v APORIZATIONIPHASE CHANGE
Heal
Transfer
Division
Section
507.9
Page 1
August 1996
MELTING
CONTENTS-SYMBOLS
Page
I. GENERAL
A.
Contents
"
_
_
1
B. Symbols
C.
D.
E.
I
References
Introduction, Applications, and Basic Physical Concepts
Predictive Methods
_
2
7
8
"
II. PREDICTIVE EQUATIONS FOR MELTlNG
A.
One-Dimensional Melting of Pure Materials without DensilY Change
I. Solutions for Materials that are Initially at
Melting Temperalure
2. Solutions for Materials that are Initially nOt at the Melting Tempera!ure
a. Slab geometry
b. Cylinder
B. One-Dimensional Melting with Density Change
C. The Qu.asi-SUl/ic Approximation
I. Examples of the Quasi-Slatic Approximation for Melting of a Slab
2. Examples of the Quasi-Static Approximation for Melling of a Cylinder
D. Estimation of Melting Time
L. Melting Time of Foodstuff
2. Other Approximations for Melting Time
9
me
9
10
10
_
I I
12
13
13
L3
14
14
15
III. SOME METHODS FOR SIMPLIFYING SOLUTION
A.
B.
The Integral Method, wilh Sample Solution for Melting of a Slab
The Enthalpy Method
15
.,
16
IV. APPLICATIONS BIBLIOGRAPHY
A.
B.
C.
D.
E.
F.
G.
H.
1.
Casting, Molding, Sintering
Mulli-Component Systems and Crystal-Growth
Welding, Soldering, aod Laser. EleclfOn-beam, and Elcclro-discharge Processing
Coating and Vapor and Spray Deposition
Ablalion and Sublimation
Medical Applications and Food Preservation
Nuclear Power Safety
Thermal Energy Slorage
Geophysical Phenomena and Melting in Porous Media
17
18
19
21
21
22
23
23
25
B.SYMBOLS
c
specific heat. Jlkg K
Ei
Ei(x);;;
u
v
.. ~ds,X>O
f
·s
V
x
~
h st
h
h
k
N su
P
[j>
q
r
R
R
latent heal of fusion. Jlkg K
enthalpy
con vecti ve heat transfer coefficienl, W Im 2s
thermal conductivity, W/m K
Stefan number, ;;; c(To . Tr)lh st
pressure, Pa
shape coefficient in Plank's equation
heat flux, J/m 2s
radius, m
radial position of the rnel{- frO!)t, m
shape coefficiem in Plank's equalion
lime. S
T
X
velocity of lhe melt-front due to density change. mfs
specific volume. m 3/kg
volume of the body. m 3
coordinate
position of the melting fronl along the x-direction. m
Greek Symbols
CI.
Thermal diffusivity, m 2/s
o Melting front growth rale constant in lhe integral
solution, Eq. (9-58)
c..Vst (Specific volume of the liquid phase) - (specific
volume of the solid phase), m 3/kg
K
E@;
Va:
Melting rate parameter, dimensionless
== p(lps
temperature. K or "C
GENIUM PUBliSHING
p
X
rectangular enclosures: Experiments and numerical simula­
tions. J. Heat Transfer, vol. 107. pp. 794.
Density, kg/m)
Concentration, kglkg
c..
Subscripts
o
a
f
I
f
sf
a
s
w
At the surface (x=O)
Ambient (fluid surrounding the melting object)
Fusion (melting or freezing)
Initial (at t=O)
Liquid
Phase-change from solid to liquid
outer
Solid
at wall
There always was an incense interest in predicting melting
phenomena as related to such applications as food preserva­
tion, climate and its control, navigation, and materials
processing, expanding with time into new areas such as
power generation and medicine. Increasingly rigorous
q uantilative 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 subsec­
tion 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 Air­
Conditioning 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.
c., Gobin.
7. Bonacina.
Comint Go> Fasano, A.. and Primicerio. M.
1974. On the esti malion of thennophysical propel1ies in
nonlinear heat conduction problems. lnt. J. Heat Mass Trans­
fer, 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. Recy­
cling 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. Ne w York.II. Chan, S. H.. Cho, D. H. and Kocamustafaogu Ilari. G. 198).
