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Engineering Analysis with Boundary Elements 16 (1995) 83-92
Copyright © 1995 Elsevier Science Limited
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Stefan's work on solid-liquid phase changes
Bolidar Sarler
Laboratory for Fluid Dynamics and Thermodynamics, Faculty of Mechanical Engineering, University of Ljubljana,
AsTcerteva 6, 61 000 Ljubljana, Slovenia
This paper presents a detailed review of Jo~ef Stefan's research on solid-liquid
phase changes published in six treatises between the years 1889 and 1891. His
achievements on this subject are related to the broader context of his interest
in transport phenomena, particularly liquid-gas phase changes and chemical
reactions in the years 1873 and 1889 respectively. Stefan's eponymous work is
placed in perspective between the present and the first experimental and analytical
attempts to describe the solid-liquid phase change by the pioneers Blake in the
17th, and Lamt, Clapeyron and Neumann in the 18th century. In honour of
Stefan's work involving moving and free boundaries, the concepts of the Stefan
problem and the Stefan number are widely used nowadays. The primary intention
of this paper is to attempt to complete and unify the information on the historical
roots which led to these two terms.
Key words: Stefan problem, Stefan number, solid-liquid phase change, analytical
solution, melting, dissolving, freezing, solidification.
when studying freezing of fuel in air and spacecraft or
ablation of bodies in re-entry. It has to be taken strictly
into account in many safety considerations involving
nuclear fission and fusion reactions. It is critical in cryosurgical treatment of tumours and in cryopreserving of
biological cells such as blood or organs for transplantation. Almost every product undergoes a solid-liquid
phase change process at some stage, so the control of
these phenomena represents a key to modem manufacturing and materials processing of metals, ceramics,
polymers, composites, and electronic components.
Typical related techniques are continuous, ingot, form,
precision, and die casting, arc, resistance, plasma,
electron beam, laser, and friction welding, meltspinning, planar flow casting, atomisation, bulk undercooling, and surface remelting rapid solidification techniques, glass forming, laser and plasma spray coating,
Bridgman or Czochralski crystal growth, electroslag
remelting purification, etc.
INTRODUCTION
Solid-liquid phase changes are encountered extensively
in nature. The solidification and convection of lava
influences tectonic acti,dty and melting or freezing of
the polar ice caps is critical for our climate. The technology associated with solid-liquid phase change
has accompanied humankind since the Neolithic. As
solid-liquid processes in nature determine glacial
periods, technology, such as casting for example,
associated with solid-liquid phase change determines
the evolution of our civilization.
In contrast to the Copper, Bronze and Iron Ages our
current era of industrial development is not defined by
its type of cast metal implements and weapons. Instead,
all aspects of modem technology are related to the level
of understanding of the solid-liquid phase change. This
knowledge has become an indispensable part of the pure
sciences such as astrophysics, geophysics and meteorology, including studies of the planet's core solidification,
magma migration, global weather modelling, etc. We
use it in mechanical and chemical engineering problems
such as latent heat storage systems, desalination of sea
water, purification of industrial service water, in civil
and mining engineering, in problems of ground freezing
for construction, permafrost, concrete hardening, and
water impact freezing in mines. It is important in electrical engineering problems like melting of safety fuses,
in the food processing industry in deep freezing and
freeze-drying of foodstuffs, in aerospace engineering
HISTORICAL ROOTS
Three basic conditions had to be satisfied in order to
establish the first analytical treatments of the solidliquid phase change. First, sufficient experimental
evidence and an understanding of sensible heat and
the heat of fusion, second, a sufficiently developed
mathematical background, and third, an idea of how
to combine these elements.
83
84
B. Sarler
The secrets of heat transfer during solid-liquid phase
change processes had been demistified by the Scottish
medical doctor, physician and chemist Joseph Black
(1728-1799) in a series of experiments performed with
water and ice at the University of Glasgow between
the years 1758 and 1762. He demonstrated that solidliquid phase change processes could not be calorically
understood within the framework of sensible heat
alone. As a consequence, he introduced the term and
the concept of latent heat.
French mathematician and physicist Jean Baptiste
Joseph Fourier (1768-1830) provided the necessary
physics and mathematics for heat conduction in his
famous 'mathematical poem' La Throrie Analytique
de la Chaleur, published in 1822.
The idea of how to analytically incorporate latent
heat in heat conduction equations was first expounded
in a pioneering paper 2s by the physicist Gabriel
Lam6 (1795-1850) and the mechanical engineer Emile
Clapeyron (1799-1864) in 1831. This work presents an
extension of the earlier work by Fourier29 which tried to
produce a rough estimate of the time which has elapsed
since the Earth began to cool from the initially molten
state; however, solidification was not taken into account.
Lam6 and Clapeyron pose the following simplified
model of the problem: the curvature of the Earth has
been neglected. The initially molten Earth has a uniform
melting temperature. The freezing process is initiated by
the abrupt temperature drop on the surface which
remains at a fixed temperature after the initiation of
the cooling. The material properties of the Earth are
constant. Convective effects are neglected.
The most important conclusion of their work is that
under the described circumstances the solid crust thickness grows proportionally to the square root of time.
The authors did not explicitly derive the equation for
determining the relevant constant of proportionality.
