Two Phase Flow, Phase Change and Numerical Modeling

110

Fig. 14. Evolution of the curvature radius along a microchannel
In the evaporator and adiabatic zones, the curvature radius, in the parallel direction of the
microchannel axis, is lower than the one perpendicular to this axis. Therefore, the meniscus
is described by only one curvature radius. In a given section, r
c
is supposed constant. The
axial evolution of r
c
is obtained by the differential of the Laplace-Young equation. The part
of wall that is not in contact with the liquid is supposed dry and adiabatic.
In the condenser, the liquid flows toward the microchannel corners. There is a transverse
pressure gradient, and a transverse curvature radius variation of the meniscus. The
distribution of the liquid along a microchannel is presented in Fig. 14.
The microchannel is divided into several elementary volumes of length, dz, for which, we
consider the Laplace-Young equation, and the conservation equations written for the liquid
and vapor phases as it follows
Laplace-Young equation

vl c
2
c
dP dP dr
dz dz r dz
σ
−=− (9)

d(A w ) d(A P )
dz dz A A
g
Asin dz
dz dz
ρ=+τ+τ−ρβ
(12)
Theoretical and Experimental Analysis of Flows and Heat Transfer
within Flat Mini Heat Pipe Including Grooved Capillary Structures

111

2
vv vv
vililvwvwvv
d(A w ) d(A P )
dz dz A A
g
Asin dz
dz dz
ρ=−−τ−τ−ρβ (13)
Energy conservation

()
2
w
wwsat
2
ww
Th 1dQ

+

++≤≤−




(15)
we get a linear flow mass rate variations along the microchannel.
In equation (15), h represents the heat transfer coefficient in the evaporator, adiabatic and
condenser sections. For these zones, the heat transfer coefficients are determined from the
experimental results (section 5.3.3). Since the heat transfer in the adiabatic section is equal to
zero and the temperature distribution must be represented by a mathematical continuous
function between the different zones, the adiabatic heat transfer coefficient value is chosen
to be infinity.
The liquid and vapor passage sections, A
l
, and A
v
, the interfacial area, A
il
, the contact areas
of the phases with the wall, A
lp
and A
vp
, are expressed using the contact angle and the
interface curvature radius by

22

=× θ


(20)

4
π
θ= −α
(21)

Two Phase Flow, Phase Change and Numerical Modeling

112
The liquid-wall and the vapor-wall shear stresses are expressed as

2
lw l l l
1
wf
2
τ=ρ
,
l
l
l
e
k
f
R
= ,

=
μ
(23)
Where k
l
and k
v
are the Poiseuille numbers, and D
hlw
and D
hvw
are the liquid-wall and the
vapor-wall hydraulic diameters, respectively.
The hydraulic diameters and the shear stresses in equations (22) and (23) are expressed as
follows

2
c
hlw
sin 2
2rsin
2
D
sin
θ

×θ−θ+


=

22sin r
2
μθ
τ=
θ

θ−θ+


(26)

vvv c
vw
222
c
4
kw d sin r
2
sin
2d 4r sin
2


μ− θ




τ=
θ

μ
(28)
where D
hiv
is the hydraulic diameter of the liquid-vapor interface. The expressions of D
hiv

and τ
iv
are

222
c
hi
c
sin 2
d4rsin
2
D
2r
θ

−θ−θ+


=
θ
(29)

vcvv

, and T
w
. The integration starts
in the beginning of the evaporator (z = 0) and ends in the condenser extremity (z = L
t
- L
b
),
where L
b
is the length of the condenser flooding zone. The boundary conditions for the
adiabatic zone are the calculated solutions for the evaporator end. In z = 0, we use the
following boundary conditions:

