Indian Journal of Engi neering & Materials Sciences Vol. 9 , February 2002, pp. 25-34

A short review of classical Stefan problem

K N Shukla Vikram Sarabhai Space Centre, T hiruvananthpuram 695 022, Ind ia

Received 18 September 2000; accepted 31 Allgllst 2001

The paper describes the sta te-of-art solution tec hnique for c lassical Stefan proble m. W ith a short di sc ussion o n the we ll -posedness o f heat conduction proble m with me lting or freezing, some analytical solutions are presented.

The change of state occurring with melting or freez­ing is associated with many of today's practical prob­lems. The solidification of castings, freezing and thawing of soils and foodstuffs, the ablati on of the skin of rockets and miss iles etc. are some of the prac­tical examples of heat conducti on with melting or freezing.

Materi als can exist in solid, liquid or gas depending on their temperature and pressure. As shown in the phase di agram presented in Fig. I, for most materi als under constant pressure there is a fi xed melting tem­perature, above which solid phase changes to liquid phase and a boiling temperature above which liquid phase changes to gas phase. Energy in the form of heat is required for the phase change from soli d to liquid and liquid to gas while heat is released in the reverse process . The amount of heat required during the phase change process is known as latent heat. The simplest and most easily observed phase change proc­ess is the melting or freezing across a mov ing bound­ary whose position is not known a priori and is to be determined as a part of the solution.

Solid

, , , , , Liquid , , , , ,

Vapour

Temperature

Fig. I-Phas~ diagram

Statement of Problem and Existence of Solution Let us consider the melting of a solid. At an ins tant ,

the solid and liquid phases are separated by a moving plane given by x = set) . The region x<s( t) represents the solid phase and the region x >s(t) the liquid phase as shown in Fig. 2. The heat balance across the sur­face of separation at x=s(t) is;

aT x = s+ ds -A. -I = pL- , L >O ax x = s. dt

=> ". ( I )

aT, aTt L ds -A.s--A.,- = p -ax ax dt

where, L is the latent heat and the subscripts s and I refer to the solid and liquid phases respecti vely. In addition, the temperature in both the liquid and solid phases at the interface must be at the melting tem­perature Till'

The first mathematica l ex position of the problem was made by J Stefan and the title 'Stefan's Problem'

Ta

set) x

Fig. 2--Tempe rature di stributi on in a me lt ing sol id

26 INDIAN 1. ENG. MATER. SCI., FEBRUARY 2002

was originated. It has, since then become an active field of research for both mathematicians and applied scientists.

Let us consider the melting solid or stationary melt in which the temperature distribution , T is described by:

aT Pc,,-= W J..VT) at ... (2)

The assumption of stationary melt ensures convective motion due to density gradient in the melt. This is just ified only for small temperature gradient in the melt. For constant values of the thennal parameters p, cp and Ie, Eq.(2) is changed to:

... (3)

for an unidimensional Cartesian co-ordinate system. Eqs (1) and (3) are supplemented by the following initial and boundary

t = 0 s(O) = 0 T, (x, t) = 7~, < Till

{ > 0

t>O

x=O T, (0, t) = To

x=a T) (x, t) = Tfl

(4) (5)

(6)

A detailed discussion of the well-posed ness of the problem is given by Rubinstein) for special cases, however there is no general well posed Stefan prob­lem.

The simplest exact solution for a planar interface moving with a constant speed U into a fluid at the melting temperature is described as

.. . (7)

It is obvious that for U> 0, T < Till in the solid, how­ever for U < 0, T> Till and the solid is said to be super heated. Similarly, the solution valid for liquid phase in which the planar interface moves into a fluid at so­lidifying temperature can be written as

... (8)

and for U > 0, T < Till the fluid is said to be super cooled.

