2.1 I. vol. 10. pp. 195-205. 246. Shamsundar, N. 1982. Formulae for freezi ng outside a 264. Wilson. D.G. and Solomon. A.D. 1986. A Stefan-IYpe circular tube with axial varialion of coolantlemperature. problem with void formalion and its explicit solution. IMA Int. J. Heat Mass Transfer, vol. 25. pp. 1614-1616. J. Appl. Malh., voL 37. pp. 67-76. 247. Solomon, A.D., Wilson, D.G. and Alexiades, V. 1983. 265. Wilson. D.G .. Solomon. A.D.. and Alcxiades. V. 1984. A Explicil solutions to phase change problems. Quan. Appl. model of binary alloy solidification. lnl. J. Num. Meth. Math., vol 41. pp. 237-243. Engng. vol. 20. pp. 1067-1084. 248, Solomon, A,D., Wilson. D.G. and Alexiades, V. 1984. The 266. Wilson. D. G .. Solomon. A. D .. and Boggs, P.T.. eds. quasi-stationary approximation for the Stefan problem with 1978. Moving Boundary Problems. Academic Press, New a convective boundary condllion. lnt. 1. Math. & Math. York. Sci., vol. 7, pp. 549-563. 267. Yoo, J. and Rubinsky. B. 1983. Numerical computation 249. Soward, A.M. 1980. A unified approach to Stefan's using finite elements for the moving IOterfacc in heal problem for spheres and cylinders. Proc. R. Soc. Lond., transfer problems wilh phase tranSformalion. Num. Heal vol. A348, pp. 415-426. Transfer. vol. 6. pp. 209-222. 250. Tao, L. C. 1967. Generalized numerical solution of 268. Yoo. J. and Rubinsky. B. 1986. A finite element method freezing in a saturated liquid in cylinders and spheres. for the study of solidification processes in the presence of AIChE J.. vol. 13. p. 165. natu ral convection. Int. J. Numerical Methods Eng.. vol. 251. Tao, L. C. 1968. Generalized solution of freezing a 23. pp. I 785- 1805, saturated liquid in a convex container. AIChE J., vol. 14. p. 269. Yu. X .. Nelson. OJ. and Vick, B. 1995. Phase change with 720. multiple fronts in cylindrical systems using the boundary 252. Tao, L. N. 1981. The exact solutions of some Stefan c1emenl method. Engng Anal. Boundary Elemenls, vol. 16. problems with prescribed heat tlux.1. Heat Transfer. vol. pp.161-170. 48, p. 732. 270. Zabaras. N. and: Mukherjee, S.. 1994. Solidifkation 253. Tamawski, W. 1976. Mathematical model of frozen problems by the boundary element method. Inl. J. Solid consumption products. Int J. Heal Mass Transfer, vol. 19. and Structures. vol. 31, DP. 1829-1846. p.15 27l. Zabaras. N. and YU GENIUM PUBLISHING Section 507.8 VAPOR~ATION~HASECHANGE Heat Page 10 FREEZING Transfer August 1996* INTRODUCTION, APPLICATIONS, AND BASIC PHYSICAL CONCEPTS Division (T) and pressure (P) during the transition from the liquid to D. INTRODUCTION, APPLICATIONS, AND BASIC the solid phase, viz PHYSICAL CONCEPTS dP his -:-- Eq. (8-1) Freezing occurs naturally, such as with environmental ice in dT Tc..vis the atmosphere (hail, icing on aircraft), on water bodies and where his is the enthalpy difference between phases (= lis ground regions at the earth's surface, and in the magma at hi < 0, the lateot heat of freezing), and c..vis is the specific the envelope of the molten earth core. It is also a part of volume difference between the phases (= Vs - Vi ). Examina many technological processes, such as preservation of tion of Eq. (8-}) shows that increasing the pressure will foodstuffs, refrigeration and air-conditioning, snow and ice result in an increase of the freezing temperature if c..vis < 0 making, manufacturing (such as casting, molding, sintering, (i.e., when the specific volume of the liquid is higher than combustion synthesis, coating and electro-deposition, that of the solid, which is a property of tin, for example), soldering, welding, high energy beam cutling and fonning, but will result in a decrease of the freezing temperature crystal growth, electro-discharge machining, printing), when 6.vlS> 0 (for water, for example). The latter case organ preservation and cryosurgery, and thermal energy explains why ice may melt under the pressure of a skate storage using solidlliquid phase-changing materials. A blade. bi bl iography for these appl ical ion s, wi th a brief in troduc In some materials, called glassy, the phase change between tion, is given in Subsection V, "Applications Bibliography." Freezing is often accompanied by melting, and the thermo the liquid and solid occurs with a gradual transition of the dynamics as well as transpon principles-and thus the physical pcopcnies, from those of one phase to those of the other. When the liquid phase flows during the process. the mathematical treatment--of the two processes are very similar. Specific discussion of melting is given in Section flow is strongly affected because the viscosity increases greatly as the liquid changes to solid. Other malerials, such 507.9 of this Databook. as pure metals and ice, and eutectic alloys, have a definite Tn simple thermodynamic systems (i.e., without external line of demarcation between the liquid and the solid, and fields, surface tension, etc.) of a pure material, freezing the transition is abrupt. This situation is easier to analyze occurs at certain combinations of temperalUre and pressure. and is therefore more rigorously addressed in the literature. Since pressure Iypicall y has a reI at ive Iy smaller in fl uence, To illustrate the above-described gcadualtransition, most only the freezing temperature is often used La identify this distinctly observed in mixtures, consider the equilibrium phase transition. phase diagram for a binary mixture (or alloy) composed of It is noteworthy that many liquids can be cooled to tempera species a and b, shown in Fig. 8-1. Phase diagrams, or tures significantly below the freezing temperature without equations describing them, become increasingly compli solidification taking place. 111is phenomenon is known as cated as the number of components increases). X is the supercooling. Water, for example, is known to reach concentration of species b in the mixture. t denotes the temperatures of about -40 C wjthout freeZing, and silicates liquid, s the solid, s~ a solid with a lattice structure of and polymers can sustain supercooling levels of hundreds of species a in its solid phase but containing some molecules degrees. To initiate freezi ng, it is necessary to form or of species b in that laltice, and sb a sol id with a Ian ice introduce a solid-phase nucleus into the liquid. Once this structure of species b in its solid phase but containing some nucleus is in trod uced, [reezi ng proceeds rapidly. Further molecules of species a in that lattice. "Liquidus" denotes information can be found in Alexiades and Solomon [II the boundary above which the mixture is just liquid, and [3], Knight [40], Perepezko and Uttomark [53], and "sol idus" is the boundary separating the final sol id mixture Pounder [55]. of species a and b from the sol id-liquid mixture zones and TIle conditions for freeZing are strongly dependent on the from the other zones of solid S~ and solid sb. concenuation when the material contains more than a single For illustration, assume thal a liquid mixture is at poin! I. species. Furthermore, freezing is also influenced by eXlernal characterized by concentration Xl and temperature T t (Fig. effects, such as elecuic and magnetic fields. in more 8-1), and is cooled (descending along the dashed line) while complex thermodynamic systems. mainlaining the concentration constanl. When the tempera The equilibrium thermodynamic system parameters during ture drops below the liquidus