()
0
ccmin
00
lv
0
vsatv
0
lv
cmin
r r (a)
w w 0 (b)
P P T (c)
P P - (d)
r


anymore in direct contact with vapor. In this case, the liquid configuration should
correspond to Fig. 14c, but actually, the continuity in the liquid-vapor interface shape
imposes the profile represented on Figure 14d. In this case, the curvature radius is
maximum. Then, in the condenser, the meniscus curvature radius decreases as the liquid
thickness increases (Fig. 14e). The transferred maximum power, so called capillary limit, is
determined if the junction of the four meniscuses starts precisely in the beginning of the
condenser.
6.2 Numerical results and analysis
In this analysis, we study a FMHP with the dimensions which are indicated in Table 1. The
capillary structure is composed of microchannels as it is represented by the sketch of Fig. 1.
The working fluid is water and the heat sink temperature is equal to 40 °C. The conditions of
simulation are such as the dissipated power is varied, and the introduced mass of water is
equal to the optimal fill charge.
The variations of the curvature radius r
c
are represented in Fig. 15. In the evaporator,
because of the recession of the meniscus in the channel corners and the great difference of
pressure between the two phases, the interfacial curvature radius is very small on the
evaporator extremity. It is also noticed that the interfacial curvature radius decreases in the
evaporator section when the heat flux rate increases. However, it increases in the condenser
section. Indeed, when the heat input power increases, the liquid and vapor pressure losses
increase, and the capillary pressure becomes insufficient to overcome the pressure losses.
Hence, the evaporator becomes starved of liquid, and the condenser is blocked with the
liquid in excess.
The evolution of the liquid and vapor pressures along the microchannel is given in Figs. 16
and 17. We note that the vapor pressure gradient along the microchannel is weak. It is due
to the size and the shape of the microchannel that don't generate a very important vapor

Two Phase Flow, Phase Change and Numerical Modeling


50 W
60 W
10 W
20 W
30 W
40 W
50 W
60 W
Evaporator Adiabatic Zone Condenser

Fig. 15. Variations of the curvature radius r
c
of the meniscus

8.00E-05
5.00E+03
1.00E+04
1.50E+04
2.00E+04
2.50E+04
3.00E+04
3.50E+04
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
z (m)
P
v
(Pa)
Evaporator Adiabatic Zone Condenser
10 W
20 W


Fig. 17. Variations of the liquid pressure P
l

8.00E-05
5.08E-03
1.01E-02
1.51E-02
2.01E-02
2.51E-02
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
z (m)
w
l
(m/s)
Evaporator Adiabatic Zone Condenser
10 W
20 W
30 W
40 W
50 W
60 W

Fig. 18. The liquid phase velocity distribution
0.0
0.5
1.0
1.5
2.0
2.5

between the temperature distribution which is obtained from a pure conduction model and
that obtained experimentally (Fig. 21).
20
30
40
50
60
70
80
90
100
110
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
z (mm)
T (°C)
Experimental 10 W
Experimental 20 W
Experimental 30 W
Experimental 40 W
Experimental 50 W
Experimental 60 W
Model
Evaporator
Adiabatic zone Condenser

Fig. 20. Variations of the FMHP wall temperature

0
20
40

when compared to those of a copper plate having the same dimensions. Reductions in the
source-sink temperature difference are significant and increases in the effective thermal
conductivity of approximately 250 percent are measured when the flat mini heat pipes
operate horizontally.
The main feature of this study is the establishment of heat transfer laws for both
condensation and evaporation phenomena. Appropriate dimensionless numbers are
introduced and allow for the determination of relations, which represent well the
experimental results. This kind of relations will be useful for the establishment of theoretical
models for such capillary structures.
Based on the mass conservation, momentum conservation, energy conservation, and
Laplace-Young equations, a one dimensional numerical model is developed to simulate the
liquid-vapor flow as well as the heat transfer in a FMHP constituted by microchannels. It
allows to predict the maximum power and the optimal mass of the fluid. The model takes
into account interfacial effects, the interfacial radius of curvature, and the heat transfer in
both the evaporator and condenser zones. The resulting coupled ordinary differential
equations are solved numerically to yield interfacial radius of curvature, pressure, velocity,
temperature information as a function of axial distance along the FMHP, for different heat
inputs. The model results predict an almost linear profile in the interfacial radius of
curvature. The pressure drop in the liquid is also found to be about an order of magnitude
larger than that of the vapor. The model predicts very well the temperature distribution
along the FMHP.
Although not addressing several issues such as the effect of the fill charge, FMHP
orientation, heat sink temperature, and the geometrical parameters (groove width, groove
height or groove spacing), it is clear from these results that incorporating such FMHP as
part of high integrated electronic packages can significantly improve the performance and
reliability of electronic devices, by increasing the effective thermal conductivity,
decreasing the temperature gradients and reducing the intensity and the number of
localized hot spots.
8. References
Angelov, G., Tzanova, S., Avenas, Y., Ivanova, M., Takov, T., Schaeffer, C. & Kamenova, L.