C. REFERENCES
6, Bernard.
Transfer
Division
MELTING
SYMBOLS, REFERENCES
Pagel
August 1996
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!t­
ing 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 deforma­
lions 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,
D.. and Maninez. F. 1985. Melling in
GENIUM PUBLISHING
Heat
Transfer
Division
VAPoroZATION~HASECHANGE
MELTING
REFERENCES
Section
507.9
Page 3
August 1996
26. Goodman, T.R. and Shea. lJ. 1960. The melting of (inite
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47. Malyarenko. V.V. 1985. Method ofcalculation ofche freeZing
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28. Hasebe, M.; Murakami, N.; Kobayashi. T. 1995. Personal
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48. Mitche[!. A. 1994. Recent developments in specialty melting
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49. Moallemi. M. K. and Viskanta. R. 1986. Ana[ysis of close·
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29. pp. 855-867.
30. Hersht, 1. and Rabin. Y. 1994. Shear melting of solid-like
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Section 5U7.9
Page 4
August 1996
v MUKlLAHUN/I:'!1.A,:)t'. LHA.Nv!::
MELTING
REFERENCES
Heat
Transfer
Division
field effect 00 !.he solid-mel! phase transformations. IEEE
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Heat
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Division
Section
VAPORIZA TlONIPHASE CHANGE
MELTING
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507.9
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August 1996
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variable grid scheme for a lransient environmenlal ice model.
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lon, MA.
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heat transfer problems wah a change of phase. J. Heat Trans­
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128. Friedman, A. 1968. The Slefan problem in several space
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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
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I 16. Chawala. T. C .. Pederson. D. R.. Leaf. G.. Minkowycz. W.1.,
and S houman. A. R. 1984. Adapli ve coUocalion method for
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I 33. Goodrich. L. E. 1978. Efficient numerical technique for one­
dimensional thermal problems wllh phase change. Int. 1. Heat
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117. Churchill, S. W. and Gupta. l. P. 1977. Approximations for
conduction with freezing or melting, Inc 1. Heat Mass Trans­
fer. vol. 20. pp. 1251· 1952.
118. Cleland, D.1.. Cleland, A.C. and Earle, R.L. 1987. Prediction
of freezing and thawing times for multi-dimensionaJ shapes by
simple formulae. Part I: regular shapes: Part 2: irregular
shapes. tnt. l. Refrig. 10: pp. 156-166: pp. 234-240.119.
119. Comini, G .. del Guidice, S.. Lewis, R. W. and Zienkiewicz, O.
C. 1976. Finite element solution of nonlinear heat conduction
proble ms w-ilh special reference to phase change. Int. 1. Nu m.
Methods Engng. vol. 8. pp. 613-624.
120. Crow Ie y, A. B. 1978. Nu mcrical solution of Stefan problems.
Int. J. Heat Mass Transfer, vol. 21. pp. 2l5-219.
121. Dilley. I.E and Lior, N. 1986. The evaluation of simple
analytical solutions fonhe prediction of freeze-up time. frccz­
ing. and melting, Proc, 81h Inlenwlional Heal TrQJ1sfer Conf..
4: J 727- J 732. S<ln FranCISCo. CA.
134. Gupta. R. S. and Kumar. A. 1985. Treatment of mulh-dimen·
sional moving boundary problcms by coordinate transforma­
lion. Int. 1. Heal Mass Transfer. vol. 28. pp. 1355-1366.
135. Hajl-Sheikh. A. and Sparrow. E. M. 1967. The solution of heat
conduction problems by probability methods. 1. Heat Trans·
fer. vol. 89, PP. 121·131
136. Hogge, M. and Gerrekens. P. 1985. Two-di mensional deform­
ing 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 immobi­
lizalion 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.
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and Mahin. K. 1986, An Inversc
Section
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VAPORIZATIONIPHASE CHANGE
MELTING
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finite element method for the analysis of stationary arc weld­
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140. Hu. H.. ArgyropouJos. A. 1995. Modeling of Stefan problems
in compLex configurations involving [wo different melals
using the enthalpy method, Modeling and Simulation in Ma­
terials Sci. and Engng, vol. 3. pp. 53-64.
