Heal vAPORIZATIONIPHASE CHANGE Section 507.9 Transfer MELTING Page 1 Division CONTENTS-SYMBOLS August 1996

Page

I. GENERAL A. Contents " _ _ 1 B. Symbols I C. References _ 2 D. Introduction, Applications, and Basic Physical Concepts " 7 E. Predictive Methods 8

II. PREDICTIVE EQUATIONS FOR MELTlNG A. One-Dimensional Melting of Pure Materials without DensilY Change 9

I. Solutions for Materials that are Initially at me Melting Temperalure 9 2. Solutions for Materials that are Initially nOt at the Melting Tempera!ure 10

a. Slab geometry - 10

b. Cylinder - _ I I B. One-Dimensional Melting with Density Change 12 C. The Qu.asi-SUl/ic Approximation 13

I. Examples of the Quasi-Slatic Approximation for Melting of a Slab 13

2. Examples of the Quasi-Static Approximation for Melling of a Cylinder L3 D. Estimation of Melting Time 14

L. Melting Time of Foodstuff 14 2. Other Approximations for Melting Time 15

III. SOME METHODS FOR SIMPLIFYING SOLUTION A. The Integral Method, wilh Sample Solution for Melting of a Slab 15 B. The Enthalpy Method ., 16

IV. APPLICATIONS BIBLIOGRAPHY A. Casting, Molding, Sintering 17 B. Mulli-Component Systems and Crystal-Growth 18 C. Welding, Soldering, aod Laser. EleclfOn-beam, and Elcclro-discharge Processing 19 D. Coating and Vapor and Spray Deposition 21

E. Ablalion and Sublimation 21 F. Medical Applications and Food Preservation 22 G. Nuclear Power Safety 23 H. Thermal Energy Slorage 23 1. Geophysical Phenomena and Melting in Porous Media 25

B.SYMBOLS

c specific heat. Jlkg K u velocity of lhe melt-front due to density change. mfs v specific volume. m 3/kg

f ·s..

V volume of the body. m3Ei(x);;; ~ds,X>OEi

x coordinate ~

X position of the melting fronl along the x-direction. m hst latent heal of fusion. Jlkg K h Greek Symbolsenthalpy h con vecti ve heat transfer coefficienl, WIm2s CI. Thermal diffusivity, m2/s

k thermal conductivity, W/m K o Melting front growth rale constant in lhe integral solution, Eq. (9-58)Nsu Stefan number, ;;; c(To . Tr)lhst

P pressure, Pa c..Vst (Specific volume of the liquid phase) - (specific volume of the solid phase), m 3/kg[j> shape coefficient in Plank's equation

q heat flux, J/m2s E@;r radius, m K Va:R radial position of the rnel{- frO!)t, m

Melting rate parameter, dimensionlessR shape coefficiem in Plank's equalion == p(lpslime. S

T temperature. K or "C

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I

Pagel MELTING Transfer

August 1996 SYMBOLS, REFERENCES Division

p Density, kg/m) rectangular enclosures: Experiments and numerical simula­

X Concentration, kglkg tions. J. Heat Transfer, vol. 107. pp. 794.

7. Bonacina. c.. Comint Go> Fasano, A.. and Primicerio. M.

Subscripts o At the surface (x=O) a Ambient (fluid surrounding the melting object) f Fusion (melting or freezing)

Initial (at t=O) f Liquid sf Phase-change from solid to liquid a outer s Solid w at wall

C. REFERENCES

There always was an incense interest in predicting melting phenomena as related to such applications as food preserva­tion, climate and its control, navigation, and materials processing, expanding with time into new areas such as power generation and medicine. Increasingly rigorous quantilative predictions started in the 19th century, and the number of published papers is in the tens of thousands. The main books and reviews on the topic, representative key general papers, as well as some of the references on appropriate thermophysicaJ and transpol1 properties, are listed as citations [ I J- [19 I] below. A funher rather exten~

sive, yet not complete, list of references is gi ven in subsec­tion rv below under the specific topics in which melting plays an important role. In addition to the identification of past work on specific topics, this extensive list of references also helps identify various applications in which melting plays a role, and the journals which typically cover the field. While encompassing sources from many countries, practically all of the references listed here were selected from the archival refereed literature published in English. Many pertinent publications on this topic also exist in other languages.

t. AJexiades, V. and Solomon, A. D. 1993. Mathem.atical Mod~

eling ofMelting and Freeting Processes. Hem isphere. Wash, inglOn, D.C.

2. ASHRAE (American Soc. of Heating. Refrigerating. and Air­Conditioning Engineers). 199D. Refrigeration. ASHRAE. Atlanta, GA.

3. ASHRAE (American Soc. of Heating. Refrigerating. and Air· Conditioning Engineers). 1993. Cooling and freezing times of foods. Ch. 29 in Fundamentals. ASHRAE, Atlanta. GA.

4. Bankoff, S. G. 1964. Heat conductionordiffusion with change of phase. Adv. Chern. Eng.. vol. 5. p. 75.

5. Barei ss, M. and Beer. H. 1984. An anal ytica! solu tion of the heat transfer process during meltmg of an un fixed solid phase change material inSIde a horizontal tube, Int. J. Heat Mass Transfer, vol. 27, p. 739.

6, Bernard. c., Gobin. D.. and Maninez. F. 1985. Melling in

1974. On the esti malion of thennophysical propel1ies in nonlinear heat conduction problems. lnt. J. Heat Mass Trans­fer, vol. 17. pp. 861--867.

8. Budh ia. H. and Kreith.. F. 1973. Heat transfer wi th me fling of freezing in a wedge. Int. J. Heat Mass Transfer, vol. 16. pp. 195-211.

9. BurldJard. R.; Hoffelner, W.; Eschenbach, R.C 1994. Recy­cling of metals from waste with thermal plasma. Resource5. Ccnservation and Recycling, vol. 10, pp.1 1- I6.

10. Clladam, 1. and Rasmussen. H., cds. 1992. Free Boundar)' Problems: Theory and Appticalions. Longman. New York.-

II. Chan, S. H.. Cho, D. H. and Kocamustafaogu Ilari. G. 198). Melli ng and solidi fication with intema! radiati ve transfer - a generalized phase change model. Int. J. Heat Mass Transfer. vol. 26. pp. 621-633.

12. Cheng, K. C. and Seki. N. eds. 1991. Free:;ing and Melting Hear Transfer in Engineering. Hemisphere, Washi ngton. D.C.

13. Cheung. F.B. and Epstein. M. 1984. Solidification and Mc!t­ing in Ruid Row. [n Adv. in Transport Processes Vol. 3, ed. A.S. Majumdar, p. 35-117. Wiley Eastern. New Delhi.

14. Cooper, A.R. 1985. Analysis of the continuous mehing of glass. 1. Non-Crystalline Sol.. vol. 73. pp. 463-475.

15. Crank. 1. 1984. Free and Moving Boundary Problems. Oxford University Press (Clarendon). London and New York.

16. Diaz. L. A. and Viskanta. R. 1986. Experimcms and analysis on the melli ng of a semi ·transparent material l1y r.:ldi ation, Wanne Stoffi.ibertragung. vol. 20. p. 311.

17. Dong. Z-F.. Chen. Z-Q.. Wang. Q.1. and Ebadian. M.A. 1991. Experimemal and analytical study of comact melting \n a rectangularcavity. J. Thennophysics Heat Transfer. vol. 5. pp. 347-354.

18. Egolf, P. W. and Manz, H. 1994. Theory and modeling of phase change materials with and without mushy regiuns. Int. J. Heat Mass Transfer. vol. 37. pp. 2917-2924.

19. Fang. Z.-H. and Chen. L.-R. 1994. Simplified treatment to calculate the melting temperature of metals under a hlgh pressure. J. Physics Condensed Matter. voL 6. pp. 6937-694.

20. Fasano, A. and Primicerio. M.. cds. 1983. Free Boundarv Problems: Theory and Applications. Pitman. London. .

21. Frenken. J. W.M. and van Piruteren, H.M. 20 1994, Surface melting: dry, slippery, wet and faceted surfaces. Surface SCI.. Yol. 307-09. pt B, pp. 728-734,

22 Friedman. A. and Boley, B.A. 1970. Stresses and deforma­lions in melting plates. 1. Spacecrafl Rockets. vol. 7. p. 324

23. Fukusako. S, and Yamada. M. 1993. Recent advances in research on water-freezing and ice,melting problems. E"p. Thennal flUId Sci.. vol. 6. pp. 90, 105.

24, Ghatee. M.H .. Boushehri. A. An analytical equation of state for molten alkali metals. [nt. 1. Thermophyslcs. vol.l6. pp.1429.1438.

25, GiIpi n, R. R.. Robertson. R. B.. and Singh, B. 1977. Radiati vc heating III ice. 1. Heat Transfer. vol. 99. pp. 227-232,

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Heat VAPoroZATION~HASECHANGE Section 507.9 Transfer MELTING Page 3 Division REFERENCES August 1996

26. Goodman, T.R. and Shea. lJ. 1960. The melting of (inite slabs. J. Appl. Mech.. vol. 27, pp. 16-24.

27. Hasan, M., Mujumdar, A.S. and Weber, M.E. 1991. Cyclic melting and freezing. Chern. Eng. Sci., vol. 46. pp. 1573-1587.

28. Hasebe, M.; Murakami, N.; Kobayashi. T. 1995. Personal database for phase diagrams of Phase Equilibria, vol. 16. pp. 396-402.

29. Herrington. E.F.G. 1963. Zone Me/ring o/OrgaJlic Com­pOlUlds. Wiley, New York.

30. Hersht, 1. and Rabin. Y. 1994. Shear melting of solid-like boundary layers in thin liquid films. 1. Non·Crystalline Solids. vol. [72·7, pt 2 pp. 857-86.

31. Hilbig, G. 1990. Three-dimensional analysis model for the straight-through flow in an all-electric glass melting furnace. Glastechnische Berichte. vol. 63, pp. 185-192.

32. HiIl,l. 1987. One-dimensional StefOll Problems: An /JltrodtK­tion. Longman Sci. Tech. Pub!.. Burnt Mill.

33. Ho. C. 1. and Viskanta, R. 1984. Heal tratlsfer during inward melting in a horizontal CUbe. In<.l. Heat MassTransfer, voL 27, p.705.

34. Ho, c.-I. and Viskanta, R. 1984. Heat Transfer during melting from an isothermal vertical wall. J. Heat Transfer, vol. 106. pp. 12-19.

35. Ho, C.-I. and Viskanta, R. 1984. Inward solid-liquid phase change heat uansfer in a rectangular cavity with conducting vertical walls. In(. J. Heat Mass Transfer. vol. 27. p. [055.

36. Hong, J. S. and Rubinsky. B. 1995. Phase transformalion in materials with nonuni form phase transition temperarures. 1. Hear Transfer, vol. I L7, pp. 803-805.

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61. Sparrow, E. M .. GUrlcheff, G. A., and Myrum. T. A. 1986. Correlation of melting results for bolh pure substances and impure substances. I. Heal Transfer. vol. 108, p. 649.

62. Sparrow. E. Moo Ramadhyani. 5 .. and Palankar. S. V. 1978. Effecl of subcooling on cylindrical mel ling. 1. Heat Transfer. vol 100, p. 395.

63. Springer. G. S. 1969. The effecls of axial heal conduclion on the freezing or melting ofcylinders. In!. J. Heat Mass Transfer, vol. 12. pp. 521-524.

64. Srefan.l. 1891. Ueber die theoriedereisbildung. insbesondere ueber die eisbildu ng im polarmeere. Ann. Phys. Chern., N.5. vol. 42. pp. 269-286.

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66. Tkaehev. A. G. 1953. Problems of Heal Transfer during a Change of State: A Collection of Articles. AEC·lr-34-05. pp. J69-178. translaled from a publicallon of the State Power Press. Moscow.

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field effect 00 !.he solid-mel! phase transformations. IEEE Trans. Magnetics, vol. 30. pI 2 pp. 1121-1123.

68. Vanfleet. R.R. and Mochel. I.M. 1995. Thermodynamics of melting and freezing in small particles. Surf. Sci.. vol. 341 . pp. 40-50.

69. Visk.ant.a. R. 1985. Natural convection in melting and solidi­fication. Narura! Convection: Fundamentals and Applica­tions, pp. 845. (S. Kakac. W. Aung, and R Vislcama, eds.). Hemisphere, Washington. O. C.

70. Webb. B. W. and Viskama. R. 1986. Analysis of beat transfer during melting of a pure metal from an isothermal venical wall. Num. Heat Transfer. vol. 9, pp. 539-558.

71. Wen.C.. Sheffield.I.W., O'Oell, M.P.. and Le!and.1-E. 1989. Analytical and experimental investigation of melting heat transfer. 1. Thennophys.. vol. 3. pp. 330-339.

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75. Zhang, G.-P.. Iiji. L. M., and Weinbaum, S. 1985. Quasi·three dimensional steady state analytic solution for melting or freeZing around a buried pipe in a semi-lOfinite medium 1 Heat Transfer vol. 107. p. 245.

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77. Chiesa. F. M. and Guthrie. R. L L. 1974. Natural convection heal transfer rates dUring the solidification and melting of metals and alloy systems. J. Heat Transfer. vol. 96. pp. 377­384.

78. Dietsche. C. and Muller. U. 1985. Influence of Bernard convection on solid-I iquid interfaces. I. Fluid Mech.. vol. 161, pp.249.

79. Epstein. M. and Cheung. F. B. 1983. Complex freeZIng­melting interfaces in (Juid now. Annu. Rev. Auid Mech., vol. 15. p. 293.

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pp.199-214.

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128. Friedman, A. 1968. The Slefan problem in several space variables. Trans. Am. Math. Soc.. vol. 133. pp. 5 J-87: Correc­tion 1969, vol. 142. p. 557.

129. Friedman, A. 1982. Variational Principles (JIld Free Bound­ary Problems. Wiley. New York.

I30. Fukusako, S. and Seki. N. 1987. Fundamental Aspects of Analytical and Numerical Melhods on Freezing and Melting Heat-Transfer Problems. Ch. 7 in ArIIlual ReviewofNwnerico I Fluid Mechanics and Heat Trai!sfer Vol. I. ed. T. C Chawla. p. 351-402. Hemisphere. Washington. D.C

131. Furzeland, R. M. 1980. A comparative study of numerical methods for moving boundary problems. 1. Inst. Math. Appl.. vol. 26. p. 411.

132. Goodman, T. R. 1964. Application of integral methods to

transient nonli near heat transfer. In Adv./n Heat Transfer Vol. I. cd. T.F. Irvine and J.P. Hartnett p. 51-122. Academic Press, San Diego. CA.

I 33. Goodrich. L. E. 1978. Efficient numerical technique for one­dimensional thermal problems wllh phase change. Int. 1. Heat Mass Transfer. vol. 21, p. 615.

134. Gupta. R. S. and Kumar. A. 1985. Treatment of mulh-dimen· sional moving boundary problcms by coordinate transforma­lion. Int. 1. Heal Mass Transfer. vol. 28. pp. 1355-1366.

135. Hajl-Sheikh. A. and Sparrow. E. M. 1967. The solution ofheat conduction problems by probability methods. 1. Heat Trans· fer. vol. 89, PP. 121·131

136. Hogge, M. and Gerrekens. P. 1985. Two-di mensional deform­ing finite element mcthods for surface ablation. AIAA l. Vol. 23, p. 465.

