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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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