Heat
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158. Meyer. G.H. 1981. A numerical method forlhe solidification
of a binary alloy. tne. J. Heac Mass Transfer. vol. 24. pp. 778­
781.
159. Mingle. 1. 1979. Stefan problem sensitivity and uncertainly.
Num. Heal Transfer. vol. 2. pp. 387 -393.
141. Hyman, J.M. 1984. Numerical methods for uacking inter·
faces. Physica D, vol. 120. pp. 396-407.
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.
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.
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.
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.
162. Ockendon. 1. R. and Hodgkins. W. R.. eds. /975. Moving
Boundary Problems in Heal Flow an4 Diffusion. Oxford
Universily Press (Clarendon). London.
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.
163. Pharo. Q. T. 1.9&5. A fast unconditionally stable tinite-differ­
ence scheme for the heat conductions with phase change. [nt.
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145. Kim. CJ., Ro. S.T. and Lee. 1.5.1993. An efficient computa·
tional technique 10 sol ve the moving boundary problems in lhe
axisymmetric geometries. lnt. J. Heat Mass Transfer. Vol. 36.
pp. 3759-3764.
164. Rabin. Y. and Korin. E. 1993. Efticient numerical SOIUllOn for
the multidimensional soliditication (or melting) problem us­
ing a microcomputer. Int. 1. Heat Mass Transfer. vol. 36. pp.
673-683.
146. Lazaridis, A. 1970. A numerical solution of the
multidimentional solidification (or melting) problem. 1m. J.
Heat Mass Transfer, vol. 13, pp. 1459-1477.
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 progres­
sion and thawing time. [nl. J. Refrig.. vol. 17. pp.374-}80.
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.
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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
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152. McDaniel, O. J. and Zabaras. N. 1994. Least-squares from­
tracking 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
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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
166. Rao. P. R. and Sastri. V. M. K. 1984. Efticienl numencal
mcthod for cwo dimensional phase change problems. Inc. J.
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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. 559­
57!.
168. RUbillshcein. L. I.. 1967. The Sfefan Problem. Translations of
Mathematical Monographs. Am. Malh. Soc .. Vol. 27. Provi­
dence, 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 of average freez­
ing and thawing temperature vs. time with infinite slab corre­
lauon. 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 multi­
dimensional 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'
GENIUM PUBLISHING
Heat
Transfer
Division
VAPORUATION~HASECHANGE
Section
MELTfNG
INTRODUCTION, APPLICATIONS, AND BAS[C PHYSICAL CONCEPTS
phase Stefan problem. Solar Energy.
YOI.
24. pp. 525-528.
176. Solomon, A.D.. Wilson, D.G. and AJexiades, V. 1983. Ex­
507.9
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D. INTRODUCTION, APPLICATIONS, AND BASIC
PHYSICAL CONCEPTS
plicit 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 boundary condition. Inc. J. Mach. & Math. Sci _. vol.
7. pp. 549-563.
178. Tarnawski. W. 1976. Mathematical model of frozen con­
sumption 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 semi­
explicit 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
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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 re­
gion 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
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187. Westerberg, K. W.. Wilclof. C. and Finlayson. B. A. 1994.
Time-<lependent finite-clement models ofphase-change prob­
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pp. 119-143_
188. White. R.E. 1985. The binary alloy problem: existenu., unique­
ness. and nu merieal approx imalion. SlAM 1. N um. Anal., vol.
22, pp. 205-244.
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 fol­
lowing freeze-preservation of foodstuffs, snow and ice re­
moval. manufacturing (such as in casting, mOlding, sinter­
iog, 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-chang­
ing materials. A bibliography for these applications. with a
brief introduction, is given in Subsection [v.
Melting is often accompanied by freezing, and the thermo­
dynamics as well as uanspon principles of the two pro­
cesses 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.
189. Wilson. D. G.• Solomon. A. D., and Boggs, P.T.. cds. 1978.
190. Yoo. J. and RUbinsky. B. 1983. Numericalcomputarion using
fini Ie elements for the moving interface in heal transfer prob­
lems 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 during directional solidification. IoU. Nu II\. Melh.
Heal Fluid Row, vol. 4. pp. 115-)29.
dP
h(
dT
T6vS(
- = -S ­
Moving BowuJury Problems. Academic Press, New York.