International Journal Heat and Mass Transfer, Vol.51, No.19-20, pp. 4637-4650
Do, K.H. & Jang, S.P. (2010). Effect of Nanofluids on the Thermal Performance of a Flat
Micro Heat Pipe with a Rectangular Grooved Wick,
International Journal of Heat and
Mass Transfer
, Vol.53, pp. 2183-2192
Gao, M. & Cao, Y. (2003). Flat and U-shaped Heat Spreaders for High-Power Electronics,
Heat Transfer Engineering, Vol.24, pp. 57-65
Groll, M., Schneider, M., Sartre, V., Zaghdoudi, M.C. & Lallemand, M. (1998). Thermal
Control of electronic equipment by heat pipes,
Revue Générale de Thermique, Vol.37,
No.5, pp. 323-352
Hopkins, R., Faghri, A. & Khrustalev, D. (1999). Flat Miniature Heat Pipes with Micro
Capillary Grooves,
Journal of Heat Transfer, Vol.121, pp. 102-109
Khrustlev, D. & Faghri, A. (1995). Thermal Characteristics of Conventional and Flat
Miniature Axially Grooved Heat Pipes,
Journal of Heat Transfer, Vol.117, pp. 1048-
1054
Faghri, A. & Khrustalev, D. (1997). Advances in modeling of enhanced flat miniature heat
pipes with capillary grooves, Journal of Enhanced Heat Transfer, Vol.4, No.2, pp.
99-109
Khrustlev, D. & Faghri, A. (1999). Coupled Liquid and Vapor Flow in Miniature Passages
with Micro Grooves,
Journal of Heat Transfer, Vol.121, pp. 729-733
Launay, S., Sartre, V. & Lallemand, M. (2004). Hydrodynamic and Thermal Study of a
Water-Filled Micro Heat Pipe Array,
Journal of Thermophysics and Heat Transfer,
Vol.18, No.3, pp. 358-363
Lefèvre, F., Revellin, R. & Lallemand, M. (2003). Theoretical Analysis of Two-Phase Heat

Microelectronics
Reliability
, Vol.44, pp. 315-321
Murakami, M., Ogushi, T., Sakurai, Y., Masumuto, H., Furukawa, M. & Imai, R. (1987). Heat
Pipe Heat Sink,
6
th
International Heat Pipe Conference, pp. 257-261, Grenoble, France,
May 25-29, 1987
Ogushi, T. & Yamanaka, G. (1994). Heat Transport Capability of Grooves Heat Pipes,
5
th

International Heat Pipe Conference
, pp. 74-79, Tsukuba, Japan, May 14-18, 1994
Plesh, D., Bier, W. & Seidel, D. (1991). Miniature Heat Pipes for Heat Removal from
Microelectronic Circuits,
Micromechanical Sensors, Actuators and Systems, Vol.32, pp.
303-313
Popova, N., Schaeffer, C., Sarno, C., Parbaud, S. & Kapelski, G. (2005). Thermal management
for stacked 3D microelectronic packages,
36th Annual IEEE Power Electronic
Specialits Conference (PESC 2005)
, pp. 1761-1766, recife, Brazil, June 12-16, 2005
Popova, N., Schaeffer, C., Avenas, Y. & Kapelski, G. (2006). Fabrication and Experimental
Investigation of Innovative Sintered Very Thin Copper Heat Pipes for Electronics
Applications,
37th IEEE Power Electronics Specialist Conference (PESC 2006), pp. 1652-
1656, Vol. 1-7, Cheju Island, South Korea, June 18-22, 2006
Romestant, C., Burban, G. & Alexandre, A. (2004). Heat Pipe Application in Thermal-Engine

