The heat conduction problem, in the presence of a change of state, was solved for the case of an indefinitely long cylindrical layer cavity. As boundary conditions, it is imposed that the internal surface of the cavity is maintained below the fusion temperature of the infilling substance and the external surface is kept above it. The solution, obtained in nondimensional variables, consists in two closed form heat conduction equation solutions for the solidified and liquid regions, which formally depend of the, at first, unknown position of the phase change front. The energy balance through the phase change front furnishes the equation for time dependence of the front position, which is numerically solved. Substitution of the front position for a particular instant in the heat conduction equation solutions gives the temperature distribution inside the cavity at that moment. The solution is illustrated with numerical examples.

1.
Lamé
,
G.
, and
Clapeyron
,
B. P.
, 1831, “
Mémoire sur la solidification par refroidissement d’un globe liquide
,”
Ann. Chim. Phys.
0365-1444,
47
, pp.
250
256
.
2.
Weber
,
H.
, 1912,
Die partiellen Differentialgleichungen der mathematischen Physik
,
5th ed.
,
Vieweg & Sohn
,
Braunschweig, Germany
, Vol.
2
.
3.
Stefan
,
J.
, 1889, “
Über einige probleme der theorie der wärmeleitung
,”
Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche, Klasse Abteilung 2a
,
98
, pp.
473
484
.
4.
Stefan
,
J.
, 1889, “
Über die diffusion von säuren and basen gegen einander
,”
Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche, Klasse Abteilung 2a
,
98
, pp.
616
634
.
5.
Stefan
,
J.
, 1889, “
Über die theorie der eisbildung, insbesondere über die eisbildung im polarmeere
,”
Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche, Klasse Abteilung 2a
,
98
, pp.
965
983
.
6.
Stefan
,
J.
, 1889, “
Über die verdampfund und die auflösung als vorgänge der diffusion
,”
Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche, Klasse Abteilung 2a
,
98
, pp.
1418
1442
.
7.
Stefan
,
J.
, 1891, “
Über die theorie der eisbildung, insbesondere über die eisbildung im polarmeere
,”
Ann. Chim. Phys.
0365-1444,
42
, pp.
269
286
.
8.
Koshkin
,
N. I.
, and
Shirkevich
,
M. G.
, 1982,
Handbook of Elementary Physics
,
4th ed.
,
Mir
,
Moscow
.
9.
Yao
,
L. S.
, and
Prusa
,
J.
, 1989, “
Melting and Freezing
,”
Adv. Heat Transfer
0065-2717,
19
, pp.
1
95
.
10.
Fox
,
L.
, 1974, “
What Are the Best Numerical Methods?
,”
Moving Boundary Problems in Heat Flow and Diffusion
, Vol.
210
,
University of Oxford
,
New York
, pp.
25
27
.
11.
Carslaw
,
H. S.
, and
Jaeger
,
J. C.
, 1986,
Conduction of Heat in Solids
,
2nd ed.
,
Clarendon
,
Oxford
.
12.
Rubinstein
,
L. I.
, 1971, “
The Stefan Problem
,”
Translations of Mathematical Monographs
, Vol.
27
,
American Mathematical Society
,
Providence, RI
.
13.
Elliot
,
C. M.
, and
Ockendon
,
J. R.
, 1982,
Weak and Variational Methods for Moving-Boundary Problems
(
Research Notes in Mathematics
),
Pitman
,
London
.
14.
Crank
,
J.
, 1984,
Free and Moving Boundary Problems
,
Clarendon
,
Oxford
.
15.
Hill
,
J. M.
, 1987, “
One-Dimensional Stefan Problems: An Introduction
,”
Pitman Monographs and Surveys in Pure and Applied Mathematics
, Vol.
31
,
Longman Scientific & Technical
,
Essex, England
.
16.
Alexiades
,
V.
, and
Solomon
,
A. D.
, 1993,
Mathematical Modeling of Melting and Freezing Processes
,
Taylor & Francis
,
Washington, DC
.
17.
Abramowitz
,
M.
, and
Stegun
,
I. A.
, 1965,
Handbook of Mathematical Functions
,
Dover
,
New York
.
You do not currently have access to this content.