An analytical model for the prediction of the transfer bar temperature in a roughing mill

Abstract

An online model is presented for the prediction of temperature distributions in the bar in the roughing mill of a hot strip mill. The model consists of an analytic model for the prediction of temperature distributions in the interstand zone, and a semianalytic model for the prediction of temperature distributions in the bite zone. The prediction accuracy of the model is examined through comparison with predictions from a finite element model.

List of symbols

f _s

forward slip,

h _air

heat transfer coefficient due to air cooling, at the bar surface

h _lub

heat transfer coefficient at the roll/bar interface

h _rad

heat transfer coefficient due to radiation, at the bar surface

h _rw

heat transfer coefficient due to water cooling, at the roll surface

h _wat

heat transfer coefficient due to water cooling, at the bar surface

H ₁

bar inlet thickness

H ₂

bar outlet thickness

thermal conductivity

l _d

contact length at the roll/bar interface

bar length in the interstand zone

reduction ratio

roll radius

shape factor

T _air

air temperature

T _R

roll temperature at the roll/bar interface

T _s

bar temperature at the roll/bar interface

T _wat

water temperature

T ₁

bar inlet temperature

T ₂

bar exit temperature

u ⁿ

normal component of the velocity vector

u _x, u _y

x, y component of the velocity vector

V _it

bar velocity in the interstand zone

V _R

roll tangential velocity

V _s

bar velocity at the roll/bar interface

V ₁

bar inlet velocity at the bite zone

V ₂

bar outlet velocity at the bite zone

Γ_c

roll/bar interface

emissivity of bar surface

effective strain

effective strain rate

coefficient of Coulomb friction

penalty constant

ρc _p

heat capacity

Stefan–Boltzmann constant

flow stress

σ _n

normal stress

σ _t

tangential stress

σ ₁

back tension, in a stress unit

σ ₂

front tension, in a stress unit

roll angular velocity

plastic deformation zone(bite zone) in the bar

Introduction

In hot strip mills and plate mills, the bar, while moving along the processing line, is subject to complex cooling and reheating, including heat loss due to air cooling and contact heat transfer, water cooling by either the descaling water or the spray water, heat generation due to plastic deformation and friction, and, sometimes, heat supply form the reheating device such as a bar heater. Since product quality – product dimensions, metallurgical and mechanical properties of the strip, and production economy – production speed, roll life, power consumption – are all critically influenced by the bar temperature, sound process control in general requires a precision model for the prediction and control of the bar temperature.

Several types of model have been proposed which may be applicable to the prediction of the bar temperature. Laasraoui and Jonas,1 based on an explicit finite difference model, calculated the temperature distribution across the thickness of the strip during hot strip rolling. Chen et al.,2 based on an integrated finite difference model and an explicit dynamic relaxation model, analysed the plastic deformation and heat transfer occurring in the strip. Devadas and Samarasekera.3 and Tseng,4 based on an Eulerian finite difference model, analysed the steady state thermal behaviour of the strip and that of the work roll. Yamada et al. 5 and Hwang et al.,6 based on an finite element (FE) model, rigorously treated the interdependence by conducting a coupled analysis of the thermomechanical behaviour of the strip and the thermal behaviour of the work roll. Recently, an analytic model7 was proposed for the prediction of temperature distribution across the thickness of the strip in the finishing mill of a hot strip mill.

In this paper, we present an analytic model for the prediction of the temperature distributions in the bar moving on the interstand zone of the roughing mill of a hot strip mill. Also presented is a semianalytic model for the prediction of temperature distributions in the bite zone. Then, it is demonstrated that the two models may be combined to produce an online model for the prediction and control of the bar temperatures while being processed in the roughing mill. The prediction accuracy of the proposed model is examined though comparison with the predictions from a FE process model.

FE model

The FE model used for the examination of the prediction accuracy of an online model consists of four basic FE models: a model for the analysis of steady state thermo–viscoplastic deformation of the strip (Model A), a model for analysis of steady state heat transfer in the strip (Model B), a model for the analysis of steady state heat transfer in the work roll (Model C) and a model for analysis of non‐steady state heat transfer in the strip (Model D). As shown in Fig. 1, interaction between the thermal behaviour of the work roll and that of strip caused by roll–strip contact, as well as interaction between the thermal and mechanical behaviour of the strip, are taken into account by iterative solution schemes. Details regarding the process model and its solution accuracy are given elsewhere.8 The predicted temperature distribution in the strip as well as in the roll is illustrated in Fig. 2. Clearly seen is the effect of heat transfer from the strip to roll at the roll/strip interface, as well as the effect of heat generation in the strip due to plastic deformation.

