a model for heat conduction through the oxide layer of steel during hot rolling
TRANSCRIPT
A model for heat conduction through the oxide layerof steel during hot rolling
MartõÂn Torresa, Rafael ColaÂsb,*
aGalvak, S.A. de C.V., San NicolaÂs de los Garza, N.L. Mexico, MexicobFacultad de IngenierõÂa MecaÂnica y EleÂctrica, Universidad AutoÂnoma de Nuevo LeoÂn, A.P. 149-F, 66451 Universitaria Cd, Mexico
Received 12 February 1999
Abstract
A heat conduction model developed to predict the temperature distribution within the oxide layer of carbon steel being rolled is
presented. This model takes into account the different physical properties of the three oxide species, and the parabolic growth of the layer.
The thickness of the layer is divided into 40 nodes or elements of which 36 are considered to be of wustite, three of magnetite and only one
of hematite to comply with their proportions. It is found that this particular model is too complicated when the aim is to evaluate the effect
of the oxide crust in thin oxide layers, such as those encountered during strip rolling, because similar results can be obtained using a single
node model based on the properties of wustite; whereas with the behaviour of thick layers, such as those encountered after reheating or
during roughing passes, is modelled, the present model becomes valuable. The thermal gradients predicted by the model can be employed
to predict the integrity of the oxide layer. # 2000 Elsevier Science B.V. All rights reserved.
Keywords: Hot rolling; Oxidation; Carbon steel; Heat transfer; Modelling
1. Introduction
During the hot rolling of steel, as well as in other hot
working operations, an oxide layer grows on top of the free
surfaces of the metal, modifying the cooling rate of the plate
or strip. The effects caused by this layer have to be con-
sidered while the plate or strip is in air, during which heat is
lost by convection and radiation to the surrounding media,
and while it is being deformed, when the predominant heat
losses are due to conduction to the work-rolls, since heat will
¯ow through the oxide [1±5].
At temperatures above 5608C, i.e. in the range of interest
for hot rolling, the layer is formed by three distinctive oxides
[6,7]: wustite (FeO), magnetite (Fe3O4) and hematite
(Fe2O3), each one with its own thermophysical properties
[8±10] and temperature-dependent growth rates [7,11,12].
Some authors [13,14] do not take into account the exis-
tence of the oxide layer while modelling the hot rolling of
steel, while others [1±5] assume a layer made only of
wustite. The aim of this work is to present the results
obtained with a model developed to compute heat transfer
within a layer formed by the three different oxide species
while a piece of steel is subjected to hot rolling conditions,
and compare these results with those obtained when only
one species is employed.
2. Model
Heat losses in plate or strip of steel in air can be assumed
to occur as shown in Fig. 1, in which it is assumed that the
heat ¯ows in only one direction (through the thickness) and
that the contact between the different layers is perfect, i.e.
without any pores or cavities within their boundaries, and,
therefore the heat that ¯ows into one species should leave it.
The amount of heat lost is given by
q
A� ÿk dT
dx(1)
where q is the heat transfer rate, k the thermal conductivity
of the material, A the cross-sectional area, T the temperature
and x the thickness co-ordinate.
The temperature fall due to conduction is then given by
dT
dt� a
d2T
dx2(2)
Journal of Materials Processing Technology 105 (2000) 258±263
* Corresponding author. Tel.: �52-8-3294020, ext. 5770;
fax: �52-8-3320904.
E-mail address: [email protected] (R. ColaÂs).
0924-0136/00/$ ± see front matter # 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 5 6 9 - 0
where t is the time and a the thermal diffusivity of the
material,
a � kCpr
(3)
where Cp is the heat capacity and r the density.
Heat losses during air cooling are caused by convection
and radiation, the former can be described by
q
A� h�Ts ÿ T1� (4)
where h is the convective coef®cient, and Ts and T1 are the
temperatures on the surface and that of the surrounding
media. Heat losses due to radiation are calculated by
q
A� es�T4
s ÿ T41� (5)
where e is the emissivity and s�5.6699�10ÿ8 W/m2 K4 is
the Stefan±Boltzmann constant.