line. solidification Slarts, phase transition can be calculated from the knowledge that creating a mixture of liquid and of solid sa. Such a two the partial molar Gibbs free energies (chemical potentials) phase mixture is called the mushy zone. At point 2 in Lhat of each component in the two phases mus! be equfll (cf. mushy zone; for example, the solid phflse (sa) porlion Alexiades and Solomon [3], Hultgren el af [35), Kurz and contains a concentration X2.s;; of component b, and Ihe Fi sher [4 IJ, Lior [45], and Pou Iikakos [54]). One imponant Iiquid phase portion contains a concentration Xv of result of using this principle for SImple Sl ngle-component component b. The ratio of the mass of the solid Sa to that of systems is the Clapeyron equation relating the temperature the liquid is determined by the lever rule, :lnd is GENIUM PUBLISHING Heat VAPORIZATIONIPHASE CHANGE Section 507.8 Transfer FREEZING Page 11 Division INTRODUCTION, APPLICATIONS, AND BASIC PHYSICAL CONCEPTS August 1996* (X2.l - X2)/{ X2 - X2,5) at point 2. Further cooling to below inception of-and during-melting and freezing (cf. the solidus line, say to point 3, results in a solid mixture (or Cheung and Epstein (17), Yao and Prusa [72] and refer alloy) of Sa and SJ,. containing concentrations X3,so and X3.Sb ences (79)-[152]). The flow may be forced, such as in the of species b, respectively. The ratio of the mass of the solid freezing of a liquid flowing through or across a cooled pipe, Sa to that of Sb is again determined by the lever rule, and is andlor may be due to natural convection that arises when (X3.Sb - X3)1( X3 - ;(3.s) at point 3. ever there are density gradients in the liquid, here generated by temperature and possibly concentration gradients. It is T noteworthy that the change in phase usually affects the original flow, such as when the liquid flowing in a cooled pipe gradually freezes, and the frozen solid thus reduces lhe flow passage, or when the evolving dendritic structure gradually changes the geometry of the solid surfaces that are in contact with the liquid (cf. Cheng and Wong [85], Cho and Qzisik [88], Epstein and Cheung [91, 92], Epstein l +s, and Hauzer [93), Gilpin (lOl ,\02). Hirata and Hanaoka s, [106], Hirata and Ishihara (107), Hwang and Tsai [107]. Kikuchi et 01 (112), Kuzay and Epstein [115], Lee and S.+Sb Hwang [116), Madejski [119], Sampson and Gibson [129, 3 __ ..... __ ._ ...... ~l;5.~ . 130), Seki er of (135), Thomason ef at (144), Weigand et al (147), Zerkle and Sunderland {I52]). Under such circum o X, x 1 stances, strong coupling may exist between the heat transfer and fluid mechanics, and also with mass transfer when more Figure 8-1. A LUjuid-Solid Phase Diagram ofa Binary than a single species is prescnt, and the process must be Mixture mOdeled by an appropriate set of continuity, momentum, A unique situation occurs if the initial concentration of the energy, mass conservation, and stale equations, which need liquid is x.e: upon constant concentration cooling, the liquid to be solved simultaneously. fonns the solid mixture Sa + Sb having the same concentra tion Xe and without the formation of a two-phase zone. Xe E. PREDICTIVE METHODS is called the eutectic concentration, and the resulting solid mixture (or alloy) is called a eutectic. The mathematical description of the freezing process is characterized by non-linear partial differential equations, It is obvious from the above that the concentration distribu which have anaJytical (closed-form) solutions for only a tion changes among the phases, which accompany the few simplified cases. As explained above, the problem freezing process (Fig. 8-1), are an important factor in lhe becomes even less tractable when flow accompanies lhe composition of alloys, and are lhe basis for freeze-separa process, or when more lhan a single species is presenl. A tion processes. very large amount of work has been done in developing The presence of a lwo-phase mixture zone with tempera solution melhods for the freezing problem (sometimes also lure-dependent concentration and phase-proportion obvi called the Stefan problem, after the seminal paper by Stefan ously complicates'heat transfer analysis, and requires the [62]), published both as monographs and papers, and simultaneous solution of both the heat and mass transfer included in the list of references to this Section (Alcxiades equations. Furthennore, the liquid usually does not solidify and Solomon (3). Bankoff [6), Chadam and Rasmussen on a simple planar surface between the phases. Crystals of [ 12]. Cheng and Seki [16], Crank [22J, Fasano and the solid phase are fanned at some preferred locations in the Primicerio [26L Hill [33], Qzisik [51), Tanasawa and Lior liquid, or on colder solid surfaces inunersed in the liquid, [64J, Yao and Prusa (72), and references [I52J - [273), with and as freezing progresses the crystals grow in the fonn of emphasis on the reviews by Fox [175J, Friedman [178], intricately-shaped fingers, called dendrites. This compli Fukusako and Seki fl79J, Meinnanov (209). Ockendon and cates the geometry significantly and makes mathematical Hodgkins (219), Rubinshtein [233], and Wilson et al [266] . modeling of the process very difficult. An introduction to Generically, solutions are obtaioed either by 1) Iinearizi ng these phenomena and further references are available in the original equations (e.g., perturbation methods) where Chalmers [13], Colucci-Mizenko et al (20], Flemings {27], appropriate, and solving lhese linear equations, or 2) Kurz and Fisher [41), Murray et al (49). Poulikakos, [54], simplifying the original equations by ncglcl:ting terms. such Trivedi and Kurz (65), Gilpin [98]-[100J, Prescott et al as the neglection of lhennaJ capacity in lhe "quasi-static (124) and Glicksman et at and Hayashi and Kunimine (both method" described in Subsection IV below, or 3) using the cited in Subsection V. B.). "inlegral method," which satisfies energy conservation over Flow of the liquid phase often has an imponant role in the the entire body of interest, as well as the boundary condi- GENIUM PUBLISHING Section 507.8 VAPORIZATIONIPHASE CHANGE Heat Page 12 FREEZING Transfer August 1996* INTRODUCTION, APPLICATIONS, AND BASIC PHYSICAL CONCEPTS Division tions, but is only approximately Correct locally inside the this class of problems. In this Section we deal with cases in body (Subsection IV. A. below), or 4) employing a numeri which the densities of both phases are the same, and in cal method. which the freezing liquid does not flow, thus also ignoring, Many numerical methods have been successfully employed for simplification, the effects of buoyancy-driven convec in the solution of freezing problems, both of the finite tion that accompanies the freezing process when a tempera difference and element types, and many well-tested ture gradient ex.ists in the liquid phase. As stated in Subsec software programs exist that include solutions for that tion I.E., the effects of natural convection may sometimes purpose. A significant difficulty in the formulation of the be significant, and information about this topic can be found numerical methods is the fact that the liquid-solid interface in the references quoted in that Section. Freezing of non moves and perhaps changes shape as freezing progresses opaque media may also include internal radiative heat (making this a "moving boundary" or "free boundary" transfer, which is