In order to examine the stability of the interface over small disturbances, consider Eq. (I) for a non­planar boundary defined by F(x, t) = O. If the normal direction of the surface into the fluid is n, then an en­ergy balance gi ves

-J..- Dt= pL8n [ aT]X= S+

an x= S. .. . (9)

which may be written in the form

x- S aF [J..VT] - + f..F = pL-

x= s. at ... (10)

If we define the phase boundary F=x-s(t, y) and the liquid is assumed at constant melting temperature Till, Eq. (10) becomes

J.. aT _ J.. rH as _ as a Ix=s. a Ix=s. a - p a x y y (

. ... (11)

Let us assume a perturbation in the position of phase boundary given by an explicit relation

x = S(t,y)= Ut + E em sin ny ... (12)

where I: « I and n > O. Correspondingly, let us as­sume the solution in the form

T - T", = [ :, ) [1 -exp l ~ (VI - x) 11 . . . (13)

+ E sin ny g( x, f), x < Ut

The temperature distribution T must satisfy the two­dimensional heat Eq. (2), resulting into

ag = a. (a2

~ - /12 g 1 at a x

... ( 14)

Eq. (14) admits a solution of the form

g = A e m +lII(x .ut) ... (15)

SHUKLA: A SHORT REVIEW OF CLASSICAL STEFAN PROBLEM 27

if

a ( m2 - ,/ ) = ( eJ - mu) ... (16)

From Eq. (13) , we can write

( J [ ( )2 1 LVI PC p V 2 T-TIII= - --(Vt-x)-, -- (Vt-x) ..

cp a 2. A

. m 2

[ 2 1 + AE SIO ny eaT 1+ m(x - Vt) + 2T (x - Vt) + ... .

Thus the condition T=TI11 at the interface implies that

[L~ + A] E eat sin ny = O( E2 ) cp a

LV =>--+A=O

cp a

.. . (17)

Similarly, by using the condition at interface de­scribed by Eq. (11),we get

{LpV (1 + ~ (Vt - X) ) + EA sin ny mA eaT+

lII(x .Ut ) J x= s .

- { A n2 E

2COS

2 nYe2aT } x= s = L{V + EeJ eat sin ny J

[ LV 2 1 => -~ p + mAA - LpeJ E eat sin ny = O( E2 )

LV 2

=> --- + rnAA - LpeJ = 0 a

. .. (18)

Eliminating A from Eqs (17) and (18), we get

. . . (19)

and eliminating (J from Eq. (16), we gel

2 2 I [I ry ] III -11 =- - - v - +2mV a a

or (m + ~ V r = n"

Hence a for positive root m + Via = n, we get from Eq. (19), (J = -V n. Thus, for large values of Vt-x, the perturbation g expressed by Eq. (15) is small as com­pared to the unperturbed solution for T. Hence, the planar interface is stable to small disturbances if V > o and the solid is not superheated. With V < 0, the solid is superheated and the interface is unstable. This results into an ill posed Stefan problem.

State-or-Art Solution Technique or Phase Change Problem

The problem posed in the preceding section is non­linear and the solution cannot be obtained in general by principle of superposition. The exact solutions are available only for specific cases , e.g., Mening and Ozisik2

, Ozisik3 and Ozisik and Uzzel4, Ku and Chan5

developed a generalized Laplace transform technique for phase change problems. Tritscher and Broad­bridge6 obtained a similarity solution for a multi phase Stefan problem. Varga et al. 7 studied the fundamental of melting when a shell of phase change material ride on a horizontal cylinder. However, when exact solu­tions are not available, approximate semi- analytic and numerical methods can be used for the solution of these problems. It is known that a physical system undergoing a transformation has a tendency to move to a more probable state, a state of greater entropy. In classical thermodynamics, this principle requires that the Helmholtz thermodynamic potential be a mini­mum at equilibrium. Although Biot8

.9 applied the

variational technique to many problems, Chambers 10

was first to show its applicability to heat conduction under the assumption of no motion of the medium. He defined a function F over a volume V and its surface area A as

( ) 2 [ 1 ~ 1 aT a I 2 f = -fpc , - dV -- -fA(VT ) dV

2 \1 I at at 2 v

f aT f aT - Q-dV+ q-dA v at A at

... (20)

where, Q is the rate of heat generation per unit vol­ume and q is the normal heat flow across the surface A. The variation was performed with respect to (X = aT/at) and it was shown that the vani shing of the variation produced the equation of heat conduction, Eq. (2) modifi ed by the heat generation term Q. Lardener ll used the vari ational technique for the so­lution of transient phase change problem. Zyszkow-

28 INDIAN 1. ENG. MATER. SCI., FEBRUARY 2002

ski 12 applied the method to the transient phase change problem with non-linear boundary conditions.