Hollander [1] assumed that heat loss by convection
in a strip cooling from 1200 to 9008C are only about
4±6% of the total loss and, therefore, is negligible. Other
authors [2±5] use an empirical equation to calculate the
effect of convection and radiation on a cooling piece of
steel,
H � a� bTs � c�Ts � 273�4 (6)
where H is the heat lost per unit area, a�6746 W/m2,
b�21.2 W/m2 8C, c�4.763�10ÿ8 W/m2 8C4, and Ts is
expressed in 8C.
Other phenomena that the model has to comply with are
the growth of the layer and the distribution of the different
oxide species. In the present work it is assumed that the
growth follows a parabolic regime [7,15],
Dx � kpt0:5 (7)
where Dx is the oxide thickness at a given time (t), and kp the
growth coef®cient, which depends on the type of oxide and
temperature. Eq. (7) implies that the growth is very rapid on
oxide-free surfaces, but decreases as the oxide builds
up. During actual rolling the layer is removed with high
pressure water jets (descaling), exposing the free surface of
the metal to the surrounding media.
The proportion of the different species within the whole
layer is taken from experimental work [6,7,11] which report
that the wustite represents around 90% of the layer, whereas
the magnetite is around 8% and the hematite occupies only
2% of the full layer.
Heat conduction within the oxide layer and strip or plate is
approximated by ®nite differences in one dimension, Fig. 2.
To start with, the oxide thickness is divided into 40 nodes or
elements, of which 36 are considered to be wustite, three
magnetite and one hematite, in order to comply with the
reported proportions [6,7,11], while half the thickness of the
strip is divided into 20 nodes, this being done after assuming
symmetrical cooling on the top and bottom surfaces [5]. A
thermal pro®le or a constant temperature value can be
speci®ed on both the strip and the oxide layer. An initial
oxide thickness of 1 mm (the thickness of each node is set to
2.5�10ÿ8 m), and constant temperature within the layer are
assumed when the case of a plate or strip coming from
descaling is considered. Since the size of the elements on
layer and strip are different (normally the thickness of the
elements in the strip will be around 6�10ÿ4 m), an implicit
®nite difference method [16,17] was chosen.
Once the size of the elements in the layer and strip is
established, a stability time, which assures that the thermal
gradient does not penetrate more than one node per iteration,
is calculated in a way that will be shown later. With the
stability time it is possible to calculate a dimensionless
parameter (Z) for the strip,
Zs �ks Dtm2
l2(8)
Fig. 1. Heat transfer through the three different oxide species.
Fig. 2. The nodes used to compute heat ¯ow.
M. Torres, R. ColaÂs / Journal of Materials Processing Technology 105 (2000) 258±263 259
where ks is the thermal conductivity of the steel, and m�20
the number of nodes into which half the thickness (l) of the
strip is divided. Heat ¯ow can then be calculated by [16],
�1ÿ Zs�Ti; j�1 ÿ 12Zs�Tiÿ1; j�1 ÿ Ti�1; j�1�
� �1ÿ Zs�Ti; j ÿ 12Zs�Tiÿ1; j ÿ Ti�1; j� (9)
where the sub-index j indicates that the computations are
being conducted at the jth time interval, Ti, j is therefore the
temperature at the ith node and jth interval. Special care
has to be taken at the centre of the strip, where heat ¯ow
is zero, and at the surface, where heat ¯ow is calculated
by Eq. (6).
The temperature pro®le in the oxide layer is obtained by
assuming that the temperature at the surface of the strip is
that at the interface steel±wustite, and a set of equations
similar to those described by Eq. (9), but with Zw replacing
Zs is used,
Zw �aw Dt
Dy2(10a)
where aw is the thermal diffusivity of the wustite and Dy the
distance between nodes. The temperatures in the magnetite
and hematite nodes are obtained in a similar way, but with
Zm and Zh de®ned as
Zm �am Dt
Dy2(10b)
Zh �ah Dt
Dy2(10c)
where am and ah are the thermal diffusivities of magnetite
and hematite.
