6 Mass Transfer
where T
k
is the kth Chebyshev polynomial defined as
T
k
(ξ)=cos[k cos
−1
(ξ)]. (31)
The derivatives of the variables at the collocation points are represented as
d
a
f
i
dη
a
=
N
∑
k=0
D
a
kj
f
i
(ξ
k
),
d
a

i
(ξ
k
), j = 0,1, . . ., N (32)
where a is the order of differentiation and D
=
2
L
D with D being the Chebyshev spectral
differentiation matrix (see for example, Canuto et al. (1988); Trefethen (2000)). Substituting
equations (29 - 32) in (17) - (20) leads to the matrix equation given as
A
i−1
X
i
= R
i−1
, (33)
subject to the boundary conditions
f
i
(ξ
N
)=
N
∑
k=0
D
0k
f

=
⎡
⎣
A
11
A
12
A
13
A
21
A
22
A
23
A
31
A
32
A
33
⎤
⎦
, X
i
=
⎡
⎣
F
i

1
), ,f
i
(ξ
N−1
), f
i
(ξ
N
)]
T
, (36)
Θ
i
=[θ
i
(ξ
0
), θ
i
(ξ
1
), ,θ
i
(ξ
N−1
), θ
i
(ξ
N

),r
1,i−1
(ξ
1
), ,r
1,i−1
(ξ
N−1
),r
1,i−1
(ξ
N
)]
T
, (39)
r
2,i−1
=[r
2,i−1
(ξ
0
),r
2,i−1
(ξ
1
), ,r
2,i−1
(ξ
N−1
),r

1,i−1
D
2
+ a
2,i−1
D, A
12
= −D, A
13
= −N
1
D (42)
A
21
= b
2,i−1
, A
22
= D
2
+ b
1,i−1
D, A
23
= D
f
D
2
, (43)
A

in such a way that the modified matrices A
i−1
and R
i−1
take the form;
430
Advanced Topics in Mass Transfer
Successive Linearisation Solutionyof Free
Convection Non-Darcy Flow with Heat and Mass Transfer
7
A
i−1
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜

00··· 0100··· 0000··· 00
00··· 0010··· 0000··· 00
A
21
A
22
A
23
00··· 0000··· 0100··· 00
00··· 0000··· 0010··· 00
A
31
A
32
A
33
00··· 0000··· 0000··· 01
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟

⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

2,i−1
(ξ
N−2
)
r
2,i−1
(ξ
N−1
)
0
0
r
3,i−1
(ξ
1
)
.
.
.
r
3,i−1
(ξ
N−2
)
r
3,i−1
(ξ
N−1
)
0

⎟
⎟
⎟
⎟
⎟
⎠
(46)
After modifying the matrix system (33) to incorporate boundary conditions, the solution is
obtained as
X
i
= A
−1
i
−1
R
i−1
. (47)
431
Successive Linearisation Solution of
Free Convection Non-Darcy Flow with Heat and Mass Transfer
8 Mass Transfer
4. Results and discussion
In this section we give the successive linearization method results for the main parameters
affecting the flow. To check the accuracy of the proposed successive l inearisation method
(SLM), comparison was made with numerical solutions obtained using the MATLAB routine
bvp4c, which is an adaptive Lobatto quadrature scheme. The graphs and tables presented in
this work, unless otherwise specified, were generated using N
= 150, L = 30, N
1


(0) for different values of Gr
∗
at different orders of
the SLM approximation using L
= 30, N = 150 when Le = 1, N
1
= 1, D
f
= 1, S
r
= 0.5
Table 3 and 4 represent the numerical values of the local Nusselt number and Sherwood
number, respectively, for various buoyancy ratios
(N
1
). We observe that the local Nus selt
number and Sherwood number tend to increase as the buoyancy ratio N
1
increases. Increasing
the buoyancy ratio accelerates the flow, decreasing the thermal and concentration boundary
layer thickness and thus increasing the heat and mass transfer rates between the fluid and the
wall.
Gr
∗
2nd order 3rd order 4th order 6th order 8th order 10th order
0.5 0.49979076 0.49970498 0.49970494 0.49970494 0.49970494 0.49970494
1.0 0.45388857 0.45388333 0.45388333 0.45388333 0.45388333 0.45388333
1.5 0.42518839 0.42518709 0.42518709 0.42518709 0.42518709 0.42518709
2.0 0.40449034 0.40448985 0.40448985 0.40448985 0.40448985 0.40448985

