K_ZW_2013_2.pdf

at SciVerse ScienceDirect

Applied Thermal Engineering 58 (2013) 241e251

Contents lists available

Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

Three-dimensional temperature distributions of strip in continuousannealing line

Zong-Wei Kang, Tei-Chen Chen*

Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC

h i g h l i g h t s

3-D temperature distributions of strip in CAL were calculated by two methods. Crown of rolls has a significant influence on the transverse temperature distribution of strip. Phase transformations have a significant influence on the longitudinal temperature distribution of strip. 3-D temperature distributions of strip in CAL can be used to predict the residual stress and warpage of strip.

a r t i c l e i n f o

Article history:Received 4 December 2012Accepted 31 March 2013Available online 17 April 2013

Keywords:Continuous annealing line (CAL)Finite element method (FEM)Energy balance method (EBM)Thermal contact resistance

* Corresponding author. Department of MechanicalKung University, No. 1 University Road, Tainan 70102757575; fax: 886 6 2352973.

E-mail address: [email protected] (T.-C. C

1359-4311/$ e see front matter 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.applthermaleng.2013.03.06

a b s t r a c t

In this study, the three-dimensional (3-D) temperature distributions of strip in the whole continuousannealing line (CAL) were evaluated by using the techniques of energy balance method (EBM) and finiteelement method (FEM). The results show that both the effects of ferriteeaustenite phase transition of thesteel strip and the thermal contact resistance between the strip and taper rolls have very significantinfluence upon the distributions of temperature. These taper rolls tend to introduce the non-uniformdistributions of the temperature and plastic deformation along both the width and thickness of thestrip which are closely related to the phenomenon of warping during punching process. Although thecomputational time by EBM is very short compared to that by FEM, the results evaluated by these twomethods are well consistent.

2013 Elsevier Ltd. All rights reserved.

1. Introduction

The continuous annealing line (CAL) is characterized by fasterdelivery and higher thermal efficiency than conventional batchannealing and can provide a sound heat treatment on the stripmaterials with higher quality. Sound control of the temperaturealong the continuous annealing furnace is important to guaranteethe physical properties of the final product and to save energy ofoperation. In the CAL, the crowns with various profiles are given tohearth rolls for the purpose of preventing strip snaking. However,these crowns easily introduce non-uniform distributions of tem-perature along thewidth whichmay result in the local plastic straindue to both the thermal and mechanical loadings. These localplastic strains of the strip material will then be accumulated andfinally may lead to the residual stress and warpage of the strip

Engineering, National Cheng1, Taiwan, ROC. Tel.: 886 6

hen).

All rights reserved.2

during punching process. Since the direct contact temperaturemeasurement is difficult to perform due to the high speed of thestrip and possible damage to the steel. On the other hand, indirectmeasurement as pyrometers is also imprecise in consideration ofcomplicated radiation interaction between several surfaces in thefurnace. It would be helpful to establish a physical model to observethe temperature history of strip and to gain knowledge of thedetailed mechanisms of heat transfer between each component.

Prieto et al. [1] developed a powerful stepwise thermal model toestimate the 1-D temperature history in CAL without taking ther-mal contact between roll and strip into account. Ho and Chen [2,3]extended Prietos model to evaluate the 2-D temperature distri-butions of strip in preheating section (PHS) along the longitudinaland transverse directions by taking the thermal contact resistancebetween the strip and the rolls as well as view factor of radiationinto account. In the CAL, the rolls with crown are necessary to avoidthe strip snaking [4]. A schematic diagram of roll is shown in Fig. 1.However, the crown generates not only the non-uniform distribu-tion of tensile stress in the transverse direction but also the non-contact between the taper roll and outside of strip which lead to
mailto:[email protected]://crossmark.dyndns.org/dialog/?doi=10.1016/j.applthermaleng.2013.03.062&domain=pdfwww.sciencedirect.com/science/journal/13594311http://www.elsevier.com/locate/apthermenghttp://dx.doi.org/10.1016/j.applthermaleng.2013.03.062http://dx.doi.org/10.1016/j.applthermaleng.2013.03.062http://dx.doi.org/10.1016/j.applthermaleng.2013.03.062

Nomenclature

A surface area, m2

cp specific heat at constant pressure, J/kg-Ke strip thickness, mE Youngs modulus, GPaEr least square errorF view factorFi traction, N/m2

