DE IN REHATING ROLLING MIILLS

1. Investigation of the slab heating characteristics in a reheating furnace with
the formation and growth of scale on the slab surface
Jung Hyun Jang a
, Dong Eun Lee a
, Man Young Kim a,*, Hyong Gon Kim b
a
Department of Aerospace Engineering, Chonbuk National University, Chonbuk 561-756, Republic of Korea
b
CAMSTech Co., Ltd., Chonbuk 561-844, Republic of Korea
a r t i c l e i n f o
Article history:
Received 12 September 2009
Received in revised form 6 May 2010
Accepted 6 May 2010
Available online 10 June 2010
Keywords:
Steel slab
Reheating furnace
Scale
Radiative heat transfer
Transient conduction
a b s t r a c t
In this work, the development of a mathematical heat transfer model for a walking-beam type reheating
furnace is described and preliminary model predictions are presented. The model can predict the heat
flux distribution within the furnace and the temperature distribution in the slab throughout the reheat-
ing furnace process by considering the heat exchange between the slab and its surroundings, including
the radiant heat transfer among the slabs, the skids, the hot combustion gases and the furnace wall as
well as the gas convection heat transfer in the furnace. In addition, present model is designed to be able
to predict the formation and growth of the scale layer on the slab in order to investigate its effect on the
slab heating. A comparison is made between the predictions of the present model and the data from an
in situ measurement in the furnace, and a reasonable agreement is found. The results of the present sim-
ulation show that the effect of the scale layer on the slab heating is considerable.
Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction
The reheating furnace process, that is the midway step between
the continuous casting process and the hot-rolling process, is com-
monly used to raise the temperature of the slab, and, thereby, the
plasticity of the slabs so that the subsequent hot-rolling process
runs on wheels. Since the reheating furnace process should have
lower energy consumption and combustion-generated pollutant
emissions, the analysis of transient heating characteristics of the
slab in the reheating furnace has attracted a great deal of interest
during the past few decades. Furthermore, because the attainment
of uniform temperature distributions inside the slab and the target
temperature of the slab at the furnace exit determine the quality
and productivity of the steel product, the reheating furnace process
must be analyzed accurately and rapidly. However, experimental
approach for analyzing a real reheating furnace process is greatly
limited by the complex three dimensional structures and their
influence on the furnace process. Therefore, models and methods
to predict the furnace combustion and heat transfer processes
are in high demand.
These analytical studies can be classified into following two cat-
egories. The first one [1–5] is to solve the full Navier–Stokes and
energy conservation equations governing the hot gas flow and
combustion process in the furnace, where the thermal radiation
acts as an energy source term via the divergence of radiative heat
flux. Although these full CFD analyses make it possible to accu-
rately predict the thermal and combusting fluid characteristics in
the furnace, they necessitate long computational time and result-
ing much cost because of such difficulties as the treatment of so
many governing equations and the complexity of the furnace
structure as well as the uncertainty of the models. The second
method [6–12], which is simple but can reasonably simulate the
thermal behavior of the slab, focuses on the analysis of the radia-
tive heat transfer in the furnace and the transient heat conduction
within the slab. The model suggested in this work can be also cat-
egorized as the second approach.
Meanwhile, the reheating furnace is filled with hot-oxidizing
combustion gases that mainly consist of H2O, CO2, O2, and N2.
The steel slabs are heated up by these gases. As the steel surface
temperature rises, however, it reacts with the furnace gases result-
ing in the formation of iron oxide layer that is generally termed
scale. The thickness of the scale layer depends on slab residence
time in the furnace, its surface temperature, temperature and
aggressiveness of the furnace gas. The formation of scale causes
physical loss of the slab. In addition, since the thermal conductivity
of scale is very small compared to that of steel, the existence of
scale on the slab can greatly affect the heat transfer behavior. How-
ever, the previous works to predict thermal heating characteristics
of the slab in a reheating furnace have not ever considered the
existence of scale.