137. Hsiao. l.S. and Chung. BT.F. 1986. Efficient algorithm for finite element solution to two-dimensional heat lfansfer with melung and freezing. 1. Heat Transfer, vol. 108. pp. 462---464.

138. Hsu. C. F.. Sparrow. E. M. and Patan!<ar. V. 198 I. Nu merical solution of moving boundary problems by boundary immobi­lizalion and a conlrol- volume based. finite difference scheme. Int. J. Heat Mass Transfer. vol. 24, pp. 1335·1343.

1)9. H~u. Y. F.. Rubinsky. n.. and Mahin. K. 1986, An Inversc

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finite element method for the analysis of stationary arc weld­ing processes. J. Heat Transfer. voL 108. pp. 734-741.

140. Hu. H.. ArgyropouJos. A. 1995. Modeling ofStefan problems in compLex configurations involving [wo different melals using the enthalpy method, Modeling and Simulation in Ma­terials Sci. and Engng, vol. 3. pp. 53-64.

141. Hyman, J.M. 1984. Numerical methods for uacking inter· faces. Physica D, vol. 120. pp. 396-407.

142. Kern, J. 1977. A simple and apparently safe solution 1O the generalized Stefan problem. loe. 1. Heat Mass Transfer. vol. 20. pp. 467-474.

143. Kern, J. and Wells, G. 1977. Simple analysis and working equations for the solidification of cylinders and spheres. Mel. Trans.• Vol. 8b, pp. 99-105.

144. Kim, c.J. and Kaviani. M. 1992. A numerical method for phase-change problems with convcclioll and diffusion. [nl. J, Heat and Mass Transfer. Vol. 36, pp.457-467.

145. Kim. CJ., Ro. S.T. and Lee. 1.5.1993. An efficient computa· tional technique 10 solve the moving boundary problems in lhe axisymmetric geometries. lnt. J. Heat Mass Transfer. Vol. 36. pp. 3759-3764.

146. Lazaridis, A. 1970. A numerical solution of the multidimentional solidification (or melting) problem. 1m. J. Heat Mass Transfer, vol. 13, pp. 1459-1477.

147. Lee. R.-T .. Chiou, W.-Y. Finite-element analysis of phase-change problems using multilevel techniques. Num. Heat Transfer B. vol. 27, 1995, pp. 277-290.

148. Lee. SL and Twng, R. Y. 1990. An enthalpy formulation lor phase change problem with large thermal diffusivity jump across the interface. Int. 1. Heal Mass Transfer. vol. 33. pp. 1237-1247.

149. Li. D. 1991. A 3-dimensional hyperbolic Stefan problem with discontinuous temperature. Quart. Appl. Math., vol. 49. pp. 577-589.

150. Little, T.O. and Showalter. RE. 1995. Super-Stefan problem. Int. J. Engng Sci., vol. 33, pp. 67-75.

151. Majchrzak, E., Mochnacki, B. Application of the BEM in Ihe thermal lheory of foundry. Engng Analysis with Boundary Elements. vol. J6, 1995, pp. 99- 121.

152. McDaniel, O. J. and Zabaras. N. 1994. Least-squares from­tracking finite element meulod analysis of phase change wilh na,tural convection. Int. 1. Num. Meth. Engng, vol. 37. pp. 2755-2777.

153. Meirmanov. A.M. 1992. The Stefan Problem. Walter de Gruyter. Berlin-New York.

154. Mennig. J. and Ozisik. M. N. 1985. Couplcd integral cquallon approach for sol ving melli ng or solidi tication. !01. J, Heal Mass Transfer. vol. 28, p. 1481.

155. Meyer, G. H. 1973. Multidimensional Stefan problems. SIAM J. Num. Anal., Vol. 10. pp. 522-538.

156. Meyer. G.H. 1977. An application of the method of lines 10

multidimensional free boundary problems, 1. [nsl. Malh Appl .. vol. 20, pp. 317-329.

157. Meyer. G.H. 1978. Direct and iterative one-dimensIonal fronl tracking methods for the two·di menslonal Slefan problem Mum Heac Transfer. vol. I. pp, )5 '-364

158. Meyer. G.H. 1981. A numerical method forlhe solidification of a binary alloy. tne. J. Heac Mass Transfer. vol. 24. pp. 778­781.

159. Mingle. 1. 1979. Stefan problem sensitivity and uncertainly. Num. Heal Transfer. vol. 2. pp. 387 -393.

160. Morgan. K., Lewis. R.W.• and Zienkiewicz. O.c. 1978. An improved algorithm for heat conduction problems with phase change. Ine. J. Num. Methods Engng, vol. 12. pp. [19 1- 1/95.

161. Mori. A. And Araki, K. 1976. Methods for analysis of the moving boundary-surface problems. Jill Chern. Engng.. vol. 14, No.4. pp. 734-743.

162. Ockendon. 1. R. and Hodgkins. W. R.. eds. /975. Moving Boundary Problems in Heal Flow an4 Diffusion. Oxford Universily Press (Clarendon). London.

163. Pharo. Q. T. 1.9&5. A fast unconditionally stable tinite-differ­ence scheme for the heat conductions with phase change. [nt.

J. Heat Mass Transfer. vol. 28. p. 2079.

164. Rabin. Y. and Korin. E. 1993. Efticient numerical SOIUllOn for the multidimensional soliditication (or melting) problem us­ing a microcomputer. Int. 1. Heat Mass Transfer. vol. 36. pp. 673-683.

165. Ramos. M., Sanzo P.D.. Aguirre-Pucme. J. and Posado. R. 1994. Use ofche equivalent volumetric enchalpy variation in non-linear phase.change processes: freezing-zone progres­sion and thawing time. [nl. J. Refrig.. vol. 17. pp.374-}80.

166. Rao. P. R. and Sastri. V. M. K. 1984. Efticienl numencal mcthod for cwo dimensional phase change problems. Inc. J. Heat Mass Transfer. vol. 27. pp, 2077-2084,

167. Raw. M. J. and Schneider. G.E. 1985. A new implicit solution procedure for muhidimensionaltinite-<lifference modeling of the S'efan problem. Numerical Heat Transfer. vol. 8. pp. 559­57!.

168. RUbillshcein. L. I.. 1967. The Sfefan Problem. Translations of Mathematical Monographs. Am. Malh. Soc .. Vol. 27. Provi­dence, R.t 02904.

169, Rubi nsky, B. and Shi tzer, A. 1978. Analyl ical solulions 0 f Ihe heat equlllions invol vi ng a moving bounda ry with appl ication to lhe change of phase problem (The inverse Stefan problem), J. Heat Transfer. vol. 100. p. 300-304.

170. Rubiolo de Reinick. A.C. 1992. Estimacion ofaverage freez­ing and thawing temperature vs. time with infinite slab corre­lauon. Computers in IndUStry. vol. 19. pp. 297-303.

171. Saitoh. Too Nakamura. M. and Gorni. T 1994. Time·space method rQr multidimensional melting and free1:ing problems. 1m. J. Num. Meth. Engng. vol. 37. p~. 1793-1805.

172. Saunders. B. V. 1995. Boundary confonning grid generation syslem for Inlerf3Ce traCking. Computers Math. Appl . vol. 29. pp. 1-17.

173 Sharosund:lr. N. and Sparrow, E. M. 1975. Analysis of multi­dimensional conduction phase change via the cnlhal py model. J. Heal Transfer. vol. 97. pp. 333-340.

174. Shyy. Wei. Rao. M.M .. Udaykumar. H.S. Scaling procedure and finile volume computation> vf phase'change problems wuh convection. Engng Analysis with Boundary Elements. vol. 16.1995.pp, 123·147,

17S Solomon. A. D. 19110, I\n casil y-compUlablc solution 10 J t \VO'

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phase Stefan problem. Solar Energy. YOI. 24. pp. 525-528.

176. Solomon, A.D.. Wilson, D.G. and AJexiades, V. 1983. Ex­plicit solutions to phase change problems. Quan. Appl. Mach.. yol 41. pp. 237-243.

177. Solomon, A.D., Wilson. D.G. and Alexiades. V. 1984. The quasi-stationary approximation for the Stefan problem with a convective boundarycondition. Inc. J. Mach. & Math. Sci _. vol. 7. pp. 549-563.

178. Tarnawski. W. 1976. Mathematical model of frozen con­sumption products.lnt. J. Heat Mass Transfer. vol. 19. p. 15.

179. Taylor. A. B. 1974. The mathematical formulation of Scefan problems. In Moving BOlUldary Problems in Heal Flow and

Diffusion. J. R. Ockendon and W.R. Hodgkins. cds.. p. 120. Oxford University Press. London.

180. Udaykumar. H.S.. Shyy. w. 1995. Grid-supponed marker particle scheme for interface tracking. Num. Heat Transfer. Pan B Fundamentals, vol. 27, pp. 127-153

181. Verdi, C. and Visintin, A. 1988. Error estimates for a semi­explicit numerical scheme for Stefan-lype problems. Numerische Mathemalik. vol. 52, pp. 165-185.

182. Vick. B. and Nelson. D. 1993. Boundary element method applied to freeZing and mehing problems. Num. HeacTransfer B, vol. 24, pp. 263-277.

183. Voller. V. R. and Cross. M. 1981. ACCUTalesolutionso(moving boundary problems using the enlhalpy method. Int J. Heac Mass Transfer. vol. 24, pp. 545-556.

184. Voller, V. R.. Cross, M .. and Walton. P. 1979. Assessment of weak solucion lechniques for solving Slefan problems, (cd.).

Numerical Melhods in Thermal Problems, p. 172. Pinendge Press.

185. Voller, V.R. and Prakash. C. 1987. A fixed grid numerical modeling methodology for convection-diffusion mushy re­gion phase-change problems. Inc. 1. Heat Mass Transfer. vol. 30, pp. 1709-1719.

/86. Weinbaum. S. and Jiji, L. M. 1977. Singular perturbalion lheory for melting or freezing in finite domains not al the fusion temperature. 1. App/. Mech. vol. 44. pp. 25-30.

187. Westerberg, K. W.. Wilclof. C. and Finlayson. B. A. 1994. Time-<lependent finite-clement models ofphase-change prob­lems with moving heat sources. Num. Heal Transfer B. Yol. 25. pp. 119-143_

188. White. R.E. 1985. The binary alloy problem: existenu., unique­ness. and nu merieal approx imalion. SlAM 1. Num. Anal., vol. 22, pp. 205-244.

189. Wilson. D. G.• Solomon. A. D., and Boggs, P.T.. cds. 1978. Moving BowuJury Problems. Academic Press, New York.

190. Yoo. J. and RUbinsky. B. 1983. Numericalcomputarion using fini Ie elements for the moving interface in heal transfer prob­lems with phase transformation. Num. Heat Transfer. vol. 6, pp. 209-222.

191. Zhang, Y.: Alexander, J.l.D.: Quauani. J. 1994. Chebyshev collocalion method for moving boundaries. heat transfer. and convection duringdirectional solidification. IoU. Nu II\. Melh. Heal Fluid Row, vol. 4. pp. 115-)29.

D. INTRODUCTION, APPLICATIONS, AND BASIC PHYSICAL CONCEPTS

Melting occurs naturally, such as with environmental ice and snow, and with the magma in the earth core. It is also a part of many technological processes. such as thawing fol­lowing freeze-preservation of foodstuffs, snow and ice re­moval. manufacturing (such as in casting, mOlding, sinter­iog, combustion synthesis, coating and electro-deposition. soldering. welding, high energy beam cutting and forming. crystal growth. electro-discharge machining, printing). thawing of cryo-preserved or cryosurgically treated organs. and thermal energy storage using solidlliquid phase-chang­ing materials. A bibliography for these applications. with a brief introduction, is given in Subsection [v.

Melting is often accompanied by freezing, and the thermo­dynamics as well as uanspon principles of the two pro­cesses are very similar. Their mathematical treatment is therefore also similar. Specific discussion of (reezing is given in Section 507.8 of the Databook.

In simple thermodynamic systems (i.e .. without external fields, surface tension, elc.) of a pure material, melting of a solid occurs at cenain combinations of temperalure and pressure. Since pressure cypically has a relatively smaJler inOuence, only the melling (or "fusion") temperature is often used to identify lhis phase transition.

The conditions for melting are strongly dependent on the

concentration when che material conta.ins more than a sJOgle species. Funhennorc. melting is also sensitive to extemal effecLS, such as electric :lnd magnelic fields. in more complex chermodynamic systems.

The equilibrium thermodynamic system parameters during phase transition can be calculaced from lhe knowledge chat the partial molar Gibbs free energies (chemical pOlentials) of each component in the two phases must be equal (cc. Alell;iades and Solomon (I]. Hultgren el a1 {37]. Kechin [40J, Liar [46J, Poulikakos [54}). One imponant result of using this principle for simple single-componenc systems is the Clapeyron equation relating the cemperature (n and pressure (P) during the transition from the solid to the liquid phase. viz.

dP hS( -=-­dT T6vS( Eq. (9-1)

where hs[ is lhe enthalpy difference becween the phases (=

h l - hs > O. the latent heat of melting) and 6.v<J. is lhe specific volume difference between the phases (= v( - vs)­

Exantination of Eq. (9-l) shows that increasing the pressure wi II result in an rise of the melti ng tcmperature if 6 Vs( > 0 (i.e .. when the specific volume of lhe liquid is higher than that of the solid. which is a propeny of (in. for example). but will result in a decrease of che meliing temperalure when 6Vsl < 0 (for waler. for example)_ The latter case

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explains why ice may melt under the pressure of a skate blade.

In some materials, called glassy, me phase change between the solid and liquid occurs with a gradualrransition of the physical properties, from those of one phase to those of the other. When the liquid phase flows during the process, the flow is strongly affected because the viscosity decreases greatly as the solid changes to liquid. Other materials, such as pure metals and ice, and eutectic alloys, have a definite line of demarcation between the liquid and the solid. the transition being abrupt. This situation is easier to analyze and is merefore more rigorously addressed in the literature.

To illustrate the above-described gradual transition, most distinctly observed in mixtures, consider the equilibrium phase diagram for a binary mixture (or alloy) composed of species a and b, shown in Fig. 9-l. Phase diagrams, or equations describing them, become increasingly compli­cated as the number of components increases. X is me concentration of species b in the mixture, t denotes the liquid, s the solid, Sa a solid with a lattice structure of species a in its solid phase but containing some molecules of species b in that lattice, and Sb a solid with a lattice structure'of species b in its solid phase but containing some molecules of species a in that lallice. "Liquidus" denotes the boundary above which the mixture is just liquid, and "solidus" is the boundary separating the final solid mixture of species a and b from the solid-liquid mixture zones and from the other ZOnes of solid Sa and solid Sb.

T )

• LiquidUS

Liquid !

..X~._s'..... " .. " tI

o X, x

Figure 9-/. A Liquid-Solid Phase Diagram ofa Binary Mixture

For illuSlIation (Fig. 9-1), assume that a solid mixture or alloy of components So and Sb containing concentrations (X I. Sa and XJ. st» of species b, respecti vely is at point I and thus at temperature T, (Fig. 9-l). TIle above concentrations are those identified by the intersections of the horizontal dot-dash line passing through point I, with the left and right wings of the solidus Ii ne. respectively. The ratio of the mass of the solid Su 10 Ihat of sh is detemlined by Ihc lever rille.

and is <xl. Sb - X t)/(X I - Xl. •.,) at point 1. This solid is then heated, ascending along the dashed line corresponding 10

XI. When the temperarure rises above the solidus line. inelting starts, creating a mixture of liquid and of solid sa. Such a two-phase mixture is caJled the mushy zone. At point 2 in that musby zone, for example. the solid phase (so)

portion contains a concentration X2. SA of component b, and the liquid phase portion contains a concentration Xl. ( of component b. The ratio of the mass of the solid So to that of the liquid can again be detennined by the lever rule, and is (xu. - Xl)/(X2 - X2. SA) at poim 2. Further heating to above the liquidus line, say to point 3, resuLts in a liquid mixture having concentration XI.