G Gebhart factorsh heat transfer coefficient, W/(m2-K)heq equivalent heat transfer coefficient, W/(m2-K)H strip thickness, mLS side length of strip, mn number of surfaces in enclosureNu Nusselt numberPr Prandtl numberq heat flux, W/m2_Q heat transfer rate, WqC conductive heat flux, W/m2

qR convection/radiation heat flux, W/m2

r radial distance from the center, mR radiative exchange factors for application of Gebhart

methodRa Rayleigh numberRc thermal contact resistance, (m2-K)/WRe Reynolds numberSF surface, m2

t time, sT temperature, KTN atmosphere temperature in furnace, Ku strip speed, m/sui displacement vector, mU displacement, mV control volume, m3

w strip width, mX,Y,Z; x,y,z Cartesian coordinatesa phase of ferriteath thermal expansion coefficient, 1/KaR thermal diffusivity of roll, m2/sg phase of austenite in steel material emissivityij strain tensorq coordinate in circumferential directionq0 angular width of the periphery in contact with strip

l thermal conductivity, W/(m-K)m dynamic viscosity, Pa-sn Poissons ratior density, kg/m3

s StefaneBoltzmann constant, W/(m2-K4)sij stress tensor, Pasn contact pressure, MPasy yield strength, Pau angular speed of roll, rad/sd Kronecker deltaDt time interval, s

Subscriptsce ceilingcond conduction mechanismconv convection mechanismen enclosuresfl floorhp heating platehp-l to indicate heat transfer rate from heating plate to

enclosure placed on lefthp-r to indicate heat transfer rate from heating plate to

enclosure placed on righti,j generic surface indexin refers to enclosure inlet conditionsloss losses through wallsout refers to enclosure outlet conditionsrad radiation mechanismR rollss strips-a to indicate heat transfer rate from strip to enclosure

located aboves-b to indicate heat transfer rate from strip to enclosure

located belows-l to indicate heat transfer rate from strip to enclosure

located on lefts-r to indicate heat transfer rate from strip to enclosure

located on rightsw side wallsS-H strip in horizontal positionS-V strip in vertical positionw wallsws inner surfaces of wallsNs thermal conditions in surroundings of furnaceNen thermal conditions in enclosure atmosphere of furnace

Fig. 1. Schematic diagram of hearth roll and strip.

Z.-W. Kang, T.-C. Chen / Applied Thermal Engineering 58 (2013) 241e251242

the non-uniform temperature distributions of the strip along thetransverse direction. In addition, when the tension of strip isdecreased, the snaking of strip may occur easier. Therefore, thetension of the strip should keep at a high level, and this situationmay induce the heat bucking in the transverse direction.

As reported in the previous study [3], the strip in PHS is stilldeformed within the elastic range. However, as the strip temper-ature is significantly increased in heating section (HS), the plasticdeformation will take place due to the decrease of yield strength ofstrip at high temperature. Based on the FeeC phase diagram, theferriteeaustenite phase transition occurs near 727 OC, where thecrystal structures of the phases of ferrite a and austenite g are BCCand FCC, respectively. The volume change due to this phase tran-sition can be considered by modifying the value of thermalexpansion coefficient, while the effect of latent heat accompaniedwith can be accommodated by the curve of specific heat. The

Fig. 2. Simplified schematic diagram of CAL.

Z.-W. Kang, T.-C. Chen / Applied Thermal Engineering 58 (2013) 241e251 243

phenomena of phase transformations involving austenite, ferrite,pearlite, bainite, and martensite are very important in the heattreatment of steel.

To the authors knowledge, there exist very few studies relevantto the 3-D temperature distribution of strip in CAL. In this study, the3-D temperature distributions of strip in the CAL, composed of PHS,HS, soaking section (SS) and cooling section (CS), were theoreticallyevaluated and discussed under some specific operational condi-tions. Both the techniques of energy balance method (EBM) andfinite element method (FEM) were utilized to deal with the thermaland mechanical models of the problems. The results werecompared and discussed. The surface temperatures of rolls and thecontact pressure between strip and roll were first evaluated byenergy model of roll and mechanical model of strip through finiteelement simulation, respectively [5]. And then the correspondingthermal contact resistance between the roll and strip was deter-mined via the contact pressure. Finally, the longitudinal andtransverse temperature distributions of strip in thewhole CALwereevaluated iteratively under different operational conditions byeither EBM or FEM.

2. Mathematical model

2.1. EBM scheme to evaluate strip temperature

A simplified scheme of components in CAL considered in thepresent study is illustrated in Fig. 2, which includes PHS, HS, SS, andCS. In EBM scheme, the individual dimensions of PHS, HS, SS, and CSwere 17.828 3 2.3 m, 22.62 11.9 2.4 m, 22.62 6.75 2.4 mand 25.02 3.8 3.2 m, and were divided into 7, 30, 13 and 7enclosures, respectively. The strip is fed into the furnace from the

Table 1Materials used for ceiling, side walls and floor [1].