0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijheatmasstransfer.2010.05.061
* Corresponding author. Address: Department of Aerospace Engineering, Chon-
buk National University, 664-14 Duckjin-Dong, Duckjin-Gu, Jeonju, Chonbuk 561-
756, Republic of Korea. Tel.: +82 63 270 2473; fax: +82 63 270 2472.
E-mail address: manykim@jbnu.ac.kr (M.Y. Kim).
International Journal of Heat and Mass Transfer 53 (2010) 4326–4332
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt

2. Thus, the current work would suggest a mathematical heat
transfer model to predict the formation and growth of the scale,
and find the heat flux impinging on the slab surface and tempera-
ture distribution inside the slab considering it. The furnace is mod-
eled as radiating medium with spatially varying temperature and
is filled with hot combustion gases that consist of H2O, CO2, O2,
and N2, and have highly spectral radiative characteristics. Accord-
ingly, the weighted sum of gray gas model (WSGGM) [13] is used
to consider the non-gray behavior of the combustion gases. In
the following sections, after describing the methodology adopted
here for the prediction of furnace processes within the reheating
furnace, the formation and growth of scale and its effect on the
heat transfer characteristics and thermal behavior of the slab are
investigated. Finally, some concluding remarks are given.
2. Theoretical models
2.1. Reheating furnace process and furnace model
The walking-beam type reheating furnace modeled in this work
is shown in Fig. 1. This furnace, currently run in the steel industry,
has about 35 m in length and 11 m in width, and the highest fur-
nace roof is about 5 m inside. There are five zones in the reheating
Nomenclature
C specific heat, J/(kg K)
I radiation intensity, W/m2
N/; Nh discretized number of each radiation direction
~n unit normal vector on control surface
qC
slab convective heat flux, W/m2
qR
slab radiative heat flux, W/m2
qT
slab total heat flux, W/m2
r position vector
~s unit direction vector
T Temperature, K
kp parabolic rate constant
R universal gas constant, J/(mol K)
Q activation energy, J/mol
Greek symbols
q density of slab or scale, kg/m3
b0 extinction coefficient, =ja+rs, mÀ1
ja absorption coefficient, mÀ1
rs scattering coefficient, mÀ1
r Stefan–Boltzmann constant, =5.67 Â 10À8
W/(m2
K4
)
U scattering phase function, srÀ1
X solid angle, sr
Non-firing
Zone
Charging
Zone
Preheating
Zone Heating Zone Soaking Zone
Furnace Length = 39.2m
Slab
Burner
Furnace Wall
y
x
a
b
Furnace Wall
Slab
F F F
M M
Symmetry Line
Fig. 1. Geometry of the reheating furnace: (a) longitudinal and (b) transverse sections.
J.H. Jang et al. / International Journal of Heat and Mass Transfer 53 (2010) 4326–4332 4327

3. furnace as shown in Fig. 1(a); non-firing, charging, preheating,
heating, and soaking zones. The fixed and moving skids are ar-
ranged in the furnace as shown in Fig. 1(b), and the slabs are sup-
ported and moved in the furnace by the fixed and moving beams,
respectively. Namely, after the slab is supported and heated on
the fixed skids for a certain time, it is moved on the next fixed
beam by the cyclic movement of the moving skids, which consists
of sequential upward, forward, downward, and then backward
movements. The slab is a high carbon steel, whose carbon content
is 0.35–0.55%. The slab is 1.1 m in width, 0.23 m in height and 8 m
in length. There are 29 slabs with 0.1 m interval between them in
the reheating furnace. The slab is assumed to be isothermal of
26.8 °C when charged into the furnace and the slab residence time
is about 180 min. The thermophysical properties of the slab are gi-
ven in Table 1. Those values shown Table 1 is obtained from the
POSCO technical RD center. Although the temperatures and the
concentration distributions of the gases within a real furnace vary
according to conditions of combustion and flow at each location,
the mean temperature and mean mass fraction based on experi-
mental data, listed in Tables 2 and 3, respectively, is used in this
work. Furthermore, the temperatures of the furnace wall and skids
used are listed in Table 2.