The integral method which dates back to well­known Von Karman and Pohlhausen who used it for approximate analysis of boundary layer equations was applied by Goodman l3,14 and his co-author Shea l5 to solve a one-dimensional melting problems. The method has subsequently been applied by Cho and S d I d 16. 17 P 18 19· . 20 un er an , oots', Tlen and GeIger and Yuen21 in a variety of cases of phase change prob­lems. As compared to the exact solution, the accuracy in predicting the location of interface by the integral method varies from 5-10 percent. The perturbation method has been used by several researchers22.3o. However, the analysis becomes very complicated if higher order solutions are to be determined. It is also difficult to use the solution for multidimensional problem. Hwang et al.31 applied the perturbation technique to study the effects of wall conduction and interface thermal resistance on the phase change boundary. An accuracy of 8-10 percent can be achieved in the location of phase change boundary and the freezing time. Kern32 developed a simple and safe solution to the generalized Stefan 's problem. Prudhomme et al.33 derived a general recurrence rela­tion for the series solution of the solidification of slab cylinder and sphere. Chuang and Szekely34 hav~ treated the phase change problem as moving heat source problem and applied the technique of Green's function for solution . Budhia and Kreith35 applied the method for solving the phase change problem in a wedge. Chung and Szekely36 developed an integral equation for the solution of solid-liquid interface in a phase change problem. Rubenstein I developed an in­tegral eq uation for the so lution of solid-liquid inter­face in a phase change problem. Bolei7.38 introduced an embedding technique to solve the melting problem of a slab . The method develops a general starting so­lution and is versatile to solve one, two or multidi­mensional phase chanae problems. Lederman and

N b . .

Boley used the embedding techl1lque to obtain an analytical short time and numerical full time solutions for an axi-syl11metric melting or solidification of cir­cular cylinders. The location of interface as well as the melt time could be achieved to an accuracy of 5 percent. A large number of purely numerical solutions arc available because of availability of high speed computers. Baxter40 developed a lumped formulation for the fusion time of slabs and cylinders. Bonacina el al. 4 1 worked out a three time level implicit scheme unconditionally stable and convergent for phase

change problem with temperature dependent thermal conductivity. Chen and Lin42 coupled the finite differ­ence technique with Laplace transform and solved the Stefan's problem with radiation-convection boundary conditions. The finite difference solution to a phase change problem in a sphere was obtained by Cho and Sunderland43. Crowley and Ockendon44 developed an explicit finite difference formulation for solution of an alloy solidification problem. Dusi nberre45 also applied finite difference technique to solve phase change problem. Huang et al.46 used a body fitted coordinate to solve the phase change problem. Springer47 consid­ered the ax i-symmetric case of melting or freezina in

I· d 4849 . b a cy III er. Tao' obtained some generalized nu-merical solution of freezing in cylinder, sphere and convex container. Sparrow and Chuck50 developed an implicit-explicil numerical scheme for phase change problems. The temperature distribution and the phase change can be located within an accuracy of 3 per­cent.

Solidification of a Semi-infinite Liquid Let us consider the solidification of a semi-infinite

liquid at a uniform temperature T>Tm. The free sur­face of the body is brought to a temperature T<Tm at t = 0 and maintained at that temperature for t > 0 . As a result, the solidification starts at the surface x = 0 and the solid/liquid interface moves in the positive direc­tion (Fig. 2). We also assume that the thermal proper­ti es of the two regions on the two sides of the phase boundary are different but do not change with tem­perature. We also exclude convection in the liquid phase so that we model the process as a pure heat conduction problem in both phases. For mathematical description, it is convenient to choose the x-co­ordinate of the solid phase at the surface of the solidi­fied layer. The Fourier heat conduction equation can be written as

and

aL _ a2 L ---a--at a x 2

O<x<s(t),t>O

aT, a2 T, --=a--at a x 2 '

s(t) < x < 00

The initial conditions are

t = 0, s(O) = 0,

Tdx, t)= T"

... (21)

... (22)

. .. (23)

SHUKLA: A SHORT REVI EW OF CLASSICAL STEFAN PROBLEM 29

The boundary conditions are stated as follows:

(>0, x=o, T.JO,t)=To

( > 0, X -7 00, T I (x, t) = T a

(24)

(25)

when the freezing boundary moves from s to s+ds in time interval dt, it liberates the amount of enthalpy of transformation p L ds per unit area where L is the spe­cific enthalpy of melting (latent heat). The energy balance at the interface x =s, described by Eq. (I) holds. In addition, the continuity of temperature di s­tribution at the interface requires

T,Jx,t) = Tdx,t) . . . (26)

Solutions to the set of Eqs (21-26) can be written as

erj x

T.,-To 2~a.J (27) = ...