Eq. (9-1)
where hs[ is lhe enthalpy difference becween the phases (=
h l - h s > O. the latent heat of melting) and 6.v<J. is lhe
specific volume difference between the phases (= v( - v s )­
Exantination of Eq. (9-l) shows that increasing the pressure
wi II result in an rise of the melti ng tcm perature 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 6V sl < 0 (for waler. for example)_ The latter case
GENIUM PUBLISHING
Section 507.9
Page 8
August 1996
VAPORIZATIONIPHASE CHANGE
MELTING
PREDICTIVE METHODS
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 compli­
cated 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'..... " .. "
..
o
!
t
I
X,
x
Figure 9-/. A Liquid-Solid Phase Diagram of a 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 X J. 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.
Heat
Transfer
Division
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 concentra­
tion 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 distribu­
tion 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-separa­
tion processes.
The presence of a two-phase mixrure zone with tempera­
ture-<lependent concentration and phase-proportion obvi­
ously 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 cominu­
ity. 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
GENIUM PUBLISHING
Heat
Transfer
Division
VAPOIDZATION~HASECHANGE
Section
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
[64 D, 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]. Generi­
cally, 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 proper­
ties 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 numeri­
cal 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 materi­
als 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 solu­
tions 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, [12 J, [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
507.9
Page 9
August 1996
MELTING
PREDICITVE EQUATIONS FOR MELTING. ONE DrMENSIONAL
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 solu­
tions 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 simplifica­
tion, 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 tempera­
cure 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
GENIUM PUBliSHING
Section 507.9
Page to
August 1996
VAPORIZATIONIPHASE CHANGE
MELTING
ONE-DIMENSIONAL MELTING WITHOm DENSITY CHANGE
Liquid
T
EXAMPLE
Solid
The temperature of the vertical surface of a large
volwne of solid paraffin wax used for heat storage,
initially at lhe fusion temperature, T j 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 kglm 3 , 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)
1 Pbas<:~hange
v~ace
T
J
Heat
Transfer
Division
= =
=
T,=Tr = Ti
N
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_
SI~1
= (2.14 kJ I kg"C)(5S"C - 28"C)
241. 2 kJ I kg
=0.266.
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
T([ X(t)]:::. Tf for t > 0,
_kl(~Tt)
oX
"" Pihs[ dX(t)
(X(tIJ
for I> 0,
d1
Eq. (9-5)
1.2,--------------------,
The analytical solution of this problem yields the tempera­ ture distribution in the liquid as.
x)
[2..ja[t
,
0.8
'A'
erf - ­
Tt(x,t)=To-(To-T/)
.
erfA
1.0
0.6
for t>O, Eq.(9-6)
0.4
where erf stands for the error function (described and
tabulated in mathematical handbooks), and 'A.' is the
solution of the equation
A'e'" 2 erf(A.') =
0.2
0.0
+---,--,---,--r---r-----r--r-----r-,...---j
o
~.( ,
2
Eq. (9-7)
with NSI~[ being the Stefan Number, here defined for the
liquid as
Eq. (9-8)
3
4
5
Ns tc ,
Figure 9-3. The Root
A' oj Eq. (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
Equation (9-7) can be solved to find the value of A' for (he
magnitude of N SlC(' 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)
temperalure, conductive heat transfer takes place in born
phases. Consider a semi-infinite solid initially at a tempera­
ture T; lower than lhe melting temperature Tf(Fig. 9-4). At
=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).
time I
=
GENIUM PUBliSHING
=
T
Section
VAPO~ATION~HASECHANGE
Heat
Transfer
Division
Liquid
~
.l
507.9
Page 11
August 1996
MELTING
ONE-DIMENSIONAL MELTING WITHOUT DENSITY CHANGE
Solid
Eq. (9-14)
~
T( (x,t)
l
b. Cylinder
Very few exact solutions exist for melting of bodies that are
shaped
differently than slabs. This is an available exact
,~T (
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
____ T,'" T;
pipe inside an insulator). Ie should be noted that natural
~ PfJase-<,hange
convection of melt was found to have an impo.nant role in
asx-4 oo
~ inlerface
such problems (d. Yao and Prusa (74)), and the solution
shown can only be used for rough evaluation. Other phase­
X (t)
o
change problems in non-planar geometries are solved by
approximate and numerical methods (Alexiades and
Figure 9-4. Melling of II semi-inftniie solid initially tU a below­
Solomon (1], Cheung and Epstein [13], Yao and Prusa [74],
~
V
melting temperature. Heat conduction Ia1us place in both
phases.