Material t, mm t, mm

Ceiling and side walls PHS (ce/sw) HS (ce/sw)Rock wool 30/30 30/30Rigid rock wool 50/50 50/50Ceramic fibre (96 kg/m2) 35/35 35/35Ceramic fibre (128 kg/m2) 37/37 37/37Steel sheet 3/3 3/3

Floor PHS HSCalcium silicate 150 150Insulating fire brick JM23 115 115

left side and moves alternately upward and downward. The totalnumbers of major rolls in PHS, HS, SS, and CS are 1, 13, 7 and 1,respectively. As the steel strip is moving through the taper rolls, thethermal contact conductance between the strip and rolls should betaken into account. Indirect radiative heating tubes are utilized inthe PHS, HS, and SS, while turbulent jets of air are installed in CS toquench the strip. These heating tubes are situated between bothsides of the strip. Combustion of coke oven gas or propane takesplace inside these heating tubes. Since a great number of heatingtube rows in types of multiple U shape andW shape are arranged inthe PHS, HS, and SS of CAL, they can be satisfactorily considered asheating planes in thermal model. These heating planes supply thethermal energy to the strip, the walls and the furnace atmosphere.The atmosphere of the furnace is made up of a mixture of nitrogenand hydrogen (93%N2 and 7%H2). Thematerials used for the furnacewalls are described in Table 1, including the ceiling, the side wallsand the floor. The ceiling and the side walls are made of the su-perimposition of the first fivematerials, and the floor is made of thelast two materials. The detail dimensions in various furnaces areshown as in Table 1. The input data of computational model includethe furnace dimension, the strip dimension (1204 0.503mm), thephysical properties of the walls [1], strip velocity (3.3 m/s), stripdensity (7860 kg/m3), and the temperature at entry (298 K). Thetemperature of air is not uniform throughout the furnace. The at-mosphere temperatures at different zones experimentallymeasured by thermocouples are listed in Table 2, in which thezones divided in each section are shown in Fig. 2. The physicalproperties of the atmosphere are listed in Table 3 [2,3]. The sizes ofrolls are shown in Table 4. In addition, the view factors betweentwo any components of enclosure can be calculated by the formulain the article [6].

Fig. 3 shows a representative enclosure to illustrate the energybalance for each of the components in the enclosures, including theheating plate, the walls, the strip and the enclosure. The associatedrelations of energy balance at each following component in thefurnace can be expressed as [2]:

(1) Heating plate

Energy balance should be remained at the heating plate. In otherwords, the heat supplied by the heating plate should be equal tototal heat moving out of the heating plate via the convection andthe radiation, as shown in Fig. 3, and can be written as

_Qhp _Qconv;hpen X

j ce;sw;fl;s_Q rad;hpj (1)

where _Qhp denotes the heat supplied by the heating plate,_Qconv;hpen represents the heat flowing out of the heating plate to

the environment via the convection, whileP

j ce;sw;fl;s_Q rad;hpj

t, mm t, mm l, W/(m-K)

SS (ce/sw) CS (c/sw)96/48 96/48 2 107T2 2 105T 0.0239162/81 162/81 2 107T2 2 105T 0.0239114/57 114/57 2 107T2 2 105T 0.0132120/60 120/60 107T2 8 105T 0.00148/4 8/4 0.015T 9SS CS283 283 107T2 2 105T 0.0464217 217 2 108T2 5 105T 0.0993

Table 4Size of roll in CAL.

Size of roll (mm) A L1 L2 D1 D2 h

PHS 500 210 340 750 749 0.5HS 500 210 365 750 748 1.0SS 500 575 300 750 749 0.5CS 500 210 365 750 748 1.0

Table 2Atmosphere temperatures at different zones of CAL.

Temperature (K) PHS HS SS CS

Zone 1 470 1096 1089 578Zone 2 469 1116 e 526Zone 3 e 1126 e 518Zone 4 e 1136 e 514


shows the heat moving out of the heating plate to the ceiling, theside wall, the floor, and the strip via the radiation.

(2) Ceiling of the furnace

Energy balance is also satisfied at the ceiling of the furnace. Itmeans the total heat absorbed by the ceiling through the convec-tion and the radiation should equal to the heat loss to the sur-rounding of the furnace, as shown in Fig. 3.

_Q loss;ce _Qconv;ceen X

jhp;sw;fl;s_Q rad;cej (2)

where _Q loss;ce denotes the heat loss from the ceiling to the sur-rounding of the furnace, _Qconv;ceen represents the heat absorbedby the ceiling from the environment via the convection, whilePjhp;sw;fl;s

_Q rad;cej shows the heat absorbed by the ceiling from the

heating plate, the side wall, the floor, and the strip via the radiation.

(3) Side walls of the furnace

Energy balance is also satisfied at the side wall of the furnace. Itmeans the total heat absorbed by the side wall through convectionand radiation should equal to the heat loss to the surrounding of thefurnace, as shown in Fig. 3.

_Q loss;sw _Qconv;swen X

jhp;ce;fl;s_Q rad;swj (3)

where _Q loss;sw denotes the heat loss from the side wall to the sur-rounding of the furnace, _Qconv;swen represents the heat absorbedby the side wall from the environment via the convection, whilePjhp;ce;fl;s

_Q rad;swj denotes the heat absorbed by the side wall from

the heating plate, the ceiling, the floor, and the strip via theradiation.

(4) Floor of the furnace

Energy balance is also satisfied at the floor of the furnace. Itmeans the total heat absorbed by the floor through the convectionand the radiation should equal to the heat loss to the surrounding ofthe furnace, as shown in Fig. 3.

