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: rcolas@ccr.dsi.uanl.mx (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.

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Fig. 8. Temperature pro®le within the oxide layer of the out-going stock

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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

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M. Torres, R. ColaÂs / Journal of Materials Processing Technology 105 (2000) 258±263 263