Shi, P.Z., Chua, K.M., Wong, S.C.K. & Tan, Y.M. (2006). Design and Performance
Optimization of Miniature Heat Pipe in LTCC,
Journal of Physics: Conference Series,
Vol.34, pp. 142-147
Sun, J.Y. & Wang, C.Y. (1994). The Development of Flat Heat Pipes for Electronic Cooling,
4
th

International Heat Pipe Symposium, pp. 99-105, Tsukuba, Japan, May 16-18, 1994
Tao, H.Z., Zhang, H., Zhuang, J. & Bowmans, J.W. (2008). Experimental Study of Partially
Flattened Axial Grooved Heat Pipes,
Applied Thermal Engineering, Vol.28, pp. 1699-
1710
Tzanova, S., Ivanova, M., Avenas, Y. & Schaeffer, C. (2004). Analytical Investigation of Flat
Silicon Micro Heat Spreaders,
Industry Applications Conference, 39
th
IAS Annual
Meeting Conference Record of the 2004 IEEE, pp. 2296-2302, Vol.4, October 3-7, 2004

Two Phase Flow, Phase Change and Numerical Modeling

120
Xiaowu, W., Yong, T. & Ping, C. (2009). Investigation into Performance of a Heat Pipe with
Micro Grooves Fabricated by Extrusion-Ploughing Process,
Energy Conversion and
Management
, Vol.50, pp.1384-1388
Zaghdoudi, M.C. & Sarno, C. (2001). Investigation on the Effects of Body Force Environment
on Flat Heat Pipes,

conditions continues to play the paramount role for the fundamental approach of the whole
solidification process. Steel properties are critical upon the solidification behaviour.
Different chemical analyses of carbon steels alter the solidus and liquidus temperatures and
therefore influence the calculated results. Shell growth, local cooling rates and solidification
times, solid fraction, and secondary dendrite arm-spacing are some important metallurgical
parameters that need to be ultimately computed for specific steel grades once the heat
transfer problem is solved.
2. Previous work and current status
Solidification heat-transfer has been extensively studied throughout the years and there are
numerous works on the subject in the academic and industrial fields. Towards the
development of continuous casting machines adapted to the needs of the various steel
grades a great deal of research work has been published in this metallurgical domain. In one
of the early works (Mizikar, 1967), the fundamental relationships and the means of solution
were described, but in a series of articles (Brimacombe, 1976) and (Brimacombe et al, 1977,
1978, 1979, 1980) some important answers to the heat transfer problem as well as to
associated product internal structures and continuous-casting problems were presented in
detail. The crucial knowledge-creation practice of combining experiments and models
together was the main method applied to most of these works. In this way, the shell
thickness at mold exit, the metallurgical length of the caster, the location down the caster
where cracks initiate, and the cooling practice below the mold to avoid reheating cracks
were some of the points addressed. At that time, the first finite-element thermal-stress
models of solidification were applied in order to understand the internal stress distribution
in the solidifying steel strand below the mold. The need for data with respect to the

Two Phase Flow, Phase Change and Numerical Modeling

122
mechanical properties of steels and specifically creep at high temperatures as a means for
controlling the continuous casting events was realized from the early years of analysis
(Palmaers, 1978). In a similar study, the bulging produced by creep in the continuously cast

secondary dendrite arm-spacing is about one-half of the primary one. The effect of cooling
rate on zero-strength-temperature (ZST) and zero-ductility-temperature (ZDT) was found to
be significant (Won et al, 1998) due to segregation of solute elements at the final stage of
solidification. The calculated temperatures at the solid fractions of 0.75 and 0.99
corresponded to the experimentally measured ZST and ZDT, respectively. Furthermore, a
set of relationships that take under consideration steel composition, cooling rate, and solid
fraction was proposed; the suggested prediction equation on ZST and ZDT was found in
relative agreement with experimental results. In a monumental work (Cabrera-Marrero et al,
1998), the dendritic microstructure of continuously-cast steel billets was analyzed and found
in agreement with experimental results. In fact, the differential equation of heat transfer was
numerically solved along the sections of the caster and local solidification times related to
microstructure for various steel compositions were computed. Based on the Clyne-Kurz
model a simple model of micro-segregation during solidification of steels was developed