According to Heinrich Weber's note on page 122,
in his publication3° of Georg Friedrich Bernhard
Riemann's lectures on partial differential equations,
Franz Neumann (1798-1895) solved in detail a problem
similar to the one solved by Lam6 and Clapeyron
with the initial temperature above the melting point.
Neumann presented the solution to the problem in his
unpublished Kfnigsberg lectures in the early 1860s.
Jo~ef Stefan was the one who, in addition to Lamr,
Clapeyron and Neumann, contributed importantly to
establishing the roots of the subject in the last century.
Many facts relating to Stefan's work on solid-liquid
phase changes are not known to the majority of the
computational moving boundaries community. On the
other hand, the author did not find any existing
complete bibliography or review of Stefan's work on
that subject. Therefore, it seems appropriate to publish
a paper about his activities in this special issue of Engineering Analysis with Boundary Elements devoted to the
Stefan problems.
SHORT BIOGRAPHY
Jo~ef Stefan was born on 24 March 1835 at St Peter/
Sv. Peter, nowadays Klagenfurt/Celovec, and died on
7 January 1893 in Vienna. Klagenfurt and Vienna were
part of the Austro-Hungarian empire at that time, and
are nowadays part of Austria. Stefan's nationality was
Slovene. His name has been written as Stefan or Stephan
and his first name as Josef.
Several expositions of Stefan's work and life have
already been published, cf. Stefan's biography by
(~ermelj,9 written in Slovene with a brief English
summary. More recent is Stefan's biography by Sitar 1°
published in Slovene as well, but including a more
comprehensive English and German summary. Two
other booklets by Strnad should be mentioned as well.
The first one was published on the occasion of the
150th anniversary of Stefan's birth II and the second
one 12 is dedicated to the centenary of his passing. This
latter volume is written in Slovene and English. These
four sources have been used in preparing the following
brief resum6 of the famous scientist.
Stefan completed secondary school in Klagenfurt and
matriculated at the University of Vienna in 1855. He
finished his studies of mathematics and physics in two
years and was awarded a doctorate. A year later he
Fig. 1. Jo~,efStefan, portrait and signature. From the annals of
the Vienna Academy of Sciences from 1893.
Stefan's work on solid-liquid phase changes
became a lecturer, and in the year 1860, when only 25
years old, he acquired the title of Professor. In the
same year he was appointed associate, and 5 years
later, became a full member of the Academy of Arts
and Sciences in Vienna. From 1875 to 1885 he served
as secretary of the Section for Mathematics and Natural
Sciences, and from 1885 to his death he was vicepresident of the Academy. At the age of 30 he became
the director of the Instil;ute of Physics. His outstanding
scientific achievements brought him many domestic
and foreign decorations and memberships in several
distinguished scientific societies.
Jo~ef Stefan is considered one of the most gifted
researchers in history, tte was an outstanding theoretician and experimentalisl:, and at the same time an excellent lecturer. His research is amazing in its profundity
and originality combined with unique diversity and
breadth. Stefan's wide-ranging scientific opus includes
solid mechanics, acoustics, thermodynamics and transport phenomena, optics, and electricity and magnetism,
expounded in around 100 treatises. Today, the Stefan
radiation law and the Stefan radiation constant have
become standard content in any high school physics
and engineering textbook.
Many of you probably do not know that Stefan's
energy was not focused only on science. Prosody and
lyrics in the Slovene language were another part of the
humanities that attracted him, particularly during his
youth.
Let us now turn to Stefan's work on solid-liquid
phase change which is of most interest to the readers
of this journal. Stefan explores the solid-liquid phase
change in a series of wolrks. 2'4-8 This part of his research
could be placed in the broader context of his interest
in moving boundary problems such as for example
diffusional transport of material in a reaction zone 3 or
experimental 1 and theo:retical 6 aspects of evaporation.
PHYSICAL AND MATHEMATICAL FRAMEWORK
Let us first postulate the common assumptions in
Stefan's treatises on solid-liquid phase changes.
(1) One-dimensional planar symmetry, measured by
the Cartesian coordinate x.
(2) Constant specific heats cv, thermal conductivities
kv, and diffusivities av, of the solid 79 = S and
the liquid phase 7:' = £, together with the constant
heat of fusion h~.
(3) Material properties of the phases could differ in
general. The exception is the constant and equal
density p = Ps = Pz: of the solid and liquid phases.
(4) Heat flux F v in phase P is governed by the Fourier
constitutive relation
Fv = - k v
0
Tv
(1)
85
with T~, representing the temperature of the
phase P.
(5) The temperature field T~(x, t) at time t at point x
is governed by the Fourier equation
0 T~,(x, t) = a v
o2
0-7
Tp(x, t)
(2)
(6) The solid-liquid interphase conditions at interphase point x~a(t) at time t are
Ta4 = rs(xca , t) = rr.(xaa, t)
(3)
d
0
ph~4 dtt x ~ ( t ) = - k £ -~x T£(xa4' t)
0
+ ks -~x Ts(xa4' t)
(4)
(7) The temperatures of the phases are assumed to be
in the general form
(x-x_0
Ts = As + Bs erf tk(4as(t _ to))l/z]
(.
x -_xo
T£ = Ac + Bc erf ~(4al:(t - to))l/2]
(5)
(6)
with the coefficients As, Bs, Ac, BL, Xo and to,
which have to be determined through the specific
initial and boundary conditions.