Till -To elf8

and

x

... (28) T{/ -TI = ----'-----'-

eif8

Eqs (27) and (28) contain a term 0 which is yet to be determined. Substituting for aT/ax and aT/ax from the Eqs (27-28) into Eq. (I), we get

I

I ( T il -Till) (AI) (a.,.)2 exp8 2 elf8 + Till - To ..1.., -;;;

I x--------~~-

ex p( 8 2 cxJ a l )elfc( 8.,[cZ1 al) ... (29)

=.J7i L 8 c",J T III - To)

Let us introduce the Stefan number 51] for the phase transition as

L 5,,=----­

c"JTIII-To)

Also let

K = ..1.." k = as e = T{/ - Till ). , a '

}'" al Till - To

The Eq. (29) then becomes

I I e ---, -- + K ). K{/ 2 --------1-exp8-erj8 ,

exp( 8 - K {/ )eifc( 15K (/ 7. )

=.J7is,,8 ... (30)

Eq. (30) is a transcendental equation detell11ining 0 as a function of the four dimensionless groups: 5", K), Ka ll2

, e and OK/12 . It is also known as the Neumann 's solution for the problem under consideration. Stefan51

obtained solutions for the temperature profile and the interface velocity for two cases, a step input tem­perature at the boundary and a specific heat flux which would produce a constant interface velocity. Thus, the Stefan's work was a special case of Neu­mann's solution. Although Franz Neumann ( 1798-1895) derived the exact solution of the melting prob­lem and presented in lectures in the 1860's, however the first publication of these lectures appeared only in 1912 (Ref. 52) after Stefan. The treatise of Carslaw and Jaegar53 describes results attributed to Neumann on solidification from a plane wall. Muehlbauer and Sunderland54 presented a concise review of the prob­lem of heat conduction with freezing or melting.

Axi-symmetric melting

Let us consider the axi-symmetric melting on a hollow cylinder of inner radius R and the wall thick­ness t:..R (Fig. 3). We assume that the heating takes

Ta

Fig. 3-Temperature distribution during ax i-sy mmetric melting

30 INDI AN J. ENG. MATER. SCI., FEBRUARY 2002

place through the wall of the hollow cylinder uni­fo rmly and the film coefficient that operates on the hea tcd side is u. The overall heat transfer coefficient fo r a flat plate is written as

u ;:; ---­ex + !J.R1A/II

where, 1I/ is the thermal conductivity of the walloI' the cy I i nder.

Let the mclting takes place uniformly around the ho ll ow cylinder and after a time I , the phase boundary moves to a distance s. The energy balance at I can be written as

2n

... (31)

Wc now introduce a dimensionless variable '1 = siR and the following dimensionless numbers

in the Eq . (32), which finally becomes

SII [_I. + In(l + 17)J (I + 1]) d17 = 1 BI. dFo

.. . (32)

where SII is the Stefan number for melting analogous to the solidification defined earlier. The integration of Eq . (32) with the initial condition

Fo = 0, Tj = 0

leads to Eq . (33).

SIl r 2 Fo = - L(I + 1]) In (l + 1]) -

2

17(2+1]{~- ~i)J17~1 ... (33)

To describe the process of melting inside the cylin­der, a similar equation for the energy bala.nce as de­scribed by Eq . (33) with changing R+s by R-s and

In(R+s)/R by -In(R-s)/R can be written. Thu s on inte­gration it gives to

Sn J Fo = - [(I-17 ;- In(I-1] ) +

2

( I I) 1 Bi +"2 (217 - 1fJ.1] ~ I

Sphcrical mclting

. .. (34)

For melting on a thin spherical shell, the energy balance at l can be wri tten as

whi ch is written in dimensionless form as

[ I [ I 11 1 ell7 SIl - . + 1--- (I +17t-= I

BI i + 1] dFo . . . (36)

Eq . (36) , on integration becomes

Fa = S;' [( 1+ ~i )c(l + 1])3 - I) - % «(I + 17 / - 1],1] ~ I ... (37)

A similar equation can be derived for melting inside a thin spherical shell as

Fa= Sn [( ~-l) (1_(1+17)3) 3 l Br.

