ISIJ I nternational, Vol. 38 (1998) , No. 11, pp. 1262- 1269

Finite Element Analysis of Hot Rolled Coil Cooling

Seong -Jun PARK , Byung- Hee HONG , Seung Chul BAIK1) and Kyu Hwan OH

Division of Materials Science and Engineering, Seoul National University, Seoul 151-742, Korea. E-mail: kyuhwan@plaza.snu.ac.kr1 ) Technical Research Laboratory, Pohang Iron & Steel Co. Ltd., Pohang, 790-785,

Korea.

(Received on February 20, 1998; accepted in fina/ form on Ju/y 23, 1998)

A new unit layer model for the equivalent thermal conductivity of layered steel strips has been proposed. The equivalent thermal conductivity is a function of strip thickness, surface characteristics and compressive stress. The modeled equivalent thermal conductivity corresponds well to the experimental data. Finite element analyses (FE M) fo r cooling of hot rol led coil have been carried out under various cooling conditions using the equivalent thermal conductivity as the thermal conductivity in radial di rection of hot rol led coi l. A new calculation procedure using ABAQUS has been developed , where the radial compressive thermal stress is taken into account for calculation of the equivalent thermal conductivi tι or the orthotropic stress dependent thermal conductivity. The calculated cooling curves using the orthotropic stress dependent thermal conductivity have been compared with data calculated using isotropic and orthotropic stress independent conductivity and with experimenta l data. The cooling curves calculated using equivalent thermal conductivity as rad ial thermal conductivity are in better agreement with experimental data.

KEY WORDS: unit layer model; equivalent thermal conductivity; strip thickness; surface characteri stics; compressive thermal stress; orthotropic stress dependent condu ctivity.

1. Introduction the voids. 2) Therefore, the surface characteristics and

radial stress of hot ro l1ed coil ll1ay inflllence on this heat now at thc co ntact interface. A few attempts havc bccn made to derivc thc cq uivalcnt thermal conduclivily as a funclion of sLrip thickness 0 1' thickness of ai1' layer and to make heat transfer analysis in the coi l. 8 - t 이 However, lhc e(fect of surface characteristics and rad ial thermal stress on the equivalent thermal conductivity has not bccn taken into account in Lhc previo us investiga tions.

Greenwood and Wi ll iamson developcd a thcory which

\‘스

A hot rolled coil is a coiled stecJ strip in holJow cyli ndrical fo1'm schematically shown in Fig. l. Tt is produced by hot ro ll ing and coiling at 540 720oC. After coil ing process, it is cooled down to 1'00111 te ll1perature at the coil storagc fìeld bcfore pickling process. It takes 4- 5 days for natural a ir cooling. 1) Rapid cooling of the hot ro l1cd coil can savc limc and storage field . Therefore, various methods for the rapid cooling, such as the forced ai r cooling or thc watcr coo ling, are used 0 1' tried in steel produclion companies nowadays .

As thc cooling timc o f Lhc hol rolled coil is dependent on many variables such as ambient tempcraLurc, coolant, cooling method , strip thickness, coil weight , eIC. , a nU l11erical analysis of heat transfer is hclpfu l for thε

expecta tio n o f cooling Lime and design of rapid cooling proccss. [n the nU ll1erical analysis, it is rcq uircd to detcrmine the thcrmal conduclivity of thc hOl rolled coil.

combines the elas tic behavior of individual asperities with 、

a statistical model to describe the asperi ty pop lI lation. 3)

N ivikov dcrived thc rclatiollship betwccn thc lhcrmal

A hot ro lled coil can be considered as a periodicall y laminaled matεri a l of steel layer and interface layer in radial direction. Therefore, thc thcrll1a l rcsistance of intcrface layεl' bclwccn contacted steel layer is impor­tant and has been investigated by many authors. 2

-7

)

Gcnera l1y, a free surface of steel layers is not smooth but has asperities in microscale. A real contact interface cOllsislS of actllal contact spots bClwccn asperities on each surface and voids among the aClual conlact spots. Heat transfcr across thc contact interface has various I11cchan ism: cond llction through the actual conlact spots, conduction through thc air in voids, and radiation across

‘ ) 1998 ISIJ 1262

Axial

一 _j_ _ _ I | - -키~‘

I --...、_-+- _ _ I " .... 、i

H익~p_ .Di'r…

Fig. 1. Schcmatic diagram of hot rollcd coil

ISIJ International, Vol. 38 (1998) , No. 11

resistance and thc extcrnally applied load by pcrforming a statistical analysis, assuming Ga ussian distribution of height of spherical asperities.4J κ1ik ic considered the Gaussian distribution of surfacc profìlc heights and slopes, and dcrivcd an cxprcssion for the intcrfacial conductancc that is proportional to p O