The deviations from the defined paragraphs will
be made clear whenever they appear. It should be
pointed out that the constructed framework on the
one hand considerably shortens our passage through
Stefan's work, but on the other hand, produces several
differences in form (but not in sense) between his
original equations and the more generalized analytical
representation used in this text.
REVIEW OF STEFAN'S WORK ON
SOLID-LIQUID PHASE CHANGES
On some problems in the theory of heat conduction
First problem
In his first solid-liquid phase change paper 2 Stefan
solved the problem which could be considered as
the melting complement of the Lam6 and Clapeyron
problem.
The semi-infinite half space (x0, oo) consists of a
material which may exist in either a liquid or solid
phase. Initially the material is in the solid phase at
freezing temperature Tos = T ~ . At time t > to the
surface at x 0 is heated and maintained at a given
constant temperature above that of the freezing temperature. At time t > to the problem is to determine
the thickness of the liquid layer x~4(t)-Xo and the
temperature distribution in the liquid phase.
86
B. Sarler
temperature variation in the liquid phase
LIQUID I SOLID
3
T£ = Tr
(11)
Tr - Tos (x - Xo)
2C(t t0) U2
with the constant C given by
C= (kz(Tr-- T°s)-• 1/2
\
2
]
and the same solid-liquid interphase growth law (9) as
the general case.
After being sure of the limiting expressions, Stefan
discussed the original Lam6 and Clapeyron problem
and proposed an experiment with the immersion of a
cold sphere into a large reservoir of water at T g in
order to check the growth rate of the ice around
the sphere. Stefan's tendency to experimentally confirm
his theoretical developments and vice-versa can be
perceived straightforwardly from his works.
[sl
1
llllIllIlllllllll
0
2phM
(12)
Fig. 2. Illustration of the solution to thefirst problem. Difference
between isotherms is 0.1(K). x0=0.0(m), t0=0.0(s),
T0s=0-0(K), Tr = I'0(K), p = l(J/(kgK)), kc = I(W/
(mK)), c£ = 1 (J/(kgK)), T~ = 1 (K), hM = 1 (J/kg),
C = 0"62006263.
Stefan used the same assumptions in solving the
problem as did Lam6 and Clapeyron, however, his
discussion was more complete. For temperature distribution in the liquid phase occupying the space (x0, xM), he
derived eqn (6) with the constants
Second problem
Infinite space ( - ~ , o o ) is assumed to consist of a
material which may exist in either a liquid or solid
phase. Initially it is assumed that the region ( - ~ , x 0 )
occupies the liquid phase at some known uniform
temperature T0c above the freezing temperature, while
the region (x0, c~) is assumed to be occupied by the
solid phase with a known uniform temperature Tos
below the freezing point. At time t > to the problem is
to determine the position of the boundary separating
the solid and the liquid phases and the temperatures in
both phases.
Tos- Tr
AL = Tr,
B£
erf(Ca~)/2 )
(7)
The constant C is the solution of the transcendental
equation
phMC = -k£B£(C)Tr -1/2 exp (-C2o~l)oL~ 1/2
(8)
The position of the solid-liquid interphase was found
to be
x~4(t) = Xo + 2C(t - t0) U2
BL = T o s - Tr
.2
(9)
The problem discussed is nowadays called a 'one-phase
problem', because heat conduction takes place in a
single phase only.
As usual in physics, Stefan tried to find the limiting
expressions of his formulae. The first limiting expression
considered was the sensible heat dominated problem
with hM = 0 (J/kg), and the second limiting expression
presents the latent heat dominated problem h ~ >>
cc(rr - To.s).
The first case gave him the well known Fourier solution (6) for heating a semi-infinite liquid slab occupying
the space (x0, oo) with the constants
Ac = Tr,
LIQUID} SOLID
(10)
and in the second case he determined the linear
t s]
1
0
0
i
x[m]
2
3
Fig. 3. Illustration of the solution to the second problem.
Difference between isotherms is 0.1(K). x0=l.5(m),
to = 0.0 (s), T0s = 0.5 (K), T0L -----1.3 (K), p = I (J/(kgK)),
k s = 1 (W/(mK)), Cs = 1 (J/(kgK)), k£ = 1 (W/(mK)), cc =
1(J/(kg K)), T• = 1(K), hM = 1 (J/kg), C = -0-10037600.