+ % (I - (1 -1] )2) J .. . (38)

The inward solidification of cylinders and spheres has been considered by Riley et ai.3o.

Dynamics of melt growth and axi-symmctric mclting Consider a vertical cylindrical tube of radius ro sur­

rounded by a bulk of pure homogenous solid (Fig. 4). The surface of the tube is just above the melting tem­perature of the surrounding solid T m, thus heat is transferred from the tube. The solid is initially at tem­perature T « Tm and the melt is allowed to form at the wall. The melting front advances readily outward smoothly and uniformly, its motion being determined by the rate at which the excess heat energy of the melting solid is conducted through the surrounding melt. The driving force for the process is the differ-

SHUKLA: A SHORT REVIEW OF CLASSICAL STEFAN PROBLEM 31

let < Tm ~-

; Under coo led solid / /

/ /

/ /

I I

I I r

I \ \ \ \

\ \

\

'\ , , ,

/'

"-"- --

------ .l. / - , -.... . ./. Fusio n

/ " , To~T~

\ . , . \/'---

I

\ \ \ \ ,

\

r I

I I

/ /

I

Fig. 4--Schematic of me lt growth around a horizontal cylinder

ence between the free energy available in the two phases and the total energy per unit length in the sys­tem,

!1 Grill := - !1G n ( R2 - R/ ) + 2n R y . .. (39)

The first term of the RHS of Eq. (39) denotes the heat of fusion and the second term denotes the surface energy in the system. t:,.G denotes the change in free energy of system. y is the surface energy and R is the radius of the molten material as described in Fig. 4. For sufficiently small R, the second term dominates so that t:,.G10 1 is positive; for sufficiently large R, the first term dominates and then t:,.Gro l is negative. For the stable growth of the melt, t:,.GIOI exhibits a maximum value and the corresponding R is given by

R = y/t:,.G ... (40)

where the difference of the energy between the two phases is equal to the pressure drop in the melt.

!1G := p( Tn, ) - P r (41 )

and thus

P(T",)- PI?:= y/R ... (42)

where p(Tm) is the pressure at the melting point and PR is the equilibrium liquid pressure at the temperature of

interface. Eq. (42) is a simple model describing the equilibrium of the melt growth which states that the pressure drop due to deviation in melting temperature from the interfacial temperature is equivalent to the work performed against surface tension . Heat is con­ducted through the melt by diffusion.

The melt layer is confined in an annular region with the moving boundary due to phase change. The gov­erning equations are the continuity, momentum and energy equations for the liquid in the melt and the energy equation for the solid. It is reasonable to as­sume uniform temperature of the melt during initi al phase so the energy equation for the melt needs no consideration. Thus, the system of equations left for analysis consists of the continuity, momentum fo r the liquid in the melt and the energy equation for the sub cooled solid. Considering the azimuthal symmetry along the pipe the continuity and momentum equa­tions for an incompressible fluid (melt) are

a -(nt}=O ar

and

... (43)

... (44)

By using Eq. (43), the viscous term of Eq. (44) van­ishes and it reduces to

all all I ap -+I.t-=---at ar p ar . .. (45)

where ;/ is the sum of all the normal stress and it is expressed in terms of static pressure and the normal friction as

I au I.t p =p+p-=p-J-I-

dr r 0 •• (46)

Substituting for pI from Eqo (46) into Eq. (45) and using the continuity Eqo (43), we get

au I ap 2 p U u 2

-=---- --+-at p ar p r2 r

... (47)

Let R be the radius of the melt at an instant I, the con­tinuity Eqo (43) on integration becomes

32 INDIAN 1. ENG. MATER. SCI., FEBRUARY 2002

ru=RR . .. (48)

where it is assumed that the liquid adjacent to the melt surface moves with velocity same as that of the melt surface. Thus

RR au RR+ R2 u=-, -=

r at r

Thus, Eq. (47) becomes

RR + R2 = _ � ap _ 2 JJ- � + u 2

r par p / r ... (49)

Integrating Eq. (49) between r = ro to r, one gets a dynamical equation of the interface

.. ·2 ro 1 (RR + R )In-= - -(peT m) - PRJ R p

1 .. (1 1) + - RR( RR - 2v) - --2 R2 2

rO

Energy equation

... (50)