Meanwhile, the presence of the skid structure considerably dis-
torts heat transfer to the slab surfaces, both as a result of radiative
shielding of the slab bottom surface and conduction of energy
across the slab/skid contact area. It is reported that conduction
heat loss to the skid is found to be two orders of magnitude less
than the reduction in radiative heat transfer to the slab and then
the dominant factor in the formation of skidmarks is the radiative
shielding by skid structures [6]. On the basis of the fact, it is as-
sumed that conduction heat loss to the skid is negligible. There-
upon, although the fixed skids are contact with the slab bottom
surface in the real furnace, it is assumed that there is an interval
of 0.005 m between the slab and fixed skid.
2.2. High temperature oxidation of metal
The reheating furnace is filled with hot-oxidizing combustion
gases consisting of H2O, CO2, O2, and N2. In such a high tempera-
ture environments, a metal easily reacts with the combustion gases
through the processes of (a) a initial oxygen adsorption, (b) a
chemical reaction to form a surface oxide and oxide nucleation,
(c) a growth of continuous oxide film, and (d) a cavity/micro-
crack/porosity formation within the film shown as shown in
Fig. 2. As a result, an oxide layer consists of wustite, magnetite
and hematite as shown in Fig. 3, forms on the slab surface. This
can best be illustrated by the well known Ellingham/Richardson
diagram [14]. Wustite, FeO, is the innermost phase of the scale
which forms next to the metal and is the most iron rich. Magnetite,
Fe3O4, is the intermediate phase. Hematite, Fe2O3, is the outer-
most layer of the scale and has the highest oxygen content. How-
ever, at temperatures higher than approximately 600 °C, the per-
cent composition of the three oxides, wustite, magnetite and
hematite are about 96%, 4% and 1%, respectively [15,16]. Therefore,
in this work, it is assumed that the scale layer consists of only one
component, wustite. The thermal properties of wustite by Torres
and Colas [17] are listed in Table 1. As shown in Table 1, since
the thermal conductivity of wustite is very small compared to that
of slab, it is expected that the existence of the wustite on the slab
surface can affect the heat transfer behavior.
Meanwhile, in most cases, the rate of scale growth at high tem-
perature follows a parabolic regime. In this case, the thickness of
scale can be represented as follows:
Table 1
Thermal properties of steel slab and wustite.
Temperature (°C) Conductivity
(W/m K)
Specific heat
(J/kg K)
Density
(kg/m3
)
Slab
30 26.89 299.0 7778
400 25.44 401.6
600 22.70 512.0
800 20.89 542.8
1000 23.69 478.9
Wustite [FeO]
– 3.20 725.0 7750
Table 2
Temperature conditions used in this work (°C).
Zone Tg,upper Tg,lower Tg,skid
Non-firing 950 930 730
Charging 950 930 730
Preheating 1000 980 780
Heating 1200 1170 970
Soaking 1180 1160 960
Table 3
Concentration distribution of the furnace combustion gases.
Furnace gas
H2O CO2 O2 N2
0.111 0.177 0.015 0.697
Metal
O2
O2
O2
Metal
O2
O2
O2
Metal
O2
O O
Metal
O2
O O
Metal
O-2 e-
O2
M+2
O
Metal
O-2 e-
O2
M+2
O
Metal
O2
M+2
O
Metal
O2
M+2
O
(a) adsorption (b) nucleation and growth
(d) cavity, microcrack, and
porosity formation
(c) film growth
Fig. 2. Schematic representation of the oxide film formation in gaseous environ-
ments at high temperature environments: (a) adsorption, (c) nucleation and
growth, (c) film growth, and (d) cavity, microcrack, and porosity formation.
Oxidizing atmosphere, 2O
Hematite, 2Fe 3O
Magnetite, 3Fe 4O
Wustite, FeO
Steel, Fe
Fig. 3. Illustrative example of the typical scale layers on the slab surface exposed to
oxidizing environment.