.94, whcre P is lhe

app1icd compressivc strcss. 5J Taucherl el al. experimen­tally measured the therma l resistance of layered speci­mcns undcr normal stress.6) McWaid e1 a/. measurcd the lhcrmal rcsistancc of layered specimens under thc nonnal strcss in vacuum , then compared the measured data with models for thc thcrmal rcsista nce.7

) Most thcories and experiments relate thermal resistance of contact inlerface with applicd compressive strcss, because compressive stress increascs real contact area th us rcduccs thc thcrmal resislance of conlact interface.

In this papcr, the cquivalent thermal conductivity of hot rolled coil in the radial direction was modeled using thermal resistance of a new unit layer model, which can takc into account the strip lhickness, the surface charactcristics and thc comprcssive strcss. Thc proposcd modε1 was comparcd with cxpcrimCnlal dala. The rcsults of fìnite elcment calculation for hot rolled coil cooling were compa rcd with experimental data. Then lhe effects of lhc radial thcrmal strcss on thc cooling of hOl rolled coil are discussed.

2. Equivalcnt Thcrmal Conductivity

An objcct thal contains layers of ditl‘crenl thcrmal conductivity has anisolropy in thermal conduclivi ly. In casc of the hot rolled coil, the axial and hoop direction arc the directions pa없l

bc잉따twec이cn s tee려I s있tnψp. 1 1미n those dilπrcc따t디ionsι， lhe ‘c찌:씨q u비ll V씨a띠l니Ic띠cnt 1까 t

t야thc야rma씨:tI니I con띠ducαtlV、v씨11μty is almost the same with thc thcrmal conductivity of steel. On the other hand, the radial dircclion is thc di rection normal to the contact interface which prohibilS lhe hcat fl ow. Thlls the equivalent thcrmal conductivity may be quite diffcrent from that of steel. In order to derivc the equivalent lhcrmal conduc­

\ι tivity in radial direction, a set of laminated steel, oxide and interface layer is modcled as a unit layer of lhe hot rolled coil.

Figurc 2 shows a schematic diagram of lhc modelcd unit layer of lhe hot ro lled coil. The lotal thcrmal resistance of the unit layer,RT, becomcs,

RT = , t = R‘ + Ro+ R; ................... (1) ’ ‘ e‘’

where t, R" Ro, R; are the unit layer thick ness, the thermal resistances of meta l, oxide and in terfacc layer, rcspec­tively. Thc therl11al resistances of melal and oxide laycrs can bc cxpresscd by thc thickncss and lhermal rcsistancc

0 1' each layer as follows.

기

/ ‘ , ‘ 、

c -’ κ

--R

2L Ro= ~ u .. .... ............... ..... ... . (3)

k。

whcre k‘ ’ "0 arc thc thcrmal conductivities of the steel and oxide layer, and 1" '0 arc thc thickncsses of steel and oxidc laycrs, respcctively. The mean thickncss of oxidc layer was mcasured as 7μm using scanning electron microscope. l) Slowik el al. have mcasured lhc therma l conductivity of ox ide layer. l l) (See Tablc 1.)

At the interfacc layer, heal fl ows simultaneously through contact points and voids fìlled with air. As the voids betwecn hot rolled strips are in micrometer scale, the heat conduction through voids by air is dominant rather than convcction. I n this work, thrce modcs of heat transfer arc ∞nsidcred to contributc to the heat transfer in the in terface layer; condllction th rough contact spots, conduction lhrough air in voids and radiation through voids. Then the thermal resistance of interface layer becomes

1 1 1 1 ---;:-=-~ -+-~-+-:::- ............ ...... (4) R; R‘;<I .S R‘; <1 ‘l R

wherc RCd,s> RCd.a and Rrd arc thermal resislances of conduction lhrollgh contact points, co nduction through air and radiation through voids, respectively.

Equations (1) and (4) can give the eqlliva lent thermal conductivity, kcq, as fo ll ows.

kcq= -

R.+ R‘+ • v 1 1 1

- + - + -Rμ1， 8 R “1.0 Rrù

... (5)

M ikic proposed simplc expressions for contact Con­dllctance as a function of the surfacc roughness, a, and the nominal compressive stress, P, assuming Gaussian distribution of surface height in cases that the sllrface

S 1 lf 2

Oxide ιι~//“0"?//-ιι~ß I l。

Interface 鐵鐵옳騙 li

Oxide

St eel

Fig. 2. Schcmalic diagra l1l 0 1' lInit laycr 0 1' laycrcd hot 1 이Icd

stnps

Tablc 1. Inpllt variablcs for calclllating efTcctivc thermal conductivity.