4 0.31170490 0.31170917 0.31170917 0.31170917 0.31170917 0.31170917
5 0.32980517 0.32980715 0.32980715 0.32980715 0.32980715 0.32980715
10 0.39608249 0.39638607 0.39638637 0.39638637 0.39638637 0.39638637
Table 3. Values of the Nusselt Number, -θ

(0) for different values of N
1
at different orders of
the SLM approximation using L
= 30, N = 150 when Gr
∗
= 1, Le = 1, D
f
= 1, S
r
= 0.5
N
1
2nd order 3rd order 4th order 6th order 8th order 10th order
0 0.43723544 0.39927412 0.37520193 0.36285489 0.36273086 0.36273086
1 0.45378321 0.45388312 0.45388333 0.45388333 0.45388333 0.45388333
2 0.51638584 0.51639273 0.51639273 0.51639273 0.51639273 0.51639273
3 0.56500365 0.56500444 0.56500444 0.56500444 0.56500444 0.56500444
4 0.60522641 0.60522641 0.60522641 0.60522641 0.60522641 0.60522641
5 0.63977204 0.63977193 0.63977193 0.63977193 0.63977193 0.63977193
10 0.76607162 0.76609949 0.76609963 0.76609963 0.76609963 0.76609963
Table 4. Values of the Sherwood number, -φ

(0) for different values of N
1

3.0 0.21828033 0.21825939 0.21825951 0.21825951 0.21825951 0.21825951
4.0 0.19258954 0.19260872 0.19260881 0.19260881 0.19260881 0.19260881
5.0 0.17369571 0.17369607 0.17369607 0.17369607 0.17369607 0.17369607
Table 5. Values of the Nusselt Number, -θ

(0) for different values of S
r
at different orders of
the SLM approximation using L
= 30, N = 150 when Gr
∗
= 1, Le = 1, D
f
= 1, N
1
= 1
433
Successive Linearisation Solution of
Free Convection Non-Darcy Flow with Heat and Mass Transfer
10 Mass Transfer
S
r
2nd order 3rd order 4th order 6th order 8th order 10th order
0.0 0.57981668 0.51916309 0.50219360 0.49634082 0.49632751 0.49632751
0.5 0.45386831 0.45388333 0.45388333 0.45388333 0.45388333 0.45388333
1.5 0.35029249 0.35029514 0.35029514 0.35029514 0.35029514 0.35029514
2.0 0.36241935 0.36241908 0.36241908 0.36241908 0.36241908 0.36241908
3.0 0.37807254 0.37803636 0.37803657 0.37803657 0.37803657 0.37803657
4.0 0.38517909 0.38521742 0.38521761 0.38521761 0.38521761 0.38521761
5.0 0.38839482 0.38839561 0.38839562 0.38839562 0.38839562 0.38839562

= 1, Le = 1, Sr = 0.5, N
1
= 1
D
f
2nd order 3rd order 4th order 6th order 8th order 10th order
0.0 0.32610052 0.32342809 0.33553652 0.33829291 0.33829308 0.33829308
0.5 0.38480751 0.38479579 0.38479579 0.38479579 0.38479579 0.38479579
0.8 0.42201071 0.42200998 0.42200998 0.42200998 0.42200998 0.42200998
1.2 0.49527611 0.49527429 0.49527429 0.49527429 0.49527429 0.49527429
1.4 0.55343701 0.55343690 0.55343690 0.55343690 0.55343690 0.55343690
1.8 0.84855160 0.84854722 0.84854722 0.84854722 0.84854722 0.84854722
Table 8. Values of the Sherwood Number, -φ

(0) for different values of D
f
at different orders
of the SLM approximation using L
= 30, N = 150 when Gr
∗
= 1, Le = 1, S
r
= 0.5, N
1
= 1
0 2 4 6 8 10
0
0.1
0.2
0.3