4328 J.H. Jang et al. / International Journal of Heat and Mass Transfer 53 (2010) 4326–4332

4. x2
¼ kpt ð1Þ
where x is the scale thickness, kp is parabolic rate constant, and t is
the oxidation time. The parabolic rate constant is exponentially
dependent on temperature, expressed as:
kp ¼ k0eÀQ=RT
ð2Þ
where kp is a constant, Q is the activation energy of iron oxidation, R
is the gas constant, and T is absolute temperature. The current work
used the parabolic rate constant, expressed as [18]:
kp ¼ 6:1eÀ169;452=RT
ð3Þ
2.3. Governing equations
The two dimensional transient heat conduction equation to pre-
dict the temperature distribution within the slab including the
scale is,
qC
@T
@t
¼
@
@x
k
@T
@x
þ
@
@y
k
@T
@y
ð4Þ
where q, C, and k represent density, specific heat, and conductivity
of the slab or the scale, respectively. The boundary condition of Eq.
(4) is the total heat flux on the slab or scale surface, qT
slab, which can
be obtained from the sum of the convective and radiative heat flux
as following,
qT
slab ¼ qC
slab þ qR
slab ð5Þ
where qC
slab and qR
slab are the convective and radiative heat flux,
respectively. The convective heat transfer between the furnace gas
and the solid surface is evaluated by using the equation,
qC
slab ¼ HcðTgas À TslabÞ ð6Þ
where Hc is the gas convective heat transfer coefficient at the sur-
face of the slab of 7.8 W/m2
K [8]. Also, Tgas and Tslab are the temper-
atures of the furnace gas and the slab surface, respectively.
The radiative heat flux on the slab or scale surface is calculated
from the following equation,
qR
slab ¼
Z
X¼4p
Ið~rw;~sÞð~s Á~nwÞdX ð7Þ
where Ið~rw;~sÞ is the radiation intensity at the slab surface ~rw and
directions ~s. ~nw is the outward unit normal vector at the slab sur-
face, and X is the solid angle. For a radiative active medium, the
radiation intensity at any position ~r along a path ~s through an
absorbing, emitting and scattering medium can be given by the fol-
lowing radiative transfer equation (RTE),
1
b0
dIð~r;~sÞ
ds
¼ ÀIð~r;~sÞ þ ð1 À x0ÞIbð~rÞ þ
x0
4p
Z
X0
¼4p
Ið~r;~sÞUð~s0
!~sÞdX0
ð8Þ
where b0 = ja + rs is the extinction coefficient, ja is absorption coef-
ficient, rs is scattering coefficient, x0 = rs/b0 is the scattering albe-
do, and Uð~s0
!~sÞ is the scattering phase function of radiative
transfer form the incoming direction ~s0
to the scattering direction
~s. Ibð~rÞ is the blackbody radiation of the medium. This equation, if
the temperature of the medium, Ibð~rÞ, and the boundary conditions
for intensity are given, provides a distribution of the radiation
intensity in the medium. For a diffusely emitting and reflecting wall
with temperature Tw, the outgoing intensity at the which is the
boundary condition of Eq. (8) can be expressed as the summation
of the emitted and reflected ones like,
Ið~rw;~sÞ ¼ ewIbwð~rwÞ þ
1 À ew
p
Z
~s0Á~nw0
Ið~rw;~s0
Þj~s0
Á ~nwjdX0
ð9Þ
where ew is the wall emissivity and Ibw ¼ rT4
w=p is the blackbody
intensity of the wall.
2.4. Solution method
As mentioned in the previous, the model developed in this work
consists of coupled two parts, i.e., the radiation model and the heat
conduction model. At certain time step, the radiation model solves
the RTE using the slab surface temperature that resulted from the
conduction model at the previous time step. And then, the conduc-
tion model solves the heat conduction equation using the radiative
heat flux on the slab surface that resulted from the radiation model
at current time step. The above process is repeated every time step.