ignored in the equations presented below. problem). This requires continuous monitoring of the Infonnation about such problems is contained in references interface position during the solution sequence, and [14] and (31]. adjustment of the numerical model cell or element proper The solutions presented below can be found in many books ties to those of the particular phase present in them at the and reviews that deal with melting and freezing (cf. refs. [3], time-step being considered (cf. Alexiades and Solomon [3], (6), [22], [33], [41 J, [45], [72], and [179J, [209], [233]) and in Crank [22], Dilley and Lior (171 J, Fasano and Primicerio textbooks dealing with heat conduction (cf. [51] and [54]). (26], Fox (175], Friedman (178J, Furzeland [180], Gupta and Kumar [185], Hsu et at [189], Hyman [192]. Kim et al Solutions fo[" Materials that are Initially at the (197), Meyer [212, 213]. Mon and Araki (218), Ockendon 1. and Hodgkiss [219], Rubinsky and Shitrer [234], Saunders Freezing Temperature (239], Siegel et al (245), Tsai and Rubinsky [255). If the liquid to be frozen is initially at the freezing tempera Udaykumar and Shyy [256], Vick and Nelson [258J, Wilson ture throughout its extent. as shown in Fig. 8-2, heat et al [266], Yoo and Rubinsky [268], Yu et at [269], transfer occurs in the solid phase only. This somewhat Zabaras and Mukherjee [270], Zhang et at [273]). Several simplifies the solution and is presented first. formulations of the original equations were developed to Solid LiqUId simplify their numerical solution. One of them is the T popular "enthalpy method" discussed in more detail in Subsection iV.B. below. The predictive equations provided below are all for materi als whose behavior can be characterized as being pure. This T, (x.t) would also apply to multi-component material where changes of the freezing temperature and of the composition during the freezing process can be ignored. General Phase -<: hangc interlace solutions for cases where these can not be ignored are much more difficult to obtain, and the readers are referred to the / literature; some of the key citations are provided in refer ences (3), [27). [41], [77J, [2651 and Subsection V.B. o X (I) Furthermore, the solutions presented here by closed-fonn equations are only for simple geometries, since no such Figure 8-2. Freezing ofa Semi-infinite Liquid Initially at solutions are available for complex geometries. Simplified the FusiQn Temperature. Heat cOllaucrioll rakes place consequently in olle ofthe phases only. expressions. however, are presented for freezing times also in arbitrary geometries. Consider a liquid of infinite extent is to the right (x> 0) of the infinite surface at x = 0 (i.e., semi-infinite). described in II. PREDICTIVE EQUATIONS FOR Fig. 8-2. initially at the freeZing temperature '0. For timc FREEZING WITHOUT FLOW t> 0 the temperature of the surface (at x ::: 0) is lowered to To < h and the solid consequently Slarts to freeze there. In A. ONE-DIMENSIONAL FREEZING WITHOUT this case the temperature in the liquid remains constant, DENSITY CHANGE Ts =Tf so the temperature distribution needs to be calcu lated only in the solid phase. It is assumed that Ihe liquid Examination of the simplified one-dimensional case remains motionless and in place. The initial condition in the provides some important insights into the phenomena, solid is identifies the key parameters, and allows analytical solu tions and thus qualitative predictive capability for at least ~(x,t)=Tf in x>O. at £=0, Eq. (8-2) GENIUM PUBUSHING Heat VAPO~TION~HASECHANGE Section 507.8 Transfer FREEZING Page 13 Division INTRODUCTION, APPLICATIONS, AND BASIC PHYSICAL CONCEPTS August 1996* the boundary condition is EXAMPLE for Eq. (8-3) T.(O,t)=1Q t>O, The temperature of the vertical surface of a large and the liquid-solid interfacial temperature and heat flux volume of liquid paraffin used for heat storage, initially continuity conditions, respectively, are at the freezing temperarure. 7,. =Tt = 60 ·C. is suddenly lowered to 30·C. Any motion in the melt may be TAX{t)] = T for t> 0, Eq. (8-4) f neglected. How long would it take for the paraffin to solidify to a depth of 0.1 m? Gi ven properties: O'.s = -ks(~T.) 7 = p,his dXd(t) for t > 0, Eg. (8-5) (1.09)10- m2/s, Ps = Pt =814 kg/m), his =-241 kJlkg, oX [X(I)] t Cs =2.14 kJlkg T. To find the required time we use Eq. The analytical solution of this problem yields the tempera (8-9), in which the value of A' needs to be detennined. ture distribution in the solid as 'A' is calculated from Eq. (8-7) or determined from Fig. 8-3, which requires knowledge of Nsle,. From Eq. (8-8) _ (2.14kJ I kg 'C)(30'C - 60 ·C) er{20) N - 0.266. S,e, -241.2kJ I kg T (x, t) =To + (T - To) for t > 0, Eg. (8-6) s J en A' The solution of Eq. (8-7) as a function of Ns,cs is given where erf stands for the error junction (described and in Fig. 8-3, yielding 'A' "" 0.4. Using Eq. (8-9). the time tabulated in mathematics handbooks), and A' is the solution of interest is calculated by of the equation 1= [X{t)]2 _ (O.lm)2 Eg. (8-7) 4 'A.' 20:, - 4(O.4)2[(l.09)1O-7 m 2 Is] 5 with =(1.43)10 5 =39.8h. NStet being the Stefan Number, here defined for the solid as 2. Solutions for Materials that are Initially not at the _ cs {To -1J) Freezing Temperature N SI<, = I' Eg. (8-8) Its If, initially, the liquid to be frozen is above the freezing temperature, conductive heat transfer takes place in both recalling that the latent heal of freezing, his. must be entered phases. Consider a semi-infinite liquid initially at a into the equations as a negative value. Equation (8-7) can be tern perature T; higher ilian the freezi ng temperature 1). (Fig. solved to find the value of A' for the magnitude of N srec 8-4). At time t =0 at the liquid surface temperature at x = which is calculated for the problem at hand by using Eq. ° is suddenly lowered to a temperature To < T ,and main (8-8). The solution of Eq. (8-7), yielding the values ofA' as t tained at that temperature for I > 0. Consequently, the liquid a function of Ns,e, for 0:5 NSre :5 5, is given in Fig. 8-3. starts to freeze at x = O. and the freezing interface (separat 1.2 ..,------, ing in Fig. 8-4 the solid to its left from the liquid on its right) located at the position x =XCI) moves gradually (0 the 1.0 right (in the posilive x direction). 0.8 'A' T Solid Liqu,d ~ 0.6 ___--Tl T, asx-)oo 0.4 0.2 T, (x.l) T, 0.0 -f-----,,....-----r-..----,--r--..------.---.-----.,....--I -+ o 2 3 4 5 PIus The interface position(which also is the freezing front o X(l) progress) is defined by Eq. (8-9) Figure 8-4. Freezing ofa Semi-infinite LUjuid Initiolly at Eq. (8-9) an Above-freezil/g Temperaiure. Heat conduction takes place in bOI}, phases. GENIUM PUBLISHING Section 507.8 VAPORIZATIONIPHASE CHANGE Heat Page 14 FREEZING Transfer August 1996* INTRODUCTION, APPLICATIONS, AND BASIC PHYSICAL CONCEPTS Division The analytical solution of this problem yields the tempera of the liquid and solid phases differ, motion of the phase ture distributions in the liquid and solid phases, respec change interface is not only due to the phase change tively, as process, but also due to the associated volume (density) change. A reasonably good analytical solution for small (- ±10%) solid-liquid density differences is available (Alexiades. and Eq. (8-10) Solomon (3D for the semi-infinite slab at x ~ 0, initially solid at Ti > Tf> frozen by imposing a constant temperature To < T at the surface x =0 (Fig. 8-4). £t is assumed that and f Pi> Ps, Cl ,cs' kt, ks, his, and Tr are constants and positive. The freeze front X(c) starts at x = X(O)-="",() and advances to the right. Buoyancy-driven convection is ignored, but the Eq. (8-11) volume expansion of the solid upon freezing is considered, in that it pushes the entire liquid volume rightward (Fig. 8 4) without friction, at unifonn speed u(t) without motion where erfc is the complementary error function; "is a inside the liquid itself. The temperature distributions are, in constant, obtained from the solution of the equation the solid and liquid phases erf[2k) Eq. (8-15) ~(x,t) = - ertA'" Eq.