Stefan's work on solid-liquid phase changes
Stefan found the temperature distribution in the solid
phase, occupying the sp~Lce(x~(t), c~) to be in the form
(5) with the constants
AS =
1 - erf (Cd$/2)
later in this paper,
xAa(t)~Xo+ ['Ph1
T~ - T0s erf (Ca -1/2)
87
cs ~.-~-__~r--~o) (t-to)
hM
3
(13)
(16)
and the temperature distribution in the liquid phase,
occupying the space (-,x~, xM(t)) to be in the form (6)
with the constants
for predicting the measurements of the thickness of
polar ice. The formula includes the provision for variation of the boundary temperature with time. Since the
time to, when the freezing process was initiated, is not
known, Stefan used the relative formula
TA4 -- Tc~5 err (Cot -1/2)
Bs= Tos1 - erf (Cas 112)
Az: =
xA4(t2) - xA~(tl)
TA4 -- TO/;e r f ( C a -1/2)
1 - erf(CalY 2)
(14)
T~ - Tot. erf ( Ca -1/2)
B~ = -To£ q
1 - erf(Ca~ 1/2)
h~
The constant C represents the solution of the
transcendental equation
phMC = -k£B£( C)~ --1/2 exp (-C2ot£1)o~£ 1/2
+ ksBs(C),x -1/2 exp ( - C 2 a 8 1 ) ~ s 1/2
(15)
The solid-liquid interphase moves as prescribed by eqn
(9).
After the derivation of the formulae Stefan commented that the solution with h ~ = 0 can be reduced to
the solution of a single phase heat conduction problem
with the jump of material properties from the values S
to the values/3 when the temperature exceeds TM.
The final remark concerning the discussed problem
taking place in the spac~ (-oo, oo) was that it has little
meaning from the experimental heat conduction point
of view, but could be much easier practically verified
in the context of species diffusion.
On the theory of ice formation, particularly on ice
formation in the Arctic seas
From his colleague Johann Hann, Stefan received hints
that English and German expeditions had made
measurements of the t]fickness of polar ice as a function of time and measurements of the temperature of
the ice as a function of its depth. The existence of
these measurements was very stimulating to Stefan
because of their possible analytical description and
interpretation .4,5
Third problem
First Stefan convinced himself from the measurements that, at least at the beginning of the winter,
the temperature in the ice rises approximately linearly
with the depth of the ice. Stefan used the following
approximate formula which is developed and explained
3
(17)
for correlating the measured data in the Gulf of
Boothia (1829-1832), Assistance Bay (1850-1851),
Port Bowen (1824-1825), Walker Bay (1851-1852),
Cambridge Bay (1852-1853), Camden Bay (18531854), the Princess Royal Islands (1850-1851), Mercy
Bay (1851-1853), and data provided by the Deutsche
Nordpolfahrt (1869-1870).
Since the quantities xM (t) and Tr (t) have been determined experimentally, Stefan could calculate the mean
of the coefficient
-2ks
- ~ 1.168 x 10-8 (m2~
kksJ
phA4
(18)
From this coefficient he subsequently calculated
mean thermal conductivity of the polar ice to be
1-756(W/(mK)). Nowadays the value 2-240 (W/(mK))
for pure ice at pressure 1 (bar) and at melting point
is usually used. 26 This indicates the relatively good
agreement with the value determined by Stefan.
Stefan goes on to give the derivation of the theoretical
basis for the formula (17) used for correlating these
measurements.
Fourth problem
Stefan posed the following problem pertinent to the
description of ice formation: the semi-infinite half
space (x0, oo) consists of a material which may exist in
either a liquid or solid phase. Initially the material is
in the liquid phase at freezing temperature T0z: = T•.
At time t > to the surface at x0 is cooled and has the
surface temperature T r ( t ) < T~. At time t > to the
problem is to determine the thickness of the solid layer
x ~ ( t ) - X o and the temperature distribution in the
solid phase.
Stefan first assumed that the surface has been cooled
by an abrupt change Tr = constant and thus obtained
88
B. Sarler
the Lam6 a n d Clapeyron problem, which has the
explicit solution (5) for temperatures in the solid phase
occupying the space (x0, xM(t)) with the constants
As = Tr,
Bs
To£ - Tr
erf(Casl/2 )
to be
(x_z x___0)2f , + (x - Xo)4f,,+
2! a s
4! a-----~S
"'" + (x - xo)g
F =fq
(19)
+ ( x - x o)3 g'4 ( x - xo)
a------7-3!
the constant C determined from
phMC = ksBs(C)rr -1/2 e x p ( - - C 2 a s 1 ) a S 1/2
(20)
and the moving of the phase interface given by eqn (9).
The linear approximation of the temperature distribution in the ice gives
Ts = Tr
Tr - r0c
2C(t 7"-o)
V i ( x - x°)
(21)
(xM-z X--o)2ft q (X34 -- Xo)4f,!
TM = f q
2!a s
+...
(23)
CS
He found the solution of the problem in region
(Xo, X~(t)) to be in the form
phMx~
ks
2(x~_~__zXo).f, 4(x.M
2!a s
+
4!a2x°)3 f ', + . . . + g
(30)
Function g could be eliminated from eqn (29) by first
multiplying eqn (30) by x~4 - x0 and second subtracting
it from eqn (29)
~f
(25)
ha4,
cs
Phz4x~
(31)
ks
(xA4 -- X0)2.f,
2! as
3(XM -- Xo)4f,,
4! a 2
2(x M - x0) 3 g,
3!as
and the constants
Bs_
(29)
3(xA4 - Xo) 2 gt _1_5(x.M - Xo) 4 gtt...