The analytical solution of this problem yields the tempera­
ture distributions in the solid and liquid phases, respec­
tively, as
T
Eq.(9-1O)
and
source q'
Eq. (9-11)
where erfc is the complementary errorjwtclion, X is a
constant. obtained from the solution of the equation
Eq. (9-12)
where Ns rt ( 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.
Figure 9-5. Outward Melting of an Initially Solid Cylinder at
Temperalure T, < TI , 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
T(r,r)
lim
r --+
= Tf .
Eq. (9-16)
O¢
The heat transfer energy balance at the line heat source is
expressed by
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
lim
r --?
[
0
GENIUM PUBUSHING
2 k
-'{(r
d7t(r.f)] =q>O
,
(ar
.
Eq. (9-17)
VAPOREATION~HASECHANGE
Section 507.9
Page 12
August 1996
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
Heat
Transfer
Division
MELTING
ONE-DIMENSIONAL MELTING, THE QUASI-STA TIC APPROXIMAnON
+L[Ej(-~}£i(-A."1)]
47tk(
4a
l
(
Eq. (9-19)
in 0 < r < R( t) for t > 0,
~(r,()=To+(Tf-To)
Ei( _r 2 14a.c)
.(
,,2
El -A. at /
as
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 < T r, melted by imposing a constant temperature
To> T r 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
)
Eq. (9-20)
in R(t):5 r fort> 0,
respectively. where Ei is a function defined by
Ei(x)
=f e~s
ds. x> O.
in 0 S x ::;: X(t) for t > 0,
erfc[
Eg. (9-21)
_
(
T.(X,I)- To + T! - To
Eq. (9-23)
J-
(I -I-l )cl
2 a,l
)
enc
m
]
Eg.
('")
~)(A
(9-24 )
and where A" is the root of the transcendental equation
in X(t) S x for l> O.
The location of the melt ing fron t is
Eq. (9-22)
Solutions of Eq. (9-22) can. in general, be obtained rela­
tively easily by a variety of commonly-used software
packages for the solution of nonlinear algebraic equations.
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
B. ONE-DIMENSIONAL MELTING WITH
DENSITY CHANGE
,_1
A.'"e"
erfA.'"
Eg. (9-27)
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 densi­
ties 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.
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
P
IJ=_I
p,
Eq. (9-28)
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
GENIUM PUBLISHING
VAPORQATION~HASECHANGE
Heat
Transfer
Division
Section
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 I} and Prusa and Yao [54}, and
additional results are given by Shamsundar and Sparrow (59J.
r
507.9
Page 13
August 1996
MELTING
THE QUASI-STATIC APPROXIMATION
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
0, and it otherwise overestimates the values of both
X(f) tand T(x, t). While the errors depend on the Specl'fjIC
problem, they are confined to about 10% in the
above-described case (Alexiades and Solomon (1 J.)
NSf( ~
For the same melting problem but with the boundary
condition of an imposed time-dependent heat flux qo(t),
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.
_kl(dTf
dx
=qo(r) for [>0,
)
Eg. (9-31)
0.1
the quasi-static approximate solution is
,
X(t) =:
-l-f
qo (t)df fort>
phsl
0,
Eq. (9-32)
o
Tl(x,f) = T1 + qo
kf
in 0 :::; x
~
[.!iLl -xJ
Eq. (9-33)
phsl
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
.
dx
)
OJ
= h{TQ(I) - TlCO.f)] for r ~ 0,
Eq. (9-34)
the quasi-static approximale solution is
It is assumed here that the problems are one·dimensional. and
that the material is initially at the freezing temperarure Tr.
Eq. (9·35)
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) > T r 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. (9-36)
in 0
s x S X(r) for r > O.
2. Examples of the Quasi-5tatic Approximation for
Melting of a Cylinder
Eq. (9-29)
Eq. (9-30)
in 0 ::: x :::; X(t) for t 2: O.
