Journal of Materials Processing Technology 153–154 (2004) 894–899

Computer simulation of the thermal behaviour of a workroll during hot rolling of steel strip

A. Péreza,∗, R.L. Corrala, R. Fuentesb, R. Colásca Centro de Investigación en Materiales Avanzados S.C., Div. De igenieria y procesos de Manufactura,

Miguel de Cervantes # 120, Complejo Industrial Chihuahua, Chihuahua, Mexicob CIDIM, S.A. de C.V., Monterrey Nuevo León, Mexico

c Facultad de Ingenier´ıa Mecánica y Eléctrica, Universidad Autónoma de Nuevo León, San Nicolás de los Garza, NL, Mexico

Abstract

The basis of a mathematical model able to predict the thermal response of a work roll in a continuous hot strip mill, under typical workingconditions, is presented in this work. The model considers that the steady state is reached from dynamic conditions in such parameters asthe length of the strip, the temperature variation along the length of the strip, and the temperature variation from the beginning to the endof the process. The phenomena taking place are considered at three different levels: (i) the independent cycle of the roll, (ii) rolling of astrip-rest, and (iii) a whole campaign. The three levels and their effects are then superimposed.© 2004 Elsevier B.V. All rights reserved.

Keywords:Simulation; Hot rolling steel; Heat transfer; Thermal profile

1. Introduction

In industrial production, clear and exact information aboutthe process parameters is needed in order to manipulate andcontrol product quality requirements. The development ofadvanced computation systems and a better understandingof engineering processes have led to the development ofpractical simulators tailored to each industrial process withthe major goal of reproducing the whole or some part ofthe process at every stage under given circumstances, notonly with respect to product quality but also with respect toprice. This is achieved by means of a prediction algorithmwith which it is possible to isolate the effects of operationalparameters, separating variables that act in a direct formfrom the casual ones, optimizing the process and facilitatingthe simulation of new operational conditions (mechanical,thermal, electric properties, etc.).

In the case of the hot rolling process, Hollander[1] wasone of the first to study the process in these lines. To date,most works have been focused on the development of newlamination process models which attempt to obtain a betterfit to reality.

∗ Corresponding author. Tel.:+52-614-439-1101;fax: +52-614-439-1158.E-mail addresses:antonino.perez@cimav.edu.mx (A. Perez),leticia.corral@cimav.edu.mx (R.L. Corral).

In this work a model has been developed with the follow-ing features:

(i) It is based on a mathematical solution using Fourierseries with normalization of temperature results to pro-vide simulation flexibility. With it, a robust databasewas generated that enabled the reproduction of a wholecampaign.

(ii) It has a space-temporal (r, θ(t); t) formulation that al-lows the calculation of the complex temperature profileof the work roll.

(iii) It allows a temperature variation along the rolling pro-cess of a strip-rest, as well as interaction with differenttemperature profiles between one piece and another,just as it occurs in real operation.

(iv) It is convenient to perform sensitivity analysis bymanipulation of operation parameters (such as dispo-sition of cooling headers, velocity and percentages ofdeformation, variation of temperature on each stripto deform, etc.), and physical properties of the workroll.

(v) It is unique because it is easy to understand, and doesnot require complex mathematical techniques.

Introduction of this work is important since it is the firstto include all of the above issues, in contrast to other workstypically found elsewhere[2–14].

0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.matprotec.2004.04.103

A. Perez et al. / Journal of Materials Processing Technology 153–154 (2004) 894–899 895

Nomenclature

An coefficient in the Fourier seriesA0 average temperature in the

work roll and coefficient of the Fourierseries

Bn coefficient in the Fourier seriesCn coefficient in the Fourier seriesC0 coefficient in the Fourier seriesCpb specific heat of the back-up roll

(J kg−1 K−1)Cpr specific heat of the work roll (J kg−1 K−1)Dn coefficient in the Fourier seriesf, f(�) heat flow (Wm−2)h, h(θ) normalized heat transfer coefficient in

function of the boundary conditions(Wm−2 K−1)

kb thermal conductivity of the back-up roll(Wm−1 K−1)

kr thermal conductivity of the work roll(Wm−1 K−1)

Pn coefficient in the Fourier seriesP0 coefficient in the Fourier seriesQn coefficient in the Fourier seriesr radial coordinate (m)t time or temporary coordinate in the

interval (0, tr] (s)tr rotation period of the work roll (s)Ta ambient temperature (K)T2π temperature in the surface of the work roll

before carrying out the contact withthe strip (K)