3[ as
5! a2s
with the position of the interphase boundary
C8
(X,M -- X0) 3 gt
as well as the second (4) interphase condition
(24)
A s = T ~ +h__~_~,
xo)g4
r : . - ( x ~ - xo)
T s = A s + Bsex p ( - C ( x - )Co) + asC2(t - to))
xM(t ) = x 0 + aC(t - to)
+
4 ( x 5 , a ; 0 ) 5 g" - • -
Fifth problem
Stefan then assumed the cooling of the surface by the
exponential law
Tr = TM -ha4 (exp ( B r ( t - to) ) - 1)
-41"-.aYs
m
(22)
Next Stefan evaluated the movement of the phase interface given by the exact (15) and the approximate value
(22) of the constant C and concluded that the approximate formula gives around 3% thicker ice for the temperature drop at the boundary T~ - Tr = 30 (K).
4(xM - x0) 5 g,,...
5!a 2
(32)
C = nnlr/ 2 a s-1/2
(26)
This case presents the special closed-form solution with
linear advancing of the interphase boundary corresponding to the special exponential decay (23) of the
boundary temperature Tr.
A second equation which does not involve g could be
directly obtained from eqn (28)
0
O-'ttTM = 0
=f,
+
2! as
J
4l a 2
+'"+(x~a-xo)g'+
Sixth problem
Stefan assumed the cooling of the surface :by the general
law
Tr(t) = f ( t )
(27)
and proposed the general solution of the problem
(28)
with the functions f and g being functions of time only.
This general solution has the property of identically
satisfying the boundary condition. This solution
should also satisfy the first (3)
with the constant C given by
C = (ks(Toe - Tr)) 1/2
\
2ph~
,,
a------if5! s g "'"
(x~ - x0) 5 g ,,,
"
s
(XM -- Xo) 3
3!as
g I!
(33)
By multiplying eqn (33) by (x M -Xo)2/(3as) and
adding it to eqn (32) an equation without g' is
89
Stefan's work on solid-liquid phase changes
obtained
Tan - (xan - Xo) phanx ~
ks
~f
(xan - x0')2f, -{ (xan - x0)4ft,
6as
24a~
. . . . -~ (xan -- x0) 5 ,,
45,~
g "'"
(34)
Stefan subsequently truncated all terms with second
order derivatives to obtain
Tan - (xan - Xo) Ph----~-~:x~-J--~
- f - (xan - x 0 ) Z f
ks
6as
(35)
coefficients, and secondly, to indirectly experimentally
verify the result of his second problem from Ref. 2.
The latter, which is of principal interest in the
context of this paper, could be justified by assuming
parallels between Fourier's and Fick's constitutive
laws for heat and species diffusion, the heat and species
diffusion equations, and between the thermal initial and
boundary conditions of the second problem and the
species conditions in the described experimental configuration. If the chemical interface x~(t) moves in
the direction of the basic compound, the interface
concentration conditions are
Ct~(xn, t) = Cm,
Q4(xn, t) = 0 (kmol/m 3)
(37)
and the interface species flux conditions are
and made two approximate integrations of the truncated
general solution, the first one by assuming f ' = 0
d
0
Ct~o~ xT~(t ) = - a A -~x CA(x~' t) - c~t3~ Ct~(x~, t)
(38)
( 2 k s ( l ~ _ T r ( t ) + Tr(to).)(t_to)l/2
xan(t) ~ Xo + \phan
2
(36)
and the second one by assuming f ' ¢ O. The latter
assumption gives the formula (16).
Stefan concluded the discussion in this paper by
showing how higher order approximations could be
obtained and what their importance is. He evaluated
the effects of the successive corrections by using the
example f ( t ) = (Tan - a) sin (bt) with Tan - a =30 (K)
and bt = 7r/4, 7r/2, 37r/2.
On interdiffusion of acid and basic compounds
In his work 3 Stefan first discusses some experiments
with diffusion in the presence of a chemical reaction.
The experiments were based on two 0.12 (m) long (one
end closed) glass tubes with inner and outer diameters
of 0.007(m) and 0"023 (m) respectively. One tube was
filled with a diluted acidic and the other with a diluted
basic compound. Both ends of the tubes were brought
together and the chemical reaction between acid and
base, producing salt and water, began at the contact
area. The contact area :shifted in one or other direction
depending on the relative concentration of the two compounds. By placing litmus in the tube and by engraving
a length scale on the glass, the volumes occupied by the
basic compound (litmus coloured blue) and the acidic
compound (litmus coloured red) could be measured as
a function of time. Stefan performed several experiments
with different combinations of chemical agents such as
hydrochloric acid and ammonium hydroxide or acetic
acid and sodium hydroxide.
Stefan's measurements show that the position of the
basic-acid interface move is proportional to the
square root of time. His interest in these experiments
was twofold. First, to describe the reactive diffusion
process as such and to measure the species diffusion
where C denotes concentration, a the species diffusion
coefficient, subscript .A acidic and subscript/3 the basic
compound, and CB0 the constant. The description of
the experiments gives an analogous experimentally
verified solution to his second problem which is thus
indirectly proved.
At the end of the paper, Stefan also experimentally
and analytically treats the species diffusion equivalent
of the Lam6 and Clapeyron problem. He arranged the
experiment by putting into contact the chemically reacting contents of a glass tube and that of a much larger
reservoir.