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
(Fig. 9·6)
'0
GENIUM PUBUSHING
Section 507.9
Page 14
August 1996
VAPORlZA
no NIPHASE CHANGE
Heat
Transfer
Division
MELTING
ESlTIvl"ATION OF MELTING 11MB
due to a temperature imposed at the internal radius ri, i.e.
T((r;,t) ~ To(t) > Tf for l > O.
Eq. (9-37)
The solution is
2R(ri In R(t)
r;
=
J[R(t/ ( 1- ~kl
hI";
lo[rl R(t)]
TtCr,t) ~ Tf + [ To(f) - Tf ] [
{ )J
In 'i / R t
J[
I
r/]+
4k(
phsl
T,,(f) - Tf Jdr. Eq. (9-45)
o
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
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.
I
2R(r)2 In R(f) = R(I)2 Ii
".2 + ~f[To(I) - TI jdt.
D. ESTIMATION OF MELTING TIME
Eq. (9-39)
phst
o
lEthe melting for the same case occurs due to the imposition of a heat flux qo at rj,
-kl(dTtJ =qo(t»O fort>O.
dr '1. 1
Eq. (9-40)
the solution is
L Melting Time of Foodstuff
T. (r,f): T - qo(r)r; In_r_
t
I
kl
R(t)
in
ri ::; r
Eq. (9-41)
S R(t) for t> O.
1/2
R(t)= r/+2..:Lf' q,,(/)dt
phs (
(
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.
o
(ort>O.
Eq. (9-42)
]
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
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 T a, can be
approximated by Plank's equation
7j,
Eq. (9-46)
_kl(dT{)
dr 'i.T
=h[T,,(t)-71(r;,f)] > 0
for
I>
O. Eq.(9-43)
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.
The solution is
in
Table 9-1
Shape factors for Eq, (9-46) from ASHRAE (3J
r,. S; r ::; R(/) at ( > 0,
Fonns
with R(t) calculated from the transcendental-integral
equation
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 I able,
To use Eq. (9-46) for melting. k and p should be the values
for the food in its (hawed state,
GENIUM PUBLISHING
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
with the heating
is surrounded by air at Ta =
perfonned with a convective heat transfer coefficient h
= 68 W/m 2 K. For the fish, Tf = -I·C. hst.::: 200 kJlkg, 1'5
= 992 kg/m J , and k., = 1.35 W/m K.
Using Table 9-1, the geometric coefficients for the
cy Ii ndncal 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
n·c.
=
= 200000, 992 (1/4(0.1) + 1/16(0.1)2) = 68665 = I. 9h.
I( -I) - 231
68
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 (132 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. 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.
n.
EXAMPLE
I
507.9
Page 15
August 1996
MELTING
SIMPLIFIED SOLUTIONS-THE INTEGRAL METIIOD
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."
r
Section
VAPORIZATTONfPHASE CHANGE
Heat
Transfer
Division
1.35
o
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)
at :: a, dXI
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)
&I.
(9·48)
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.
_k,(OT,]
=ph,tdX(I)
dX ! x(I)1
dl
Eq.(9·51)
forl>Oo
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.
GENIUM PUBLISHING
Section 507.9
Page 16
August 1996
VAPORIZATIONIPHASE CHANGE
MELTING
SIMPLIFIED SOUITIONS-TIfE ENTHALPY METHOD, BIBLIOGRAPHY
_I
C1. t
i.[Xf(t~(X'I)dx
_TfX(t)]
dl
o
~_ph« dX(l) _ aT (O,t).
kl
dt
Eq. (9-53)
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
Heat
Transfer
Division
identica1 for NS/C'l up to about 0.5, with me integral solution
overestimating the value of Q by only about 4% when N su(
'= 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
=
-
b~
hlX +(Ta
X
2
- Tf )
Eq. (9-55)
h
.
c
Substitution of the polynomial {Eqs. (9-54), (9-55») into Eq.
(9-53). results in the ordinary differenlial equation
T=
TI
for
for
h - hs(
for
e
h
~
(solid)
cTf
eTf < h < eTI
+ nSl
h;:: eTf +Jv.