_Q loss;fl _Qconv;flen X

jhp;ce;sw;s_Q rad;flj (4)

Table 3Properties of atmosphere [2,3].

Property Function Range

r, kg/m 325.38/T 400 < T(K) < 1000cp, J/kg K 1.096 104T2 5.499 102T 1054.84 400 < T(K) < 1000l, W/m K 6.358 105T 1.299 102 400 < T(K) < 1000m, Pa s 2.966 108T 1.011 105 400 < T(K) < 1000

where _Q loss;fl denotes the heat loss from the floor to the sur-rounding of the furnace, _Qconv;flen represents the heat absorbed bythe floor from the environment via convection, whilePjhp;ce;sw;s

_Q rad;flj shows the heat absorbed by the floor from the

heating plate, the ceiling, the side wall, and the strip via radiation.

(5) Vertical strip

The total heat absorbed by the vertical strip should equal to theheat transferred to the left and right sides of the strip through theconvection and the radiation, as shown in Fig. 3. This net absorbedheat leads to the increaseof thestrip temperature, as shown inEq. (5).

_Qconv;sl _Q rad;sl _Qconv;sr _Q rad;sr _QSV_QSV rsewucp;SV

TSV ;out TSV ;in

(5)

where _Qconv;sl and _Qconv;sr denote the heat transferred to the leftand right sides of the strip through convection, respectively;_Q rad;sl and _Q rad;sr denote the heat transferred to the left and rightsides of the strip through radiation, respectively; _QSV representsthe heat absorbed by the vertical strip, rs, e, w, u and cp,S-V denotethe density, the thickness, the width, the speed and the specificheat of the vertical strip, respectively; TS-V,out and TS-V,in are thetemperature of the vertical strip at the outlet and the inlet,respectively.

(6) Horizontal strip

The total heat absorbed by the horizontal strip should equal tothe heat transferred to the upper and bottom sides of the stripthrough the convection, the radiation and the contact heat con-duction of the roll, as shown in Fig. 3. This net absorbed heat leadsto the increase of the strip temperature, as shown in Eq. (6), inwhich

Fig. 3. Enclosure surfaces and heat transfer rates in EBM.


_Qconv;sa _Q rad;sa _Qconv;sb _Q rad;sb _Qcond;sR _QSH

Fig. 4. Mechanical model of strip.

_QSH rsewucp;SH TSH;out TSH;in_Qcond;sR TR Ts=Rcsn

(6)

where _Qconv;sa and _Qconv;sb denote the heat transferred to theupper and bottom sides of the strip through the convection,respectively, _Q rad;sa and _Q rad;sb denote the heat transferred to theupper and bottom sides of the strip through the radiation,respectively, _QSH represents the heat absorbed by the horizontalstrip, TS-H,out and TS-H,in are the temperature of the horizontal stripat the outlet and the inlet, respectively, _Qcond;sR represents theheat transferred to the strip through the contact heat conduction ofthe roll, TR and Ts are the surface temperature of the roll and thestrip, respectively, Rc denotes the contact heat resistance betweenthe roll and the strip that is dependent upon the contact pressure,sn, between these two bodies.

(7) Furnace atmosphere

The total heat absorbed by and supplied to the surroundingcomponents through the convection should be balanced, as shownin Eq. (7):

Xjhp;ce;sw;fl;s

_Qconv;jen 0 (7)

whereP

jhp;ce;sw;fl;s_Qconv;jen denotes the total heat transfer be-

tween the surrounding and the heating plate, the ceiling, the sidewall, the floor, and the strip via the convection.

Consequently, totally seven unknown parameters in eachenclosure, including six unknown temperatures for the heatingplate, the ceiling, the side wall, the floor, and the strip in the hor-izontal and the vertical positions, as well as one unknown heatingpower for the heating plate, can be completely determined bysolving the equations from (1) to (7).

The radiative heat transfer rates, which are calculated using thetechnique of surface to surface approach, canbe expressed byEq. (8).

_Q rad;i Pnj1

Ri;jT4i T4j

Ri;j Aisidi;j Gi;j

Gi;j Fi;jj

di;j 1 j

Fi;j

(8)

where n is the number of surfaces surrounding the surface i; j is ageneric subscript for each of the surfaces that proceed heat ex-change by radiation; Ti and Tj are the temperature at the surfaces iand j, respectively; Ri,j are the exchange factors that depend on thesurface emissivities and the view factors; Fi,j, Ai is the surface area; sis the StefaneBoltzmann constant; i is the emissivity; di,j is theKronecker delta and Gi,j denotes the Gebhart factor proposed bySiegel and Howell [7]. The view factors can be referred to the workof Gross et al. [6] under the condition of rectangular surfaces on theparallel or the perpendicular planes.