Figure 1.

Integrated FE model for analysis of thermomechanical behaviour of strip

Figure 2.

a temperature distributions in roll near bite zone, b temperature distributions in bar at bite zone, predicted from FE process simulation. Process conditions: carbon per cent of the bar material = 0·155; T ₁ = 1100°C; ω = 2·718 rad s⁻¹; R = 650 mm; H ₁ = 211 mm; H ₂ = 179 mm

Analytic model for prediction of bar temperature in interstand zone

On emerging from the reheating furnace, the bar is subject to high pressure water jet for descaling, losing a substantial amount of heat. After descaling, the bar again starts to lose heat by air cooling, before and after passing through the roll gap. In general, the thermal history of a cross‐section of the bar is different from that of another cross‐section away from it, and consequently, the temperature distributions in the bar are not only time dependent but also three‐dimensional.

Neglecting the variation of the temperature in the transverse direction, the present problem may be transformed into an initial boundary value problem, as follows:

heat equation (1) where k is the thermal conductivity (W mm⁻¹ °C⁻¹) and ρc _p is the heat capacity (J mm⁻³ °C⁻¹):

(2) (3) Define

Then, equation (3) becomes (4) h _rad (5) (6) initial condition (7) It may be shown that the solution of the above initial boundary value problem is given by (8) where, T _w is ambient temperature, h _HTC is heat transfer coefficient at the bar surface, h is a bar thickness, k is the thermal conductivity of a bar, ρc _p is the heat capacity per unit volume of a bar, and λ _n are obtained by solving (9) An iterative solution process is necessary, since h _rad is not known. Iteration may start with h _rad calculated from the initial temperature of the bar. The temperature distributions predicted from the analytic model are in excellent agreement with the prediction from the FE model, as shown in Fig. 3, illustrating the validity of the analytic model.

Figure 3.

Temperature distributions in bar in interstand zone: T _air = 20°C, h _air = 6·96×10⁻⁶ W mm⁻² °C⁻¹

Approximate model for prediction of heat flux and heat generation in bite zone

The bar temperature varies in the bite zone, mainly due to contact heat transfer and heat generation induced by plastic deformation. Let P _d, P _f and P _r (10) (11) (12) An approximate model which is used for the prediction of P _d, P _f and P _r in the present investigation is given elsewhere.8

The average heat generation occurring in a particle flowing though the bite zone may approximately be given by (13) where y′ denotes the normalised distance from the centreline, with y′ = 1 representing the surface, and the integration is performed along the streamline y′ = constant.

By examining the actual distribution of predicted from the FE simulation, which is illustrated in Figs. 4 and 5, it may be deduced that the distribution may be approximated by (14) where, (15)

Figure 4.

Average heat generation in bite zone of R1 stand, pass 1 (Process conditions are shown in Table 1)

Figure 5.

Average heat generation in bite zone of R2 stand, pass 3 (Process conditions are shown in Table 1)

Table 1.

FE process conditions

Variables	R1 pass 1	R1 pass 2	R1 pass 2	R2 pass 1	R2 pass 2	R2 pass 3
H _{1, mm}	250	214	177	140	101	67
H _{2, mm}	214	177	140	101	67	43
V _{R, mm s} ⁻¹	1783·33	1916·67	1766·67	2566·67	2816·67	3150
R, mm	650	650	650	650	650	650
h _air, W mm⁻² °C⁻¹	0·000001–0·00001
h _spray, W mm⁻² °C⁻¹	0·001–0·005
_{, kN mm} ⁻²	0·1375% carbon steel
T _w, °C	20
h _lub, W mm⁻² °C⁻¹	0·1
h− _rw, W mm⁻² °C⁻¹	0·011667
k _r, W mm⁻¹ °C⁻¹	0·027
ρc _pr, J mm⁻³ °C⁻¹	0·004248
k, W mm⁻¹ °C⁻¹	0·03
ρc _p, J mm⁻³ °C⁻¹	0·00688
T _slab, °C	1179, uniform bar temperature

Integration of equation (14) (16) where, (17) (18) It follows that (19) Recently, Kim et al. 9 derived a theory that relates any parameter describing the thermomechanical behaviour of the strip with the process parameters. According to the theory, the parameter A may in general be expressed by a function of eight dimensionless variables, or (20) where C ₁ and C ₂ are constants, and P′^* denotes the deformation energy of the ideal, homogenous deformation in the roll gap.