(interface)
(liquid)
Eq. (9-60)
Eq. (9-56)
which, using the initial condition Eq. (2-49), gives the
solution
X(I)
=26 ( a(t ) 11l ,
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
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.
GENIUM PUBliSHING
Heat
Transfer
Division
V APORlZATIONIPHASE CHANGE
MELTING
APPLICA nONS BIBLIOGRAPHY
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 superconductors. Row of the molten material. the course of its solidification (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 manufacturing of materials for superconductors.
Section
507.9
Page 17
August 1996
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 alia y 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. I nt. 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 male rials in
e;>;truders. Pressure dislribution along the length of {he
extruder screw. Fibre Chem.. vol. 16. pp. 362-365.
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.
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.
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.
Rauwendaal. C. 1989. Improved analytical melting theory. Adv,
Polymer Technol.. vol. 9. pp. 331-336.
Bose. A. Technology and commercial status of powder-injection
molding. 10M. vol. 47. pp. 26-30.
Rauwendaal. C. 1991n. Melting theory for temperature-dependem
fluids. C)(aCI analytical solution forPQwer-law fluids. Adv.
Polymer Technol., vol. I!. pp. 19-25.
Chiru vella. R. V.: J aluria. 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.
Rckhson. S.M.. Rekhson. M .. Ducroul:. l.-P.. and Tarakano v , S,
1995. Heal transfer effects in glass processing. Ceramic
Engng Sci. Pro<;.. voL 16. pp. 19-37.
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.
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
Page 18
August 1996
VAPORIZATIONfPHASE CHANGE
MELTING
APPLICATIONS BIBLIOGRAPHY
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
on
The bibliography in this section primanly focuses
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 significance 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 properties must be considered.
One of the most prominent applications is alloy-making.
and the last several decades have seen large and increasing
in vol vement with crystal growth, primarily for the c leetranics 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 multicomponent 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 multicomponent 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 Handbook 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 solidliquid 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
Heat
Transfer
Division
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 ofcon vecli 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. J 995. 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 of a 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
VAPORJZATIONIPHASE CHANGE
MELTING
APPLICATIONS BIBLIOGRAPHY
Heat
Transfer
Division
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 I icon 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
Sc~tion
507.9
P~ge 19
August
1996
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. N umencal 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 Srirn vasan.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 me lied 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, 3Cline. 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..1 model·
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
Page 20
August 1996
VAPORlZATION/PH..\SE CHANGE
MELTING
APPLICA TIONS B JB UOGRAPHY
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
welding. Nucl. Metall.. vol. 20. pp 1160-1170.
(0
arC
Heal
Tr:wsfcr
Division
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.
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.
R3Jurkar. K. and Pandit. S. 1986. Formallon and ejeclion 01 ED,',.I
debns J Engng Industry. vol. 108. pp_ 22-2(
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.
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. I nsl. evlech. Eng n~'. Pan B: Journa I 0(
Engng ;vlanur'aclure, vol. ~09. pp 7S-8~.
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.
Rubin. BT. 1975 TheoreucaJ model of the submerged arc welding
process. ,"fela!1. Trans. B. vol. 68. pp. 175-) 32.
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.
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.
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.
Surface Coatings Techno!.. vol. 76-77. 1995. pp. 589-594
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
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).
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 c haract<::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
Spoerer. 1.. Igenbergs, E. Treatment of the surrace layer ot Sled wilh
hIgh energy plasma pulses.
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 u rg
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
VAPORIZATION/PHASE CHANGE
MELTING
APPLICATIONS BrB LIOGRAPHY
HeJ.t
Transkr
Divi~iol1
S':Clion
507.9'
Pag<:: 21
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.
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.
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.
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.
D. COATING, AND VAPOR AND SPRAY
DEPOSITION
Layadi. N.: Roca I Cabarrocas. 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.
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.
Sidfens. H ·D.: :-.!assell$tew. K. 1994 R.:cem developme:ll$ in
slOgle- WI re vacuu marc spraylOg. J. Therm<ll SprJ y Tech nol .
vol. 3. DD. -l12...oli7.
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. 271394.
Zaal. J. H. 1983. A quaner ot' 3 century of pi asma sprayi ng. rllln. Rev
Mar. Sci.. vol. 13. PD. 9-.12.