A unique situation occurs for heating along the line of concentration X~: the liquid fonned has the same concentra­tion as that of the solid mixture of Sa + S/>. and a two-phase (mushy) zone is not fonned during the melting process. x. is called the eutectic concentration, and the solid mixture (or alloy) having that concentration is called a eutectic. --

It is obvious from the above that the concentration distribu­tion changes among the phases. which accompany the melting process (Fig. 9-l), are an important factor in the composition of alloys. and are the basis for freeze-separa­tion processes.

The presence of a two-phase mixrure zone with tempera­ture-<lependent concentration and phase-proportion obvi­ously complicates heal transfer analysis, and requires the simultaneous solulion of both the heat and mass transfer equations. Furthermore. the solid usually does not melt on a simple planar surface between the phases. This complicates the mathematical modeling of the process significantly. Further references to melting of multi-component systems are provided in Subsection IV.B.

Flow of the liquid phase often has an important role during melting (cf. Cheung and Epstein f l3], Viskanla [69], Yao and Prusa [74J, and references [76] - [106]). The flow may be forced, associated with the removal of the melt, and/or may be due to natural convection that arises whenever lhere are density gradients in the liquid. here generated by temperature and possibly concentration gradients. Under such circumstances, strong coupling may exist between lhe heat transfer and Ouid mechanics. and also with mass transfer when more than a single species is present. and the process must be modeled by an appropriate set of cominu­ity. momentum. energy, mass conservation, and Slate equations, which nee<! to be solved simultaneously.

E. PREDICTIVE METHODS

The mathematical description of the melting process is characterized by non-linear partial differential equations. which have analytical (closed- form) solutlons for only a few simplified cases. As explaIned above. (he problem

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becomes even less tractable when flow accompanies the process, or when more than a single species is present. A very large amount of work has been done in developing solution methods for the melting problem (sometimes also called the Stefan problem. after the seminal paper by Stefan [64D, published both as monographs and papers, and included in'the list of references to this Section (Alexiades and Solomon [Il. Bankoff [4], Chadarn and Rasmussen [10), Cheng and Seki [12), Crank OS], Fasano and Primicerio [20J. Hill f32J, Ozisik [50J. Tanasawa and Lior (65J. Yao and Prusa [74], and refereuces [107] - (l91J, with emphasis on the reviews by Friedman [129], Fukusako and Seki (130]. Meirmanov (153], Ockendon and l{odgkins [162], Rubinshtein (168), and Wilson et aJ [189]. Generi­cally, sol utions are obtai ned either by (l) linearizing the original equations (e.g.• perturbation methods) where appropriate, and solving these linear equations, or (2) simplifying !.he original equatioos by neglecting terms, such as the neglection of thermal capacity in tile "quasi-static method" described in Subsection m. below, or (3) ,using the "integral method," which satisfies energy conservation over the entire body of interest. as well as the boundary condi· tions. but is only approximately correct locally inside !.he body (in Subsection ill.A. below), or (4) employing a numerical method.

Many numerical methods have been successfully employed in the solution of melting problems, both of tile finite difference and element types, and many well-tested software programs exist that include solutions for that pUllJose. A significant difficulty in the formulation of the numerical methods is the fact that the liquid-solid interface moves and perhaps changes shape as melting progresses (making this a "moving boundary" or "free boundary" problem). TIJis requires continuous monitoring of the interface position during the solution sequence, and adjustment of the numerical model cell or element proper­ties to those of the particular phase present in them at the time-step being considered. Several fonnutations of the original equations were developed to simplify their numeri­cal solution. One of them is the popular "enthalpy method" discussed in more detail in Subsection m.B. below.

The predictive equations provided below are all for materi­als whose behavior can be chracterized as being pure. This would also apply to multi-component material where changes of the melting temperature and of the composition during the melting process can be ignored. General solu­tions for cases where these can not be ignored are much more difficult to obtaJn. and the readers are referred to the literature; some of the key citations are provided in the general reviews [I J, [12J, [28]. {37], and [6S]. and are listed under the Multi-Component Systems and Crystals heading of Subsection TV.B.

Funhermore, the solutions presented here by c1osed-fonn equations are only for simple gcometries, since no such

solutions are available for complex geometries. Simplified expressions. however, are presented for melting times also in arbitrary geometries.

II. PREDICTIVE EQUATIONS FOR MELTING

A. ONE-DIMENSIONAL MELTING OF PURE MATERIALS WITHOUT DENSITY CHANGE

Examination of the simplified one-dimensional case provides same important insights into !.he phenomena. identifies the key parameters, and allows analytical solu­tions and tilus qua!i(ative predictive capability for at least this class of problems. In this Section we deal with cases in which the densities of both phases is the same, and in which the melt does not flow, thus also ignoriog, for simplifica­tion, the effects of buoyancy-driven convection which accompanies the melting process when a temperature gradient exists in the liquid phase. As stated in Subsection 1.0., the effects of natural convection may sometimes be significant., and infonnation about this topic can be found under the references quoted in mat Subsection. Melting of non-opaque solids may also include intemal radiative heat transfer, which is ignored in the equations presented below. Information about such problems is contained in references [II j, [16), [25), [3/]. and (74]. The solutions presented below can be found in many books and reviews that deal with melting and freezing (cf. refs. [I]. [4]. [IS], [32], (46], (74], [130], [153]. and [168] and in textbooks dealing with heat conduction (d. [50J, and [52]).

1. Solutions for Materials that are Initially at the Melting Temperature

If the solid to be melted is initially at the melting tempera­cure throughout its extent, as shown in Fig. 9-2, heat transfer occurs in the liquid phase ooly. This somewhat simplifies the solution and is presented fIrst.

Consider a solid of infinite extentAo the right (x > 0) of the infinite surface at x = 0 (i.e.. semi-infinilc). described in Fig. 9-2, initially at the fusion temperature Tf For lime t> 0 the tempcrarure of the surface (at x =0) is raised to To> Tf , and the solid consequently starts to melt there. In this case the temperature in the solid remains constant, Ts = Tf so the temperature distribution needs to be calculated only in the liquid phase. It is assumed that the liquid formed by melting remains motionless and in place. with the initial condition

T,(X.I)=Tj inx>O,att=O. Eq. (9-2)

the boundary condition is

T(O.l) '" To for I ;> O. Eq. (9-3)

and the liquid-solid interfacial temperature and heat flux continuity conditions

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LiquidT Solid

1 Pbas<:~hange

v~ace

TJ T,=Tr = Ti

o X (t) X Figure 9-2. Melting ofa semi-infinite liquid initially at the fusion temperature. Hem condzutlon takes place consequently in the liquid pJw.se only_

T([ X(t)]:::. Tf for t > 0,

_kl(~Tt) "" Pihs[ dX(t) for I> 0, Eq. (9-5) oX (X(tIJ d1

The analytical solution of this problem yields the tempera­ture distribution in the liquid as.

[ x)erf - ­2..ja[t

Tt(x,t)=To-(To-T/) , for t>O, Eq.(9-6) . erfA

where erf stands for the error function (described and tabulated in mathematical handbooks), and 'A.' is the solution of the equation

A'e'" 2erf(A.') = ~.( , Eq. (9-7)

with NSI~[ being the Stefan Number, here defined for the liquid as

Eq. (9-8)

Equation (9-7) can be solved to find the value of A' for (he magnitude of NSlC(' which is calculated for the problem at hand by using Eq. (9-8). The solution of Eq. (9-7), yielding tIle values of 'A.' as a function of Nsu. for °~ N5t~ ~ 5, is given in Fig. 9-3.

The interface position (which is also the melting front progress) is defined by

Eq. (9-9)

EXAMPLE

The temperature of the vertical surface of a large volwne of solid paraffin wax used for heat storage, initially at lhe fusion temperature, Tj =T( =28T, is suddeuly raised to 58"C. Any motion in the meh may be oeglected. How long would it take for the paraffin to solidify to a depth of 0.1 m? Given properties: al= (1.09)10-7 ml/s, Ps = p( = S14 kglm3, hsl =241 kJfkg, cl

= 2.14 kJfkg"C. To find the required time we use Eq. (9-9), in which the value of 'A.' needs to be determined. A' is calculated from Eq. (9-7), which requires the knowledge of NSI~r From Eq. (9-8)

N = (2.14 kJ I kg"C)(5S"C - 28"C) =0.266. SI~1 241. 2 kJ I kg

The solution of Eq. (9-7) as a function of NSI~l is given in Fig. 9-3, yielding A <>$ 0.4. Using Eq. (9-9), the time of interest is calculated by

1.2,--------------------,

1.0

0.8

'A' 0.6

0.4

0.2

0.0 +---,--,---,--r---r-----r--r-----r-,...---j

o 2 3 4 5 Nstc,

Figure 9-3. The Root A' ojEq. (9-7)

2. Solutions for Materials that are Initially not at the Melting Temperature

a. Slab geometry

U, initially. the solid to be melted is below the melting temperalure, conductive heat transfer takes place in born phases. Consider a semi-infinite solid initially at a tempera­ture T; lower than lhe melting temperature Tf(Fig. 9-4). At time I =0 at the solid surface temperature at x = 0 is suddenly raised to a temperature To> Tf , and maintained at that temperature for t > O. Consequently, the solid stans to melt at x =O. and the melting interface (separating in Fig. 9-4 the liquid to its left from the solid on its right) located at lhe position x =X{r) moves gradually to the right (in the positive x direction).

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T Liquid

~ Solid

.l ~

T( (x,t) l

~T(, ~ ..

~ ____ T,'" T; ~ PfJase-<,hange ~ inlerface asx-4 oo

V o X (t)

Figure 9-4. Melling ofII semi-inftniie solid initially tU a below­melting temperature. Heat conduction Ia1us place in both phases.

The analytical solution of this problem yields the tempera­ture distributions in the solid and liquid phases, respec­tively, as

Eq.(9-1O)

and

Eq. (9-11)

where erfc is the complementary errorjwtclion, X is a constant. obtained from the solution of the equation

Eq. (9-12)

where Nsrt( is the Stefan number defined by Eq. (9-8).

Solutions of Eq. (9-12) are available for some specific cases in several of the references (cf. (l], [50], and can, in general. be obtained relatively easily by a variety of commonly-used software packages used for the solution of nonlinear algebraic equations.

The transient position of the melting interface is

Eq. (9-13)

where A is the solution of Eq. (9-12), and thus the expres· sian for the rate of melting, i.e. the velocity of the motion of the so lid -liq uid interface, is

Eq. (9-14)

b. Cylinder

Very few exact solutions exist for melting of bodies that are shaped differently than slabs. This is an available exact solution for the somewhat practical problem of the melting of an axially-symmetric cylinder due to a line heat source in the center (Fig. 9-5), sayan electric wire or beat-carrying pipe inside an insulator). Ie should be noted that natural convection of melt was found to have an impo.nant role in such problems (d. Yao and Prusa (74)), and the solution shown can only be used for rough evaluation. Other phase­change problems in non-planar geometries are solved by approximate and numerical methods (Alexiades and Solomon (1], Cheung and Epstein [13], Yao and Prusa [74],

T

source q'

Figure 9-5. Outward Melting of an Initially Solid Cylinder at Temperalure T, < T

I, Having a Line Heal Source of Intensity q'

(W/m) Along its Axis

Caldwell and ehiu [115], Gupta and Kumar [1341, Kern and Wells (143], Kim and Ro [ 145], Laz.aridis (146], Li {l49], Meyer [156-158]. Rabin and Korin (164], Raw and Schneider [ 167], and Shamsundar and Sparrow (173]).

For an infinite cylinder having consLant properties, in(ually solid at To < T{, with a line heat source of intensity q' (W/m) at r= 0, the initial conditions are

R(O) = O. T(r.O) = Yo < Yr, Eq.(9-15)

and the far-field boundary condition is

lim T(r,r) =Tf . Eq. (9-16)

r --+ O¢

The heat transfer energy balance at the line heat source is expressed by

d7t(r.f)] ,2lim -'{(rk =q>O[ (ar . Eq. (9-17)

r --? 0

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Section 507.9 VAPOREATION~HASECHANGE Heat Page 12 MELTING Transfer August 1996 ONE-DIMENSIONAL MELTING, THE QUASI-STA TIC APPROXIMAnON Division

The analytical solution of this problem (by similarity) gives the position of the outward-moving melt front as

R( t). = 2A."(a(1 )112 • Eq.(9-18)

and lhe temperature distributions in the liquid and solid phases as.

T«(r,t)= Tf +L[Ej(-~}- £i(-A."1)]47tk( 4a

l( Eq. (9-19)

in 0 < r < R(t) for t > 0,

Ei(_r2 14a.c)~(r,()=To+(Tf-To) .( ,,2 )

El -A. at / as Eq. (9-20)

in R(t):5 r fort> 0,

respectively. where Ei is a function defined by

-Ei(x) =fe~s ds. x> O. Eg. (9-21)

and where A" is the root of the transcendental equation

Eq. (9-22)

Solutions of Eq. (9-22) can. in general, be obtained rela­tively easily by a variety of commonly-used software packages for the solution of nonlinear algebraic equations.

B. ONE-DIMENSIONAL MELTING WITH DENSITY CHANGE

For most materials the density of lhe liquid and solid phases is somewhat different, usually by up to about 10% and in some cases up to 30%. Usu.a1ly the density of the liquid phase is smaller than that of the solid one, causing volume expansion upon melting and shrinkage upon freezjng. Solid metals and plastic materials that are confined would increase their volume during melting and may lhus burst their enclosure. Water is one of the materials in which the density of tbe liquid phase is ltigher than that of the solid one. and thus the volume of the water is smaller than thaI of the ice from wltich it was fonned by melting. 1f the densi­ties of the liquid and solid phases differ. motion of the phase-change interface is not only due to the phase change process. but also due to the associated volume (dcnslly) change.

A reasonably good analytical solution for small (- ±t"O%) solid-liquid density differences is available (Alexiades and Solomon (l)) for the semi-infinite slab at x ~ 0, initially solid at Tj < Tr, melted by imposing a constant temperature To> Tr at the surface x = O. It is assumed that PI. < PSt c/.. <:S. k(, Ies. !lsi, and T r are constants and positive. The melt front X(t) starts at X(O)-=-O and advances to the right (Fig. 9-4). Buoyancy-driven convection is ignored, but the liquid volume expansion upon melting is considered. in that it pushes the entire solid body also rightward without friction. at uniform speed u(t) without motion in the liquid itself. The temperature distributions are. in the liquid and solid phases

in 0 S x ::;: X(t) for t > 0, Eq. (9-23)

erfc[ J -(I -I-l)clm

]

_ ( ) 2 a,l Eg.T.(X,I)- To + T! - To ('")

enc ~)(A

(9-24 )

in X(t) S x for l> O.