A free boundary problem is posed in which a part of energy is transferred from melt into the fusion in the form of latent heat and the remaining is conducted into the sub cooled solid. Thus, the energy equation is written in cylindrical co-ordinates in the presence of azimuthal symmetry taking into account the phase change boundary as

aT RR aT a ( aT) -+--=- r- R(t)�r<oo at r ar r ar

The boundary conditions are

aT r=R(t), Aa;:-= RpsL

r�ooT= T�

The initial condition is

t ---1 0, T = T �

... (51)

(52a)

(52b)

... (52c)

The energy Eq. (51) is complicated by the presence of a moving boundary at the left hand side. The term

can be formally eliminated by introducing the La­grange co-ordinate h, as

1 ? 2 h = � ( r- - R ), 'r = t , 2 ... (53)

V(h, 'r )= T(r, t )

the above co-ordinate system transforms Eq. (50) into

. . . (54)

Further, by introducing a time variable defined as

and a new temperature variable VI given by

� Vdh,'r/)= f (T�-V(h,'r)]dh

Iz

The above Eq. (54) reduces to

... (55)

Let us assume that the temperature gradient is appre­ciable only in a thermal boundary layer about the solid of thickness 8="0.1). Thus, we may neglect the terms of second and higher orders in 81R. Thus, we may write

... (56)

The magnitude of O/R may be estimated as follows: At a time, t, when the melt radius is much greater than a, the difference between the temperature in the sub­cooled solid, T and the melt solid interface TR is slightly less than T m -T. This temperature drop is quite large in a thin layer in the solid region, whose thick­ness is approximately given by the diffusion length "CA., t). The flow of heat per unit time into an unit length of the solid is given by

SHUKLA: A SHORT REVIEW OF CLASSICAL STEFAN PROBLEM 33

Q = ..1.( Till -T � ) 2nR rat

On the other hand, the heat required per unit time of melting is given by

d 2 Q = - [ n( R - a) pLJ = 2n( R - ro )RLp dt

. .. (57)

Equating these two relations and on integration, we get

0 0 . (58)

Expressing In (Rlro) as In (1- (R-ro)/ro) and consider­ing the first two terms in the series expression, we obtain a relation for the thermal boundary layer thick­ness as

I 8 [ pL ]2-'-- < (at)4 R 4roP-,cp(TIIl-T�)

0 0 ' (59)

By way of example, on a tube of radius I cm for ice at 273 K, ,&/R = (O.l278)I..JTf25. This justifies the assumption made in simplifying Eq. (56).

Thus, with an assumption of the thin thermal boundary layer in the solid region, the solution of Eq. (55) at the interface is obtained as

0 0 . (60)

The temperature distribution described by Eq. (60) is a function of the radius of the molten material R which is to be determined by Eq. (50). The pressure drop can be expressed by the term equivalent for de­pression in melting56. Using the Gibbs-Thomson's equation for pressure temperature relationship57

( ) T R -Tm L PR - P Till = p-, Tm

... (61)

Thus, a direct relation describing the effect of under­cooling on the melt growth is obtained as

(RR + R2 ) In ro = .!..- p , L (TrTm ) R p . l Tm

+ - RR( RR - 2v) - - -I . . [ 1 1 ) 2 R2 a2

... (62)

For a more general treatment of Eq. (62) of dimen­sionless variables is introduced as

Eq. (62) is transformed as

(PP + ji )P In P = A [8111 - 8 R ] - !( pp - C)( I - P) 8111+B 2

where

A = 4( ro / P., L, B= Cp T�,C=4� a p L a

... (63)

The expression for interfacial temperature is also transformed as

. .. (64)

Thus, the three important parameters for the growth analysis, P, P' and 9R are described by a set of simul­taneous Eqs (63) and (64). Numerical solutions for the radial growth of the melt and interface temperature are performed by Shukla58 for n-octadecane and ice­water systems. Boger and Westwater59 considered the effect of buoyancy on the melting and freezing proc­ess. Halle and Viskanta60 considered the phase change problem with interfacial motion in materials cooled or heated from below.

Conclusions The classical Stefan's problem has been briefly

reviewed. Various solution techniques are discussed. Although numerical solutions are available for variety of phase change problems, analytical solutions are constantly being developed to understand the physics of the problem.

34 INDI AN J. ENG. MATER. SCI., FEB RUA RY 2002

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