The brilliance of Stefan's experiments is shown in the
care with which he avoided undesired effects and treated
the ostensibly unimportant details of density change
due to the chemical reaction and the temperature or
influence of the natural convection processes.
Experiments on evaporation and on evaporation and
dissolution as diffusion processes
In Refs 6 and 7 Stefan complemented his liquid-gas
experiments on evaporation I of ether into air from
narrow vertical tubes with the solid-liquid dissolution
equivalent. This shows that more than 15 years after
the evaporation experiments, Stefan was still obsessed
with the mysteries of the phase-change. His principal
observation from the evaporation experiments was that
the liquid ether level drops in proportion to the square
root of time and that the evaporation does not depend
on the cross-section of the tubes.
The dissolution experiment was initiated by
immersing 6'7 a sodium chloride crystal of rectangular
dimensions 0.007 (m) x 0-009 (m) x 0.030 (m) in water
with the longest axis parallel to gravity. The crystal
was covered with glass except at the upper end. Stefan
measured the amount of dissolved salt as a function of
B. Sarler
90
time. One of these experiments started on 24 June 1893
and finished on 9 July 1889, when 0"0250 (m) of the salt
was dissolved. The intermediate measurements after 1, 4
and 9 days were 0.0063 (m), 0"0126(m) and 0-0188 (m)
respectively. He found the same, already mentioned
laws, as in the evaporation equivalent.
In order to check the influence of the convective
effects in addition to the pure diffusive effects a similar
crystal was put into water with the uncovered end
facing downwards. In this case the linear upwards
movement of the solid-liquid interface over time was
observed. Natural convection took place since the
density of the salt is greater than the density of pure
water.
In the continuation of the paper Stefan analytically
treats diffusion in gases, experiments on evaporation,
diffusion in liquids, and experiments on dissolution.
The first dissolution problem, corresponding to the
mentioned experiment, was defined as follows.
Seventh problem
The semi-infinite half space (x0, c¢) consists of a
material V which dissolves in contact with material W.
At time t > to the surface at x0 is put into contact
with pure material W. Stefan assumed the following
boundary conditions at x0 at time t > to
Cv(x0, t) = 0 (kmol/m 3)
(39)
C v ( x M , t) = C v w
(40)
d
0
Cv°(1 - CvwVv ) dt x ~ ( t ) = Otv--~xCv
(41)
Eighth problem
Stefan substantially relaxed the constraints as defined at
the beginning of his fourth problem. He assumed the
temperature of the water to be above the melting point
and the variation of the density of the water during
the solid-liquid phase change. He assumed the position
of the ice to be fixed and allowed the movement of
the water due to phase change. This left him with the
following governing equations for ice
a
X
-- X 0
.,~-/
\(4C~v(t- to) ) V ' /
.
.
.
Cvw
erf(Ct~vV2 )
Ts
o
02
and the constant C solved from
Cv0(1 - Cvwvv)Cctv 1/2 = ¢r-I/2Bv( C ) exp (-C2ct~l).
(44)
T£
(46)
with Vxz representing the velocity of the water. Stefan
related this velocity to the phase-change density change
Vx£ - Pc - Ps d
dt x ~ ( t )
(47)
Since Pc > Ps for water, bulk water moves away from
fixed bulk ice during freezing. He also explicitly
developed the modified second interphase boundary
condition
X
(t) = hMPr.
--
(43)
(45)
Ot Tc + Vxz -~x T£ = ac ~
(42)
of species V in the region (Xo,Xzn(t)), with xM(t )
advancing as (9) with the constants
By _
o2
and water
hMps
{
Av = 0 (kmol/m3),
A paper s was published in the year 1890 on the
prominent first pages of the first edition of the first
volume of the Monatshefte far Mathematik und Physik.
It presents Stefan's last paper with respect to solidliquid phase change. The paper 5 was actually published
in Wiedeman's Annalen a year later, but Stefan had
written it in 1889. The paper 8 first repeats the discussion
of the fourth, fifth and sixth problems.
a
The newly introduced quantities Cvo, Cvw and Vv stand
for the volumetric concentration of the solute V in the
crystal, the saturated volumetric concentration of the
solute in solvent W, and for the volume occupied by
the solute, respectively.
Since the diffusion process of species 1; in medium W
was found (after lengthy derivation!), to be approximately governed by the species diffusion equation, the
result in the already well known form (5) was obtained
for the concentration field
.
On the theory of ice formation
Ot Ts = as ~
and interphase conditions at xM(t)
Cv = Av + B v e r f ,
Stefan concluded the discussion in this paper with
the analytical expressions for the generalized case with
initial material being a solution of materials V and W
in contact with a solution of the same type but different
proportions for its ingredients.
x
0
(t) -
rc + ks
0
rs
(48)
Stefan did not try to solve the posed problem.
Ninth problem
Instead, he solved the fourth problem with the initial
temperature of the liquid phase T0c set arbitrarily
above the melting point. Stefan found the explicit
solution of this problem to be in the same general
form as his second problem with the solid phase occupying (x0, x ~ (t)), the liquid phase occupying (xM (t), c¢ )
Stefan's work on solid-liquid phase changes
Mathematical contributions to this subject have
progressed in three main areas: approximate analytical
methods, numerical techniques, and qualitative results
such as existence and uniqueness. 17-19'2°'22-24 Extensive
progress on these types of problems has been
made25-27 as well in all other areas of science and engineering, covering the various problems listed in the
Introduction. Surely, Stefan would be surprised to see
his name associated with such numerous and diverse
spectra of activities.