Modeling Solidification Phenomena in the Continuous Casting of Carbon Steels

123
(Won & Thomas, 2001). In this way, the secondary dendrite arm spacing can be sufficiently
computed with respect to carbon content and local cooling rates. In another study (Han et al,
2001), the formation of internal cracks in continuously cast slabs was mathematically
analyzed with the implementation of a strain analysis model together with a micro-
segregation model. The equation of heat transfer was also numerically solved along the
caster. Total strain based on bulging, unbending, and roll-misalignment attributed strains,
was computed and checked against the critical strain. Consequently, internal structure
problems could be identified and verified in practice. The unsteady bulging was found to be
(Yoon et al, 2002) the main reason of mold level hunching during thin slab casting. A finite
difference scheme for the numerical solution of the heat transfer equation together with a
continuous beam model and a primary creep equation were developed in order to match
experimental data. A 2D unsteady heat-transfer model (Zhu et al, 2003) was applied to
obtain the surface temperature and shell thickness of continuous casting slabs during the

problem for different carbon steel grades from the metallurgical point of view is maybe

Two Phase Flow, Phase Change and Numerical Modeling

124
what makes this purely computational study more intriguing and specific in nature. Critical
formulas that bind the heat transfer problem with the various solidification parameters and
strains in the slab are presented and discussed.
3.1 The heat transfer model applied
The general 3D heat transfer equation that describes the temperature distribution inside the
solidifying body is given by the following equation (Carslaw & Jaeger, 1986) and (Incropera
& DeWitt, 1981):

P
T
CkTS
t
ρ
∂
=∇⋅ ∇ +
∂
(1)
The source term S, in units W· m
-3
, may be considered (Patankar, 1980) to be of the form:

CP
SS ST=+⋅ (2)
that is, by a constant term and a temperature dependent term and can be related to
correspond to the latent heat of phase change. Furthermore, T is the temperature, and ρ, C

ρ

∂∂∂ ∂∂

=++


∂∂ ∂ ∂ ∂


(4)
The boundary conditions applied in order to solve (4) are as follows:
Heat flux in the mold is equalized to the empirical equation used by other researchers (Lait
et al, 1974),

md
qt
65
2.67 10 2.21 10=×−× (5)
The mold heat-flux (q
m
) is given in W/m
2
, and t
d
(in seconds) is the dwell time of the strand
inside the mold. Involving an expression for the local heat-transfer coefficient inside the
mold (Yoon et al, 2002) a more realistic formula was derived that exhibited good results in
the present study:


−
(8)

()
cc env
qhTT=⋅− (9)
where
h
s
, h
r
, and h
c
are the heat transfer coefficients for spray cooling, radiation, and
convection, respectively,
T
w0
is the water temperature, T
env
is the ambient temperature, σ is
the Stefan-Boltzmann constant, and
ε is the steel emissivity (equal to 0.8 in the present
study). Natural convection was assumed to prevail at the convection heat transfer as
stagnant air-flow conditions were considered due to the low casting speeds of the strand
applied in practice. The strand was assumed to be a long horizontal cylinder with an
equivalent diameter of a circle having the same area with that of the strand cross-sectional
area, and a correlation valid for a wide Rayleigh number range proposed by (Churchill &
Chu, 1975) was applied, written in the form proposed by (Burmeister, 1983):

DD

T
hW
0.55
0
1 0.0075
1570
4
−
=⋅⋅
(11)
where W is the water flux for any secondary spray zone in liters/m
2
/sec, and h
s
is in
W/m
2
/K. At any point along the secondary zones (starting just below the mold) of the
caster the total flux q
tot
is computed according to the following formula, taking into account
that q
s
may be zero at areas where no sprays are applied:

tot s r
qqqq=++ (12)
In mathematical terms, considering a one-fourth of the cross-section of a slab assuming
perfect symmetry, the aforementioned boundary conditions can be written as:


tot y x m
q
at
y
W x W z L
T
k
y
q at y W x W z L
,0 ,0
,0 ,
=≤≤≤≤

∂

−=

∂
=≤≤>


(14)