The location of the melt ing fron t is

Eg. (9-25)

and the speed of the solid body motion due to the expansion is

u(t) = (1-I-l)A.'"{(i;fi. Eg. (9-26)

In Eqs. (9-23) - (9-26) 'A'" is the root of lhe equation

,_1A. '"e" erfA.'"

Eg. (9-27)

which can. for the specific problem parameters, be SOlved numerically or by using one of the many software packages for solving nonlinear algebraic equations. The remaining parameters are defined as

PIJ=_I Eq. (9-28)p,

Because of the approximate nature of this analytical solution It is expected thaI Eq. (9-25) slighlly overestimates [he posllion or the melt frool. If PI > p" melting will cause

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Division THE QUASI-STATIC APPROXIMATION August 1996

the volume of formed liquid to become smaller !han !.he volume formerly occupied by the solid, and may thus move the melling solid leflwaW. in a direction opposite to that of the melting interface. and the solution represented by Eqs. (9-15) - (9-20) would not be valid. Approximate, but less accurate solutions for Ihis and other cases are described by Alexiades and Solomon rI} and Prusa and Yao [54}, and additional results are given by Shamsundar and Sparrow (59J.

C. THE QUASI-STATIC APPROXIMATION

To obtain rough estimates of melting processes quickly, in cases where heat transfer takes place in only one phase, it is assumed that effects of sensible heat are negligible relative to mose of lalenl heat (NSte ~ 0). This is a significant simplification. since the energy equation then becomes independent of time, and solutions to the steady-state heat conduction problem are much easier to obtain. At the same time, the transient phase-change interlace condition [such as Eq. (9-5)] is retained, allowing the estimation of me transient interface position and velocity. lbis is hence a quasi-static approximation, and its use is shown below. The simplification allows solution of melting problems in more complicated geometries. Some solutions for me cylindrical geometry are presented below. More details can be found in refs. (I], (15). (32/, (117], (J 30]. and (177].

It is important to emphasize mat these are just approxima­tions, without full infonnation on the effect of specific problem conditions on the magnitude of the error incurred when using them. In fact, in some cases, especially with a convective boundary condition, they may produce very wrong results. It is thus necessary to examine the physical viability of the results, such as overall energy balances, when using these approximations.

It is assumed here that the problems are one·dimensional. and that the material is initially at the freezing temperarure Tr.

1.� Examples of the Quasi-5tatic Approximation for Melting of a Slab

Given a semi-infinite solid (Fig. 9-2), on which a time· dependent temperature To(t) > Tr is imposed at x = 0, the above-described quasi-static approximation yields the solution for the position of me phase-change fron! and of the temperature distribution in the liquid. respectively, as

Eq. (9-29)

Eq. (9-30)

in 0 ::: x :::; X(t) for t 2: O.

The heat flux needed for melting, q(x, f). can easily be detennined from the temperature distribution in the liquid (Eq. (9-23)J.

This approximate solution is equal to the exact one when NSf( ~ 0, and it otherwise overestimates the values of both

t� 'fjX(f) and T(x, t). While the errors depend on the Specl IC

problem, they are confined to about 10% in the above-described case (Alexiades and Solomon (1 J.)

For the same melting problem but with the boundary condition of an imposed time-dependent heat flux qo(t),

_kl(dTf ) =qo(r) for [>0, Eg. (9-31)

dx 0.1

the quasi-static approximate solution is

,

X(t) =: -l-f qo (t)df fort> 0, Eq. (9-32)phsl o

Eq. (9-33)Tl(x,f) = T1 + qo [.!iLl -xJ kf phsl

in 0 :::; x ~ X(t) for I > O.

For the same case if the boundary condition is a convec­tive heat flux from an ambient fluid at the transient temperature_TaU), characterized by a heat rransfer coefficient h,

_kf(dTl ) = h{TQ(I) - TlCO.f)] for r~ 0, Eq. (9-34)

. dx OJ

the quasi-static approximale solution is

Eq. (9·35)

Eq. (9-36)

in 0 s x S X(r) for r > O.

2.� Examples of the Quasi-5tatic Approximation for Melting of a Cylinder

It is assumed in these examples mal the cylinders are very long and that the problems are axisymmetric. Just as in the Cartesian coordinate case, the energy equation is reduced by the approximalion to its steadY-Slate form.

Consider the outward-dire<:ted melting of a hollow cylinder with internal radius ri and outef radius '0 (Fig. 9·6)

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Section 507.9 VAPORlZAnoNIPHASE CHANGE Heat Page 14 MELTING Transfer August 1996 ESlTIvl"ATION OF MELTING 11MB Division

due to a temperature imposed at the internal radius ri, i.e. 2R(ri In R(t) =

T((r;,t) ~ To(t) > Tf for l > O. Eq. (9-37) r; The solution is

lo[rl R(t)]� TtCr,t) ~ Tf + [To(f) - Tf ] [ { )J�In 'i / R t Eq. (9-38)

in 'i S; r R(t) for t > 0,

and the transient position of the phase front. R(f). can be� calculated from the transcendental-integral equation�

I

2R(r)2 In R(f) = R(I)2 - ".2 +~f[To(I) -TI jdt. Eq. (9-39) Ii phst

o

lEthe melting for the same case occurs due to the imposi­�tion of a heat flux qo at rj,�

-kl(dTtJ =qo(t»O fort>O. Eq. (9-40)dr '1. 1

the solution is

T. (r,f): T - qo(r)r; In_r_ Eq. (9-41) t I k

l R(t)

in ri ::; r S R(t) for t> O.

1/2

R(t)= r/+2..:Lf' q,,(/)dt (ort>O. Eq. (9-42) ( phs( ]

o

If the melting for the same case occurs due to the imposi­tion of a convective heat flux from a fluid at the transient temperature Ta(t), with a heat transfer coefficient h, at 7j,

_kl(dT{) =h[T,,(t)-71(r;,f)] > 0 for I> O. Eq.(9-43) dr 'i.T

The solution is

in r,. S; r ::; R(/) at ( > 0,

with R(t) calculated from the transcendental-integral equation

I

( 1- ~kl J[R(t/ - r/]+ 4k( J[T,,(f) - Tf Jdr. Eq. (9-45)hI"; phsl

o

The solutions for inward melting of a cylinder, where heating is applied at the outer radius r 0> are the same as the above-described ones for the outward-melting cylinder. if the replacements ri ~ ro, qo ~ -qo. and h ~ - hare made. If such a cylinder is nOt hollow then 'l == 0 is used.

D. ESTIMATION OF MELTING TIME

There are a number of appro~imate formulas for estimating the freezing and melting times of different materials having a variety of shapes. A brief introduction will be gi ',len here; other formulas, proposed by Alexiades and Solomon, [I], all applicable equally well {Q both freezing and melting times, can be found in Subsection n.D. of "Freezing" 507.8.

L Melting Time of Foodstuff

The American Society of Heating. Refrigerating, and Air­Conditioning Engineers (ASHRAE) provides a number of approximations for estimating the freezing and thawing times of foods (ASHRAE. (3]). For example, if it can be assumed that the freezing Or thawing occur at a single temperature, the time to freeze or thaw, 1(, for a body thai has shape parameters 8'and R (described below) and thennal conductivity k, initially at the fusion temperature h and which is exchanging heat via heal u-ansfer coefficient h with an ambient at the conSUlnt temperature Ta, can be approximated by Plank's equation

Eq. (9-46)

where d is the diamctcr of thc body if it is a cylinder or a sphere, or the thickness when it is an in{inite slab, and where me shape coefficients fJ' and R for a number of body foons are gi ven in Table 9-1 below.

Table 9-1 Shape factors for Eq, (9-46) from ASHRAE (3J

Fonns fl' R

Slab 1/2 1/8

Cylinder 1/4 1/16

Sphere 1/6 In4

Shape coefficien(s for other body fonns are also a vai Iable, To use Eq. (9-46) for melting. k and p should be the values for the food in its (hawed state,

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In face, freezing or melting of food typically takes place over a range of temperatures. and approximate Plank-type fonnulas have been developed for various specific food­stuffs and shapes to represent reality more closely lhan Eq. (9-46) (ASHRAE (3). Cleland er a1 [I! 8J. Rubiolo (170), Tamawski [l78}, and some of the references cited under Subsection rv.F. "Medical Applications and Food Preserva­tion."

EXAMPLE

USING PLANK'S EQUATION (9-46)� FOR ESTlMATING MELTING TIME�

Estimate the time needed to thaw a fish. !he shape of which can be approximated by a cylinder 0.5 m long having a diameter of 0.1 m. The fish is initially at its freezing temperature, and during the thawing process it is surrounded by air at Ta = n·c. with the heating perfonned with a convective heat transfer coefficient h = 68 W/m2K. For the fish, Tf = -I·C. hst.::: 200 kJlkg, 1'5 = 992 kg/mJ, and k., = 1.35 W/m K. Using Table 9-1, the geometric coefficients for the cy Iindncal shape of the fish are fJ' = 1/4 and R =1/16, while d is the cylinder diameter, =0.1 m. Substituting these values into Eg. (9-46) gives

r = 200000, 992 (1/4(0.1) + 1/16(0.1)2) = 68665 = I. 9h. I I(-I) - 231 68 1.35

2.� Other Approximations for Melting Time

As stated above. the freezing chapter in the Darabook. 507.8, Subsection U.D., shows a number of easily-comput­able approximate equations for estimating the time needed to freeze or melt a simple-shaped liquid volume initially al the the freezing temperature Tr.

Validating by comparison to the results of an experimen­tally-verified two-dimensional numerical model of melting of lake-shore water (with a mildly-sloped adiabatic lake bottom) initially at the freezing temperature, Dilley and Lior (121 j have shown that for a constant beat flux qo from the ambient to the tOp surface of the ice, with rela­tively negligible heat fluxes in the water under the ice. the relationship between the depth of melting X(l) and time is linear, viz.

Eg. (9-47)

IlL SOME METHODS FOR� STh1PLIFYING SOLUTION�

A.� THE INTEGRAL METHOD, WITH SAMPLE� SOLUTION FOR MELTING OF A SLAB�

A simple approximate technique for solving melting and freezing problems is the heal balance integral merhod (Goodman (132n. which was found to give good results in many cases. The advantage of this method is that it reduces the second-order partial differer.tial equations. describing the problem to ordinary differential equations which are much easier to solve. This is accomplished by guessing a temperature distribution shape inside lhe phase-change media, but making it satisfy the boundary condHions. These temperature distributions are then substituted imo the partial differential energy equal ions in the liquid and solid. which are then integrated over the spatial parameter(s) (here just x)

in the respective liquid and solid domains.o 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 the media. This technique is introduced here by applying it to a useful case. and lhe reader can thus also learn to apply it to other cases.

Revisiting the above-analyzed melting case of a semi­infinite solid (descnbed in Fig. 9·2)., (he equations descnb­ing this problem are the heat conduction equation

aT, (.0) a2T( (x,l) &I. (9·48)at :: a, dXI

in 0 < x < X(t), for I> 0,

with the initial condition

7{ (X. 1) = T1 in x ;> O. at 1 = 0, Eq. (9·49)

the boundary condition

7f{O.t} = To for t> O. Eq. (9-50)

and the liquid-solid imerfacial temperature and heat nux continuity conditions

~[X«()I=TI fort>O. Eq.(9·51)

_k,(OT,] =ph,tdX(I) forl>Oo dX !x(I)1 dl Eq. (9·52)

Equation (9·48) represents heat balance (energy conserva· tion),. and it is now integra led wi th respect to x over the entire liquid-phase domai n. 0 ~ x ~ X(I). in combination with the in ilia] and boundary conditions (Eqs. (9-49) - (9­52)J. to obtain

with an error within only a few percent for the rust 40 hours.

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Section 507.9 VAPORIZATIONIPHASE CHANGE Heat Page 16 MELTING Transfer August 1996 SIMPLIFIED SOUITIONS-TIfE ENTHALPY METHOD, BIBLIOGRAPHY Division

_Ii.[Xf(t~(X'I)dx_TfX(t)] C1. t dl

o Eq. (9-53)

~_ph« dX(l) _ aT (O,t). k l dt ax

This equation has two unknowns, T(x, 1) and XC!). To solve it, some temperature profile is guessed, made to satisfy the boundary conditions, and substituted into the equation. Integrating the left-hand side of the equation with Ibis temperature distribution, and taking the derivative of the distribution on the right side of the equation, reduces the original nonlinear partial differential equation system composed of Eqs. (9-48)-(9-52) to one ordinary differential equation in which the only unknown is then X(I), and the independent variable is t. This equation is thus much easier to solve, but at lhe same time lhe adequacy of me solution depends on !be quality of the chosen temperature distribu­tion. Experience from previous successful solutions or experimental results naturally improves the choice of this temperature distribution.

Say that a quadratic polynomial is chosen as the tempera­ture distribution,

Eq. (9-54)

The coefficients hi, h2' and bJ of this polynomial are calculated from the boundary conditions. and thus

Eq. (9-55)hlX +(Ta - Tf ) b~ = 2 . - X

Substitution of the polynomial {Eqs. (9-54), (9-55») into Eq. (9-53). results in the ordinary differenlial equation

Eq. (9-56)

which, using the initial condition Eq. (2-49), gives the solution

11lX(I) =26( a(t ) , Eq. (9-57)

where

Eq. (9-58)

Comparison of the inlegral solullon [Eq. (9-57») with (he exact one (Eq. (9·9)1 has shown Iha\ Ihey arc nearly

identica1 for NS/C'l up to about 0.5, with me integral solution overestimating the value of Q by only about 4% when Nsu( '= 2.8. An even better agreement is obtained if the tempera­ture distribution is estimated as a third, instead of second, degree polynomial.

Many integral solutions yield good results, with errors within a few percent The accuracy. as mentioned above. depends on the closeness of the chosen temperature distributions to the real ones. Experience from previous successful solutions or experimental results naturally improves this choice. Additional infonnation about this melhod can be found in refs. (132), [50), [108]. [154].

B. THE ENTHALPY METHOD

It is noteworthy that one of the biggest difficulties in numerical solution techniques for such problems is the need to track the location of the phase-change interface continu­ously during the solulion process, so that the interfacial conditions can be applied there. One popular technique that alleviates this difficulty is the enthalpy method (refs. [I], (741, (lO9), (124- L25], (130), (140), (173), and [183-184), in which a single partial differential equation. using the malerial enthalpy instead of the temperature. is used to

represent the entire domain, including both phases and me interface. Based on the energy equation, the one-dimen­sional melting problem is Ihus described by

Eq. (9-59)

where me temperature-enmalpy relationship is expressed by

h for h ~ cTf (solid)

c

T= TI for eTf < h < eTI +nSl (interface)

h - hs(

e for h;:: eTf +Jv. (liquid)

Eq. (9-60)

The numerical computation scheme is rather straightfor­ward: knowing the temperature, enthalpy and thus from Eq. (9-60) the phase of a cell at lime step j, the enthalpy at time step 0+ I) is compu ted from the discret ized version 0 f Eq. (9-59), and then Eq. (9·60) is used to detennine \he lem­perature and phase at that new time. If a computational cell i is in the mushy lOne, (he liquid fraclion is simply hi I nSt.

Care must be e,.;ercised in the use of the enthalpy method when the phase change occurs over a very narrow range of lemperatures. Oscillating non-realistic solutions were obtained in such cases. but several modifications to the numerical fonnulalion were found 10 be reasonably success­ful (d. [I]. [109], [124-125], and [183-1841l.