1263 ‘) 1998 ISIJ

ISIJ International, Vo l. 38 (1998) , No. 11

짧

2

ξSiNEγ。←

-

@gaz-ω@α -mE」@ζiF

5

4

3

•

E $ • 40

” ” @ z 잉 .t:: t­"0 。

>I m @

E

Experimental Data

-- t =42.7X10-6e-005P a

• 50

30

20

10

14

Compressive Stress, MPa Thcrmal rcsistanccs ofmctal, oxidc and inlcrfacc laycr as a funClion of comprcssivc slrcss.

12 10 8 6 4 2 O

O

Fig_ 4.

120

Compressive Stress, MPa Mcan lhick ncss of interface layer as a function of com prcsslve Slress

100 80 60 40 20 o

o

Fig. 3.

where the uni ts of P and l. are M Pa a nd m, respectively. T he thermal resistancc of radiation in voids can be

cxprcsscd as

n” -/1l、

… :J -M

l

-씨

---4”、

= ω

R

asperi ties deform plastically o r elastically.5) Fo r a rough surface, the mode of as얘pe잉r띠c않s dεf“ormatlo l아’I'l‘~ dc야pends on ll1atcria l p roperties and the sllrface characterist ics.3•5) Mikic also proposed a dimensionless para ll1clcr, y, which is p roportiona l to the ratio ofyield stress to contact stress to deter ll1ine the mode of asperit ies deforll1ation as fo llows. where ε，(J and Tare the emissivity, thc Stcfan- Boltzll1ann

conslan t and the te ll1peratu rc of hot rolled coil, re­spectively ‘

Authors proposed the eq ui valent therma l conductivity, using Eqs. (2), (3), (5) a nd (8)- ( 11 )

2J-I(I - v2)

y = - " .. ..... .... ,… .. ……… (6) E tan 0

kc‘l=1/ {C + 었 +[l l3 ksr1O AO(4

+ (l - A) 42.7 x 10 6팩- 5x 1O - 2P) + 4( I - A)텍 I}

The equivalen t therll1a l condllctivity in Eq . ( 12) depends on the material properties, the nominal cO ll1pressive stress norll1a l to contact in terface, the tell1pera ture and the surface characteristics.

Figure 4 shows the therll1al resistances of ll1etal, oxide and interface layer as a fllnc ti on of compressive stress AII the vallle o f para ll1eters llsed in calculation of the thermal resistances arc given in Tablc 1. Al low COIl1-

pl‘ cssivc stress rangc, thc therll1al resistance of intcrlace layer, }인， is the largest and decreascs as cO ll1pressive st rcss increases. Above 8 M Pa , the therll1al resistance of metal, Rs, is the la rgest. The therll1a l resista nccs o(‘ ll1etal and oxide layers are not function 01 cO ll1 prcssivc stress as can be secn in Eqs. (2) and (3). The therll1al resistancc of oxide layer is c10se to zero duc to thc small th ickness. Since the three ther ll1al resistances constitute the total thermal resistance in serial connection as shown in Fig. 2, thc in terface layer plays a signifìcant ro le in whole heat transfer in low cO ll1 prcssivc strcss range.

1 n order to invcstigatc thc contribll tion of hea t transfcr 1l10des in interface layer, the thermal resistances of condllction through con tact spo ts, cond llc tion through a ir in void s and radiation thro llgh voids arc shown in Fig. 5 as a fllnction 01 cO ll1prcssivc strcss. The thermal rcsistancc of conduction thro llgh contact spo ts, Rcd.s> is very la rge a t low compressive strcss below 1 MPa and decreases as compressive stress increases, bccallse the

where H , ι， v and tan 0 are the ll1icrohardness, the mod lllllS 01 clasticity, the Po isson ’s ratio a nd the ll1ean ofabsolllte slopc ofa profì le, respectively. Mikic rcportcd that thc dcfor ll1ation mode of aspcri ties wi ll bc pre­domi nantly plastic when ì' 드 0.33, and wi ll bc prc­dom inan tly elastic when y;::: 3. Sincε the typical value of y of hot rolled stccl strip is abollt 0.1 2, it can bc assumed that thc asperities will defor ll1 plastica써1. Pll llcn a nd Will ia ll1so n proposed a n a pproximate factor of the actual contact area, A , as follows. 12)

•. ..•. ..•..•••.•.•.•••• ,… .. (7)

U sing Eq. (7), M ikic proposed an equation of contact resistancc through solid as follows.5)

p

-바

R‘ù.s =찮뚫~Ð ( 단폼_)0 . 94 .. .. .... .. .. (8)

whcrc (Jp, ks a nd P arc the standard deviation of prolìlc height, thc thennal cond llctivity of stccl a nd thc nom inal cOll1pressive stress, rcspectively.