Two Phase Flow, Phase Change and Numerical Modeling

126
where z follows the casting direction starting from the meniscus level inside the mold;
consequently, the mold has an active length of L
m
. W

TT t z xW
y
W
0
at 0 (and 0), 0 , 0 == =<<<< (17)
The thermo-physical properties of carbon steels were obtained from the published work of
(Cabrera-Marrero et al, 1998); the properties were given as functions of carbon content for
the liquid, mushy, solid, and transformation temperature domain values. The liquidus and
solidus temperatures were obtained from the work of (Thomas et al, 1987):

L
TCSiMnPS
Ni Cr Cu Mo V Ti
1537 88(% ) 8(% ) 5(% ) 30(% ) 25(% )
4(% ) 1.5(% ) 5(% ) 2(% ) 2(% ) 18(% )
=− − − − −
−− −− −−
(18)

S
TCSiMnPS
Ni Cr Al
1535 200(% ) 12.3(% ) 6.8(% ) 124.5(% ) 183.9(% )
4.3(% ) 1.4(% ) 4.1(% )
=− − − − −
−−−
(19)
At any time step the simulating program computes whether a given nodal point is at a
lower or higher temperature than the liquidus or solidus temperatures for a given steel
composition. Consequently, the instantaneous position of the solidification front is derived,


()
() ()
{}
x
P
W
and =
5
5
2
2cosh tanh 2
cosh
π
αψψψψ
πψ
=−−

(23)

Modeling Solidification Phenomena in the Continuous Casting of Carbon Steels

127
Some important parameters are included in the expressions: ℓ
P
is the roll pitch in the part of
the caster under consideration,
u
C
is the casting speed, t

the meniscus level; it is clarified that the maximum value of H
5
can be around the caster
radius (27).

nm
C
CPP
Q
At
RT
0
exp( )
εσ
=⋅ ⋅⋅ −
(25)

P
gH
5
σρ
= (26)

C
HR
5,max
= (27)
For the steel slabs produced at Stomana, the constitutive equations for model II (Kozlowski
et al, 1992) were applied after integration (T in Kelvin=T
P

So, after appropriate integration of the strain rate (28), the following expression was applied
for the primary creep that exhibited better results than the correlations of (Palmaers, 1978)
specifically for the Stomana slabs, probably due to their much larger size compared to the
size of the slabs produced at Sovel:

()
nm
PCKP
Cm Q T t
1
,
/( 1) exp( / )
εσ
+
=+⋅− ⋅⋅
(32)
The unbending strain was computed according to equation (33) where R
n-1
, R
n
are the
unbending radii of the caster, (Uehara et al, 1986) and (Zhu et al, 2003).

Sy
nn
WS
RR
1
11
100 ( )

the carbon equivalent value (36) and the Mn/S ratio, as this could cause internal cracks
during casting (Hiebler et al, 1994). It should be pointed out that low carbon steels with high
Mn/S (>25) ratios are the least prone for cracking during casting.

eq C
C C Mn Ni Si Cr Mo
,
(% ) 0.02(% ) 0.04(% ) 0.1(% ) 0.04(% ) 0.1(% )=+ + − − − (36)

%Carbon Temperature
range, ºC
A
0
m n Q
C

(kJ/mol)
0.090 (low carbon) < 1000 0.349 0.35 3.1 150.6
0.090 (low carbon) 1000-1250 2.422 0.33 2.5 146.4
0.090 (low carbon) > 1250 6.240 0.21 1.6 123.4
0.185 (medium carbon) < 1000 141.1 0.36 3.1 211.3
0.185 (medium carbon) 1000-1250 1.825 0.37 2.5 144.3
0.185 (medium carbon) > 1250 1.342 0.25 1.5 102.5
Table 1. Data used for primary creep
3.1.2 Solid fraction analysis
The solid fraction values f
S
are very important especially at the final stages of solidification
in which soft reduction is applied in many slab casters in an attempt to reduce or minimize
any internal segregation problems. The following expressions extracted from the work of