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Heat VAPORlZATIONIPHASE CHANGE Section 507.9 Transfer MELTING Page 17

Division APPLICAnONS BIBLIOGRAPHY August 1996

IV. APPLICATIONS BIBLIOGRAPHY

An extensive-yet by no means complete-bibliography

identifying papers and books mat treat melting in the main

areas in which it lakes place. is presented below. The

classification is by application, and the internal order is

alphabetical by author.

A. CASTING, MOLDING, SINTERlNG

Melting is a key process-component in casting, molding,

and production of solid shapes from powders by processes

such as sintering and combustion synthesis. The malerials

include mecals. polymers. glass. ceramics. and superconduc­

tors. Row of the molten material. the course of its solidifi­

cation (including volume changes due to phase transition.

and intemal stress creation). and the evolving surface and

interior quality. are ail of significant industrial importance.

In production of parts from powders, the conditions

necessary for bonding of the panicles by melting and

resolidifcation are of importance. Such diverse processes as

glass-making. spinning and Wire-making are included.

Much attention has lately been focused on me manufactur­

ing of materials for superconductors.

Alman. D. E.: Hawk. 1. A.: Peuy. A. V. Jr.; Rawen;. 1. C. 1994, Processing ifllcrmetalhc composites by self-propagating. high-temperature synthesis. JOM, vol. 46. pp. 31-35.

Alymov. M.1.: Mahina. E.!.; Stepanov. Y.N. 1994. Model ofinitial stage of ulcrafine mela! powder sintering. Nanoslrocture<l Materials. vol. 4. pp. 737-742.

Bose. A.� Technology and commercial status of powder-injection molding. 10M. vol. 47. pp. 26-30.

Chiru vella. R. V.: Jaluria. Y.: Abib, A.H. 1995. Numerical si mulation of fluid flow and heat transfer in a single-screw extruder with different dies. Polymer Engng Sci .• vol. 35. pp. 261-273.

Dupret, F. and Dheur. L. 1992. Modeling and numerical simulation of heal transfer during the filling stage of injection mold­mg. In Heat and Mass Tronsjerin Materials Processing, I. Tanasawa and N. Lior, cds.. pp. 583-599. Hemisphere. New York. N.Y.

Gau. C. and Viskama. R. 1984. Melting and solidification of a metal system in a rectangular cavil)'. In!. J. Heat Mass Transfer. vol. 27. p. J 13.

Gau, C. and Viskanla. R. 1986. Melting and Solidification of a pure mellli on a venical wall. J. Heat Transfer. vol. 108. p. 174.

Gelder. D. and Guy. A. C. 1975. Current problems in the glass industry. Moving Boundary Problems in Heat Rowand Diffusion. (1. R. Ockendon and W.R. Hodgkins, cds.) Oxford Univ. Press (Clarendon). London and New York.

Gupta. G S,; Sundararajan. T,: Chakraboni. N. 1995, Induclion smelling process part I malhcmalical formulation. Iron makl ng and Steel maki ng vol. 22. pr, I37· \47.

Hieber. C.A. 1987. fnjection and Compression Molding Fundamen­IOls.. A. L lsayev. ed., Marcel Dekker, New York.

Lian. Sh.-Sb. and Chueh, Sh. -eh. 1995. Maki ng of new alloys with plasma melting furnace. Internal iona! J. Materials & Prod­uct Technology, vol. to, pp. 587-595.

Lindt. J.T. 1985. Mamemalical modeling of melting of polymers in a single-screw extruder: a critical review. Polymer Engng Sci.. voL 25, pp. 585-588.

Nemec, L 1994. Analysis and modeling 0 f glass melting. Ceramics - Silikaty vo\. 38 pp. 45-58.

Noskov. A.S.. Nelcrasov, A.V .. Zhuchkov, v.I.. Zav'ya!ov, A.1. and Rabinovich. A.Y. 1992. Mathematical mode! forthe melt­ing of a piece 0 f a ferrous aliay during circulatory motion of lhe liquid metal in (he ladle. Melts. vol. 4. pp. 425-432.

Pampuch. R.. Raczka. M. and Lis. I- 1995. Role of liquid phase in sol id combustion synthesis ofTill3SiOn. Int. J. Materials & Product Technology vol. 10. pp. ) 16-324.

Peifer. W. A. 1965. Levitation melting. a survey of the stale oran. J, MC{.• vol. 17. p. 487.

Pervadchuk. V.P.. Trufanova. N.M .• and Yankov. V.1. 1984. Math­ematical model of the melting of polymer materials in enruders.lnvestigation oflhe form of me interface and of melt velocity profiles. Fibre Chem., vol. 16. pp. 358-361.

Pervadchuk. V.P., Trufanova. N.M., and Yankov, V.l. 1984. Math­ematical model of the melting of polymer malerials in e;>;truders. Pressure dislribution along the length of {he extruder screw. Fibre Chem.. vol. 16. pp. 362-365.

Pervadchuk. V.P.. Trufanova. ;..l.M .. and Yankov. V.l. 1985. Math­emalical model and numerical analysis of heaHransfer processe~ associaled with the melting of polymers in plaslic3ting. extruders. J. Engng. Phys., vol.~. pp. 60-64.

Rauwendaal. C. 1989. Improved analytical melting theory. Adv, Polymer Technol.. vol. 9. pp. 331-336.

Rauwendaal. C. 1991n. Melting theory for temperature-dependem fluids. C)(aCI analytical solution forPQwer-law fluids. Adv. Polymer Technol., vol. I!. pp. 19-25.

Rckhson. S.M.. Rekhson. M .. Ducroul:. l.-P.. and Tarakanov, S, 1995. Heal transfer effects in glass processing. Ceramic Engng Sci. Pro<;.. voL 16. pp. 19-37.

Roychowdhury. A.P. and Srini vasan. 1. 1994. Modeling of radiation heat lransfe r in foreheater urn ts in glass melting. Waerme­U11d Stoffuebenragung. vol. 30. pp. 71-75.

Sun. C. and Song. L. 1995. Three dimensional marhematical model of a fioal glass tank furnace. Glass Technology vol. 36. pp. 213-216.

Ungan. A. and ViskaJl!3, R. 1987. Three-dimensional numerical modeiJng 0 rei rculalion and heat transfer in a glass melting tank: Pan I. Mathemalical formulation. Glastechnische Berichle. vol. 60. pp. 71-78.

Viskanla. R. 1994. ReVIew of three-dImensional mathematical mod­eling of glass melting. 1. Non.Crystailine Solids. vol. 177 pI I. PP 347-362.

Volkov, A.E.. Shalimov. A.G. and Laktionov. A.V. 1995. Method ror continuous c1ecu-oslag melting 0 r non compact materi· a1s. In!. J. Materials & Product Techool.. vol. 10. pp, 541-544

GENIUM PUBliSHING

Section 507.9 VAPORIZATIONfPHASE CHANGE Heat

Page 18 MELTING Transfer

August 1996 APPLICATIONS BIBLIOGRAPHY Division

Whittemore, 0.1. 1994. Energy usage in firing ceramics and melti ng glass. Ceramic Engng Sci. Proc. Yol. 15. pp. 180-185.

Zhang. Y., Stangle, G. C. 1995. Micromechanistic model of micro­SlJ'Ucture development during the combustion synthesis process. I. Materials Res., vol. 10, pp. 962·980.

Zhang, Y. and Stangle, G. C. lui 1995. Micromechanislic model of the combined combustion synthesis-<1ensi fication process. J. Materials Res., vol. 10, pp. 1828- I 845.

Zhukov, A.A., Majumdar, J.• Outta. and Manna, I. 1995. Induction smelting process part I mathematical fonnulation. J. M.a­terials Sci. Lett., vol. 14, pp. 828-829.

B.� MULTI·COMPONENT SYSTEMS AND� CRYSTAL-GROWTH�

The bibliography in this section primanly focuses on melting of multi-component systems, but crystal growth

also includes pure crystals. As compared with the melting

of single component systems, multi-component system

melting is accompanied by a change of composition as

discussed in Subsection I, a phenomenon of great signifi­

cance in the formation of the liquid material The analysis

and prediction of the process are thus also made more

complex in that the species diffusion process and the effeCl

of the concentration on the melting point and other proper­

ties must be considered.

One of the most prominent applications is alloy-making.

and the last several decades have seen large and increasing

invol vement with crystal growth, primarily for the c leetran­

ics and optical industries. Crystals are typically grown by

melting the feedstock and letting it solidify in the form of a

crystal. Cryscals may be made of either pure or multi­

component materials, but even when pure crystals are made.

. much research has been done on the effect of impurities

introduced during the manufacturing process. This in effect

renders even the pure crystal to be considered as a multi­

component system. Crystal growth is accomplished by a

variety of processes. inclUding Czochralski, Bridgman,

Float-Zone, and thin film deposition. A more extensive list

of references on these subjects can be found in the Hand­

book chapter 507.8, "Freezing."

Abrams, M. and Viskanta. R. 1974. The effects or radiat ivc heal transfer upon the melting and solidification of semi-trans­parent cryStals. 1. Heat Transfer. vol. 96, p. 184.

Apanovich. Yu.V. 1984. Analysis of heal and mass processes in� growing crystals by the zone·melting method. J. Appl.� Mech. Tech. Phys.. vol. 25. pp, 443-446.�

Bennon. W.O. and Incropera. F.P. 1987. A continuum model for� momentum heat and species transport in binary solid­�liquid phase change systems - l. Model formulation; [I.� Application to solidification ill a rectangulatcavlly, Int. J.� Heat Mass Transfer. vol. 30. pp, 2161·2170. 2171· 2187�

Bennon. W.O. and Incropera. F.P. 1988. Numerical analysis of binary solid-liquid pbase change using a conti nuous model. Num. Heat Transfer. vol. 13. pp. 277-296.

BrOWI). R.A. 1992. Perspecti ves on integrated modeling of transPOrt processes in semiconductor crystal growth. In Heat and Mass Transfer in Morerials Processing. ed. L Tanasawa and N. Lior. pp. 137-153. Hemisphere, New York.

Carey, V. P. and Gebhan. B. 1982. Transpon near a vertical ice surface melting in saline water: experiments at low salini· ties. J. Auid Mech., vol. 117. pp. 403-423. .

Carey. V. P. and Gebhan. B. 1982. Transpon near a venical ice surface melting in saline water: Some numerical calcula­lions. 1. Fluid Mech., vol. I 17. pp. 379-402.

Carruthers,l. R. 1976. Origins ofconvecli ve temperature osci Ilations in crystaJ gtQwth melts. J. Cryslal Growth. vol. 32. pp. 13-26.

EllOU ney. H. M. and Brown. R. A. 1982. Elfect 0 f heal transfer on meltlsolid imerface shape and solule segregation in edge· defined film-fed growth: Finite Element anaJysis. Journal of Crystal Growth. vol. 58. pp. 313-329.

Fang. L.J.. Cheung, F.B.. Linehan, 1.H. and Pedersen. D.R. 1982. An experimental sNdy of melt penetration of materials with a binary eutectic phase diagram. Trans. Am. Nuel. Soc.. vol. 43. pp. 5[8-519.

Flemings. M.e. and Nereo. G.E. 1967. Macrosegregation. Part I. Trans. A[ME. vol. 239. pp. 1449-1461.

Glebovsky. V.G. and Semenov. V.N. J995. Growing single crystals of hlgh-purilY refractory metals by electron-beam zone melting. High Temperature Materials and Processes. vol. 14. pp, 121-130.

Glicksman. M E.. Coriell. R.. and McFadden. G. B. 1986. Interaction of nows with lhe crystal-mel! interface. Anllu. Rev, Ruid Mech.vol. 18. p. 307.

Glicksman. M. E., Smith. R.N.. Marsh. S.P.. and Kuklinski. R. 1992. Mushy zone modeling. In Hear and Mas:r Transfer in Marerials Puxessing. ed. l. Tanasawa and N. Lior. pp. 278-294. Hemisphere. New York.

Grabmeier, J.. cd. 1982. Crystals Growth.. Properties and Applica. f ions. Springer-Verlag, New York.

Himeno. N. Hijikata. K.. and Sekikawa. A. 1988, Latent heatlhennal energy storage ofa binary mixture - Rowand hcatlransfer characteristics ina horizontal cyl inder. [nt. 1. Heat Mass Transfer. vol. 3 l. pp. 359-366.

Kou. S,. Poirer. D.R. and Remings. M.e. 1978. Macroscgregation in rotaled remelted ingots. Mel. Trans. B. vol. 9B. pp, 7 I I . 719.

Lacey. A,A. and Tayler. A.B. 1983. A mushy region in a Stefan problem. IMA J. of Appl. Math .. vol. ]0. PP. 303·313.

Lan. X.K .. Khodadadi. 1.M.. Jones. P.D. and Wang. L. 1995. ,'\lumerical study o( melting of large-diameter cryslals us i ng an orbi tal solar concefllrator. J. Solar Energy Eng ng. vol. I 17.pp. 67-74.

Muhlbauer. A.. Erdmann. W" and W. Keller, 1983. Electrodynamic convection in silicon floating zones. J. Cryslal Growth. vol. 64. pp 529.

GENIUM PUBLISHING�

Heat VAPORJZATIONIPHASE CHANGE Sc~tion 507.9 Transfer MELTING P~ge 19

Division APPLICATIONS BIBLIOGRAPHY August 1996

Muhlbauer. A. and Keller. W. 1981. Flaming Zone Silicon. Marcel Dekker. New York.

Nguyen. c.T.. Bazzi. H.. and Om. J. 1995. Numerical invesugation of the effects of rotal ion on a gennani om float zone under microgravily. Num. Heat Transfer A. vol. 28. pp. 667·685.

Ostrach. S. 1983. Fluid mecharucs in cryslal growth. J. Fluids Engng. voL 105, pp. 5-20.

Prann. w.e. 1966. Zone Me/ring, 2nd ed.. Wiley. New York.

Rosenberger. F. 1979. FundameTllals o/Crystal Growr". Springer. Verlag. Berlin.

Sammakla. B. and Gebhan. B. 1983. Transpon near a vertical ice surface melting in water of vanous Salinity levels, InL J. Heat Mass Transfer. voL 26. pp. 1439-1452.

Sekhar. J.A .. Kou. S. and Mehrabian. R. 1983. Heat flow model for surface melting and soliditlC<llion of an alloy, Metall. Trans. A. vol. l~A.

Tiller. W.A. 1991. The Science of Cry$lalli:arion: Microscopic Interfacial Phenomena. Cambridge Univ. Press.

Uwaha. M,. SaitO, Y. and Saw. M. 1995. FluclUalions and instabihnes of steps in the growth and sublimation ofcrystals. J. Crystal Growth. vol. 146. pp. I 64-170.

Wh ite. R. E. 1985. The bi nary al loy probl em: exis((~nce. unique ness. and numerical approximation. SIAM J. Num. Anal.. vol. 22. pp, 205-244.

Yeckel. A.. Salinger.A. G. and Derby.1, 1,1995, Theoretical analysis and design considerauons for noat-zone refinement 0'­electronic grade si Iicon sheets. J. Crystal Growth. vol. 15~.

pp.51-64,

C.� WELDING, SOLDERING, AND LASER ELECTRON-BEAM, AND ELECTRO- ' DISCHARGE PROCESSING

The bibliography in this section is related to processes

associated with melling that are used for jOining, CUlling.

shapi ng and property-modificalion of solids. Among other

applicalions, laser energy is used for cutting and surface

propeny modification. a melting process followed by

solidification. [n the electro-discharge machining process.

sparks are generaled by the application of a voltage between

an eleclrode (the 100l) and the workpiece 10 be machined.