(8-l2) To +(Tf To) where N is the Stefan number defined by Eq. (8-8). sre s in 0 S;x S; X(t) for I >0, Solulions of Eq. (8-12) are available for some specific cases in several of the references (cf. [3] and [51 D, and can, in m general, be obtained relatively easily by a variety of erfc[-X--(I-Il)KA2-jCi;i ] commonly-used software packages used for the solution of Tt(x,t)=T,-(T;-T ) (.~~ ) m erfc 1l1V\.'" nonlinear algebraic equations. The transient position of the freezing interface is in x;:: X(r) for I> O. Eq.(8-16) 1!2 The location of the freezing front is X(t) =21.. ( as') , Eq. (8-13) where A. is the solution of Eq. (8-12), and thus the expres Eq. (8-17) sion for the rate of freezing, i.e. the velocity of the motion and the speed of the liquid body motion due to the expan of the solid-liquid interface, is sion is dX(t) =Aal!2t-112, Eq. (8-14) v(r) = (I -Il)1..'"-ja;/i. Eq.(8-18) dt s Jn Eqs. (8- 15) - (8-18) A.'" is the Toot of the equation B. ONE·DIMENSIONAL FREEZING WITH NSr., NS'~l r= (J.lJ(A"')e(Il!U.~f DENSITY CHANGE A."'e)..-2ertA'" - erfc(IlKA "') ='V 1t, For most materials the density ofthc liquid and solid phases is somewhat different, usually by up to about 10% and in Eq. (8-19) some cases up to 30%. Usually the density of the liquid which can, for the specific problem parameters, be solved phase is smaller than that of the solid one, causing volume numerically or by usi ng one of the many soft ware packages expansion upon melting and shrinkage upon freezing. TItis for solving nonlinear algebraic equations. The remaining phenomenon causes, for example. a manufacturing problem parameters are defined as in that metals and plastic materials filling a mold in their liquid phase shrink when solidified, forming voids in the E c.(To - Tf ) solid and a poorly-conduCting gas layer bet ween the mold (or container) wall. Water is one of the materials in which his the density of the liquid phase is higher than that of the Eq. (8-20) solid one, and thus ice floats on water, and pipes tend to K=~' burst when water confined in them freezes. If the densilies GENIUM PUBUSHING Heat VAPORIZATIONIPHASE CHANGE Section 507.8 Transfer FREEZING Page 15 Division INTRODUCTION, APPLICATIONS, AND BASIC PHYSICAL CONCEPTS August 1996* with his taking a negative value in this freezing problem. Ts(X,I)=TO(r)+[Tf-TO(l)]X~t) fOrl~O, Because of the approximate nature of this analytical inO::;x::;X(t) solution it is expected that Eq. (8-17) slightly overestimates the melt front position. If p, < Ps. freezing would cause the Eq. (8-22) solid to shrink, moving the liquid leftward, in a direction respecli vel y. .. opposite to that of the freezing interface, and the solution The heat flux released during freezing, q(x,t), can' easily be represe nted by Eqs. (8- 15) - (8-20) would not be valid. determined from the temperature distribution in the liquid Approximate but less accurate solutions for this and other [Eq. (8-22)J. viz. cases are described by Alexiades aod Solomon (3]. Eq. (8-23) C. THE QUASI-STATIC APPROXIMATION To obtain rough estimates of melting and freezing processes This approximate solution is exact when Sres ~ 0, and it quickly, in cases where healLTansfer takes place in only one otherwise overestimates the values of both XU) and TCx.I). phase, il is assumed thai effects of sensible heat are negli While the errors depend on the specific problem, they are gible relali ve to those of latent heat (Sle ~ 0). This is a confined 10 about 10% in the above-described case significant simplification, since the energy equation lhen (Alexiades and Solomon [3]). becomes independent of time. and solutions to the steady For the same freezing problem bUl with the boundary state heat conduction problem are much easier to obtain. At condition of an imposed time-dependent negative beat the same time, the transient phase-change interface condi flux (cooling) -qo(t), tion [such as Eq. (8-5)) is retained, allowing the estimation of the transient interface position and velocity. This is hence _ks(dTs ) =qo(l) for c>O, Eq. (8-24) a quasi-static approximation, and its usc is shown below. dx 0.1 The simplification allows solution of freezing problems in more complicated geometries. Some solutions for the the quasi-static approximate solution is cylindrical geometry are presented below. More details can ( be found in refs. f3), [22J, [33], f [66], [179 J, and [2481· X(I) =-'-Iqo(t)dt for t.>O, Eg. (8-25) It IS important to emphasize that these are jusl approxima phls lions, without full information on the effecl of specific o problem conditions on the magnitude of the error incurred when using lhem. lo facl, in some cases, especially Wilh a T,(x,t) = Tf + qo (..!lJL I-.t) in 0 ::; x $ X(t) for t > O. convective boundary condition, they may produce very ks phls Eq. (8-26) wrong results. It is lhus necessary 10 examine the physical viabi lity of the results, such as overall energy balances, when using these approximations. Note thaI both his and qo must be entered into the equations as negative values. It is assumed here that the problems are one-dimensional, and lhat the material is initially al the freezing temperalUre T1 2. Examples of the Quasi-Static Approximation for Cylinder 1. Examples of the Quasi--5tatic Approximation for a h is assumed in these examples that the cylinders are very Slab long and that the problems are axisymmetric. Just as in the slab case, the energy equatlon is reduced by the approxima Given a semi-infinite liquid on which a lime-dependent tion (0 its steady state form. temperature To(t) < Tr is imposed at x = 0, (Fig. 8-2), the above-described quasi-SIalic approximation yields the Consider the outward-directed freezing of a hollow solution for the position of the phase-change front and of cylinder of liquid with internal radius T, and Ouler radius To the temperature distribution in the solid as (Fig. 8-5) due to a low temperature imposed at the internal radius ri, I ]1/2 r.(r;.I}=TO(I) GENIUM PUBLISHING Section 507.8 VAPO~TION~HASECHANGE Heat Page 16 FREEZING Transfer August 1996* _ INTRODUCTION, APPLICATIONS, AND BASIC PHYSICAL CONCEPTS Division lower temperature TaCt), with a heat transfer coefficient In[rl R(t)] T,(x,t) =T - [T - To(t) ] [ ()] Eg. (8-28) h at I), the heat flux boundary condition there is f f In Ii I R t s =h[Ta(t)-~(Ii,t)]>O in Ii :5 r :5 R(t) for t > 0, _ks(dT ) fort>O, Eq.(8-33) dr ';" and the transient position of the freezing front, R(t), can be and the solution is calculated from the transcendental equation In[r / R( t)] I T r t =T - T -T r s( ,) f [ f a()] In[r; I R(r)]-kJ hr; = s 2R(t)2 1n R(t) R(t)2 -'-? + 4k I[ JO{t) - Tf ]dt. 