As a result, the slab temperature could be calculated at each loca-
tion in the furnace.
The transient heat conduction equation expressed in Eq. (4) is
discretized by using the finite volume method (FVM) [19]. A cen-
tral differencing scheme is used for the diffusion terms in the x-
and y-directions, while the unsteady term is treated implicitly.
The resulting dicretized system is then solved iteratively by using
the TDMA (tridiagonal matrix algorithm) algorithm until the tem-
perature field in the slab satisfies the following convergence
criterion:
maxðjTn
i;j À TnÀ1
i;j j=Tn
i;jÞ 6 10À6
ð10Þ
where TnÀ1
i;j is the previous value of Tn
i;j in the same time level.
Meanwhile, the RTE expressed in Eq. (8) must be analyzed in or-
der to compute the radiative heat flux on the slab or scale surface
as shown in Eq. (7). In this work, the finite volume method (FVM)
for radiation suggested by Chui and Raithby [20], and developed by
Chai et al. [21] and Baek et al. [22] is adopted to discretize the RTE.
More detailed information on the FVM can be easily found in the
literature [10,20–22].
3. Results and discussion
The reheating furnace heat transfer model developed in this
work was used to investigate several aspects of furnace behavior,
especially focusing on the prediction of the scale formation and
its effects on the heating characteristics and thermal behavior of
the slab.
As mentioned previously, in order to consider the non-gray
characteristics of the combustion gases, the WSGGM, which postu-
lates that total emissivity and absorptivity may be represented by
the sum of gray gas emissivities weighted with a temperature
dependent factor, is used, and further information on the model
0 5 10 15 20 25 30 35
0
200
400
600
800
1000
1200
Temperature(
o
C)
z (m)
Experiment
Tg (upper zone)
Tg (lower zone)
Tcenterline, W.Scale
Tcenterline, W/O.Scale
Fig. 4. Comparison of the predicted and experimental results of the centerline
temperature of the slab.
J.H. Jang et al. / International Journal of Heat and Mass Transfer 53 (2010) 4326–4332 4329

5. is easily found in Smith et al. [13] On the other hand, it is assumed
that there is no scattering, i.e., rs = 0, and the emissivity of the fur-
nace wall is 0.8 and the emissivities of the skid structure, slab and
scale are set to 0.5. In fact, the exact surface emissivity of the scale
layer has not been reported, and it is only expected to be equal to
or slightly higher than the surface emissivity of the slab. In this
work, therefore, it is assumed that the emissivity of the scale is
equal to one of the slab. Also, because of its symmetry, a half of
the furnace is modeled in order to reduce computing time. The spa-
tial mesh systems used in this study is (Nx  Ny) = (156  151) and
angular systems of (Nh  Nu) = (4  12) for 2p sr.
Firstly, in order to validate the model developed in this work,
the results from the present model are compared with the experi-
mental data provided by POSCO. Fig. 4 shows the longitudinal tem-
perature profile of the centerline temperature of the slab. As
shown, the temperature of the slab with scale is overall higher than
one of the slab without scale. And, it can be seen from the figure
that the temperature difference gradually decreases as the slab is
heated and is 10 °C at the exit of the furnace. Also, at comparison
with experimental data, the result considering the scale effect is
a better agreement with the experimental data than the result
not considering the scale effect. This means that the formation/
growth of scale is one of the important matters which must be con-
sidered in order to more exactly investigate the thermal behavior
of the slab in the reheating furnace.
The reheating furnace is filled with hot-oxidizing combustion
gases and the slabs are heated up by these gases. In addition, these
gases react with the steel slabs resulting in the formation of iron
oxide layer that is generally termed scale. As above mentioned,
scale growth rate of most metals at high temperature follows the
0 20 40 60 80 100 120 140 160 180
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0
200
400
600
800
1000
1200
ScaleThickness(mm)
Time (min)
Sth(upper surface)
Sth(lower surface)
Tg (upper zone)
Tg (lower zone)
Temperature(
o
C)
Fig. 5. Scale growth on the top and bottom surfaces of the slab.