Useful references to the Stefan problems are presented
in the bibliographies by Tarzia 13'14 and Sarler. 15'16 The
first two include some interesting statistical data on
the development of Stefan problems, and the latter
two include a glossary of key words.
Another measure of respect in honour of Stefan's
work on solid-liquid phase change is the definition of
the dimensionless number
SOLIDI LIQUID
3
2
t[s]
1
0
0
3
~AT
hM
Ste= Fig. 4. Illustration of the solution to the ninth problem.
Difference betwen isotherms is 0.1(K). T0z=l'3(K),
Tr=0"0(K), ps= l(J/(kgK)), ks= l(W/(mK)), cs=
1 (J/(kgK)), k£ = 1 (W/(mK)), c£ = 1 (J/(kgK)), T~ =
1 (K), h~ = 1 (J/kg), C = 0.52026175.
and the constants being
TM -
As = Tr,
BS
-
Tr
erf(Casl/2 )
(49)
Toc
Ac = 1 - erf(Ca~ 1/2)
TM - Toz erf ( Ca-z 1/2)
B£ = T0£ -
91
(50)
1 - eff(Ca~ 1/2)
The constant C is dete:rmined from eqn (15), and the
phase interface moves as in eqn (9). The problems of
the second and ninth type are nowadays called 'twophase problems' due to the conduction process taking
place in both phases.
CONCLUSIONS
The works described present the culmination of the
initial period in the research of the solid-liquid phase
changes. It should be mentioned that after Stefan, the
first serious publication upgrading his work did not
appear until 1931.31 This illustrates how exceptional
his research was. Stefan's essential contribution even
today influences the very content and style of solidliquid phase changes research.
In honour of Stefan's work on problems involving
phase changes or free and moving boundaries, such
problems are now loosely classified as Stefan problems.
(51)
This measures the ratio between the sensible and the
latent heat during the solid-liquid phase change process
and is called the Stefan number. In eqn (51) ~ stands for
the average specific heat, AT the temperature difference,
and /~M the average latent heat of fusion during the
process. Unfortunately, the author could not reliably
determine who first used the now-standard terms
Stefan problem and Stefan number.
Since this special EABE issue deals primarily with the
numerical solution of Stefan problems by discrete
approximative methods based on Green's functions, it
is appropriate also to remember the historical roots of
this particular technique. A class of analytical solutions
to Stefan problems could be obtained by the decomposition of heat conduction problems with phase change
into problems without phase change but with a
moving heat source. The temperature distribution due
to such a moving source can be obtained by Green's
functions. This solution strategy was recognized by
Lightfoot in 1929 and presents one of the theoretical
bases for this edition of the journal. The first numerical
solution of a Stefan problem based on Green's functions
is attributed to Chuang and Szekely 32 in 1971. A comprehensive review of other achievements made so far
in this field can be found in Ref. 33.
ACKNOWLEDGEMENTS
This paper presents a part of the lecture that was given
at the Jo~ef Stefan Institute, Ljubljana, Slovenia, on 22
March 1993, honouring the occasion of the centenary of
Jo~ef Stefan's death. The author would like to acknowledge the Ministry of Science and Technology of the
Republic of Slovenia for support.
92
B. Sarler
REFERENCES
Selected books on Stefan problems
Stefan's work on solid-liquid phase changes
17. Datzef, A. Sur le probl6me lin6aire de Stefan. M~morial
des Sciences Physiques, 1970, No. 69, Gauthier-Villars,
Paris.
18. Rubinstein, L. I. The Stefan problem. Translations of
Mathematical Monographs, Vol. 27. American Mathematical Society, Providence, RI, 1971.
19. Yamaguchi, M. & Nogi, T. The Stefan Problem. SangyoTosho, Tokyo, 1977.
20. Elliot, C. M. & Ockendon, J. R. Weak and variational
methods for moving boundary problems. Research Notes
in Mathematics, No. 59, Pitman, London, 1982.
21. Crank, J. Free and Moving Boundary Problems. Oxford
Science Publications, Clarendon Press, Oxford, 1984.
22. Tarzia, D. A. The two-phase Stefan problem and some
related conduction problems. SB-MAC, Gramado, Vol. 5,
1987.
23. Meirmanov, A. M. The Stefan problem, de Gruyter
Expositions in Mathematics, VoL 3. Walter de Gruyter,
Berlin, 1992.
24. Hill, J. M. One-dimensional Stefan problems: An
introduction. Pitman Monographs and Surveys in Pure
and Applied Mathematics, VoL 31. Longman Scientific and
Technical, Harlow, 1987.
25. Wrobel, L. C. & Brebbia, C. A. (eds) Computational
Methods for Free and Moving Boundary Problems in
Heat and Fluid Flow. Elsevier Applied Science, London,
1993.
26. Alexiades, V. & Solomon, A. D. Mathematical Modeling of
Melting and Freezing Processes. Hemisphere Publishing
Corporation, Washington, DC, 1993.