The temperature at the wustite±magnetite interface is
obtained by
1� Zm
zwm
� Zw
zmw
� �Tk;n�1 ÿ Zm
zwm
Tk�1;n�1
� 1ÿ Zm
zwm
ÿ Zw
zmw
� �Tk;n � Zm
zwm
Tkÿ1;n � Zm
zmw
Tkÿ1;n
(11)
where zwm and zmw are de®ned as
zwm � 1� nw
nm
; zmw � 1� nm
nw
(12)
and nw and nm are calculated by
nw � kw
aw
; nm � km
am
(13)
in which the sub-index `w' and `m' indicate that the thermal
properties are those for wustite or magnetite, respectively.
Table 1 summarizes the values of the different thermo-
physical coef®cients employed by the model. Only the
properties of the steel were considered to be temperature
dependent [18], since it was not possible to ®nd sensible
relationships for the temperature dependence of the different
properties of the oxides [9,10].
A further dimensionless parameter (Zr) can be deduced
from the heat-transfer conditions at the surface of the oxide,
Zr �Zhe
khn(14)
where e is the thickness of the oxide layer and n�40 the
number of nodes into which the thickness is divided. The
stability criterion is then drawn from the parameter,
14� bZrTo � cZrT
4o (15)
where the constants b and c are those from Eq. (6) and To is
the temperature at the surface of the oxide. In this model
then, the stability time is chosen in such a way that the
condition given by Eq. (15) is ful®lled.
Once the thermal pro®le within the strip and oxide for one
cycle is obtained, the growth of the layer is calculated using
Eq. (7). The value of Dy in Eqs. (10a)-(10c) is updated,
which results in a change of the time increment (Dt), which
is now used to obtain the thermal gradients within the
material at t�Dt.
3. Results
Fig. 3 shows the temperature evolution of a 254 mm thick
slab. The three different curves represent the progression at
the centre and surface of the slab (i.e. at the metal±oxide
interface), and at the surface of the oxide layer. These curves
were obtained after assuming that the slab was coming out
Table 1
Thermophysical properties employed [9,10,18]
k (W/m K) r (kg/m3) Cp (J/kg K)
Austenitea 16.5�0.11T 8050ÿ0.5T 587.8�0.068T
FeO 3.2 7750 725
Fe3O4 1.5 5000 870
Fe2O3 1.2 4900 980
a Temperature is expressed in 8C.
Fig. 3. Cooling predicted by the model for a 254 mm thick slab; the
temperatures shown correspond to the centre of the slab, the metal±oxide
interface and the oxide surface.
260 M. Torres, R. ColaÂs / Journal of Materials Processing Technology 105 (2000) 258±263
from a reheating furnace, after being heated uniformly to
12508C. It was also assumed that the oxide layer thickness
was equal to 6.25 mm at the moment at which the slab leaves
the furnace.
The temperature pro®le developed within the oxide layer
after 40 s is shown in Fig. 4, in which the individual
thickness of the different oxide species is marked. It is
interesting to note the sharp gradients that develop in such
a small distance, as a result of the low thermal conductivity
of the compounds [8±10]. The ®nal thickness of the oxide
layer (6.30 mm) was calculated employing Eq. (7), with the
coef®cients [5,7,15] corresponding to the average tempera-
ture in the individual layers.
Simulation of the thermal gradients which develop in a
25.4 mm thick plate after descaling is shown in Fig. 5: as
with Fig. 3, the curves plotted correspond to the temperature
evolution at the centre and surface of the plate (the metal±
oxide interface) and the oxide surface. The calculations were
conducted assuming a homogeneous temperature distribu-
tion within the plate at time zero, and a descaled oxide-free
surface. This last condition implies that the oxide layer will
grow at fast rate following Eq. (7). The temperature gradient
within the oxide layer formed on top of the plate after 40 s is
shown in Fig. 6. The computed ®nal thickness of 67 mm was
obtained in a manner similar to that for computing the
growth on the slab.