The convective heat transfer rates for the surfaces are calculated by

_Qconv;i AihiTi TNen (9)

where TNen is the temperature in the enclosure atmosphere. For thecase of the convective heat transfer with respect to the furnace

surrounding, the temperature in the enclosure atmosphere TNen, asshown in Eq. (9), should be replaced by the temperature of thefurnace surrounding, TNs. The convective heat transfer coefficients,hi, are calculated using the Nusselt number, Nu. Moreover, thecontact heat resistance between the roll and the strip, Rc(sn), andthe surface temperature of each roll, TR, can be determined by themechanical model of strip and energy model of roll introduced asfollows, respectively.

2.1.1. Mechanical model of strip [3]The mechanical formulation is based on the elasto-plastic finite

element formulation that is one of the extreme principles proposedby Hill [8]. The weak form of this principle leads to the followingequation in terms of the arbitrary variation of the displacement [9]:

ZV

sijdijdV ZSF

FiduidS 0 (10)

where V is the control volume limited by the surface SF on whichthe traction Fi is prescribed, sij, ij, and ui denote the stress tensor,strain tensor, and displacement vector, respectively. A coupledthermal elasto-plastic theory is adopted in themechanical model ofthe strip. Only half roll and strip in width are established due to thegeometrical symmetry, as illustrated in Fig. 4. The strip meshedwith quadratic quadrilaterals shell element (8 nodes) is plotted as

Fig. 5. Physical properties of strip [13,14].


shown in Fig. 4(a). The side length of strip, LS, as shown in Fig. 4(b),is assumed as p times the diameter of roll for taking the heattransfer program of finite element model into account [3]. Contactelements are prescribed between the strip and the roll to ensure asuitable contact condition between them. The contact stress of strippassing through each roll can be determined by using the tem-perature of strip obtained by FEM. The roll is assumed to be rigidand fixed at its center, i.e., Ux UY UZ 0 at X Y Z 0.Moreover, the displacement of strip at the central line along the Z-direction, UZ, should be also equal to zero. The boundary conditionsand constraints are shown in Fig. 4(b). Taking a rigid displacementas load is more reasonable than a uniform tension in the actualprocess [10]. A small displacement is applied at both ends of strip tocreate the suitable tension (5000 10% N) performed actually bythe factory. The contact pressures between the strip and the roll areevaluated by the finite element method and then converted to thethermal contact resistances by using the following relation [11]

Rcsn 0:33 1:175e0:521sn

103

m2 K

.W (11)

where Rc is the thermal contact resistance corresponding to theapplied contact pressure sn. This relationwas obtained by using themethod of least squares under the tests performed on two steelspecimens with as-rolled clean surfaces without loose mill scale atcontact pressure 0e30 MPa and maximum temperatures 450e700 K. Moreover, the measured roughness of the contact surfaces iswithin the range of 1.3e7.0 mm. This relation of the contactpressure-dependent thermal contact resistance between twoblocks, described by Eq. (11), is also in good agreement with theexperimental data reported by the literature [12]. The thermalcontact resistances converted from the thermal contact pressuresare substituted into the energy balance equations. Consequently,the revised temperature of strip can be obtained. One can repeatthese procedures until the strip temperature is converged within asatisfactory tolerance.

2.1.2. Energy model of roll [3]This energy model is used to determine the surface tempera-

tures of rolls which are continuously contacting with strip andexposing to the furnace atmosphere for a long time period.Consequently, these rolls, initially having the same temperature asfurnace atmosphere, are heated over one portion through themechanisms of heat convection/radiation by exposing to hotfurnace atmosphere and cooled over the remaining portion of theperiphery through contact heat conduction by the cold strip. Boththe Lagragian and Eulerian descriptions can be used in thermalanalysis of roll. For the Lagragian one (observer fixed to the strip orroll), the thermal field must be represented as time-dependent.However, after a short transient, the thermal field becomesquasi-stationary. On the other hand, for the Eulerian one (observerfixed to the laboratory), the temperature field becomes stationary.In this work, Lagragian description rather than Eulerian descrip-tion was adopted in thermal analysis of roll, since an additionalconvective term, which is difficult to deal with in ANSYS, shouldbe included for the latter description. The thermal boundaryconditions of contact heat transfer with respect to strip andconvective/radiative heat transfer with respect to furnace atmo-sphere were individually prescribed at the rotating speed of rollalong the specific periphery of roll. The transient temperature ofroll, TR, can be obtained by solving the energy balance equationgiven by

v2TRvr2

1rvTRvr

1r2

v2TRvq2

1aR

vTRvt

(12)

with initial condition at t 0 and thermal boundary conditions atr R

TRx; y; z; t TN; t 0 (13)

lRvTRvr

TR TRcsn; ut q q0 ut (14)

lRvTRvr

heqTR TN; q0 ut q 2p ut (15)

where aR (kR/rRcp-R) is the thermal diffusivity of roll; kR, rR, and cp-R denote the thermal conductivity, density, and specific heat ca-pacity of roll, respectively; u is the angular speed of roll; while q0denotes the angular width of the periphery in contact with strip.