The closely controlled sparks meh small craters in {he

workpIece, and the melt is continuously removed.

Anthony.� T.R. and Cline. H.E. 1977. Surface rippling induced by surface tension gradients during laser surface melting and alloying J, Appl Phys" vol. 48. pp, 3888-3894.

Bakish. R, Oct. 1994. Electron beam meltlng and relining. slate oflhe art 1993. 10M. vol. 46. pp, 25-27

Bakish. R. 1995, Eleclron beam mcl::ng and refining. elevenlh conference update (with special focus on eastcm Europe and Russia). Induslrial Healing. vol. 62. pp. 40·45.

Gass. M .. ed, 1983 Lasttr Malttl'ltJI.\ ProceSJu1 5 '1onh Holland

8 asu, B. and Dale. A. W. 1990. Num<::rical study ot" stC3<./Y Sla<e ;lnd transient laser melt ing problems - I. Char.J.~terisl ic~ 0 f flow tleld and heallmnsfer. Int. J. Heat Mass Transfer. vol. J3. pp.1149-1163

Basu. B. and Dale, A. W. 1990. Numencal study ot" SlC3dy state and lransienllaser melling probkms - [I. Effect oflhe process p~elcrs. [nl. J, Heal Mass Transfer. \'01. }3. pp. /165-1 J7S.

Basu. B. and Srirnvasan.1. 1988. Nu merical study ofsleady Slate laser melling problem. Int. 1. He:!l Mass Transfer. vol. 31. pp.

2331-2338.

Belaschenko. O.K. 1986. Mathemalic;l/ analysIs of IhaonlaCt melt­ing process, Russtan Melal [urg.y. vol. 6. pp, 66-75.

Breinan. E, M .. Kcar. B. H.. and Banas. C M. 1976. Process in!! malerials with lasers, Phys. Today. vol. 29. p. 44. ~

Breslavskii. P. V. and M;v.hukln, V I, 1990. ,'vlat!JemaliclIl modeling of processes of pulsed melting and vaponDllion 013 melal with elearl y separated phase boundanes J. Eng. Phys.. vol. 57.pp.817-823

Chan. CL. and Mawmder. J. 1987. One-dimensional sleady-St3le model for damage by vaponlatlotJ arJd liquid expulsion during laser-melal interaction. J. Appl. Phys.. vol. 62. pp.

4579-4586.

Chan. CL.. Mazumder. J. and Chen. M.M, 1983 AppliCaiions of Lasers in Marerials ProcesslfIg. E.A. Metzbower. cd..

ASM. p.ISO.

Chan. C ..� Mazumder. 1. and Chen. M.M. 1984, Three·dimenslOnal lransienl model f()f convec<ion in laser melied pool. Mel. Trans., vol. 15A. pp. 2175·218.\,

Chan. C.L. Mazumder. J. and Chen......U.-1. 1987, A Ihree-<:Ji men· sional a.~isymme[ric model for convection In las<:r mehed pool Mat Sci. Engng. vol. 3. pp 306· 31 I

Chande. T. and Mazumder. J 1981. Lasers '" Mera/lurgy?, K MUkherjceand J, Malumder.eds. pp. )65·178. Melull. Soc

AIME. Warrendak, PA.

Chaode. T. and Mazu mder. 1 1984. Esli mall ng effects of process iog conditions and variable properties upon pool shape. cool· ing rales and absorption coefficlcru in laser Welding. J. Appl. Phys.. p. 198 I.

Churchill. D. J.. Snyder. J. 1. Solder ball developmeOl and growlh. Circuits Assembly, vol. 6. 1995. p, 3-

Cline. H.E. and Anthony. T.R. 1977. Heat (rcatmenl and melting material wilh ;1 scanning laser or eleCtron beam. J. App!.

Phys.. vol. 48. pp. 3895-3900,

DIBilonlo. D.D.. Eubank. PT. Patel. M.R, and Barufet. M ,A. 1989. Theorctical models of [he electric disch3rge machining process, I. A Sl mple cJthode erosion mod..::l. J. Appl. Phys,.

vol. 66. pp. 4095-4103.

Dowden. J.. Davis. M .. and Kapadia. P, 1983. Some aspects of [he Iluid dynamics of laser wc!ding. J Fluid Mech .. vol. 126. p. 12l

DuPont.l.N.: Marder. A.R. 1995. Thermal efficiency of arc wc!ding pron'<~es, Weldtng J.. vol. 74. pp. 406-5.4 16'$.

DUlla. Pradip: Joshi. Y_: Janaswamy. R. May 1995. Thenn..1model· ing of g<lS tungsten arc weldin~ process wilh nonaxlsymmelnc boundary condillons. Numerical I Ie.;t

Transfer A. vol ~7 pr. .199·518

GENIUM PUBLISHING�

S,;:ction 507.9 VAPORlZATION/PH..\SE CHANGE Heal

Page 20 MELTING Tr:wsfcr

August 1996 APPLICATIONS BJB UOGRAPHY Division

Elmer. J.W.: Newton. M.A.: Smuh. A.C. Jr. 1994. TransformatIon hardcnj ng of steel usi ng hi gh-energy electron beams. W~ Id·

. ing 1.. vol. 73. pp. 291·s-299-s.

Franc IS. A. and Cral ne. R. E. 1985. On a model lor fricl ion ing stage in fnclion welding o{lhln lUbeS. Int. J. Heat Mass Transfer. vol. 28. p. 1747.

Fnedman. E. 1976. On lhe caJculallon of temperatures due (0 arC welding. Nucl. Metall.. vol. 20. pp 1160-1170.

Gicdt. W. H. 1992. Improved electron beam weld deSIgn andcorJ,trol with beam current prof! k measurement [n Hear alld .HUS$

Transfer in Materials Processillg. cd. I. Tanasawa and N Lior. pp. 64-80. Hemisphere. New York.

Gkbovsky. V.G.: Semenov. v.~ 199.3-1994. EleCtron-beam llu3l· 109 z:one melting of refractory metals and alloys an :lnd science. Inl. J. Refr:J.ctory Melals and Hard Malerials. vol

12. pp. 295·30 l.

Goldschmled. G .. T,chegg, F.K. and Schuler. A. 1990. Elc([ron beam surface melllng..'1umerica! calculallon of the mell geometry and comparison with expenmenta! resulls. J Phys. D: App!' Phys.. vol. 23. pp. 1686-1694.

Gngorpoulos.� CP.. Dutcher. W. F. Jr. and Emery. A F 1991­Experimemal and compUlational analysis oflaser meillog o. thin silicon films. 1. Heat Transier. vol. 113. pp. 1-9.

Hokamoto. K.. Chlba. A.. FUJita. M. and [wma. T. 1995. Single.shOl ~l(plosi ve weldIng technique for the fabricalloll of mull!­layered metal base composites; effect of welding paramo eters leading to optimum bondIng condlllon. Composites Engng. vol. 5. pp. 1069·' 079.

Hsu. Y F. Rubinsky. B . and Mahin. K. 1986 An Inverse flilite element method lor the analySIS of stationary arc welding processes. J. Heat Transfer. vol. 108. pp. 734-741

Kou. S. and Sun. D.K. 1985. Fluid Ilow and weld penWJtion In

St3lJOnary arc welds. Mel. Trans.. vol. 16A. Pl'. 203·213

Kou. S. and WlUlg. J.H. \986. Three·dimensional convection in laser melted pools. MetaJl. Trans. vol. 17A. pp. 2265-2270.

I~ambrakos, S.G. and Milewski. J. 1995. Analysis and refinement of a method [Of numencaJly modeling deep-penetrallon weld· Ing processes using geometnc constraints. J. Malerials Engng Performance. vol. 4. Pl" 717- 729.

Landram. C.S. 1983. Measurement or' fusion boundary energy transport dunng arc Welding. J. Heat Transfer. vol. 105. p. 550.

LI n. S. and LIn. Y. Y 1978. Heat Iran sfer charact<::nstICs 0 f ultra pulse welding. Proc. Inl. Heal Transfer ConL. vol. 4. Pl'. 79-84

Lopez. V.: Bello. J .M.: Ruiz. j : Fernandez. B.J. 1994. Surface las~r

treatmCIll or ductile irons J MatcnaJs SCI .. vol 29. pp. 4216·4224.

."l~~huk.in. V.. Smurov, I.; FlJmanl, G.: Dupuy. C. 1994. Pcculian· Ii es of laser me III ng and evaporation 0 f su pe rconduwng c.::ramics. Thin Solid Films. vol 24. pp. 109-113

Muhukln. V.. Smurov. l.: Dupuy. c.. Jcandel. D 1994. Simulallon or' last:[ melling and evapor,l(ion of superconducling cc· ramics. Num. Hea( TranSlt:f A, vol. 26 pp. 587 -600.

:Vfazumder. J. and KaI. A Transpon phenomena In laser matcnal processing. In Heal WId Mms Transfer "I Motennls Pm·

(.<'HIIl~. cd. I. T.lna,JW;1 .lnJ '\; LJlJL PI' g t . 106 '·k01I

'ril~r~. ,,~W Y"rk

Mazumder, 1. and Steen, W.M. 1980. Heal lransfer mOdel for CW bser matenals processing. J. App!. Phys.. vol. 51. pp. 941-947.

~ehka. A.H .. cd. 1971. Electron Beam Weldmg . PrinCIples and Pract ice. McGraw-HI II. Pl'. 1- 16.

Oreper. G. :VI. and Sz:ckely. J. 1984. Heal and flUid flow phenomena in weld pools. J. Fluid Mech .. vol. I·n. 1'.53.

Patel. .VI.R.. Bandel. M .. Eubank. P T. and Di Bitonto. D.D. 1989. Theorellcal models of of the electrical discharge machin· Ing process. II. TIle anode erosion model. ). Appl. Phys..

vol. 69. PI'. 4104-4111.

R3Jurkar. K. and Pandit. S. 1986. Formallon and ejeclion 01 ED,',.I debns J Engng Industry. vol. 108. pp_ 22-2(

R;)\'lndr:lD. K.. Srinivasan.J. and Mar:J,lhe. A.G 1995. Role: or' ~Ur(3(C

tension 00 ven conVeCllon In pulsed laser melting and solidi ficallon. Proc. Insl. evlech. Eng n~'. Pan B: Journa I 0(

Engng ;vlanur'aclure, vol. ~09. pp 7S-8~.

Rubin. BT. 1975 TheoreucaJ model of the submerged arc welding process. ,"fela!1. Trans. B. vol. 68. pp. 175-) 32.

Sa)cudean. M.. Choi. M_, and Grel f. R. 1986. A slUdy of he;}! tmnster dunng arc welding. Inl. J Heat Mass Transfer. vol. 29. pp. 21 S.

Scm:J.k. V.V.. Hopkins. l.A .. McCay. M.H. and McCay, T.D_ 1995. Mell pooldynamlcsdunng laser welding. J. Phys. D; Appl. Phy>., vol. 28. pp. 2443-2450.

Spoerer. 1.. Igenbergs, E. Treatment of the surrace layer ot Sled wilh

hIgh energy plasma pulses.

Surface Coatings Techno!.. vol. 76-77. 1995. pp. 589-594

Snnl v:J.$an.) and Basu. B. 1986. A numerical slUdyorlhermocaplllary now in :J. reclangular cavity during lJscr m<:lting. lot. J. Heat Mass Transfer. vol. 29. pp. 563-57).

St~en. P. H.: Ehrhard. P.; Schussler, A. 1994. Deplh of mell-pool and heat -af(eCled z:one 10 laser sun'ace trealillelllS. Metall urg

Maler. Trans A. vol. 25A. pp. 427--135

Swn my. V T.. Ranganat han. S.. and Challopad hya y. K. I995. Resol idi t'ic;)lion behavior of laser surface· me lied metals Jnd alloys. Surface & Coalings Techno!.. vol 71. pp. 129-1)4.

Tsukamoto. S. and lrie. H. 1994. Melling process and spiking phenomenon in electron beam welding. Welding Research

Abroad. vol. 40. pp. 10·1 S.

Ulyashin. A.G.: Gorolchuk, I.G.: Yang. Z.Q.: Bumay. Y.A.: EsepkJn. V A.:TalUr.G.A.:Alifanov.A. V.: Slepanenko.A.V. 1994. l.Alser annealing o( bulk hlgh-temperalure supercllnduc­tors. Physlca C Superconducliv;ty. vol. 235-2~O, pt 5 pp.

3005- -'006

Vliar. R . Colaco. R . Almeida. A. Dc 1995. LJ cr SUd.l.:C Ifcatmenl of tool sleels. Opt. Quant. Eleclronics. vol. 27. pp. 1273·1289.

Wang. P. Lm. 5 . and Kahawila. R. 1985. Thc 'ubl( "p!,ne integra­tion (CChOlque rorsol vlOg luslOn welding problt:ms J. Heat

Transfer. vol. 107. P 485.

XIii. 1-1 .. Kunleda. M...'hshiwal<J, N. Jnd Llor. N 1994. jv!e,!,;urcmenl of <:ncrgy dIstribution imo elcclIodcs III EDM pwc<:sscs. -ld\,(Incellleflf of IlIlefligclll Prod/l<'flf)n. E_ Usui. "(~ . r::r,,,v,,;r. 8.\1 II.JpJn ';oc. PI ·CI 1>>11 E;)~n~. J1r lit)I·(,{)6

GENIUM PUIJU,)'HING�

HeJ.t VAPORIZATION/PHASE CHANGE S':Clion 507.9'� Transkr MELTING Pag<:: 21�

Divi~iol1 APPLICATIONS BrB LIOGRAPHY AuguSl1996�

Xu. X .. Gngoropoulos. c.p, Russo, R.E. 1995 Heat transfer in excimcr laser meJting of thin polysihcon layers. 1. Heal Tr:J.ns(er. vol! 17. pp.708-715.

Z.J<:h:m<l. T.. Vilek. J.."1.. Goldak.1.A .. DebRoy. T.A.. Rappaz. M.. 8hadesn13. H. K.D.H. 1995. Modeling offundamenta/ phe. nomena In welds. Modeling and Simulallon ill Malenais Sci. Engng. \'01. 3. pp. :?-65-288.

D.� COATING, AND VAPOR AND SPRAY� DEPOSITION�

The blbliogrilphy III lhl$ seclion i~ of v:l{ious coaling� rro-:esses Hl which a ~{)Iid is melted. and the ;~quid is lhen�

deposited on il ~urface al a lemperature below lhe freeZing� pain\. on whi<.:h I( re-soliddies 10 form a coaling. TIle�

pro..:eS:> can also be <.:onllnued 10 bUIld up deslred shapes.� Some of the pro<.:esses covered in the bibliography Include� pl~ma and lhermal spray deposllion. vapor deposJllon.�

dipping in a melt. and coating by liquid films.�

Apellan. D.. Paliwal. M. Soulh. R.W. and Schilling. W.F. 1983 Melling and solidification plasma spray depositioll - a phenomenologIcal review. Inl. Mel. Rev.. vol. 28, pp. 271­

394.

B30. Y. Gawne, D.T.. and Zhang. T. 1995. Effecl of feedslock panicle size on lhe heat lransfer rates and propenies of thamally sprayed polymer eoallngs. Trans. Inst. Melal Fimslllng. vol. 73, pp. 119-124.