27. Lock, G. S. H. Latent Heat Transfer, An Introduction to
Fundamentals. Oxford Science Publications, Oxford, 1994.
1. Stefan, J. Versuche fiber die Verdampfung. Aus den
Sitzungsberichten d. kais. Akademie d. Wissenschaften in
Wien. Mathem.-naturw. Classe, 1873, LXVIII, Abth. II. a.
November 1873, 385-423.
2. Stefan, J. Llber einige Probleme der Theorie der
W~irmeleitung. Aus den Sitzungsberichten d. kais. Akademie
d. Wissenschaften in Wien. Mathem.-naturw. Classe, 1889,
XCVIII, Abth. II. a. M~irz 1889, 473-84.
3. Stefan, J. Ober die Diffusion von S/iuren und Basen gegen
einander, Aus den Sitzungsberichten d. kais. Akademie d.
Wissenschaften in Wien. Mathem.-naturw. Classe, 1889,
XCVIII, Abth. II. a. April 1889, 616-36.
4. Stefan, J. Lrber die Theorie der Eisbildung, insbesondere
fiber die Eisbildung im Polarmeere. Aus den Sitzungsberichten d. kais. Akademie d. Wissenschaften in Wien.
Mathem.-naturw. Classe, 1889, XCVIII, Abth. II. a. Juli
1889, 965-83.
5. Stefan, J. Uber die Theorie der Eisbildung, insbesondere
fiber die Eisbildung im Polarmeere. Annalen der Physik und
Chemie, G. Wiedemann, Leipzig, 1891, 42, 269-86.
6. Stefan, J. Uber die Verdampfung und die Auflfsung als
Vorg~inge der Diffusion. Aus den Sitzungsberichten d. kais.
Akademie d. Wissenschaften in Wien. Mathem.-naturw.
Classe, 1889, XCVIII, Abth. II. a. November 1889, 1418-42.
7. Stefan, J. Uber die Verdampfung und die Auflfsung als
Vorgiinge der Diffusion. Annalen der Physik und Chemie,
G. Wiedemann, Leipzig, 1890, 41, 725-47.
8. Stefan, J. Ober die Theorie der Eisbildung. Monatsheftefiir
Mathematik and Physik, 1890, 1(1), 1-6.
Biographies and bibliographies
Other references
9. Cermelj, L. Jo~ef Stefan - - Life and Work of the Great
Physicist. Mladinska knjiga, Ljubljana, 1976 (in Slovene).
10. Sitar, S., Jo~efStefan, Poet and Physicist. Park, Ljubljana,
1993 (in Slovene).
11. Strnad, J. Jo~ef Stefan, on the occasion of 150 years from
his birth. DMFA, Ljubljana, 1985 (in Slovene).
12. Strnad, J. Jo~ef Stefan, The Centenary of his Death, Jo~ef
Stefan Institute, Ljubljana, 1993 (in Slovene and English).
13. Tarzia, D. A. Una revision sobre probblemas de frontera
movil y libre para la ecuacion del calor. E1 problema de
Stefan, Universidad Nacional de Rosario, Separata de
Mathematicae Notae, 1981/82, Afio XXIX, Rosario,
147-241.
14. Tarzia, D. A. A bibliography on moving-free boundary
problems for the heat-diffusion equation. The Stefan
problem. Progetto Nazionale M.P.L, Equazioni di evoluzione
e applicazionifisico-matematiche, Firenze, 1988.
15. Sarler, B., Alujevi~, A. & Malalan, J. Bibliography on
Stefan problem - - 1989. Jo~ef Stefan Institute Technical
Report, IJS-DP-5614, Ljubljana, 1989.
16. Sarler, B. Bibliography on Stefan problem - - 1994.
LFDT, Faculty of Mechanical Engineering Technical
Report, University of Ljubljana, Ljubljana, 1994.
28. Lam6, G. & Clapeyron, E. M6moire sur la solidification
par refroidissement d'un globe liquide. Ann. Chem. Phys.,
1831, 47, 250-6.
29. Fourier, J. B. J. Extrait d'un m6moire sur le refroidissement du globe terrestre. Bull. Sci. par la Soci~t~
philomathique de Paris, 1820.
30. Weber, H. Die Partiellen Differential-Gleichungen der
Mathematischen Physik. Zweites Buch: W~trmeleitung, 2nd
edn, Chapter 49. Vordringen des Frostes, Vieweg und
sohn, Braunschweig, 1901, p. 122.
31. Leibenzon, L. S. Handbook on Petroleum Mechanics.
GNTI, Moscow, 1931.
32. Chuang, Y. K. & Szekely, J. On the use of Green's
functions for solving melting or solidification problems.
Int. J. Heat Mass Transfer, 1971, 14, 1285-94.
33. Sarler, B., Mavko, B. & Kuhn, G. Chapter 16: A survey
of the attempts for the solution of solid-liquid phase
change problems by the boundary element method. In
Computational Methods for Free and Moving Boundary
Problems in Heat and Fluid Flow, ed. L. C. Wrobel & C. A.
Brebbia. Elsevier Applied Science, London, 1993,
pp. 373-400.