(37)

()
j
j
f
CCSiMnPS 67.51(% ) 9.741(% ) 3.292(% ) 82.18(% ) 155.8(% )
′
=+ + ++

(38)

1
(1 exp( 1 / )) exp( 1 /(2 ))
2
αα α
Ω= − − − −
(39)

Modeling Solidification Phenomena in the Continuous Casting of Carbon Steels

129

R

solidification rate r and thermal gradient G in the mushy zone according to the following
formula (Cabrera-Marrero et al, 1998):

PRIM rg
nr G
11
42
λ
−−
=⋅⋅ (42)
Solidification rate r is actually the rate of shell growth:

dS
r
dt
=
(43)
and the thermal gradient G is defined as:

LS
TT
G
w
()−
=
(44)
where w is the width of the mushy zone. It is interesting to note that local solidification
times T
F
are related to the local cooling rates with the expressions:

R
CC C
CC C
0.4935
(0.5501 1.996 (% ))
0.3616
(169.1 720.9 (% )) for 0 (% ) 0.15

143.9 (% ) for (% ) 0.15
λ
−
−⋅
−

−⋅ ⋅ <≤

=

⋅⋅ >


(46)
4. Results and discussion
For the Stomana slab caster that normally casts slab sizes of 220x1500 mm x mm two
chemical analyses for steel were examined depending on the selected carbon concentrations,
as presented on Table 2.

Two Phase Flow, Phase Change and Numerical Modeling

130

line 1, and the temperature at the surface of the slab is presented by line 2. The shell
thickness S and the distance between liquidus and solidus w are presented by dotted lines 3
and 4, respectively. In part (b) of Fig. 2 the rate of shell growth (dS/dt), the cooling rate (C
R
),
and the solid fraction (f
S
) in the final stages of solidification are presented. Finally, in part (c)
the local solidification time T
F
, and secondary dendrite arm spacing λ
SDAS
are also presented.
It is interesting to note that the rate of shell growth is almost constant for the major part of
solidification. Computation results show that solid fraction seems to significantly increase
towards solidification completion. Apart from unclear fluid-flow phenomena that may
adversely affect the uniform development of dendrites in the final stages of solidification

Modeling Solidification Phenomena in the Continuous Casting of Carbon Steels

131
and influence the local solid-fraction values, the shape of the f
S
curve at the values of f
S

above 0.8 seem to be influenced by the selected set of equations (37)-(41). Fig. 3 depicts
computed strain results along the caster.
where most parameters were defined in the appropriate section and E
e
is an equivalent
elastic modulus that was calculated using the following equation:

SP
e
S
TT
E
T
4
10 in MPa
100
−
=×
−
(48)

Two Phase Flow, Phase Change and Numerical Modeling

132
Consequently, the bulging strain is computed by equation (20) in which δ
B
is substituted by
δ
B,2
. It seems that the computed results in the latter case are much higher than the ones
computed with the generally applied method as described in 3.1.1. Furthermore, the
recently presented formulation (47)-(48) was proven to be of limited applicability in most

for part (a) and 16 m for part (b) from the meniscus, respectively. %C = 0.185; casting speed:
0.80 m/min; SPH: 20 K; solidus temperature = 1479ºC; (all temperatures in the graph are in ºC) Fig. 5. Results with respect to distance from the meniscus: In part (a), lines (1) and (2)
illustrate the centreline and surface temperatures of a 220 x 1500 mm x mm Stomana slab;
lines (3) and (4) depict the shell thickness and the distance between the solidus and liquidus
temperatures; in part (b), the solid fraction f
S
, the local cooling-rate C
R
, and the rate of shell
growth dS/dt are presented; in part (c), the local solidification time and secondary dendrite
arm spacing are depicted, as well. Casting conditions: %C = 0.185; casting speed: 0.80
m/min; SPH: 20 K; solidus temperature = 1479ºC; (all temperatures in the graph are in ºC)