η
φ(η)
Gr
*
= 0
Gr
*
= 1
Gr
*
= 2
Fig. 1. Effect of Gr
∗
on the temperature and concentration profiles
434
Advanced Topics in Mass Transfer
Successive Linearisation Solutionyof Free
Convection Non-Darcy Flow with Heat and Mass Transfer
11
0 2 4 6 8 10 12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

0.5
0.6
0.7
0.8
0.9
1
η
θ(η)
N
1
= 0
N
1
= 1
N
1
= 2
0 2 4 6 8 10 12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
η

In the present chapter, a new numerical perturbation scheme for solving complex nonlinear
boundary value problems arising in problems of heat and mass transfer. This numerical
435
Successive Linearisation Solution of
Free Convection Non-Darcy Flow with Heat and Mass Transfer
12 Mass Transfer
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
1.2
η
θ(η)
D
f
= 0
D
f
= 1.5
D
f
= 3
0 2 4 6 8 10
0
0.1
0.2
0.3

0.8
0.9
1
η
θ(η)
S
r
= 0
S
r
= 1.5
S
r
= 3
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
1.2
η
φ(η)
S
r
= 0
S
r
= 1.5

Successive Linearisation Solutionyof Free
Convection Non-Darcy Flow with Heat and Mass Transfer
13
6. References
El-Amin, M.F. (2004) Double dispersion ef fects on n atural convection heat a nd mass transfer
in non-Darcy porous medium, Applied Mathematics and Computation Vol.156, 1–17
Adomian, G. (1976) Nonlinear stochastic differential equations. J Math. Anal. Appl. Vol.55, 441
–52
Adomian, G. (1991) A r eview of the decomposition method and some recent results for
nonlinear equations. Comp. and Math. Appl. Vol.21, 101 – 27
Ayaz,F. (2004) Solutions of the systems of differential equations by differential transform
method, Applied Mathematics and Computation, Vol.147, 547-567
Canuto,C., Hussaini, M. Y., Quarteroni,A. and Zang, T. A. (1988) Spectral Methods in Fluid
Dynamics, Springer-Verlag, Berlin
Chen,C.K., Ho,S.H. (1999) Solving partial differential equations by two dimensional
differential transform method, Applied Mathematics and Computation Vol.106,
171-179.
Don, W. S., Solomonoff, A. (1995) Accuracy and speed in computing the Chebyshev
Collocation Derivative. SIAM J. Sci. Comput, Vol.16, No.6, 1253–1268.
He,J.H, (1999) Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. Vol.178,
257 – 262.
He,J.H. (2000) A new perturbation technique which is also valid for large parameters, J. Sound
and Vibration, Vol.229, 1257 – 1263.
He, J.H. (1999) Variational iteration m ethod a kind of nonlinear analytical technique:some
examples, Int. J. Nonlinear Mech. Vol.34, 699–708
He, J.H. (2006) New interpretation of homotopy perturbation method, Int. J. Modern Phys.B
vol.20, 2561 – 2568.
Liao,S.J. (1992) The proposed h omotopy analysis technique for the solution of nonlinear
problems, PhD thesis, Shanghai Jiao Tong University, 1992.
Liao,S.J. (1999) A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat

1204–1212
Partha, M.K. (2008) Thermophoresis particle deposition in a non-Darcy porous medium under
the influence of S oret, Dufour effects, Heat Mass Transfer Vol.44, 969–977
Trefethen, L.N. (2000) Spectral Methods in MATLAB, SIAM
Zhou, J.K. (1986) Differential Transformation and Its Applications for Electrical Circuits,
Huazhong University Press, Wuhan, China (in Chinese)
438
Advanced Topics in Mass Transfer
20
Explicit and Approximated Solutions
for Heat and Mass Transfer
Problems with a Moving Interface
Domingo Alberto Tarzia
CONICET and Universidad Austral
Argentina
1. Introduction
The goal of this chapter is firstly to give a survey of some explicit and approximated
solutions for heat and mass transfer problems in which a free or moving interface is
involved. Secondly, we show simultaneously some new recent problems for heat and mass
transfer, in which a free or moving interface is also involved. We will consider the following
problems:
1. Phase-change process (Lamé-Clapeyron-Stefan problem) for a semi-infinite material:
i. The Lamé-Clapeyron solution for the one-phase solidification problem (modeling the
solidification of the Earth with a square root law of time);
ii. The pseudo-steady-state approximation for the one-phase problem;
iii. The heat balance integral method (Goodman method) and the approximate solution for
the one-phase problem;
iv. The Stefan solution for the planar phase-change surface moving with constant speed;
v. The Solomon-Wilson-Alexiades model for the phase-change process with a mushy
region and its similarity solution for the one-phase case;