Bennett. Teo D.. Gngoropoulos. COSlaS P.; KraJnovich, Douglas J /995 Near-threshold laser spullering of gold. J. App/. Phy~ . vol. 77. pp. 849-864.

Benagnolh. M.. Marchese. M .. 1acucci. G. Modeling of panicles impacting on a rigId subslrale under plasma spraying condItiOns. J. Thermal Spray Techno!.. vol. 4. 1995. pp. 41-49.

Clyne. T.W. and Gill. S.c. 1992. Heal (low and thermal contraction during plasma spray de posilion. In Heal GildMass Transfer in Malena/s Processing. ed. I. Tanasawa and N. Lior, pp. 33-48. Hemisphere. New York.

El· Kaddah. N.. McKelliger. J. and Szekely. J. 1984. Heallransfer and (lUld (low 10 plasma spraYIng. Mewl!. Trans.. vol. 158. pp. 59·70.

Fiszdon. J. 1979. Melting of powder grains 10 3 plasma flame. Inc J Heal Mass Transfer. vol. 22. pp. 749·761.

Groma. I. and VCIO. B. 1986. Melling of powder grains 10 plasma spraywg. 1m. J. Hem Mass Transler. vol. 29. pp 549·554.

GUlfinger. C. and Chell. W. H. 1969. HC3llrans(er wllh a movwg boundary-application to flUIdiZed-bed coaling.lnt.l. Heat ,\-1 ass Transfer. \'01. 12. pp 1097·1108.

H,jlkala. K. and MilOi. 1992. Heal uaflsfer in plasma spraying. [0

Heal and Mon Transfer in MOlen'als Processing. ed. I. Tallasa"'"a and N. Lior. pp. 3-16. Hemisphere. New York.

Kang. B.. Waldvogel. 1. and Poulikakos. D. 1995. Remelling phe­nomena In lhe proccs~ of splal solidlficalion. J. Mal SCI .. vol :il}. rp -11)12 ':9~:\

Kat. A. and Mazumder. J. 1989 Threo.:..dimension:lllr.lJ1sieOllhermal analySIS fOf laser chemical vapor depOSition on uniformly moving finile slabs. J. Appf. Phys.. vol. 65. pp. :?-923·2934.

Kukla. R.; Hoimann. D ; Ick.:s. G.; Pall.. U. 1995. Spu((cr mCla!liza­lion of plasllc pans in a jUSHn-llme COaler. Surface & C03ungs Techno\.. vol. 75. pI ~ pp. 6-18-653.

Layadi. N.: Roca ICabarrocas. P.; Gem. M.; :-Vlarine. W.; Spousla. J. 1994. Excimer-Iaser-asslsted RF glo ..... ·discharge deposi­tion ot' J.morphous and mll;rocrystaJline SIlicon thill (tlms. Appl. Phys. A. vol. 58. pp. 507·512.

Pfender. E. 1988. Fundam..:ntal sludics associaled with lhe plasma spray process. Sud. Coal. Technol.. vol. 34. pp \ -1-1.

Rykalin.~ Nand Kudinov. V V 1976. Plasm:lspfJying Pure -\ppl

Chern.. vol. -lS. pp. 229-239

Sidfens. H ·D.: :-.!assell$tew. K. 1994 R.:cem developme:ll$ in slOgle- WI re vacuu marc spraylOg. J. Therm<ll SprJyTech nol . vol. 3. DD. -l12...oli7.

Vanllecl. R.R .. MOChel. 1..\1 . Thermodynamks of mcillng Jill!

(reezing In smaJl panicles. Surf. Sci .. vol. 34 I. pp.J.0-50.

Wei. D. Y .. Apelian. D. and Farouk. B. 1989. Panlck melling III hIgh temperalUre supersonic low pressure plasmas. Melall. Trans.. vol. 20B. pp. 25 t -262.

Zaal. J. H.� 1983. A quaner ot' 3 century of pi asma sprayi ng. rllln. Rev

Mar. Sci.. vol. 13. PD. 9-.12.

E.� ABLAnON AND StiBLIMATION

Melling sometimes Ol:curs wilh ablaTIO'/' :I process in which the mollen. or otherwlse-s.::parJlCd malerial. is conlinuous!y removed from lhe solid subSlrate by aerodynamic shear forces. Ablalion is of imponance in applications such 0.5

atmospheric re-entry space vehicles. Jnd in cenaln manu­racwnng process. When heated, some solid materials

change phase directly Iota vapor. without going through the liquid phase. This process is called sublimation. and is mathematically treated in a manner simIlar to lhal of melting of solids with conlinuous removal of the mel!. replacing the latent heal of melting with lhat of sublimation. A bibliography of publica/ions dealing WIth ablalion and sublimation IS given below.

Aral. N. and Karashima. K.. I. 1979. TrJnsicm lhennal response oj ablaling bodIes. AlAA J.• vol. 17. pp. 191·195.

Biol. Ivl.A. and AgraWal. H.C. 1964. Varialional analySIS 01 eJbl;Hioll for vanable plOpenies. J. Heal Transfer, vol. 86. pp 437· 442.

BiO!. M. A. and Daughada y. H. 1962. Va ri alionaJ analySI$ of abl at ion. J. Aerospace Sci .. vol. 29. pp. 227-229.

Chen. c.J. and OSlrach. S. 1969. Mellinil ablalion for lwo·dimt:il· s/Ona! and Jxisymmelric blunt bodies WIlh a body furcc. Progr. He:ll Mass Transler. vol. 2. pp. 195.

Chung, B. T. F.. Chang. T. Y.. HSIao. J. S.. and Chang. C. T. 1983. Heal (rans fer ..... iIh ahl;t! ion Ln 3 hal f- ~ pace suhjected !O

lime· \'anant h(:ll (lv.xes. J I ~eJl Tr~n~fcr vol 105. pp :;00

G£NIUM PUBLISHING�

Section 507.9 VAPORIZATJONIPHASE CHANGE Heat

P3ge 22 MELTING Transfer

August 1996 APPLICATIONS BIBLfOGRAPHY DivIsion

Chung. 8 T F. and Hsiao. J. S. 1985. Heat Transfer with ~blallon in flmte slab subjected to lime variant heat l1uxes..o\IAA J. Vol. 23. pp. 145.

C1J./"k. B. L 1972. A parametric sludy of lhe lransiene ablation of Teflon. J. Heat Transfer. vol. 94C. pp. )47·35<1

Friedman. A. :lnd McFarland. S.L \968. Two·dimenslonallranS)(nl tlbla{ion and heat conducllon analysis for mullim;llenal thrust chamberwaJls.1. Spacecraft Rockets. vol. 5. pp. i53.

Gi 1pin. R. R. 1973. The ablation of ice by a waler Jel. Trans. Can. Soc. Mech. Engrs.. vol. 2. pp. 91·95.

Guzelsu. A.N. and Cakma. A.S. 1970. Starling solution fOr an 1111aling hollow cylinder. Int J. Solids SlruCtures. \'01. 6. pp 108:

Hogge'. ~I and G.:rrek~ns. P 1983 Two·dimenSlonal Jd'orm~n,!­finite element melhods for surface ablalion. Al.-\A J vol. ~3. pp. 465.

Hone. Y. and Chehl. S 1974. A SolUlion technique lor rnullitlimel1' si ona! abl alion problem. Jnl. J. Heat ~1 ass Trans fer. vol. I :. pp. 453.

Lin. S. 1982. An exact solution of the sublimalion problem In J porous medium. Parts I and [I. 1. Heal Transfer. vol. 103, pp. 165· 168, vol. 104. pp. 808-811.

Ivlatvcev. S.K. 1969. Unsleady heat transfer in ablallon. Heal Trans· fer· SOVIet Res.. vol. I. pp. 62.

Park, C. and Balakrishnan. A. 1985. Ablallon of Galileo probe he<lt­heal shteld models in a balllsllc range. AIA.<\ j Vo!. ~3.

pp. 301.

Park. C., Lundell. J. H., Green. M J.. Winovich. w .. Jnd Covington. M. A. 1984 ..-\blation of carbonaceous materials In 3

hydrogen- heltu m arC)el flow. A [A A J Vol 22. pp. 1-\9 I.

Prasad. A. and Agrawal. H.C. 1974. Biot's VarlallOnal priocipk tor aerodynamic ablallon of melling solids. AlAA J.. vol. 12. pp. 325-327

Reinheimer. J. ! 964. Melling or sublimation of sollds wjth volume heal source. App!. Sci. Res.. vol. A13. pp. 203­

Spaid. F.W.. Chawarl. A.F.. Redckopp, L.G. and Rosen. R, 197\. Shape evolution of a subliming surface sUbjel:ted 10 u~·

steady spatially nonuni form heal tlux. Int. J. Heal ,\-lass Transfer. vol. 14. pp. 67)·687.

Swedish. M. J.. Epslein. M.. Linehan. 1. H.. Lamben. G. A .. Hauser. G. M.. Stachyra. L. J. 1979 Surface ablalion in thc 1m· pingement region of a liquid jet. AIChE J.. vol. :!5. pp. 630·)11.

Wu. T W. and Boley. B A. 1966. Bounds In melting problems wHh arbllrary fateS of ablation. SIAM J. Appl. Math .. vol. I... pp. 306·323.

Ze!a7.0Y. S.W and Kennedy. L.A. [968. Ablation of a sph.;ncal bodY

du~ 10 nonuniform surface heal nux dl.'.lnbulion.) Spa.:e· craft Rockel.s, vol. 5. pp 874.

F.� MEDICAL APPLICATIONS AND F'OOD PRESERVAnON

The blbl iography in [hiS section covers biological .Ippl it:}· lions of m.;lring ..,u..:11 JS those used tn mt:diclnc :\00 In (nuJ

proc.:r";llli>ll l'lC III<:JI<::d :lppliC:lllon" Include Ihc.: 'h.""no:

process folioWI ng organ preserv;)tion. prcsefvaIion of tissue

cultures. cryosurgery. and freezing damage to !lve {Issue.

such as in frOSt-bite.

Food preservJlion by freezing. an anCIeot praclice. Sill!

attracts much R&D 3ttenllon. In allcmpLS 10 shonel1 rreeZJn!!. limes. reduce energy consumplion. prolong. lhe lire or food~. and minimize damage to theIr nutrHional value.

laSle. odor and appearance. The thawing phase. which lakes place when lhe food tS to be used. is imponant both in lhe tIme it lakes. and in the wJ~! H affects food quailly.

!3lonJ. G� 199.l :'-lechanlCJI propeOlcs of frozen mooJel solutions J. Food Engng. vol. 2:;. pp, 253-:?69

Budm;Jn. H.. Shitzer. A. :Jnd Dayan. J. 1995 .Analysls oflhe inverse problcm of freezing and lhawing of a binary \oluti,'n during cryosurgical pWl:<:5ses J. 8ivm<:ch. Engng.. \',11 117. pp. 193·202.

Cleland. DJ.. Cleland, A.C and Earle. R.L. 1987. Pred;euon of freelin a and tlla wi nOt imes for mu lu·d imensional Sh,lpCS by ;im~e formulae. Pan I: r<:gular sh:Jp<:s: Pan:?: ir~egular shapes. rnt. J. Refng. )0: 156- I66: :? 3.. ·240

Fennema. 0 .. Poune. W. and Martha. E. 1973. Lo'" Tempaalwe PreservallOrl oj Foods and LivinJi ,l"lallttr. Marcel Dekk<:r. New York.

Ja(ob;~n. I. A and Pegg. D. E. 19l14. Cryoprcserv;llion vI' organs_ :J

review. Cryobiology, vol. 21. pp. J7i-3S.l

Mazur. P. 1966. Phvsic:'!1 and o;hemlcal basis for Injury ill singk· celled mic~0-organism5 subjected to {reeling and IhJwlng in cryobiology (ed H T '''kryman!. ,.l,~~d~ml~ Press. London.

."lawr. P. 1970. Cryobiology: The frealng or blologJCal systems SCIence. vol. 168. pp. 939·9:j.9.

Mazur. P.. Lelbo. S. P.. Farrant. 1.. Chu. E. H. Y.. HaMa. Jr.. y!.J and Smllh. L. H. 1970. [nteracuon of (ooling fates. warming nltes and prolecil ve addilives on Ihe survival of fro~en

mammalian cells_ [n The Fro~en Cell led. G, E. W. WOISlenholmeand,'vl. O·Connor). pp. 69-85. elBA Symp . London. Churchill.

Meryman. H. T (e<!. l .. 1966. Cr"oblOlogy. AC<ldemie Press. London.

Pcgg. D. E. and Karow. A. M. 1987 The biophySICS of organ cryopreservalion. NATO ASI Ser. No. 147. Plenum Press. New York. 1987.

Polge. C.. Smith. A. V.. and Parkes. A. 1949. RevlVal of pcrmalozo3 afler vitriJicalion Jnd dchydration al low tcmpcrawres. Nalu reo vol. 164. p. 666

Rabin. Y.. Shilzer. A. Exact SolUlion 10 (he one-dimensional inverse·Slcfan problem In nonideaJ bl0!<1glcal llSSUCS. J Heae Transfer. vol 117. 1')95. pp. 425·-lJ I.

Rand. R.. Ringrel, A. and von Leden. H. 1968. CrYOSlIrge,T C.L Thomas Pub!. ..'1ew Yor~

Tamawski. W. 1976 ~alhemJllc:t1 model of (roz~n l:on umpllon products 1m. J. Heal Y1~.>s TrJmt'er. vol. 19. pp I'

W;Jtanab~. M. and Aral. S 199<1 B:tCl<:::-lal i~·e· nudeJt ion ;Jet I VI tYJ nd lIS application 10 freeze conecnlrallon vI' fre'l1 food,; for mod ifl(;all on Ol-lht.:lr prf\[)Cnl~S J. ['c, I Engll\<:~r111>;'. \vl 2~. pp J5}·":7.'

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HC.ll V APORIZATION/PHASE CHANGE S~ction 507.9 Transfer MELTING Page 2]

Division APPLICATIONS BIB UOGRAPHY Augu's\ 199(,

G. NUCLEAR POWER SAFETY

Melting plays a role in sevcral areas related to nuclear

power plant safelY and operation. The primary one IS the

risk of the reactor core melldown upon fai lure of lhe

cooling system 10 Jimillhe lemperalure nse. such as lhe

accident allhe Three Mile Island reaClor. Melting, and the

subsequenl solidificalion due to cooling, have been {he subject of numerous publications. some of which arc listed

in the btbl iography in lhis section. Funher. interaclion of

the mollen melal wilh coolanl will also cause vapor

explosions and chemical reaClions lhat may generate large

qU3milies of gas. some of which (such as hydrogen) are

explosive.

Liquid-metal cooled reactors use coolants. such as pOIJS­

sium. sodium. and their alloys. which are solid at room

temperalure or somewhal above il. A number of pUbliC3'-...,

lions on lhis topic are available. (MISSING?) and lhus need

to be melted for operation. Anolher nuclear application /.

employing melting is vitrification of nuclear waste for

longer term storage.

Arakeri. v. H,. Catton. I. and Kaslenberg. W. E. 1978. An expen men­tal study or {he molten metal/water thermal interJcllon under free and forced conditions. Nuc!. SCI. Engng. vol. 66. pp. 153-166.