T 0s temperature in the surface of the strip before

the contact with the work roll (K)T ∗

s temperature in the zone of contact ofthe work roll with the strip (K)

wr angular velocity of the work roll (rad s−1)

Greek letters�r thermal diffusivity of the work roll (m2 s−1)βs constant that depends on the physical

properties of the work roll and stripθt angular position in function of the time

t in (0, tr]λ average temperature in the roll and

coefficient in the Fourier series (K)λ, λ(θ) profile of normalized temperature of the

work roll (K)ρb density of the back-up roll (kg m−3)ρr density of the work roll (kg m−3)φ, φ(θ) temperature of the means with which the roll

carries out the thermal exchange (K)

Fig. 1. Four-high stand.

2. The lamination process in hot

The hot lamination train is constituted by six unidirec-tional mills of type four, seeFig. 1, where work rolls arehoused in. The geometric parameters of the work roll andits physical properties are shown inTables 1 and 2.

The hot rolling of steel strip uses work rolls at high tem-peratures in contact with the strip being deformed. Mostof the deformation is carried out in hot, taking advantageof smaller mechanical resistance and higher ductility of thematerial. The temperature of the work rolls is increased aslong as lamination takes place. Thus, a cooling system isrequired to avoid excessive heating. Cooling is critical tocontrol wear and dilatation of the work rolls.

Hence, the boundary conditions for heat transfer are es-tablished by:

(a) Heat loss by: (i) the cooling system[15] (type of spraynozzle, pressure of the fluid, distance between the noz-zle and the surface of the work roll, and the angle of in-cidence), (ii) convection to the surrounding media, and(iii) conduction to the back-up roll.

(b) Heat generation by conduction from strip in roll bite.

Table 1Characteristic of the work rolls

Characteristic M1–M3 M4 M5–M6

Diameter (mm)Maximum 790+2/−0 502+2/−0 502+0/−2Minimum 725 450 450

Material: cast ironExternal layer Iron high

chromiumIron highchromium

Indefinitechill

Nucleus Nodulariron

Nodulariron

Nodulariron

Net weight (kg) 9780 4000 4000

896 A. Perez et al. / Journal of Materials Processing Technology 153–154 (2004) 894–899

Table 2Physical properties of the work rolls

Physical property Nucleus External layer

Nodular present in any ofthe rolls and position thatit occupies in the mill

Iron high chromium, Hi-Criron, mills 1, 2 and 3a,b

Indefinite chill,mills 4, 5 and 6a

Young’s modulus,E (MPa)Minimum 1.70E+05 2.25E+05 1.68E+05Maximum 1.80E+05 2.26E+05 1.82E+05

Thermal conductivity,K (W m−1 K)

24.2 13.8 18.46

Specific heat,Cp

(J Kg−1 K)520 492 470

Thermal diffusivity,κ(m2 s−1)

6.62E−06 3.70E−06 5.31E−06

Coefficient of thermal expansion,α, K−1

Minimum 1.10E−06 1.00E−05 7.00E−06Maximum 1.20E−06 1.10E−05 9.00E−06

Density,ρ (kg m−3) 7.03E+03 7.53E+03 7.40E+03Reason of Poisson,ε,

dimensionless0.3 0.27

a Heart of nodular or grey iron.b The microstructure of the Hi-Cr iron is M7C3, Cr7C3.

The model also includes other parameters affectingboundary conditions: (i) work rolls velocities; (ii) tem-perature, length, and oxide layer[16,17] in the strip; (iii)physical properties of roll and strip; (iv) lamination on/offtime cycles.

3. Description of the model

The mathematical model involves a variable temperaturecondition and solves the problem by normalizing the tem-perature profile. Normalization is performed to obtain theprofile with all the changes on the surface of the rolling of astrip-rest or with the adjustment average of its behaviour toa ramp of temperature. The advantage of this model is thatit considers that the variation of temperature in the rollingof a strip deforms the strip just as it happens in real opera-tion. Our proposal is to calculate the complex temperatureprofile that takes place in a work roll during hot rolling ofsteel. Similarly, this work aims to a solution for the work rollassumed as a solid semi-infinite system subjected to a pe-riodic ambient temperature (periodic phenomenon). Hence,the work roll was divided for its study into eight angularsectors, as seen inFig. 2. These eight regions of heat trans-fer are generated in: (i) the deformation area; (ii) four freeconvection zones located to the entrance and exit of eachjet area; (iii) and (v) two jet areas generated by the coolingheaders to the exit and entry of the mill, respectively; (iv)the contact of the rolls.