The temperature-dependent mechanical properties of the steelstrip including Poissons ratio, n(0.3), thermal expansion coeffi-cient, ath, Youngs modulus, E, yield strength, sy [13] and specificheat capacity [14], were shown in Fig. 5. The effect of latent heat(76 kJ/kg) created during ferriteeaustenite phase transition can beconsidered by using the modified specific heat capacity curveadopted by Brown et al. [15] and Frewin et al. [16] in ABAQUS andANSYS, respectively.

The computational procedure of energy balance model wasdivided into two steps. First, the temperatures of heating plane,ceiling, side walls, floor, vertical and horizontal strips were deter-mined in each enclosure by solving the energy balance equationsiteratively. Secondly, the strip was divided into several imaginarynarrow strips along the width direction in each enclosure. Differentview factor was specified individually in each imaginary narrowstrip. The temperatures of strip in various imaginary narrow stripswere evaluated by the temperatures of heating plane and wallsobtained previously and thermal contact resistances distributedalong the width. A 2-D temperature distribution for the strip alongthe length andwidth was thus obtained. The flowchart, as shown inFig. 6, represents the computational algorithm to calculate thetemperature of heating plane, walls and strip by the energy balanceequations and finite element models.

Fig. 6. Flowchart of simulation and algorithm of EBM.


2.2. FEM scheme to evaluate strip temperature

Both the thermal and mechanical models were analyzed by usingthe finite element code ANSYS to investigate the influence of opera-tional parameters on the temperature distributions of the strip espe-cially along the width in CAL. This method mainly includes to establish

Fig. 7. Flowchart of simulatio

3-D mechanical model of the strip to evaluate the distributions ofcontact pressure of the strip and the surface temperatures of all the rollsin CAL, aswell as to estimate the emissivity and equivalent heat transfercoefficient of the strip in each section of CAL. Due to the coupling effectbetween mechanical and energy models of strip, mainly attributed tothe effects of contact pressure-dependent thermal contact resistance,temperature-dependent material properties and thermal strain, aneffort of coupled-field analysis should be conducted. In addition, theheat source that generated by mechanical deformation of strip is verysmall and can be satisfactorily neglected. In order to evaluate 3-D dis-tributions of temperature and stress of strip, a strip with finite length isadopted in both mechanical and energy models. This strip passesthrough the inlet and moves into the interior of CAL and then is sub-jected to the complicated convective/radiative heat flux from the hotfurnace atmosphere. The strip keeps moving and then in contact withthe roll, where a conductive heat flux due to thermal contact conduc-tance between them takes place. The strip keeps moving forward andfinally passes though the outlet of the CAL. Consequently, the thermalanalyses of strip in CAL were mainly composed of two steps:

2.2.1. Estimation of equivalent emissivity or heat convectivecoefficients of strip in each section of CAL

The methods of direct sensitivity coefficient [17] and the leastsquare error were employed to evaluate the equivalent emissivity ofstrip in each section of CAL by using the measured strip temperatureat outlet of each section and the estimated strip temperature at somespecific locations along the length by energy balance model.

2.2.2. Energy model of strip [3]When a strip with finite length is moving in the furnace, its

corresponding energy equation can be written as [18]

v

vx

lTvT

vx

vvy

lTvT

vy

vvz

lTvT

vz

rTcpTvT

vt(16)

The associated thermal boundary conditions prescribed uponthe strip are:

(a) As the strip does not contact with roll:

The total heat flux imposed on the strip is composed of heatconvection and radiation:

n and algorithm of FEM.


qR sT4N T4

h

TN T

heq

T TN T (17a)

where heq sT2N T2

TN T h (17b)

(b) As the strip is in contact with roll:

When the strip is passing through the roll, the area contacting tothe roll, denoted by the contact angle q, is subjected to the heat fluxdue to the contact conduction, qC, while the remaining region ofstrip is subjected to the convection/radiation heat fluxes, qR. The

Fig. 8. Contact pressure and thermal contact resistance of strip along the width at No.1 roll of HS.

contact angle increases from 0 to 5, 10, ., 180 after each Dt as thestrip front starts to contact with the roll; while the contact angledecreases by 5 after each Dt as the strip starts to leave the roll. Therelationship between thermal contact conduction and the tem-perature of strip can be expressed by

qC TR T=Rcsn (18)

where TR is the surface temperature of roll; T denotes the tem-perature of strip in contact to the roll; Rc is the thermal contactresistance, that is a function of contact pressure sn.

Detailed solution procedures and algorithm of FEM schemewere shown in Fig. 7. All the required input data, including the sizeand physical properties of strip and roll, as well as the parametersof CAL, should be provided as input data first. It is obvious that anumerically iterative solution procedure should be performed tothe mechanical and energy models of strip and roll. The contactpressure and thermal contact resistance with an assumed initialstrip temperature and surface temperature of roll, generally esti-mated by EBM, were evaluated first bymechanical model. A revisedtemperature distribution of strip after completely passing over theroll, that takes account of the influence of thermal contact resis-tance and surface temperature of roll, was then calculated by theenergy model of strip. After that, the revised contact pressure andthe thermal contact resistance were evaluated again by mechanicalmodel. These solution procedures were iteratively proceeded untila satisfied convergence in numerical solutions was obtained.

