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N. Lior, "Freezing", Section 507.8 in Heat Transfer and Fluid Flow Data Books, Genium Publishing Corp

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N. Lior, 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,
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