T he therll1a l resistance of condllction by air becomes as

R"ιla =-」ε­“’" (1 - A)k.

1264

… (9)

where k" and 1. arc thc thermal condllctivity of air and thc Il1can thickness o f vo ids, rcspcctivcly. T hc Il1can thickness ofintcrface layer was ll1easured exper imenta lly by Baik as a function of compressive stress. l) Figure 3 shows thc meas ured thickness as a function o f nomical comprcssive stl'ess which can be approximatcd by the following cq ua tion.

‘, 1998 IS IJ

' a= 42.7' 10 - 6 exp( - 5 x 1O - 2 p) ........... ( 10)

Pressure

l

ISIJ International, Vol. 38 (1998) , No. 11

Cooling Element ’ Thermocouple

Heating Element

Layered Sleel Slrips

‘­--•-- •-‘-•--• -- ---‘'_'_'_'_'_'_'_'_'_'_'_'_'_-"- --';'''::_'---' - '-'-'-'-'-'-'-'-- .... _---------•- --

m

mw

m

르호NEγ。← -@Qζmim-m@

￠ -mEapF

50

Insulation

Schcl1lalic di‘Igral1l of experimental systell1 0 1' meas lIr­ing eqllivalent thennal condllct ivity

Fig. 6. Therll1al rcsistances of conduction through contact spots. COlld llCt ioll through air in voids and radiatioll through voids “s a function of compressive strcss

14 4 6 8 10 12 Compressive Stress. MPa

2 O

O

Fig.5.

18

Compressive Stress , MPa

Equivalcnt thcrmal conductivi ty as a function of cOl1l preSSlve stress

·i

·

3 15 12

Experimental Data -- Equation (12)

9 6

•

O O

때

i E~>>

-a->공 그 깅 Z

。Q-mE」@ ζLZZ @ -m〉-그 σ버

50

30

20

10

Fig.7.

actual contact arca increases with increasing compressivc strcss. The thcrmal resista nce of conduction through air, Rc씨， also dccrcascs as compressive stress incrcascs, however, it is not qui te sensitive to compressive stress.

_./ The thermal resistancc of radia tion, R'd' is not sensitivc to comprcssive strcss. Bclow the compressive strcss of 1 MPa, R'd is thc lowcst and abovc 1 MPaιlμ， Rκc“d“l lowc않st. S잉incc t냐thπc tl네h매11ηrc야c t냐thc l이rrrη'ma띠l니1 1니re않s잉i떠stances const디i“tlωu따lαtc t“thc c the밍rr’τ'malre않si샌stμωa씨l nceo아finterface layer in para l1el connection as shown in Fig. 2, thc hcat transfers mainly by radiatio n through voids in low cO l11pressive stress range bclow 1 MPa and by conduction through contact spots in high cO l11pressive strcss range above 1 MPa.

values. It ca n bc secn that Eq. (12) givcs rcasonablc valucs of cq ui valcnt therl11al conductivity as a function of normal compressive stress.

Experimental Mcasurcmcnt of Equivalent Thcrmal Conductivity

3.

Finite Elcmcnt Analysis of Hcat Transfcr in Hot Rollcd Coils

4. To examine the eq uiva lent thermal conductivity in Eq.

(1 2), thc thcrl11al conductivity of layered hot ro l1ed strip was measured. 1n case of lInidirectional heat transfcr, thc thcrl11al rcsistancc, R , at steady state, can bc cx­pressed as follows