'i phis o in r, ::; r::; R(I) for I> 0, Eg. (8-34) Eg. (8-29) Phase-change with R(t) calculated from the transcendent.al equation interface Eg. (8-35) The solutions for inward freezing of a cylinder, where cooling is applied at the outer radius To, are the same as the above-described ones for the outward-fre.,:zing <:)'linder, if the replacements r. ~ roo qo~ -qo, and h ---t-11 are made. Ifsuch a cylinder is not hollow then rj :0 0 is used. D. ESTIMATION OF FREEZING TlME Figure 8-5. Outward Freezing in a Hollow Cylinder Conltlinmg Liquid Initially at the Freezing TemperaJure There are a number of approximate formulas for estimating the freezing and melting times of different materials having (TJ), Subjected at us Inner Radius (rJ to a TemperaJure a variety of shapes. To If to the freezing for the same case occurs due the imposi 1. Freezing Time of Foodstuff tion of a negative heat flux qo at rj, The American Society of Heating, Refrigerating, and Ajr Conditioning Engineers (ASHRAE) provides a number of _ks(dT,) = qo(t) > 0 for t> O. &1. (8-30) approximations for estimating the freezing and thawing dr V times of foods (ASHRAE (4]), For example, if it can be the solution is assumed that the freezing or thawing occur at a single temperature, the time to freeze or !haw, If' for a body that qo (t )rj r has shape parameters [P and R (described below) and T () r,t =T f ---In- Eq. (8-31) S ks R(t) thermal conductivity k, initially at the fusion temperature Tr. and which is exchanging heat via heat transfer coefficient inrj :5: r:5:R(t) ror\>O, II with an ambient at the constant temperature Ta, can be approximated by Plank's equation , ]112 Ret) = 1/ + 2 .....2.-f qQ (l)dt for I > 0, Eq. (8-32) [ phis Eq. (8-36) o where qo(t) and his must be entered as negalive values. where d is the diameter of the body if it is a cylinder or a sphere, or the thickness when it is an infinite slab, and If tJle freezing for the same case occurs due to the j mposi where the shape coefficients [P and R for a number of body tion of a convective heat Dux to a fluid at the transient fonns are given in Table 8-1 below. GENIUM PUBUSHING Heat VAPO~ATION~HASECHANGE Section 507.8 Transfer FREEZING Page 17 Division INTRODUCTION, APPLICATIONS, AND BASIC PHYSICAL CONCEPTS August 1996* Table 8·1. . 2. Other Approximations for Freezing Time Shape Factors for Eq. (8-36), ASHRAE [4] Alexiades and Solomon [3) provide an easily-computable approximate equation for estimating the time needed to Fonns !Y R freeze a simple-shaped liquid volume initially at the freezing temperature T . It is assumed that conduction Slab 1/2 1/8 r occurs in one phase (the solid) only, that the problems are Cylinder 114 1116 axi- and spherically-symmetric for cylindrical and spherical Sphere 1/6 1/24 bodies, respectively, and that (he freezing process for differently shaped bodies can be characterized by a single Shape coefficients for other body forms are also available. geometric parameter, r, in the body domain 0 S; r S; L, To use Eq. (8-37) for freezing, k and p should be the values using a shape factor, w, defined by for the food in its frozen state. LA w=--1. Eq. (8-38) If the initial temperature (T;) of the material 10 be frozen is V higher than T and the surface temperature To is given (or r where A is the surface area across which the heal is re NSi ~ 00), the following simple formula for estimating the s moved from the body, and V is (he body volume, to account freezing lime If of an inrmitely-long cylinder of diameter d for the specific body shape, viz. and radius ro was proposed: N Fo,.[ = (0.14 + 0.085Y )+ (0. 252 - O. 0025 Yo )hl.d. o o for a slab insulated at one end Eq. (8-37) ro = I for a cylinder (Eq. (8-39) where 2 for a sphere. N Fo,! o $; ro ~ 2 always. and ro may be assigned appropriate values for shapes intermediate between the slab, cylinder. T-T Dimensionless TemperulUre,-'_f_. and sphere. For example. a football-shaped body, some -~ Tf where between a cylinder and sphere. may be assigned W = 1,5, and a short cylinder with a large diameter-to-height [n fact, freezing or melting of food typically takes place ratio may have W = 0.5. over a range of temperatures, and approximate Plank-type For the case where the temperature To < Tr is imposed on fonnulas have been developed for various specific food the boundary at 1=0, t.he time required for complete stuffs and shapes to represent reality more closely than Eq. freezing, 1m can be estimated by the equation (8-36) (Cleland et 0([167], ASHRAE [4]). EXAMPLE Using Plank's Equation (8-36) for estimating Eq. (8-40) freezing time estimate the time needed to freeze a claimed 10 be valid with an accuracy within 10% for fish, the shape of which can be approximated by a O~Nsles~4. cylinder 0.5 m long having a diameter of 0.1 m. The fish is initially al its freezing temperature, and during Validating by comparison 10 the results of an experimen the freezing process it is surrounded by air at Ta = tally-verified two-dimensional numerical model of freezing 25 "C. with the cooling performed with a convective of lake-shore water (with a mildly-sloped adiabatic lake heat transfer coefficient h = 68 W/m2K. For the fish, bottom) initially at the freezing temperature, Dilley and 3 Tr =-1 ·C, !lSi = 200 kJlk.g, Ps =992 kg/m , and ks = Lior [170] have shown that freezing progress can be 1.35 W/m K. estimated well by the following simple equations. For a constant heat flux qo from the top surface to the Using Table 8-1, the geometric coefficients for the ambient, (he relationship between the depth of freezing X(t) cylindrical shape of the fish are [p = )/4 and and time can be expressed as R = 1/16, while d is the cylinder diameter, =0.1 ffi. Substituting these values into Eq. (8-36) gives If = 200000·992 (1/4(0.1) + 1/16(0.1)2 J=6866 s =I. 9h. -1-{-25) 68 1.35 Eq. (8-41) GENIUM PUBLISHING Section 507.8 VAPO~TION~HASECHANGE Heat Page 18 FREEZING Transfer August 1996* PREDICTIVE EQlfATIONS" FOR FREEZINGWITHOur FLOW Division with an error < 0.02% for the first 100 hours. For convective cooling at the surface by air at temperature Ta and with a convective beat transfer coefficient h , the relationship between the depth of freezing XU) and time c-. be expressed as DI. PREDICTIVE EQUAnONS Eq. (842) Ji'OR FREEZING WITH FLOW where Freezing may occur when a liquid flows through a cooled . . _ hX(r) conduit or along a cooled wall where the conduit/wall BLOt number for the solid, = -ktemperature (Tw) is below the freezing temperature of the s liquid (Tw < Tr, Fig. 8-6), The heat balance at the phase -2 change interface can be expressed as h Nsu Dimensionless time parameter, - • ( k,Pscs ' aT,) CJR(x,1) q[x, R(X,f) ] + k, -a =Pshfs a ' Eq. (8-49) ( r [::c.R(x.rJ] r valid for small values of NStes (the quasi-static approxima tion). where the fIrst and second terms on the left-hand side of the When the top surface is subject to a combination of equation account for tbe heat flow from the flowing liquid constant aDd of a convective heat flux so that the total and the frozen solid, respectively, to the phase-change heat flux, qtol' there is interface, and the teon on the right-hand side expresses the rate of latent heat release due to the increase in the frozen Eq. (8-43) layer thickness. The first tenn in Eq. (8-49), i.e. the heat transfer from the flowing liquid. can be expressed as Eg. (8-43) is applicable also for this case, if NI is expressed by q[ x. R(x, I)] = ii[ I;; (x, r) - Tf ] Eq. (8-50) Eq. (8-44) for convection from a liquid at temperature Tc(x,t) with a convective heat transfer coefficient b . In the lake-freezing simulation, this expression was found ql~.R(X.I») to represent the data well up to the time when the ice depth Liquid became 2 m. --+- I Dow. Iffreezing started when the ice layer already had a thick I: ness Xi (at time (i), the approximate solution becomes 1 WalI.Tv.' N8I,(t)=-I+[1+Bi;'X +2Bis• , +2Nr(r)p, Eq.