vi. Estimation of the diffusion coefficient in a gas-solid system;
vii. The coupled heat and mass transfer during the freezing of the high-water content
materials with two free boundaries: the freezing and sublimation fronts.
2. Explicit solutions for phase-change process (Lamé-Clapeyron-Stefan
problem) for a semi-infinite material
Heat transfer problems with a phase-change such as melting and freezing have been studied
in the last century due to their wide scientific and technological applications. A review of a
long bibliography on moving and free boundary problems for phase-change materials
(PCM) for the heat equation is shown in (Tarzia, 2000a). Some previous reviews on explicit
or approximated solutions were presented in (Garguichevich & Sanziel, 1984; Howison,
1988; Tarzia, 1991b & 1993). Some reviews, books or booklets in the subject are (Alexiades &
Solomon, 1993; Bankoff, 1964; Brillouin, 1930; Cannon, 1984; Carslaw & Jaeger, 1959; Crank,
1984; Duvaut, 1976; Elliott & Ockendon, 1982; Fasano, 1987 & 2005; Friedman, 1964; Gupta,
2003; Hill, 1987; Luikov, 1968; Lunardini, 1981 & 1991; Muehlbauer & Sunderland, 1965;
Primicerio, 1981; Rubinstein, 1971; Tarzia, 1984b & 2000b; Tayler, 1986).
2.1 The Lamé-Clapeyron solution for the one-phase solidification problem (modeling
the solidification of the Earth with a square root law of time)
We consider the solidification of semi-infinite material, represented by x 0> . We will find the
interface solid-liquid
xst()
=
and the temperature TTxt(,)
=
of the solid phase defined by

()
() ()
()
f
Txt if x st t

0, , 0
=
<>
(3)

(
)
(
)
f
Tst t T t,,0
=
> (4)

(
)
(
)
(
)
x
kT s t t s t t,,0
ρ
=
>

A (5)
Explicit and Approximated Solutions for Heat and Mass Transfer
Problems with a Moving Interface


x
Txt T er
f
st a t
f
at
0
0
(,) ( ), () 2
()
2
ξ
ξ
−
=+ =
(7)
where
k
a
c
2
α
ρ
== is the diffusion coefficient and 0
ξ
> is the unique solution to the
equation

Ste
Ex x() , 0

τ
τρ ξ
==
∫
A
2
0
() (0, ) ()exp( )
t
x
Qt kT d st
. (11)
Proof.

We have the following properties:

EE Exx(0) 0, ( ) , ( ) 0, 0
′
=
+∞ = +∞ > ∀ > . (12)
Remark 1.

From (4) we have

(
)
(
)
(
)

A
(14)
which implies that the problem (2)-(6) is always a nonlinear problem (Pekeris & Slichter,
1939).
Remark 2.

A generalization of the Lamé-Clapeyron solution is given in (Menaldi & Tarzia, 2003) for a
particular source in the heat equation. A study of the behaviour of the Lamé-Clapeyron
solution when the latent heat goes to zero is given in (Guzman, 1982; Sherman, 1971).
2.2 The pseudo-steady-state approximation for the one-phase problem
An approximated solution to problem (2)-(6) is given by the pseudo-steady-state
approximation which must satisfy the following conditions: (3)-(6) and the steady-state
equation

(
)
xx
Txstt0, 0 , 0
=
<< >. (15)
Theorem 2
(Stefan, 1989a)
The solution to the problem (15), (3)-(6) is given by

()
f
TT
Txt T x x st t
st
0


f
Ste
st kT Tt a t
22
0
() 2( )/( ) 4
2
ρ
=− =A (19)
that is

f
kT T
st t
0
2( )
()
ρ
−
=
A
(20)
Remark 3.