3. Results and discussions

The distributions of contact pressure and thermal contactresistance in width as the strip is passing the No. 1 roll in HS wereshown in Fig. 8. It can be found that a uniform contact pressure wasinduced at the central portion of the strip along thewidth direction,while an extremely high peak of contact pressure was induced nearthe edge of the crown. The value of contact pressure reduces to zeronear the two edges of the strip, where a significant gap appearsbetween strip and roll. In other words, the situation of thermalcontact resistance disappears in these regions, where both the ef-fects of thermal radiation and convection instead of thermal

Fig. 9. Surface temperatures of rolls in CAL.

Fig. 10. Temperature history of strip in CAL by FEM and EBM.

Fig. 12. Temperature distribution of strip at No. 1 roll in HS.


contact resistances should be imposed upon during numericalsimulation. This kind of contact pattern almost remains unchangedas the strip passes the rolls in each section of CAL. In addition, thedata relevant to the surface temperature of rolls in CAL as a functionof strip temperature under different atmosphere temperature offurnace are shown in Fig. 9, which is required and should bedetermined in solution procedures of FEM and EBM schemes.

The equivalent heat convective coefficients, as defined in Eq.(17b), were simplified to a constant in each section of CAL, whichactually should be temperature-dependent. The values of equiva-lent heat convective coefficient in PHS, HS, and SS determined byinverse scheme were 6.31, 1.61, and 1.61 W/(m2-K), respectively,while the values of equivalent heat convective coefficient in CScorresponding to the cooling of air jet array (temperature of cooling

Fig. 11. Effect of phase transformation on temperature history of strip in HS.

air was equal to 350 K) at central part and two sides were equal to27 and 32 W/(m2-K), respectively. Higher equivalent heat convec-tive coefficient in PHSwas due to the higher temperature differencebetween strip and furnace atmosphere. The temperature history ofstrip in each section of CAL was shown in Fig. 10, where the striplengths in PHS, HS, SS and CS are 0e42.5, 42.5e337.2, 337.2e505.8and 508e554.7 m, respectively. The outlet temperatures in varioussections were 401, 1058, 1080 and 923 K, respectively. A tempera-ture gradient is produced suddenly when the strip is in contactwith each roll, since the effect of heat conductance is much higherthan heat convection and radiation. Moreover, it was found byenergy model of strip in FEM scheme that the strip temperaturealways keeps uniform distribution in thickness in the whole CALeven under such a high moving speed. In other words, only 2-Dtemperature distributions of strip, i.e., along the wide and longi-tudinal directions, are required to be displayed and discussed.

When strip is moving in HS, the temperature of strip graduallyrises higher than 727 C and then the aeg phase transformationsoccurs. On the other hand, as the temperature is decreased lowerthan 727 C in CS, the reverse phase transformation from g to atakes place. The effect of phase transformation on strip tempera-ture is very significant. As shown in Fig. 11, a maximum discrepancyof 19.5 C of the strip temperature is produced in HS in case this

Fig. 13. Temperature distribution of strip at No. 23 roll in CS.


effect is disregard. The temperature distributions of strip at the No.1 roll in HS and the No. 23 roll of CS were shown as in Figs. 12 and13, respectively.

The temperature distributions in the width at each section wereshown in Fig.14. It can be seen that the temperature near the centerof strip is higher than the two sides due to the thermal contactconductance between the roll and the strip, as shown in Fig. 14(a),where No. 1 w 13 represent the temperature of strip just passingthrough the No. 1 w13 rolls in HS, respectively. The discrepancy oftemperature distributions between FEM and EBM schemes aresmall and acceptable. On the other hand, when the strip is movingat SS, the temperature rise near the center part of strip becomessmaller due to smaller thermal contact effect between the strip and

Fig. 14. The history of transverse temperature distributions in CAL.

the roll in SS than in HS. The maximum temperature differencealong the width of strip tends to decrease gradually and finally lessthan a few degrees, as show in Fig.14(b), where No.15w 21(SS) andExit(SS) represent the temperature distributions of strip justpassing through the No. 15 w 21 rolls and outlet in SS, respectively.Finally, as the strip reaches to CS, the temperature of strip dropssignificantly due to the forced cooling effect of air jet array. It can beseen that, the variation of transverse temperature near the center ofstrip is due to the cooling effect of thermal contact conductancebetween strip and roll, as shown in Fig. 14(b), where No. 23A(CS),No. 23T(CS) and No. 23L(CS) represent the time instant that stripjust arrives at roll, at the top of roll and leaves the roll in CS,respectively. The significant temperature drop at two sides of stripis due to the stronger forced convection of heat flow and radiationinteraction between the strip and the components in CS. Thetemperature evaluated by scheme of EBM is in good agreementwith FEM. The computational time by the former is only about5 min by personal computer (Intel(R) Core(TM)2 Duo CPU2.33 GHz), significantly shorter than the latter.