(8-45) 1 x _), where Nl is defined by Eq. (8-44). Figure 8-6. Freezing DUring Liquid Fww Over a Cold Wall When the top surface is subjected 10 a combination of constant and of a convective heat flux, where the air Freezing occurs when the the term on the right-hand side of temperature varies linearly with time as Eq. (8-49) is negative. happening if at least one of the tenns on the left hand side of the equation is negative and larger T,,(t) = T • +ml, Eq. (8-46) a 1 in its absolute value than the other. Obviously. freezing can where thus occur if the interface is cooled on both sides (when T, < T and thus q 0, and also T T/and thus the m the time-variation constant of the temperature T , f < w < a gradient in the conduction leon is negative). It can. how Ta.i the initial air temperature, ever, occur even when the flowing fluid temperature is the lotal heat flux there, qlOl' is higher than T if the cooling rate through the fro7.-en layer is f high enough. or when the tube wall temperature is higher Eq. (8-47) than T/f the cooling rate q into the flowing fluid (when T, < T ) IS large enough. The approximate expression relating X(t) to time is slill Eq. f (8-42), but with Nl defined as Since lhe flow boundary and cross section keep varying during phase-change, the nature of the flow. including its GENIUM PUBLISHING Heat VAPORQATION~HASECHANGE Section 507.8 Transfer FREEZING Page 19 Division PREDICTNE EQUAnONS FOR FREEZING WITHOUT FLOw August 1996* velocity, as we)) as the consequent effects on heat transfer, decreasing hi, and stops when the convective heat transfer also vary. For example, inward freezing in a cooled tube at the interlace is equal to the conducti ve one in the ice would progressively diminish the flow cross section and layer. The flow thus becomes laminar again. wlUch brings increase the flow pressure drop (Fig. 8-7). If, as often found anolller such freezing-melting cycle about, generates in practice, the given flow head is constant, freeZing would another ice band downstream of llle first one, and so on. In result in a gradual decrease of the flow rate. Figure 8-8 also addition to such changes in the interface shape. dendritic shows the experimentally-observed fact that some re growth of the solid phase-especially prominent when the melting of the frozen layer occurs at the ex.it from the liquid is sUbcooled-wili create interface roughness On a partially-frozen region, due to the flow expansion there. smaller scale. Some references on the effects of freezing on flow, pressure Even when the above-described interface shape variations drop, and conditions leading to complete flow stoppage in are not taken into account, no analytical solutions for the conduits due to freezing, are cited in Subsection 1. D. above. complete flow-accompanied melting/freezing problem arc available. Many numerical, experimenlal and approximate results have, however, been reponed in the literature and listed in the above mentioned reviews (especially see citations (79)-[ 152] in Subsection I. C.). One useful solution is shown here. for the case of a fluid at the radially average entrance temperature Tc > Tr flowing along a flat plate, or in a tube of jnlemal radius r" which are 1+------L,-----~- convectively-cooled on their exterior surface by a fluid at temperature T < T with a convective heat transfer coeffi FilJUre 8-7. A Freezing During Liquid Flow in a Tube a r cient 11 0 (Fig. 8-9). Neglecting heat conduction in the axial direction, an approximate collocation-type transient I Solidilicalion zone ------i"I solution. which accounts for the motion of the phase change l.u_._ interface and for the heat conduction in the frozen layer was ~._.~~. ~. • ,. -'--.:' .--;:::i Section 507.8 VAPORQATION~HASECHANGE Heat Page 20 FREEZING Transfer August 1996* SOME METHODS FOR SIMPLIFYING SOLUTION Division 11 geometry index, n = 0 for flow along a flat IV. SOME METHODS FOR wall, n = 1 for flow in a cylindrical tube SIMPLIFYING SOLUTION Bi Biot number for internal heat transfer to the phase-change interface, = h;r; / kg,dimen A. THE INTEGRAL METHOD, WITH SAMPLE sionless SOLUTION FOR FREEZING OF A SLAB dimensionless length parameter, A simple approximate technique for solving melting and freezing problems is the heat balance integral method (Goodman (l81]), which was found to give good results in many cases. The advantage of this melhod is that it reduces = the second-order partial differential equations describing lhe problem to ordinary differential equations that are much q*[R*(x*, 't)] dimensionless heat flux from lhe liquid easier to solve. This is accomplished by guessing a tempera stream to the interface, ture distribution shape inside the phase-change media, but = q[R(x,t)]r;!(p.hL,-CZs) making them satisfy the boundary conditions. These temperature distributions are then substituted into the partial q[R(x,t)] heat flux from the liquid stream to lhe differential energy equations in lhe liquid and solid, which 2 interface, =q:r;.(x,r)-T/]. W 1m . are then integrated over lhe spatial parameter(s) (here just x) in the respective liquid and solid domains. This results in ordinary differential equations having time (t) as the independent variable. The disadvantage of the method is clearly the uncertainty in the temperature distribution wilhin T the media. This technique is introduced here by applying it to a useful case, and the reader can thus also learn to apply it to other cases. Consider, as shown in Fig. 8-10, a liquid initially at an above-freezing temperature (Ti > Tr) confined in a space ~ x $ L, with the surface at x = 0 subjected for time 1 > 0 Phase o change to a below-freezing temperature To < h and the surface at x interface =L is perfectly insulated, «(JTl(Jxkl = O. Freezing thus ~ starts at x =0, and lhe freezing front, as shown in Fig. 8-9, Liquid coolant is moving rightward. This problem has no exact solution, flow (T - T ho • w )T and is thus a good example for lhe application of the • integral method described in lhis section. Figure 8-9. Sl«:tch ofthe Phase-Change Problem ofa Flowing Liquid 011 a Cooled Pkmar Wall or Imide a LiqUid Cooled Tube, with NotalUJIIs for Stephan's [l41] Solution [Eq. (8-51) Insul;>.LCd surface In this solution, the internal heat flux tenn q{R(x,t)J must be Frtoz.ing specified by the user, since the solution here does not Ftonl include consideration of the flow momentwn equations. [0 (O.l) t In the derivation and discussion, the following dimension less parameters are used: GENIUM PUBUSHING Heat VAPORQATION~HASECHANGE Section 507.8 Transfer FREEZING Page 21 Division SOMEMETHODSFORS~L~GSOLUTION August 1996* T- -To where A. is a parameter yet to be delermined. The reader can 8 s_J__, wherej=s, t, or!, j Eq. (8-52) easily prove thaI lhis is indeed the solution of Eqs. (8-55), T; -To (8-56). Solid LiqUId Eq. (8-53) and the problem is re-sketched in tenns of these dimension less parameters in Fig. 8-1 L For the example at hand, the a, It is also assumed that the position of the phase change interface is defined by an expression