If the Stefan number is very small, i.e.

f
cT T
Ste

A study of sufficient conditions on data to estimate the occurrence of a phase-change
process is given in (Solomon et al., 1983; Tarzia & Turner, 1992 & 1999).
2.3 The heat balance integral method (Goodman method) and the approximate
solution for the one-phase problem
An approximated solution for the following fusion problem (similar to the solidification
problem (2)-(6))

(
)
txx
cT kT x s t t0, 0 , 0
ρ
−
=<< > (23)

(
)
TtT t
0
0, 0 , 0
=
>> (24)

(
)
(
)
Tst t t,0, 0
=
> (25)

and the heat equation (23) by its integral on the domain
st(0, ( )) given by

st st st st
ttxx
xx x
dk
Txtdx T xtdx Tst tst T xtdx T xtdx
dt c
kk
TsttTt stTt
cck
() () () ()
00 0 0
( ,) ( ,) ((),)() ( ,) ( ,)
[ ((),) (0,)] [ () (0,)]
ρ
ρλ
ρρ
=+==
=−=−+
∫∫ ∫ ∫


(29)
that is

st
x
dk

are real functions to be determined. Firstly, we can
obtain and
α
β
as a function of s and, therefore, we solve the corresponding ordinary
differential equation for
sst()
=
.
Theorem 3.

The Goodman approximated solution is given by:

α
+
−
=
A 12 1
()
()
Ste
t
cst
,
α
β
+
=
0
2

speed
When the phase-change interface is moving with constant speed we can consider the
following inverse Stefan problem: find the temperature
TTxt(,)= and ft T t() (0,)= such
that:

xx t
k
TT xstt
c
,0 (), 0( )
αα
ρ
=<<>= (34)

Tst t t((),) 0, 0
=
> (35)

x
kT s t t s t t( ( ), ) ( ), 0
ρ
=
>

A (36)

(
)
st m s st mt() 0, (0) 0 ()=> = =

Explicit and Approximated Solutions for Heat and Mass Transfer
Problems with a Moving Interface

445
2.5 The Solomon-Wilson-Alexiades model for the phase-change process with a
mushy region and its similarity solution for the one-phase case
We consider a semi-infinite material in the liquid phase at the melting temperature
f
T . We
impose a temperature
f
TT
0
<
at the fixed face x 0
=
, and the solidification process begins,
and three regions can be distinguished, as follows (Solomon et al., 1982):
i. the liquid phase, at temperature
f
TT
=
, occupying the region xrtt(), 0;>>
ii. the solid phase, at temperature
f
Txt T(,)
<
, occupying the region xstt0(),0
<
<>;

)
txx
cT kT x s t t0, 0 , 0
ρ
−
=<< > (40)

(
)
f
TtTT t
0
0, , 0
=
<>; sr(0) (0) 0
=
= (41)

(
)
(
)
f
Tst t T t,,0
=
> (42)

x
kT s t t s t r t t((),) [ () (1 )()], 0
ρε ε

(45)
where

f
k
erf exp a
c
TT
2
0
() ( ),
2( )
γπ
μξ ξ ξ
ρ
=+ =
−
(46)
and 0
ξ
> is the unique solution to the equation

0
()
() , 0
−
⎛⎞
=>=
⎜⎟
⎝⎠

If the Stefan number is small, then an approximated solution for
ξ
and
μ
is given by:

1
2
0
0
,[1/()]
2[1 (1 ) /( )]
⎡⎤
==+−
⎢⎥
+− −
⎢⎥
⎣⎦
f
f
Ste
TT
TT
ξμξγ
γε
. (49)
2.6 The Cho-Sunderland solution for the one-phase problem with temperature-
dependent thermal conductivity
We consider the following solidification problem for a semi-infinite material


where T(x,t) is the temperature of the solid phase, ρ >0 is the density of mass, 0>
A is the
latent heat of fusion by unity of mass, c >0 is the specific heat, x=s(t) is the phase-change
interface, T
f
is the phase-change temperature, T
o
is the temperature at the fixed face x=0. We
suppose that the thermal conductivity has the following expression:
kkT k TT T T
ooo
f
() [1 ( )/( )],
β
β
=
=+− − ∈
\
. (54)
Let α
o
=k
o
/ρc be the diffusion coefficient at the temperature T
o
. We observe that if β =0, the
problem (50)-(53) becomes the classical one-phase Lamé-Clapeyron-Stefan problem.
Theorem 6
. (Cho & Sunderland, 1974)
The solution to problem (50)-(54) is given by:


where xx() ()
δ
Φ=Φ =Φ is the modified error function, for δ > -1, the unique solution to the
following boundary value problem in variable x, i.e:

ixxxxx
ii
)[(1 ( )) ()] 2 ( ) 0, 0,
)(0)0, ()1
δ
′′′ ′
+
ΦΦ +Φ= >
⎧
⎪
⎨
+
Φ= Φ+∞=
⎪
⎩
(57)
Explicit and Approximated Solutions for Heat and Mass Transfer
Problems with a Moving Interface

447
and the unknown thermal coefficients
λ
and
δ

temperature at the fixed face
We consider a semi-infinite material with null melting temperature
f
T 0
=
, with an initial
temperature C 0−< and having a temperature boundary condition B 0> at the fixed
face x 0= . The model for the two-phase Lamé-Clapeyron-Stefan problem is given by: find
the free boundary
xst()= , defined for t 0> , and the temperature TTxt(,)= defined by

2
1
(,) 0 (), 0
(,) (), 0
(,) () , 0
⎧
><<>
⎪
⎪
==>
⎨
⎪
<<>
⎪
⎩
f
f
f
Txt T if x st t

2
(0, ) 0, 0,
=
>>
(64)

f
Tstt T t
1
((),) 0, 0
=
=>, (65)

f
Tstt T t
2
((),) 0, 0
=
=>, (66)

(
)
(
)
xx
kT st t kT st t st t
11 22
(), (), (), 0
ρ
−

Txt C erfc st x t
erfc a
at
1
1
1
(,) ( ), () , 0
(/)
2
σ
=
−+ ≤ > (70)

kk
st t a a
cc
22
21
21
21
() 2 ( , )
σ
ρρ
=== (71)

where 0
σ
> is the unique solution to the following equation:

() , 0

Remark 9.
It is very interesting to answer the following question: When is the Neumann solution for a
semi-infinite material applicable to a finite material
x
0
(0, )? (Solomon, 1979).
Taking into account that
erf x for 2 x() 1
≅
≤ , we have an affirmative answer for a short
period of time because
Txt C
10
(,)
≅
− is equivalent to x
erf
at
0
1
()1
2
≈
(75)
that is

x

0
22
(0, ) =− (77)
then we can obtain the following result:
Theorem 8.
(Tarzia, 1981)
i. If
q
0
verifies the inequality

Ck
q
a
1
0
1
π
> (78)
then we have an instantaneous change of phase and the corresponding explicit solution is
given by:

x
Txt A Berf x st t
at
222
2
( , ) ( ), 0 ( ), 0
2
=

Aw C Bw
erfc w a erfc w a
1
11
11
(/)
() , ()
(/) (/)
−
== (82)

aq aq
Aw erfw a Bw
kk
20 20
222
22
() (/ ), ()
π
π
==− (83)
and w 0> is the unique solution to the equation

0
() , 0
=
>Fx x x , (84)
where

q

1
1
1
(,) (,) ( ), 0, 0
2
απ
α
=
=− + > > . (86)
Corollary 9
(Tarzia, 1981)
The coefficient
σ
that characterizes the free boundary st t() 2
σ
= of Neumann solution
(69)-(74) must satisfy the following inequality:

Bkc
erf
aCkc
22
211
()
σ
< . (87)
2.9 The Neumann-type solution for the two-phase problem for a particular prescribed
convective condition (Newton law) at the fixed face, and the necessary and sufficient
condition to have an instantaneous phase-change process
We consider the following free boundary problem: find the solid-liquid interface

txx
lll
TTxstt,(),0
α
=
>> (90)

sl f
Tstt Tstt T x st t( ( ), ) ( ( ), ) , ( ), 0
=
==> (91)

ll i
Tx T t T x t(,0) ( ,) , 0, 0
=
+∞ = > > (92)

x
ss s
h
kT t T t T t
t
0
(0, ) ( (0, ) ), 0
∞
=
−> (93)

xx
ss ll

i
f
l
i
l
TT
k
h
TT
0
πα
∞
−
>
−
(96)
Explicit and Approximated Solutions for Heat and Mass Transfer
Problems with a Moving Interface

451
there exists an instantaneous solidification process and then the free boundary problem (89)-
(95) has the explicit solution to a similarity type given by

l
st t() 2
λ
α
= (97)

s

−+
=+
+
(98)

l
liif
x
erfc
t
Txt T T T
erfc
()
2
(,) ( )
()
α
λ
=− − (99)
and the dimensionless parameter 0
λ
> satisfies the following equation

Fx x x() , 0
=
> (100)
where function
F and the b’s coefficients are given by

bx x

=
>= >
A
(102)

li f
s
s
cT T
h
bb
h
0
23
()
0; 0
πα
π
−
=
>= >
A
(103)
Proof.

Function
F has the following properties:

f
li

We consider a semi-infinite material initially in the solid phase at the
temperature
f
CT 0−< =. We impose a temperature
f
BT 0>= at the fixed face x 0= , and the
fusion process begins, and three regions can be distinguished, as follows: (Tarzia, 1990b):
Advanced Topics in Mass Transfer

452
i. the liquid phase, at temperature TTxt
22
(,) 0
=
> , occupying the region
xstt0(),0;<≤ >

ii. the solid phase, at temperature
TTxt
11
(,) 0
=
< , occupying the region xrt t(), 0>>;
iii. the mushy zone, at temperature
f
T 0
=
, occupying the region
st x rt t() (), 0
<

(,) 0 () , 0
><<>
⎧
⎪
=
≤≤ >
⎨
⎪
<<>
⎩
(106)
defined for
x 0> and t 0> , such that the following conditions are satisfied:

xx t
TT xstt
22 2
,0 (),0
α
=
<< > (107)

xx t
TT rtxt
11 1
,(),0
α
=
<> (108)


Tx T t C x t
11
(,0) ( ,) , 0, 0
=
+∞ = − > > (113)

TtB t
2
(0, ) 0, 0
=
>> (114)
Theorem 11.
(Tarzia, 1990b)
i. The explicit solution to the problem (107)-(114) is given by

xx
T x t A B erf T x t A B erf
at at
111 222
12
(,) ( ), (,) ( )
22
=+ =+
(115)

kk
st t rt t a a
cc
22
21

erf
Baa
2
2
2
22
() exp( ) ( )
2
γπ σ σ
ωωσ σ
==+ (118)
where 0
σ
> is the unique solution to the equation

12
() (), 0
=
>Kx Kx x (119)
with

kB x kB x x
Kx F F Fx
a a erfc x
aa
axx x
Kx x erf Fx
Baa er
f
x

11 1
(0 ) , ( ) , 0, 0
+
′
=
+∞ +∞ = −∞ < ∀ > , (121)
KK Kx
22 2
(0 ) 0, ( ) , 0, 0
+
′
=
+∞ = +∞ < ∀ > , (122)
and the thesis holds.
Remark 11

If the boundary condition (114) is replaced by a heat flux condition of the type (77) then we
will have an instantaneous change of phase if and only if the coefficient
q
0
that characterizes
the heat flux (77) verifies an inequality (Tarzia, 1990b).
2.11 The similarity solution for the phase-change problem by considering a density
jump
We will consider the two-phase Lamé-Clapeyron-Stefan problem for a semi-infinite material
taking into account the density jump under the change of phase. We will find the interface
sst() 0=> (free boundary), defined for t 0> , and the temperature

x t if x s t t
xt if x st t