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. I n Heal Gild Mass 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.
Rykalin.~
Nand Kudinov. V V 1976. Plasm:lspfJying Pure -\ppl
Chern.. vol. -lS. pp. 229-239
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.
E. ABLA nON AND StiBLIMA TION
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.
Fiszdon. J. 1979. Melting of powder grains 10 3 plasma flame. Inc J
Heal Mass Transfer. vol. 22. pp. 749·761.
Aral. N. and Karashima. K.. I. 1979. TrJnsicm lhennal response oj
ablaling bodIes. AlAA J.• vol. 17. pp. 191·195.
Groma. I. and VCIO. B. 1986. Melling of powder grains 10 plasma
spraywg. 1m. J. Hem Mass Transler. vol. 29. pp 549·554.
Biol. Ivl.A. and AgraWal. H.C. 1964. Varialional analySIS 01 eJbl;Hioll
for vanable plOpenies. J. Heal Transfer, vol. 86. pp 437·
442.
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.
BiO!. M. A. and Daughada y. H. 1962. Va ri alionaJ anal ySI$ of abl at ion.
J. Aerospace Sci .. vol. 29. pp. 227-229.
H,jlkala. K. and MilOi. 1992. Heal uaflsfer in plasma spraying. [0
Heal and Mon Transfer in MOlen'als Processing. ed. I.
Tallas a"'"a and N. Lior. pp. 3-16. Hemisphere. New York.
Chen. c.J. and OSlrach. S. 1969. Mellinil ablalion for lwo·dimt:il·
s/Ona! and Jxisymmelric blunt bodies WIlh a body furcc.
Kang. B.. Waldvogel. 1. and Poulikakos. D. 1995. Remelling phenomena In lhe proccs~ of splal solidlficalion. J. Mal SCI ..
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 suh jected !O
lime· \'anant h(:ll (lv.xes. J I ~eJl Tr~n~fcr vol 105. pp :;00
vol :il}.
rp
-11)12
':9~:\
Progr. He:ll Mass Transler. vol. 2. pp. 195.
G£NIUM PUBLISHING
Section
507.9
P3ge 22
August 1996
VAPORIZATJONIPHASE CHANGE
MELTING
APPLICATIONS BIBLfOGRAPHY
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. H eat ~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. A 13. pp. 203Spaid. 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. A IChE 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
PRESERVA nON
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:
Heat
Transfer
DivIsion
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_
review. Cryobiology, vol. 21. pp. J7i-3S.l
:J
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.'
G£/v'/UM PUBLISHING
HC.ll
S~ction
V APORIZATION/PHASE CHANGE
Transfer
Division
MELTING
APPLICATIONS BIB UOGRAPHY
L:Jdirat,
G. NUCLEAR POWER SAFETY
Melting plays a role in sevcral areas related to nuclear
507.9
Page 2]
Augu's\ 199(,
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.
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
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.
accident allhe Three Mile Island reaClor. Melting, and the
subsequenl solidificalion due to cooling, have been {he
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.
NucL Soc. vol. 6. pp. 155·156.
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 pOIJSsium. 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. J r..
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. 7 6l 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 penetration of
a hot liquid pool into a melting miscible subSlrate, Nuel.
Techno!., vol. 64. pp. 275.
Am.
\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.
c.:
von der Hardt. P.; Jones. A. V.: Lecomte.
TJltegr.lin. A. 199<1.
~uclear safety research the Phebus FP severe accideOl
e~perimemal program. Nuclear Safety. vol 35, pp. 1<>7·
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Wol f. J. R.: Akers, D. W.: Nei mark. L. A. 1994. Relocation of mollcn
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GENIUM PUBLISHING
Seclion 507.9
Page 24
AuguSI 1996
V A PORIZA nON/PHAJI::: CHANGE
Heat
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Div\ston
MELTING
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GENIUM PUfJL1SHINC
rp
::!5.\·~61
VAPORIZATION/PHASE CHANGE
MELTING
APPLlCATrONS BIBLIOGRAPHY
Heat
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DivisIon
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GENIUM PUBLISHING
Section
507.9
Page 26'
August 1996
VAPO~ZATION~HASECHANGE
MELTING
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