similar to Eq. (8-13), with the boundary condition Eq. (8-62) ae, - =Oat X= I, for N Fo >0, Eg. (8-57) The value of A. as well as the temperature distribution in the aX ' liquid phase are now deteJ1Tlined by the integral method, as and the initial condition follows. A thermal "boundary layer". b(NFo.) is defined at an x-location where the liquid temperature is still at its e,=linO Eg. (8-59) Eq. (8-63) and ass k( (lOt his cJ;(NFo,) Eg. (8-60) ax -k" ax =cs(1i- T,,) dN Fo, Eq. (B-64) at X=: N for N > O. S( Fa,), Fa, respectively. Note that a solution is valid only if o(NFn):s; I. Now the differential energy equation for the liquid, Eq. The next step, as explained above, is to choose temperature (8-57), is integrated in the liquid phase domain from c,(N ) distributions in the two phases. Obviously, the closer the Fo to o(N ), and the boundary conditions represented by Eqs. chosen distributions are to the actual (but unknown) ones, Fo (8-58), (8-64) and (8-65). giving the expression the better the solution would be. A reasonable guess (although other ones can be tried) in the solid phase is the exact solution obtained for freezing a semi-infinite liquid initially at an above-freezing temperature, shown in Eq. (B )1), which is here, in its dimensionless form, erf( X ) eAX~NFO,) :~, = for Npo, > 0, Eg. (8-61) Eq. (8-65) f GENIUM PUBLISHING Section 507.8 VAPORIZATIONIPHASE CHANGE Heat Page 22 FREEZING Transfer August 1996* SOME METI-IODS FOR SIMPLIFYING SOLUTION Division is the for this problem. An the valueN 1I(4A2), and [with Eq. (8-67)] that the This energy-integral equation Fos = appropriate temperature distribution must be chosen for validity of this particular integral solution is confined to completing the integration. For example, the polynomial dimensioruess limes for which 0 ~ I, corresponding to N ~ 1/(4~2). distribution Fos Many integral solutions yield good results, with errors within a few percent. The accuracy, as mentioned above, St(x,NFO,) =1-(I-e/)( ~=~J" Eq. (8-66) depends on the closeness of the chosen temperature distributions to the real ones. Experience from previous where 1/ ~ 2 is the power of the polynomial, satisfies the successful solutions or experimental results naturally boundary conditions, Eqs. (8-58), (8-63) and (8-64). It is improves this choice. Additional information about this also assumed that O(NFa ) is related to NFo through the s s method can be found in refs. [l81], [51]. and [210]. relation s( N Fo,) = 2~~ N Fa, ' Eg. (8-67) B. THE ENTHALPY METHOD with the parameter Pto be determined. It is noteworthy that one of the biggest di fficulties in The temperature distribution (Eg. (8-66)] is substituted into numerical solution techniques for such problems is the need the energy integral equation (8-65), which is integrated to track the location of the phase-change interface continu using Eq. (8-67) to yield ously during the solution process, so that the interfacial conditions could be applied there. One popular technique that alleviates this di fficu lty is the enthalpy method (refs. p_A.=Il+I[_A+ A.2+~at]. 2 1/ + I as Eq. (8-68) [3], [72], (154], [1723 174), [179J, [190J, [228), [240], [259] and [260]), in which a single panial differential Next, the temperature distributions in the solid and liquid, equation, using the material enthalpy instead of the tem Eqs. (8-61) and (8-66), respectively, are introduced into the perature, is used 10 represent the entire domain, including interfacial condition Eqs. (8-59) and (8-60), to yield the both phases and the interface. Based on the energy equa following transcendental equation for the unknown param tion, just as Eqs. (8-55) and (8-57), the one-dimensional eter A mehing problem is thus described by all a1 11 . p- = k-- In X ~ 0, for r > 0, Eq. (8·73) at ax 1 where the temperature-enthalpy relationship is expressed by where II h ~cTf (solid) Ii + 2 c ZI\=---Y+Y+--'I ( 2" J Eg. (8-70) n..f;. Ii + I T= T (i nterface) I cTr < 11 < cTt + hls Eq. (8-74) h-hl5 Ii ~ cr/ + Ills (liquid) 112 c ~; Eq.(8-71) Y=A( ) The numerical computation scheme is rather straightfor ward: knowing the temperature, enthalpy and thus from Eq. the phase of a cell at time step j, the enthalpy at lime er-I ~ Tf - T, (8-74) Eq. (8-72) step (j + 1) is computed from the discretized version of Eq. a T - To f f (8-73), and then Eq. (8-74) is used to determine the tem Solution of Eg. (8-69) manually, or easily done by one of perature and phase at .that new lime. If a computational cell many soflware packages available for solving nonlinear i is in the mushy zone, the liquid fraction is simply hi I hsi ' algebraic equations, yields the val ue of A. This and Eq. Care must be exercised in the use of the enthalpy method ~, (8·68) yield the value of and thus the transient position when the phase change occurs over a very narrow range of of the freezing front, S(NFo), can be calculated from Eq. 5 temperatures. Oscillating non-realistic solutions were (8-62), and the temperature distributions in the solid and obtained in such cases, bu1 several modi ficalions (see above liquid can be calculaled from Eqs. (8-6J) and (8-66), references) to the numerical formulation were found to be respective Iy. reasonably successful. Inspection of Eq. (8-62) also indicates Ihat lhe slab would be completely frozen when the dimensionless time reaches GENIUM PUBLISHING Heat VAPOillZATION~HASECHANGE Section 507.8 Transfer FREEZING Page 23 Division APPLICATIONS BIBLIOGRAPHY August 1996* V. APPLICATIONS BIBLIOGRAPHY interface: comparison of the model with experiments. J. Materials Synth, Processing, vol. 3,1995, pp. 203-2l1 An extensive-yet by no means complete-bibliography Coupard, D., Girol, F., Quenisser, I.M. Model for predicting the identifying papers and books that treat freezing in the main engulfment or rejection of short fibers by a growing plane areas in which it takes place, is presented below. The solidification fronl. 1. Materials Synthesis and Processing, classification is by application, and the internal order is vol. 3. 1995, pp. 191-201 alphabetical by author. DiLellio, I.A., Young, G.W. Asymptotic model of {he mold region in a cOnlinuous steel caster. Metallurg. Materials Trans. B, vol. 26, 1995, pp. 1225-1241 A. CASTING, MOLDING, SINTERING Dogan. C. and Saritas, S. Metal powder production by centrifugal Freezing (solidification) is a key component in casting. molding. atomization. [m. J. Powder Metallurgy. vol. 30, pp. 419-427 and production of solid shapes from powders by processes such as Edwards, M,F., Suvanaphen, P.K. and Wilkinson, W.L. 1979. sintering and combustion synthesis. The materials include metals, Heatlransfer in blow molding operations. Polym. Engng. polymers, glass, ceramics, and superconductors. Flow of the Sci.. vol. 19; pp. 910-916. mollen material. the course of its solidification (including volume changes due to phase transition, and internal streSs crcalion), and Gau, C. and Viskanta, R. 1984. Melting and solidification of a the evolving surface and interior quality, are all of significant metal system in a rectangular cavity. Int. 1. Heat Mass industrial importance. In production of parts From powders, the Transfer, vol. 27, p. 113. conditions necessary for bonding of the particles by melting and Gilolle, P., Huynh, L.Y., Etay, 1., H3mar, R. Shape of the free resolidifcation are of importance. Such diverse processes as surfaces of the jet in mold casting numerical modeling and spinning and wire-making are included. Much anention has lately experiments. 