-
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
Z.-W. Kang, T.-C. Chen / Applied Thermal Engineering 58 (2013)
241e251244
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.
-
Z.-W. Kang, T.-C. Chen / Applied Thermal Engineering 58 (2013)
241e251 245
_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].
Z.-W. Kang, T.-C. Chen / Applied Thermal Engineering 58 (2013)
241e251246
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.
Z.-W. Kang, T.-C. Chen / Applied Thermal Engineering 58 (2013)
241e251 247
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.
-
Z.-W. Kang, T.-C. Chen / Applied Thermal Engineering 58 (2013)
241e251248
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.
Z.-W. Kang, T.-C. Chen / Applied Thermal Engineering 58 (2013)
241e251 249
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.
-
Z.-W. Kang, T.-C. Chen / Applied Thermal Engineering 58 (2013)
241e251250
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
-
Z.-W. Kang, T.-C. Chen / Applied Thermal Engineering 58 (2013)
241e251 251
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