The heat transfer model of the work roll has been designedto simulate the real phenomenon in three degrees of difficultythat will be denominated: (i) the independent cycle of theroll or cycles; (ii) rolling of a strip or periodic evolution of

temperature; (iii) a whole campaign, showing the thermalhistory of the work roll through a periodic behaviour. Thesolution to the problem is obtained in analytic form andthe expressions are described as functions of the Fouriercoefficients, carrying out the description of the heat flow,calculating the profile of temperature later. The premises ofthe model are described as follows[18]: (a) the heat transferphenomenon occurring in the work roll is periodic and (b)the heat flow is proportional to the temperature differences.Based on such premises, we obtain the thermal profile forthe work roll:

λ = Ta + T ∗s A0 + T ∗

s

50∑n=1

e−a√

n

√n

[An cos(nθt − a√

n)

+Bn sin(nθt − a√

n)] (1)

Fig. 2. Angular sectors into which the work roll was divided.

A. Perez et al. / Journal of Materials Processing Technology 153–154 (2004) 894–899 897

where:

a =√( w

2α

)rr, wr = 2π

tr, θt = 2πt

tr,

T ∗s = T 0

s − Ta + (βs − 1)(T2π − Ta)

βs,

βs = 1 +√

(ρcpk)r

(ρcpk)b

Eq. (1)gives the solution for the case of the prescribed pe-riodic source of heat, in whichAn and Bn should be de-termined, but the temperature function should complete theboundary condition of heat flow on the surface of the workroll, which is described by the relation:

f = −kr∂λ

∂r(2)

For that reason, the Fourier coefficients proposed for thedescription of the temperature profile and the heat flow aresimilar. The heat flow is proportional to the temperaturedifferences through:

f = h [φ − λ] (3)

Since, the phenomenon is periodic; it is possible to representdifferent means of the work roll in Fourier series.

f(θ) = h(θ)

⌊φ(θ) − λ(θ)

⌋(4)

f(θ) =50∑

n=1

[An cos(nθt) + Bn sin(nθt)] (5)

h(θ) = P0 +50∑

n=1

[Pn cos(nθt) + Qn sin(nθt)] (6)

h(θ)φ(θ) = C0 +50∑

n=1

[Cn cos(nθt) + Dn sin(nθt)] (7)

λ(θ) = λ +50∑

n=1

[An − Bn√

2ncos(nθt) + An + Bn√

2nsin(nθt)

]

(8)

where each one of the expressions is a function that describesthe heat flow (f(θ)), heat transfer coefficients on the surfaceof the work roll (h(θ)), and different means of temperaturewith which the work roll carries out the thermal exchange(φ(θ)); h(θ)φ(θ), describing the product of the heat transfercoefficients and its corresponding temperature of each zone.The functionλ(θ) describes the temperature profile of thework roll.

The Fourier’s coefficientsC0, Cn, Dn, P0, Pn, and Qn

of the above equations are determined through temperaturebehaviour of different means; the heat transfer coefficientsare similarly determined. TheAn andBn are Fourier coeffi-cients that describe the temperature,λ(θ), and the heat flow,

f(θ), which are evaluated from the previously presented se-ries (simultaneous equations). Relationships on each one ofthe three previous stated points (the independent cycle of theroll, rolling of a strip-rest, and a whole campaign) establisha Fourier series that describes the normalized temperature ofthe work roll, subjected to the cooling conditions character-ized by the spray nozzles (type, disposition, and operationconditions).

Of the three Fourier series obtained the series that de-scribes a whole campaign is the only one that conserves thetermA0 or λ (Eq. (1)). This series is the carrier on which wewill superimpose the other two Fourier series necessary forthe temperature calculation. Providing that we want to knowthe behaviour of the temperature during the whole campaignonly the series carrier is used, but if we want to know therolling of a strip-rest effect, use of the series carrier and theseries that describes the periodic evolution of the tempera-ture is required and they are superimposed. Finally, if wewant the calculation from the temperature to a certain detail,cycle by cycle, superimposition of the three series becomesnecessary.