1. Engng Materials and Techno!.. vol. 117, pp. been focused on the manufacturing of materials for superconduc 82-85 tors. Grill, A., Sorimachi. K. and Brimacombe, 1, )976. Heat flow, gap formation and breakouts in the continuous casling of steel Abouta1ebi, M. R.; Hasan. M .., Guthrie. R.I.L. lun 1994. Thermal slabs. Melall. Trans. B, vol. 7B, pp. 177-l89. modelling and stress analysis in the conlinuous casting of Gupta. S.c. and Lahiri. A.K. 1979. Heal conduction wilh a phase arbitrary sections. Steel Research vol. 65. pp. 225-233. change in a cylindrical mold. Int. J. Eng. Sci., vol. 17, pp. Akiyoshi, R., Nishio. S. and Tanas3wa, I. 1992. An allempllO 401-407. produce particles of amorphous materials using steam Heggs. P.I., Houghton. 1.M.. Ingham, D.B. 1995. Application of explosion. In Heal and Mass Transfer in Materials Process• the enthalpy method to the blow moulding of polymers. ing, ed. I. Tanasawa and N. Lior, pp. 330-343. Hemisphere, Plastics, Rubber and Composites Processing and Appl., vol. New York. 23, pp. 203-210 Assar, A. M.; Al-Nimr. M. A. 1994. Fabrication of metal matrix Hieber, C.A. 1987. Injeclion and Compression Molding Funda composite by infiltration process - part I: modeling of mentals.. A. J. Isayev. ed.• Marcel Dekker. New York. hydrodynamic and thermal behaviour. 1. Composite Materi. 1acobson, L. A. and McKiIIrick, J. 1994. Rapid sol idification a1s, vol. 28, pp. 1480-1490. processing. Materials Sci. Engng R. vol. II, pp. 355-408. Beckett, P.M. and Hobson. N. 1980. The effect of shrinkage on the Khan. M.A.. Rohatgi. P. K. Numerical solution to the solidi fica• rate of solidification of a cylindrical ingot. Int. J. Heal Mass tion of aluminum in thc presence of various Iibres. J. Transfer, Vol. 23, pp. 433-436. Materials Sci., vol. 30,1995, pp. 371 1-3719 Bennett, T,; Poulikakos. D. 1994. Heat transfer aspects of splat King, A.G.: Keswani. S.T. 1994. Adiabatic moulding of ceramics. quench solidification: modelling and experiment. 1. Malerials Am. Ceramic Soc. Bull.. vol.73, pp. 96·100 Science, vol. 29. pp. 2025-2039. Kroeger. P.G. 1970. A heat transfer analysis of solidification of Bose, A. Technology and commercial status of powdcr-mjection pure metals in continuous casting process. Proe. 4th [nt. Heat molding. 10M. vol. 47, pp. 26-30 Tr. Conf.. vol. I, paper Cu 2.7. Bushko, Wit C., Stokes. Vijay K. Solidification of Ku rosaki, Y. and Satoh, J. 1992. Visualization of flow and Ihermoviscoelastic melts. Part I: Fonnulation of model solidification of polymer melt in the injection molding problem. Polymer Engng Sci., vol. 35, 1995, pp. 351-364 process. In Heat and Mass TransFer in Materials Processing, Bushko, Wit C., Stokes, Yijay K. Solidification of I. Tanasawa and N. Lior, eds., pp. 315-329. Hemisphere, New thennoviscoelastic melLS. Part II: Effects of processing York. N.Y. conditions on shrinkage and residual stresses. Polymer Engng Mangels, 1. A. 1994. Low-pressure injection moulding. Am. Sci., vol. 35, 1995. pp. 365-383 Ceramic Soc. Bull., vo1.73. pp. 37-41. Clyne, T.W. 1984. Numerical treatment of rapid solidification. Mauch. F. and laclde, J. 1994. 111ermoviscoel,lslie theory of Metall.. Trans., vol. 15B, pp. 369-381. freezing of stress and strain in a symmetrically cooled infinite Cole. G.S. and Bolling. G.F, 1965. The importance of natural glass plale.. J. Non-Crystalline Solids. vol. 170. pp. 73-86. convection in casting. Trans. Melallurgical Soc.-AlME, vol. Massey, l. D. and Sheridan, A.T. 1971. Theoretical predictions of 233, pp. 1568-1572. earliest rolling limes and solidification times of ingots, 1. Iron Cole, G.S. and Bolling, G.F. 1966. Augmented natural convection Steel [nst., Yol. 209, pp. 111-35\. and equiaxed grain su'ucture in castings. Trans. Metallurgical McDonald, R.J. and Hunt, 1.D. 1970. Convective fluid motion Soc.-MME, vol. 233, PP. 1568-1572. within the interdendritic liquid of a casting. Trans TMS Coupard, D., Girot, F., Quenissct, 1.M. Engulfment/pushing AIM E. vol. I. pp. 1787-1788. phenomena of a fibrous reinforcement at a planar solidlJiquid GENIUM PUBLISHING Section 507.8 VAPO~ATION~HASECHANGE Heat Page 24 FREEZING Transfer August 1996* APPLICATIONS BIBLIOGRAPHY Division MonOleilh, B.G. and Piwonka, T.S. 1970. An investigation of heat Metallurgica et Materialia, vol. 33. 1995, pp. 39-46 flow in unidirectional solidification of vacuum cast airfoils. J. Siegel, R. 1978. Analysis of solidification interface shape during Vacuum Sci. Tech., vol. 7, pp. S126. continuous casting of a slab. Int. J. Heal Mass Transfer, vol. Nishio. S., [namura. S. and Nagai, N. 1994. Solidified-shell 21. pp. 1421·1430. formation process during inunersion process of a cooled solid Siegel, R. 1978. Shape of lwo-dimensional solidificalJon interface surface. Trans. Japan Soc. Mech. Engrs B. vol. 60. pp. 4165 during directional solidification by continuous casting. 1. 4170 Heat Transfer, vol. 100, pp. 3-10. Niyama. E.• Anzai. K. 1995. Solidification velocity and tempera Siegel. R. 1983. Cauchy method for SOlidification interface shape lure gradient in infinitely thick alloy castings. Materials during conlinuous casting. Trans. , 1. Heat Transfer. vol. 105, Trans.• JIM, vol. 36. pp. 61-64 pp. 667-671. Ohnaka. J. 1985. Melt spinning into a liquid cooling medium. 1nL Siegel, R. 1984. Solidi fication interface shape for cont inuous J. Rapid Solidification, vol. I, pp. 219-238. casting in an offset model-two analytical methods, J. Heat Ohnaka. I. 1988. Wi res: Rapid solidi fication. in Encycl. Material s Transfer. vol. 106. pp. 237-240. Sci. Engng. Suppl. vol. L R.W. Calm, ed.• pp. 584-587.. Sri vatsan, T.S., SUdarshan. T.S., Lavernia, EJ. Processi ng of Pergamon Press. discontinuously-reinforced metal matrix composites by rapid Ohnaka, I. and Shimaoka, M. 1992. Heatlransfer in rotaling liqoid solidification. Progr. Materials Sci.. vol. 39. \995. pp. spinning process. In 317-409 Heat and Mass Transfer in Matl:rials Processing. I. Tanasawa and Szekely, 1. and Dinovo, S. T. 1974. Thermal criteria for lUndish N. Lior, eds.. pp. 315-329. Hemisphere. New York. N.Y. nozzle or taphole blockage. Metall. Trans.. vol. 5, pp. 747 O'Malley. R.I.; Karabin. M.E.; Smelser, R.E. 1994. Thc roll 754. cast ing process: Numerical and el> perimelllal resolts. 1. Szekely, l. and Stanek. V. 1970. On the heat transfer and liquid Materials Process. Manuf. Sci.. vol .3. pp. 59-72. mixing in the continuous casting of steel. Metallurgical Papathanasiou. T.D. Modeling of injection mold filling: effect of Trans.. vol. I, pp.119.. undercooling on polymer crystallization. Chern. Eng. Sci .. Thomas. B.G. 1995. rssues in thermal-mechanical modeling of vol. 50, pp. 3433-3442 casting processes. [SI] 1m.• vol. 35. pp. 737-743 Patel, P.O. and Boley. B.A. 1969. Solidification problems with Upadhya, G.K., Das. S.. Chandra. U., Paul. AJ. 1995. Modeling space and time varying boundary conditions and imperfect the investment casting process: a novel approach for view mold contact. Int. J. Eng. Sci.. vol. 7. pp. 1041- 1066. factor calculations and defect prediction. Appl. Math. Model.. PehJke. R.D., Kin, M.l.. Marrone. R.E.. and Cook. 0.1. 1973. vo1. 19, pp. 354-362 Numerical simulation of casting solidification. CFS Cast Yamanaka. A.. Nakajima. K.. Okamura. K. 1995. Critical strain Metals Research 1.. vol. 9. pp, 49-55. for internal crack formation in continuous C