It is found from a series of FE process simulation that the effect of ω and are negligible. Assuming that the coefficient of friction μ = 0·3, equation (20) may be reduced to a multivariable polynomial form (21) The coefficients may be found from the least square regression of the data predicted from a series of the FE simulation.

Selecting α = 5, the distribution of the thus predicted are in good agreement with the prediction from the FE simulation, as shown in Figs. 4 and 5.

Semianalytical model for prediction of bar temperature in bite zone

Note that heat transfer then average heat flowrate at the roll and bar interface s calculated from (22) The initial boundary value problem associate with heat transfer in the bite zone may be given by heat equation (23) boundary conditions (24) (25) where, as shown in Fig. 6, h is approximated by (26) initial condition: (27) where, T ₁(y) denotes inlet temperature distribution.

Figure 6.

Approximation of bite zone and inlet temperature distributions

We may solve the problem by using the method of eigenfunction expansion. The solution procedure is described in detail in the Appendix.

The result may be summarised as follows (28) where, φ _n(y) = cos (λ _n y), and λ _n = nπ/h

The following procedure may then be taken to predict the temperature distribution in the bite zone:

calculate P _f and P _r from the mathematical expressions derived by Lee et al. 8

ii.

calculate q _s from equation (22)

iii.

calculate P _d from the mathematical expressions derived by Lee et al. 8 and A from equation (21)

iv.

calculate heat generation, from equation (19)

calculate the temperature distribution, from equation (28).

Verification of model

A roughing mill consists of several mill stands, as shown in Fig. 7a . Because the length of the processing line as well as the bar length is extremely large, the FE simulation considering the bar as an analysis domain is impractical in the light of the computational efficiency. An alternative choice would be to divide the roughing mill into several subzones. As shown in Fig. 7b , each zone may be classified into one of the following four types: the first zone, which represents a region located in front of the first mill stand, the last zone, which represents a region located between the last mill stand and somewhere downstream, near the finishing mill for example, an interstand zone, and a bite zone. The FE simulation may be preformed for each zone in sequence, starting from first zone and employing the temperatures predicted at the current zone as the inlet boundary conditions for the next zone.

Figure 7.

a outline of roughing mill and b definition of subzones

The predictions are in excellent agreement with FE predictions, as partially illustrated in Figs. 8 and 9, and also with measurements, as shown in Fig. 10, indicating that the assumptions made for the derivation of the interstand zone model as well as for the derivation of the bite zone model are valid. It is noted that the average strip temperature may possibly be seriously underestimated if the measurement of the surface temperature is made near the roll exit. It is also noted that at the roughing mill draft temperature (RDT) zone from which finish rolling starts, the temperature difference between the surface and the centre of the bar may become as high as 150°C, as shown in Fig. 11, indicating that, if it is assumed that the bar temperature is uniform and equal to its surface temperature, the strip temperature may possibly be underestimated at the end of finish rolling.

Figure 8.

Temperature distributions at inlet of the bite zone of R1 stand pass 1 and at outlet of bite zone of R1 stand, pass 1 (Process conditions are shown in Table 1)

Figure 9.

Temperature distributions at inlet of bite zone of R2 stand, pass 3 and at outlet of bite zone of R2 stand, pass 3 (Process conditions are shown in Table 1)

Figure 10.

RDT temperatures, measurement and prediction

Figure 11.

Temperature distributions in bar at RDT zone

Conclusions

The model presented in this paper may serve as an effective tool for the precise control of production speed and pressure of water sprayed in the interstand zone, to achieve the desired RDT. The model may also serve as a sound basis for exactly deducing the temperature distributions and the average temperature of the bar from the measurement of the surface temperature made in the production line, which is vital for enhancing the predicting accuracy of the models for predicting the roll force and roll power, as well as those for predicting the metallurgical behaviour of the product.

Footnotes

Appendix

References

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