Temperature evolution on both surfaces of the oxide layer,
as well as its thickness, for the simulation of 2.54 mm thick
strip being produced in a six-stand continuous mill is shown
in Fig. 7. In this case, the temperature at the metal±oxide
interface is considered to be that calculated by a two-
dimensional ®nite difference program described elsewhere
[5], whilst the temperature on the oxide surface is calculated
by the present model. The seven temperature drops shown in
Fig. 7(a) correspond to the cooling due to descaling and
contact with the work-rolls of each stand. The change of
thickness of the oxide layer, Fig. 7(b), is computed after
assuming that the oxide is fully plastic and will deform to the
same extent as the steel during rolling, and that it will be able
to grow, when the strip is in air, following the parabolic
regime given by Eq. (7). The computed thermal gradient
within the oxide layer at the exit of the ®rst stand, when the
thicknesses of the outgoing stock and of the oxide layer are
13 mm and 6 mm, respectively, is shown in Fig. 8.
Fig. 4. Temperature pro®le within the oxide layer of the 254 mm thick
slab.
Fig. 5. Cooling predicted by the model for a 25.4 mm thick plate; the
temperatures shown correspond to the centre of the plate, the metal±oxide
interface and the oxide surface.
Fig. 6. Temperature pro®le within the oxide layer of the 25.4 mm thick
plate.
Fig. 7. Computer simulation of: (a) the temperature evolution; and (b) the
oxide growth during the production of 2.54 mm thick strip.
M. Torres, R. ColaÂs / Journal of Materials Processing Technology 105 (2000) 258±263 261
4. Discussion
The most striking effect of the thickness of the oxide
layer is the temperature difference at each side of the oxide
layer which develops after short time intervals, since this
difference is of the order of 1908C after the slab is being
cooled in air for 40 s, whereas it is only around 28C in the
plate after the same time has elapsed, Table 2. However,
when the average thermal gradient is considered, it is found
to be about the same in the slab and the plate, at around
308C/mm.
The heat ¯ow (H) to the environment affects the tem-
perature gradient within the oxide layer, since during air
cooling, Eq. (6), H will be of the order of 250 kW/m2 for the
slab and around 130 kW/m2 for the plate, but when the steel
is deformed by the work-rolls, the heat ¯ow increases to
more than 30 MW/m2 [5,19], yielding, for the case of the
®rst stand, a temperature difference of 588C across a dis-
tance of 6 mm, see Figs. 7 and 8, which is equivalent to an
average gradient of around 9.58C/mm, Table 2.
Although the isolating effect of the oxide layer can be
taken into account by the use of an external node or element
with the physical properties of wustite [1±5], reducing the
complexity and time involved in the computations, the
results shown in Figs. 7 and 8, for instance, compare well
with those obtained when modelling is conducted under the
assumption that the layer is only made of wustite [5]. The
present model can be used to obtain more information
related to the temperature distribution within the layer,
which can then be used to study its integrity [20].
A problem which comes across with the use of a unique
node or element to simulate the oxide occurs when model-
ling the initial stages after reheating, when the thickness of
the layer is greater than 6 mm; in such a case, the tempera-
ture difference calculated after 2 or 3 s of cooling can be as
high as 2008C [21], which is twice as much as that calculated
with the present model, see Fig. 3.
Although the assumption that the scale deforms to the
same extent as the steel might not be true, it is worth
remembering that the crust is made mainly of wustite and
magnetite which are fairly plastic at the temperatures
involved in hot rolling [22±24], and that this straining will
be the upper limit to which the scale will be subjected,
implying the establishment of the steepest thermal gradients
that can be expected within the oxide, and the highest
growing rates, once the steel is in air.
5. Conclusions
A computer model, which considers the three different
species of oxides, is developed to calculate the temperature
distribution within the oxide layer formed on low carbon
steel being rolled.
It is concluded that the model is too complicated when the
only aim is to calculate the isolating effect of thin layers (like
those encountered during the production of the strip),
because similar results can be obtained when the oxide is
modelled by a unique node or element; but these results can
be employed to predict the integrity of the oxide layer. When
thick layers are considered (like those found just after
reheating a slab), the model is able to provide meaningful
results.