Fig. 6. Distribution of the heat flux vector on the slab and temperature contours in the scale and slab without (left) and with (right) scale: (a) 1st slab (non-firing zone), (b) 8th
slab (charging zone), (c) 13th slab (preheating zone), (d) 22nd slab (heating zone), and (e) 29th slab (soaking zone).
4330 J.H. Jang et al. / International Journal of Heat and Mass Transfer 53 (2010) 4326–4332

6. parabolic regime of Eq. (1). Fig. 5 shows growth of the scale layer
on the top and bottom surface of the slab. As shown, since the tem-
perature is relatively low from non-firing zone to preheating zone,
the growth rate is low. On the other hand, the growth rate of the
scale increases since the heating zone where the temperature is
relatively high. Also, it can be seen from the figure that because
of the temperature difference between the upper and lower zone
of the furnace, the scale layer formed on the top slab surface is
thicker than the scale layer formed on the bottom slab surface.
At the furnace exit the thicknesses of scale on the top and bottom
slab surfaces are 1.75 mm and 1.55 mm, respectively.
As listed in Table 1, the thermal conductivity of scale is very
small compared to one of steel, while the specific heat of scale is
somewhat large. As a result, the existence of scale formed on the
slab surface can greatly affect the heat transfer behavior. Namely,
due to the very small heat conductivity of scale, it should prevent
the heat energy, which is transferred from the surroundings to the
slab surface, from conducting into the inside of the slab. And the
surface of scale should be slowly heated due to its large specific
heat. Fig. 6 shows the heat flux distribution on the surface and
the temperature distribution inside the scale and slab in each zone
of the furnace according to existence and non-existence of the
scale. It can be seen that although more heat energy is transferred
to the surface in a case of considering the scale layer, the surface
temperature rises slowly due to the specific heat of the scale. Also,
due to the thermal conductivity of the scale, the temperature gra-
dient within the slab in that case is more severe, especially near
surface.
In order to more closely examine these effects of the scale, the
scale surface temperature, the temperature of the interface
between the scale and slab and the slab surface and the surface
temperature of the slab without the scale layer is illustrated in
Fig. 7. Similarly, the temperature of the scale surface is lower than
that of the slab without scale throughout the furnace. However, the
temperature difference is prominent in the entrance region of the
furnace and decreases gradually as the slab approaches the exit re-
gion of furnace.
4. Conclusions
In this work, the development of a mathematical heat transfer
model for a walking-beam type reheating furnace is described.
The model includes firstly a sub-model for consideration of the
scale effect on the thermal behavior of the slab as well as a sub-
model for analysis of radiative heat transfer, which is extremely
important in a high temperature environment such as a reheating
furnace. The model adopted in this work can predict the growth of
the scale based on the given longitudinal furnace gas temperature.
And then, considering the scale on the slab surface, the model pre-
dicts the radiative and convective heat fluxes on the slab surface,
and finally the temperature distribution in the slab throughout
the furnace. Using the model, the effect on the thermal behavior
in the reheating furnace is investigated in this work, and the fol-
lowing conclusions can be drawn.
1. The result predicted from the present model is in a good agree-
ment with the experimental data from POSCO. This means the
present model is reasonable.
2. The growth rate of the scale increases as the temperature of the
furnace gas increases. For a specific case studied in this work,
scale thicknesses on the top and bottom slab surfaces are
1.75 mm and 1.55 mm, respectively, at the furnace exit.
3. The scale layer formed on the slab surface makes the heating
rate of the slab slow and the temperature gradient within the
slab increase. As a result, the temperature of the slab with scale
is lower 10 °C than without scale at the exit of the reheating
furnace.
Acknowledgements
This work was supported by the Korea Research Foundation
Grant by the Korean Government (MOEHRD, Basic Research Pro-
motion Fund) (KRF-2008-331-D00103). Also, the authors gratefully
acknowledge the interest and support exerted by the POSCO,
Korea.
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