Chen. W.e.. Ishii. J. and Grolmes. ,\1.A. 1976. SImple heat conduc­lion model with phase change ror reoctof fuel pin..\fucl. Sci. Engng. vol. 60. pp. 452-460.

Chun, M.Ho, Gasser. R.D . Kalimi. ;vI.S.. Ginsberg. T.. and Jones. Jr.. 0. e. OCI. 5-8. 1976. PrQC Inti. Mlg_ On Fa:;1 ReoC!or

Sa/erv and Relaid PhySICS. Vol. IV. pp. 1808-1818 ConI', No. 76l 00 I. Chi cago, lHinois.

Du ij vest ijn, G.; Reic hi in. K.: Rosel. R. 1995. Computation of flows in experimental simulalions of reaclOr core melting. Com· puters and S{ruclures. vol. 56. pp. 239·247

Eck. G. and Werle. H. 1984. Expen mental studies of penetrat ion of a hot liquid pool into a melting miscible subSlrate, Nuel. Techno!., vol. 64. pp. 275.

Epstein, M. 1973. Heal conduclion (n lhe UO,.cladding composile body With Simultaneous solidi ncallon and melling, Nucl. Sci. Engng.. vol. 5/. pp, 84·87.

Epstei n. M" 1994. NalUrai <;;OlweCllon model 0 f molten pool penet ra­tton into a melting miscible substrate. Nuclear Engng DeSign. vol. 152. pp, 319-330.

Gasser. R. D and Kazmli, M. S. 1977. Study of post.accident molten fuel downward streami ng through lhe aXlaJ shield struc· turc in the liquid·metal fast breeder reactor Nuclear Tech· nol ogy. vol. 33. p, 24 8.

Golinescu. R. P.: Kazlml. M. S. 1994. ProbabllisliC analysis of dl venor plale II fel I me I nTokamak reaCtors. FUSIon Technol .. vol. 26 pt 2. pp. 512-516

Kemen. E..'v1.) and Koning. H, 1994. AnalySIS ofloss-of-coolan! and lo~,;·of-flow aCCldCflIS in the lirst wall rooling >:'Slcm of .".':TIIlT:R, J Fusion Energy. \'01 13. pp 9- 20

L:Jdirat, c.: Boen. R.: JouJn. A.: ;\tlonlOuyoux. J.P 1995. Frellch nuclear waste VUrt (ication: State ot the Jrt and (u ture developments. Cer.1mlC Engng Scienrc Proe.. vol. 16. pp 11.1 J.

Lahoud, A. and Boky. B.A. 1975. Some conslderallons on the melting or' rear{Or fuel pl:lles and rods. "iud. Engng Jnd

Design. vol. 3::. pp. 1·19.

Lyons ..\tI.F . Ndson. R,C. Pashas. T J. .1Olj W~idenb<lum. B. 190_~.

U,O fuel rod operation wilh gross cemr::.l melting. Trans.

Am. NucL Soc. vol. 6. pp. 155·156.

\lUlll. RJ.:Chen.G.Q. 1995 Vllrificaliollotsimulaled medium-and htgh-Ievel Canadian nucleor waste in a cO/1linuous luns­(erred arc plasma meller. 1. Inst Nuclear Mmenals \lul1­agemem. vol 24. pp. jl·}li.

Olander. D.R, 1994 Materials chemistry and transporl modeling for severe Jccldent analyses In Iigtl!-wJter rClIl'lOrs I. External daddmg oxidation. ,'\uclcar Engog DeSign. "01

148, pp. 253-27

Olande!. D.R 1994 Matenals cherrustry Jnd lrans[Jon madding (or severe accident ~n<llyses 10 hght·waler reactors. II: GJp processes Jnd he.1t release. i'luclear Engng Design. vQI, l48. pp ~7J·292.

P:lshos. T J and Lyons. ~1.F. 1962. liO, performance WIth J molt..:n centraJ core. Trans. Am. Nucl, Soc" vol. 5. pp, 462.

Tong. L.S, 1968. Core cooling in a IlypothellcaJ loss of coolant aeci dent: eSli male 0 f heat trans fer in core meltdown. :-Vue 1

£ngng Design .. vol. 8. pp. 309.

Trauger. D.B.; Rublll. A.M,; Beckjord. E. 1994. Three Ml1c hl~'n<:·

new findings 15 years after the :;cc;uent. ,'iuclear $:l.lety.

vol 35. pp, ~56- 259,

Tromm. Waller. Alsmeyer. Hans. TranSl<;nl experimcOls with ther­mite mells (or a core cJtcher concept based on water addmon (fom below. NueleJr Te~'hnoL vo!. III. 1995. pp

]4J·350.

von der Hardt. P.; Jones. A. V.: Lecomte. c.: TJltegr.lin. A. 199<1. ~uclear safety research the Phebus FP severe accideOl e~perimemal program. Nuclear Safety. vol 35, pp. 1<>7·

205

Wol f. J. R.: Akers, D. W.: Nei mark. L. A. 1994. Relocation of mollcn maleriallo the TM!·210wer head. Nuclear Safety. vol. n. pp. 259·279

H.THE~~ALENERGYSTORAGE

Storage of heat in a material by melti ng, and subsequent

release of the heat by re.(reez.ing the materialal1ow$ a Lug.:

<llnouot of heat to be stored to a relatively-small mass or malerial and al a theoretically con Slant lemperalUre (lhal of

fusion). An issue Ihal has received much attention is the

way to cxchange wilh such a phase-change stOrage material

at a high rate and low inveslment III heal transfer equip­

ment. A very bro,tu range of materials has been conSIdered

and used. including watcr. inorganic salts, hydrocarbons.

polymers. <lnd melals.

GENIUM PUBLISHING�

Seclion 507.9 V A PORIZA nON/PHAJI::: CHANGE Heat

Page 24 MELTING Transfer

AuguSI 1996 APPLICATIONS BIBLIOGRAPHY Div\ston

Asako. Y.: Foghri. ,'vi.: Charmchi. M.: Bahrarni. P A. !994. Charac­terization of alkanes and paraffin waxes for applic:llion as phase change energy storage medium. Num. Heat Tr:Jnsfer A. vol. 25. pp. 191·208

Bathe1l, A. G.. Viskama. R.. and Lcidenfrosl. W. 1979 Latent heal­of-fusion energy slOrage: Expenments on heat Iransler from cylinders during meillng. J. Heat Transfer. vol. 101. pp. 453-458.

Betzel. T. and Beer. H. 1988. Solidi ficalion and meltl ng heat transfer to an umfixed phase change material (PC:vl) cnc3psulated in a horizontal concentric annulus. Warme Stofftibenragung. vol. 2~. pp. 335-344.

Charach. c.. Conti. M .. Bd lecci. C. Thermodynamics ofphase '~'hJngc

storage in senes wuh a heat engi ne, J, Sol ar Energy Engng. vol. 117. pp. 336-41.

Chot. J,.c.: Kim. S,·D. 1995. Heat tr:l.nsfer III a latent hcaHtOrJge system using MgCVn6H/120 at the mehing pOlO I. En­ergy. vol. 20. pp. lJ-25,

DeWinter. F. /990. Energy slorage of solar systems. Section /I in Solar Collectors. Energy SlOrage. and Mmenals. "'!IT Press. Cambridge. MA.

Endo. M and Hoshino. M. 1987 CryslJI liquid ice thermal storage syslem. Refrig J., vol. 715. pp. 25-30.

Fixler. S, 1966. Satellite thermal contrQI using phase-change maleri­als. J. Spacecraft Rockets .. vol. 3. pp. 1362·136&.

Goldstein. RJ. and Ramsey. J. W. 1979. He:lt lfansfer (Q a melling solid With application 10 thermal energy SlOrage "YSlcms. Heat Transfer Siudies: Fesl.<cllrifl/or E R.C Ecken. pp 199·206. Hemisphere. New York.

Green. T.F. and Vliet. G C. 1981. Transient response of a latent he:ll storage unil: an analytical and cxperi melllallOvestigation, J. Solar Energy Engng.. vol 103. pp. 275-280.

Himran. S.. Suwono. A. and Mansoori. G. A. )994 Characterization of alkanes and paraffin waxes for application as phase eh ange energy storage mediu m. Energy Sources. vol. 16. pp.117-128.

Hittle. D.C. and Smith. T R. 1994. Conlrol SlraleglC~s and energy consumption for ice stotage systems using heat recovery and cold air diStribution. ASHRAE Trans. vol. 100. pp. 122)-1229.

(naba. H.. Moma. S. and Now. S. 1994. Fundamental study of cold heat -storage sysle m 0 r OIW·I ypc em ul SlOn ha vi ng cold latent-heat dispersion material. Pan I. Heat Transfer . Japanese Rcsearch. vol. 23, pp. 292·312.

Inaba. H. and Mon ta. S 1995. Flow and cold heat· swrage ch.1 raClC r­istics of phase·change emu!slon in ~ l:oiled double-will' he:ll exchanger J Heat Tr~n<;fcr. vol. 117, pr. 4<10..146.

Jones. J. W.. Shiddapur. G.S. 1995. Evalu~lion 0'- RP·459 algorilhms for modeling external melt. Ice-on-pipe thermal storage system comp0rlenlS ASHRAE Trans,. vol 10 I. PI 2. pp, 1342·1351.

Khan. J A, and Morehouse. J. H. I99.l, ,'vIeHI! malrix for hc:at t(ansf.:r enhancement dunng phase.change processes. ASHRAE Trans. vol. 100. pp, 707·714.

LacroIx. ,'vI. 1995. :'-Jumcnc.11 slUdy o( nJlUrJ.1 convCCllon domln~lcd

mdling of.1 PC~'I wnh ":<JnJU~.11~ 'n" cd ,:on 'CCllon. Tr.ln' CJII~t.I So, :'vlcetl I' ",'n'~, ",,1 I'), pp, .155·-l(,')

Li. H. HSieh. C.K.: Goswami, D. Y. 1994. Source and sink melhod of solution of two-dimensional phase change for energy stor­age. J. Solar Energy Engng. vol. 116. pp. 100·106.

Llor. N.. Ayyaswamy. P.S .. O'Leary. J.. Kauffman. K.• Ych.H. and Lorsch, H.G. 1976. Thermal energy storage consideral)on5 for solar-thermal power genera!ion. Proc, I I th IECEC. pp. 61}-622. .

.\larsh;;11. R. H_ !978. Thermal Storage and Heat Transfer in Solar Ellergy Systems. ed. F. Kreieh et ai, pp. 61-69. ASME. New York.

,'vlenon. A. S., Weber. M. E.. and Ylu)umdar. A. S. 1983 The dynamics ofenergy Slorage for paraffin wa, in cylindrical contaiflers. Can. J, Chem. Eng.. vol. 61. pp. 647

Prasad. A. and Srngh, S.P. 1994. Conduction·colllrolkd phase­change energy storage wi th r~dialj ve heat atlduion. 1. Energy Resources Technol.. vol. 116 PP. 218-223.

Pujado. P. R.. $termole. F. 1.. and Golden. J. O. 1969, Melting of J. rr nile paraffi n slab 3.S applied !O phase-change thermo I COntrol. J. Spacecrafl Rockets. vol. 6. pp. 280.

Saiwh. T. S. and Kato. H. 1994. Elementary and 5ystem performance of cyl indncaJ lalent heat storage un iI. Proc. IECEC. vol. 3. IEEE. Piscataway. NJ. USA. 94CH3478-5. pp. 1355-1360.

Shamsundar. N. 1981. Log-mean Ie mperatu rc-dl ffercnct correction (actors for passive latent-heal storage UOltS. J. Solar Energy Engng . vol. 103. pp. ::!81-284.

Shamsundar. N. and Sparrow. E. M. 1974. Storage of thermal energy by solid-li<jUld phase change - temperature drop and heat tlu.,. J. Heal Transfer. vol. 96. pp. 541-)43.

Smith. R.N. Ebersole. T E.. and GnfflO. E. P, 1980 Exchanger perfoml3/lCeS in latene heallhermal energy storage. 1. Sol. Energy Engng.. pp. 102· I 18.

Solomon. A,D, 1979. Melt time and heat Ilu.' for a simple PCM body Solar Energy. vol. 22. pp. 251-257

Solomon. A,D. Moms. M.D.. Manm. J. and OlszewskJ. M. 1986. The development of a simulalion code for a latent heat lhermal storage syslem in a space slation. Repon ORNL­6213. Oak Ridge Nanonal LaboratOry.

Solomon. A.D. and Serbin. CA. 1980. TIS - a program for simulat­ing phase change processes. Report ORNUCSD-S I. Oak Ridge National Laboratory.

Sicwan. W E, Jr :GUle. G D.: Saunders. C. K. 1995. Ice-melung and melt Watcr dIscharge tern pera ttl re characteri st ICS of pac ked ice b.-:ds for rectangular storage lanks. ,\SHRAE Trans.. ASHRAE. pp 79-89.

SICWJfl. w. E. Jr.. GUle. G. D.: Chandrasekharan. J.: Saunders. C. K. )995, Modeling of the mellmg process of Ice stores In recrilngular (hemlal energy swrage tanks with multiple ICC

openings. ASHRAE Trans.. pp. 66-78.

Stewan, William E.: Saunders. Charles K,: Gute. G. Douglas: Slick­kr. LIsa A. 1995. ICEPAK . modeling lhe icc·f;JIi ng :11111 ice· me It ing processes of thennal eller~y storage lank s. ASHRAE Trans n I pp. 1335-(338.

SIC wan. W. E. Jr.: GOle. G. D.: Saunders, C. K,: Ch;l.ndrasekharan. J.

1995. Development of a design procedure for thennal '\toragc l::Inks utilizing technologIes th~( ,cparale the Hl.IJ)"i,,('(lJfC of icc from .he SIOLl"C <>' Ic'(O :\SHRA£ rr~n'\ rp ::!5.\·~61

GENIUM PUfJL1SHINC

Heat VAPORIZATION/PHASE CHANGE 5CClion 507_9 Transfer MELTING Page 25 DivisIon APPLlCATrONS BIBLIOGRAPHY August 1996

TUffil:(. R. H. 1978. High Tc:mperature Energy Storage. pp. 85. The Fr;}nk lin Il1S(l(U le Press. Phi ladel phi a. PA.

Vlskama. R. 1983. Phase Change Heat Transfer. Solar HeM and SI<)l'oge. Lalt:1II Hem Mal(rials. cd. G. A. Lane. CRe

Pres~. Boca Rawn. FL.

1.� GEOPHYSICAL PHENOMENA AND MELTING IN POROUS ~lEDlA

ThIS biblJographical seCllon covers melting phenomena tn nalure. Since much inlerest exists in soil freeZIng and thawing, lhav.-ing in porous media IS induded here 100. The geophysic31 phenomena wclude '-onnalion of planets. beha Vlor or the e:mh core. magma dynamics. polar Ice. fI ver and lake mdting. snow melting, and thawing of soil.

Akan, A. O. 1'194. Modc:hogenrichmem ofsoluble pollulams ,n snow covers, Waler. Air and Soil Pollullon. vol. 73. pp. 2::!5-234.

Alexiades. V,: Jacob~. G.K.; Dunbar. N.W 1994. Conslralnls on mass balance of soil mOIsture dunng in SilU vito/kation. EnvlronmenlaJ Geology (New York). vol. 23. pp. 83·88,

Ashton. G. D, 1978. RIver lc~, Ann. Rev, Fluid Mech.. vol. 10. Dr 369·392

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