4. Results

Results of the work roll (continuous mill, M1), of the ther-mal profile for one and more independent cycles of the roll,lamination of the strip and the wait for the following strip,and a whole campaign, as well as the periodic attenuation ofthe temperature are described next. In this work we presentthe results obtained during a whole campaign where therewere two shutdowns, one for a delay due to the change ofwork rolls in the mill M3, with a duration of approximately28 min, and a second shutdown programmed for the routinechange of the work rolls in the last three mills (M4 at M6),with a duration of 20 min approximately[18].

The thermal responses of a work roll in a continuous hotstrip mill, under typical working conditions, are presented inthree parts: (a) the independent cycle of the roll, simulationof one and several turns: observation of each cycle of profileof temperature in the work roll during the deformation ofthe piece; (b) rolling of a strip-rest, simulation of one andseveral strips to deform: temperature in the roll as meaneffect of the multiple thermal changes in the work roll duringthe rolling of a strip and their period of cooling during oneand other strip to deform; (c) a whole campaign, simulationof thermal behaviour of the work roll during each one of theseries of lamination of the campaign and at the end of this.

In Fig. 3we can see the thermal profile from the work rollto different penetrations for (a) one cycle and (b) strip-rest.In Fig. 4, it is shown the mean temperature by effect: (a)strip-rest train and (b) a whole campaign.

The attenuation effect of temperature to the interior of thework roll is observed inFig. 5. The phenomenon shows anegative exponential behaviour, to this respect the simula-tor allows to see as the temperature profile descends as a

898 A. Perez et al. / Journal of Materials Processing Technology 153–154 (2004) 894–899

Fig. 3. Thermal profile to different penetrations: (a) one cycle and (b) a rolling of a strip-rest.

Fig. 4. The mean temperature of the work roll for effect: (a) strip-rest train and (b) a whole campaign.

Fig. 5. Decrease in temperature into the surface of the work roll: (a) exponential negative, effect cycle and (b) exponential negative and harmonicaattenuation.

A. Perez et al. / Journal of Materials Processing Technology 153–154 (2004) 894–899 899

function of the penetration.Fig. 5 shows this informationcoinciding with the expected results and allowing us to ob-serve as the temperature decays from 539◦C in the surfaceto 90◦C in less than 3 mm. This situation coincides withthe results obtained by Patula[2], Parker and Baker[4], andPérez[5] which indicate the effect of the penetration of thethermal shock (abrupt changes of temperature).

5. Discussion

In an industrial plant it is generally observed that a workroll elevates its temperature by the end of the whole cam-paign, showing in general energy income. Our model isable to predict this effect; under typical operation condi-tions from the independent cycle to a whole campaign (seeFigs. 3 and 4).

The basic form of the curve of thermal response for awork roll is repeated “periodically”,Fig. 3(b), associatedto the angular sectors present or boundary conditions (seeFig. 2). The most important feature to observe is the veryrapid decrease in temperature on the surface, as it has beenreported elsewhere[2–7,18]. Only the skin of the roll issubjected to very high thermal gradients (thermal shock).

It is observed in a point close to the area of deforma-tion (Fig. 5) the decrease in temperature into the surface: (a)negative exponential effect and (b) negative exponential andharmonic effect. The first behaviour has been documented(assuming a cyclic phenomenon of the roll). However, thesuperposition exponential-harmonic effect has been only re-ported by Pérez[18]. These last effects have assumed thestrip-rest phenomenon and whole campaign.

6. Conclusions

The thermal steady state in a work roll during hot rollingof steel strip does not exist, although technically it is reachedwhen the changes of temperature and their effects are notsignificant.

The phenomenon of heat transfer is a harmonic process,the thermal response of a work roll that takes place is con-sidered at three different levels: (a) the skin effect: it isthe result of the changes of temperature in the work roll,it reaches their steady state after 10 cycles; (b) rolling of astrip-rest effect: it is the reflection of the thermal changesduring the deformation and the time of wait of the followingpiece to deform, their steady state is reached when effec-tive time of laminating oscillates between 800 and 1000 s(without delays and in normal operation); (c) whole cam-paign effect: it presents the evolution of temperature dur-ing the series and lamination shutdown, their steady stateis reached when effective time of laminating is greater than1300 s.

The attenuation in the temperature to the interior of thework roll shows an exponential negative and harmonic effect,

these determine the stratified domains of the thermal effectsin the work roll.

Acknowledgements

The authors acknowledge the support provided by CONA-CYT, and the facilities given by HYLSA, S.A. de C.V., dur-ing this work.

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