Acknowledgements
The authors express their thanks for the ®nancial support
given by CONACYT and the facilities provided by Hylsa,
S.A. de C.V., during this work.
References
[1] F. Hollander, Mathematical models in metallurgical process devel-
opment, Iron Steel Institute Publication 123, London, 1970, p. 46.
[2] R. Harding, Ph.D. Thesis, University of Shef®eld, UK, 1976.
[3] L.A. Leduc, Ph.D. Thesis, University of Shef®eld, UK, 1980.
[4] C.M. Sellars, Mater. Sci. Technol. 1 (1985) 325.
[5] R. ColaÂs, Mater. Sci. Technol. 14 (1988) 388.
[6] L.S. Darken, R.W. Gurry, J. Am. Chem. Soc. 68 (1946) 798.
[7] F. Lorang, Rev. Univ. Mines 17 (1961) 514.
[8] G.V. Samsonov (Ed.), The Oxide Handbook, Plenum Press, New
York, 1973.
Fig. 8. Temperature pro®le within the oxide layer of the out-going stock
(13 mm height) from the ®rst stand during the production of the 2.54 mm
thick strip.
Table 2
Results of the simulations
Slab Plate Strip
Steel thickness 254 mm 25.4 mm 13 mm
Oxide thickness 6.3 mm 67 mm 6.1 mm
DT a 187.58C 2.28C 58.08CThermal gradient 29.88C/mm 32.88C/mm 9.58C/mm
Heat flow 250 kW/m2 130 kW/m2 30 MW/m2
a Across the oxide after 40 s of simulation in the slab and plate, and at
the exit of the ®rst stand for the strip.
262 M. Torres, R. ColaÂs / Journal of Materials Processing Technology 105 (2000) 258±263
[9] R. Taylor, C.M. Fowler, R. Rolls, Int. J. Thermophys. 1 (1980) 225.
[10] J. Slowik, G. Borchard, C. KoÈhler, R. Jeschar, R. Scholz, Steel Res. 7
(1990) 302.
[11] J. Paidassi, Rev. MeÂtall. 54 (8) (1957) 2.
[12] L. Himmel, R.F. Mehl, C.E. Birchenall, Trans. AIME 197 (1953)
889.
[13] H. HoÈfgen, G. Zouhar, F. Birnstock, J. Bathelt, in: Proceedings of the
Fourth International Steel Rolling Conference on Science and
Technology of Flat Rolling, IRSID, 1987, p. B.2.1.
[14] C. Devadas, I. Samarasekera, Ironmaking and Steelmaking 6 (1986)
311.
[15] C. Wagner, Atom Movements, ASM, Metals Park, 1951.
[16] R.I. Burden, J.D. Farues, Numerical Analysis, 3rd Edition, Prindle,
Weber & Schmidt, Boston, MA, 1985.
[17] S.V. Pantakar, Numerical Heat Transfer and Fluid Flow, Hemisphere,
New York, 1980.
[18] C.M. Sellars, J.A. Whiteman, Met. Technol. 8 (1981) 10.
[19] M.P. Guerrero, C.R. Flores, A. PeÂrez, R. ColaÂs, J. Mater. Process.
Technol. 94 (1999) 52.
[20] M. Torres, R. ColaÂs, Modelling of Metal Rolling Processes, Institute
of Materials, London, 1993, p. 629.
[21] R. ColaÂs, in: J.J. Jonas, T.R. Bieler, K.J. Bowman (Eds.), Advances in
Hot Deformation Textures and Microstructures, TMS-AIME, War-
rendale, 1994, p. 63.
[22] W. Jaenicke, S. Leistikow, A. StaÈdler, J. Electrochem. Soc. 111
(1964) 1031.
[23] A.G. Crouch, J. Am. Ceram. Soc. 55 (1972) 558.
[24] T.E. Mitchell, D.A. Voss, E.P. Buther, J. Mater. Sci. 17 (1982) 1825.
M. Torres, R. ColaÂs / Journal of Materials Processing Technology 105 (2000) 258±263 263