4. Conclusion

3-D temperature distributions of strip in CAL were theoreticallyevaluated by using both the techniques of EBM and FEM, in whichthe view factors, ferriteeaustenite phase transition, and thermalcontact conductance between strip and roll were taken into account.The heat flux and temperatures of heating plane, ceiling, side walls,floor and strip were obtained as well. It can be found that the crownof roll has a significant influence on the transverse temperaturedistribution of strip, while the phase change has remarkable influ-ence on the longitudinal temperature distribution of strip in HS andCS. The numerical results obtained by both techniques were in goodagreement with the literature reported and experimental datameasured at some specific locations in factory. The strip in contactwith roll results in a remarkable temperature rise. Consequently, thecentral portion of strip has the higher temperature than two sides ofstrip especially in both PHS and HS. This 3-D temperature distribu-tion of strip can be used to predict the residual stress and warpage ofstrip during punching process.

Acknowledgements

This research was supported by the National Science Council inTaiwan through Grant No. NSC 98-2221-E-006-043.

References

[1] M.M. Prieto, F.J. Fernndez, J.L. Rendueles, Development of stepwise thermalmodel for annealing line heating furnace, Ironmaking & Steelmaking 32 (2)(2005) 165e170.

[2] C.H. Ho, T.C. Chen, Two-dimensional temperature distribution of strip inpreheating furnace of continuous annealing line, Numerical Heat Transfer,Part A: Applications 55 (3) (2009) 252e269.

[3] T.C. Chen, C.H. Ho, J.C. Lin, L.W. Wu, 3-D temperature and stress distributionsof strip in preheating furnace of continuous annealing line, Applied ThermalEngineering 30 (8e9) (2010) 1047e1057.

[4] T. Masui, Y. Kaseda, K. Ando, Warp control in strip processing plant, ISIJ In-ternational 31 (3) (1991) 262e267.

[5] C.H. Ho, T.C. Chen, Temperature distribution of taper tolls in preheatingfurnace of cold rolling continuous annealing line, Heat Transfer Engineering31 (10) (2010) 880e888.

[6] U. Gross, K. Spindler, E. Hahne, Shaperfactor-equations for radiation heattransfer between plane rectangular surfaces of arbitrary position and size withparallel boundaries, Letters in Heat and Mass Transfer 8 (1981) 219e227.

[7] R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, fourth ed., Taylor &Francis, New York, 2002.

[8] R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, London,1950.

[9] J.M.C. Rodrigues, P.A.F. Martins, Coupled thermo-mechanical analysis ofmetal-forming processes through a combined finite element-boundary


element approach, Internal Journal for Numerical Method in Engineering 42(1998) 631e645.

[10] J.B. Dai, Study on the Strip Buckling in Continuous Annealing Production Line.PhD thesis, University of Science and Technology, Beijing, 2005 (in Chinese).

[11] T.R. Tauchert, D.C. Leigh, M.A. Tracy, Measurements of thermal contactresistance for steel layered vessels, Journal of Pressure Vessel Technology,Transactions of the ASME 110 (3) (1988) 335e337.

[12] S. Fukuda, N. Yoshihara, Y. Ohkubo, Y. Fukuoka, S. Takushima, Heat transferanalysis of roller quench system in continuous annealing line, Transactions ofthe Iron and Steel Institute of Japan 24 (1984) 734e741.

[13] B.A.B. Andersson, Thermal stresses in a submerged-arc welded joint consid-ering phase transformations, Journal of Engineering Materials and Technol-ogy, Transactions of the ASME 100 (1978) 356e362.

[14] J. Goldak, M. Bibby, J. Moore, R. House, B. Patel, Computer modeling of heat-flowin welds, Metallurgical Transactions B 17B (3) (1986) 587e600.

[15] S. Brown, H. Song, Finite-element simulation of welding of large structures,Journal of Engineering for Industry, Transactions of the ASME 114 (4) (1992)441e451.

[16] M.R. Frewin, D.A. Scott, Finite element model of pulsed laser welding, WeldingJournal 78 (1) (1999) 15Se22S.

[17] A.A. Tseng, T.C. Chen, F.Z. Zhao, Direct sensitivity coefficient method forsolving two-dimensional inverse heat conduction problems by finite-elementscheme, Numerical Heat Transfer Part B 27 (3) (1995) 291e307.

[18] C. Zhang, L. Li, A coupled thermal-mechanical analysis of ultrasonicbonding mechanism, Metallurgical and Materials Transactions B 40B(2009) 196e207.
Three-dimensional temperature distributions of strip in continuous annealing line1. Introduction2. Mathematical model2.1. EBM scheme to evaluate strip temperature2.1.1. Mechanical model of strip [3]2.1.2. Energy model of roll [3]2.2. FEM scheme to evaluate strip temperature2.2.1. Estimation of equivalent emissivity or heat convective coefficients of strip in each section of CAL2.2.2. Energy model of strip [3]3. Results and discussions4. ConclusionAcknowledgementsReferences

Documents

K_ZW_2013_2.pdf