As lhc eq uivalent therl11al conductivity is a function of nominal compressive strcss, il is rcqu ircd lo consider lhc sourccs of nOl11 inal cO l11 prcssive stress in hot rolled coil cooling procedure which is radia l slrcss in hol rol lcd coils. ThcI‘ e are three contributions to nominal CO I11 -

prcssive stress: the gravity, thc back tcnsion forcc in coiling proccss and the thermal stress callscd by inho l11ogcncous thermal contraction in cooling process. Among thcsc sources, the magnitudc of thermal stress is largest up to abo ut 50 MPa, when othcrs are in k Pa order. 13) To ta ke into account the interaction between lhcrmal stress and the equivalcnt conduclivity, lhc eq uiva lenl lhcrl11al conductivity should be updated in each timc step using the radial cO l11 pOnCnl of lhermal stress and temperature. Figure 8 shows the calculation proccdure llsed in this stlldy. In thc procedure, lhe eqllivalcnl lhcrl11al conduclivity al time t+ðl is calcu­lalcd frOI11 Eq. ( 12) using radial strcss, S" and lCl11pcra­tllrc a t limc t. With the equivalcnl lhermal condllctiv­ity , the temperature and thermal stress in hot rolled coil are calculated . The radia l thcrmal stress is taken as lhc compressive stress nonnal to inlcrfacc al li l11c t + ðt, if the radial slrcss is comprcssivc. Whcn the radi­al lhermal stress is tensile, the compressivc Slrcss nor-

ðT ð x R = =

q kcq

C(l 1998 ISIJ 1265

.... ( 13)

where q, ð T and ðx arc the heat flux, the temperature diffcrcncc and lhc distance between the measured posi­tions, respectivcly. Using ð T I11casured at steady slalc, the eq uivalent thcrmal condllclivity can be obtained from Eq. (13). Figure 6 shows a schematic diagra l11 of lhc experimenlal systel11 . The IF steel strips produced in Pohang 11'0 11 & Stcel Company has been layered on cach other fo r the l11eaSll rement. The number of strips was nine and lhe dimension of each layer was 0.5 x 0.5 x 0.0033 111 . Thc bottom surface of layered steel strips was in contacl wilh hcaling elcment, and the top surfacc was in contacl with water cooling element. The perimeter of specimens was insulated. Constant load has been applied to the layered strips, using universal tensile testing l11achinc ‘ For lhc meaSllrement of tel11peraturc, thrcc K typc thcrl11oco uples were attached along the side surface al lhc same inlcrvals.

Figure 7 shows the equivalent thermal conductivity calculated from Eq. (12), comparing with experimenta l

_.)

ISIJ Inte rnational, Vol. 38 (1998) , No. 11

mal to interface is set to zero. The calclllated temper­ature and compressive stress a re used in calculation of the equiva lent thermal conductivity at next time step. Thi s procedure has been implemented in the implicit finite element codc ABAQUS (Version 5.5) by writing a user sllbroutine USDFLD. Analysis was carried out by a coupled temperature-elastoplastic analysis. Tn the Sllbroutine USDFLD, thc compressive strcss norma l to interfacc calculatcd from elastoplastic analysis is dcfìn­ed as a neld variable to be used in detcrmina tion of eq uivalent thermal conductivity to calculate the tempera­ture distribution. This procedure is used when the radial equivalent thermal conductivity is calculated as a function of radial stress.

Figure 9 shows the finitc element mesh of hot rolled

Fi딩. 8. r low charl of calculalion proccdure for cooling of hol rolled coil ‘

coil. The dimensions of the hot rolled coils are listed in Table 2. In the FEM calcula tion, two-dimensional axisymmetric element has been used. The heat transfer coefficients of air and watcr spray cooling conditions are 17.5 and 2300Wjm2K , rcspectively.B) The thermal and mechanical properties of low carbon steel are given in Table 3. 14.15)

In this work, thrcε different radial thcrmal cond llc­tivities were used in FEM calcula tion . The fìrst caSe is callεd ‘ isotropic conductivity', where the radial thcrmal conductivity is same with the axial thermal conductivity. The second case is called ‘orthotropic stress indepcndent conductivity' , whcre the radial thermal conductivity is differcnt from the axial thcrmal conductivity and is not dependent on radial thennal stress in hot rollcd coi l. Tn this case, the radial thermal conductivity is calculated from thc equation of eqllivalent thermal conductivity proposed by Mazur et al. B) Thc third case is called ‘orthotropic stress depcndent conductivity', whcrc the cquivalent thermal conductivity proposed in this work is used for rad ial thermal conductivity.

Ouler Radius

Center Region

Inlermediate Region

Surface Region

Fig. 9. Finite elcmcnt mesh 01' hot rollcd coil

Tablc 2. Dimensions of thc hot rollcd coils‘

Coil Thickness Inner Outcr Width Coiling Coolant

No. Inm Radius, mm Radius, mm n대m TcmperalUrc, 'C

Coil l Air 2.7 381 959 1238 720

Coil2 Air, Water 2.7 381 958 1430 720

Coil 3 Water 3.0 381 976.5 1243 723

Table 3. Physical and mcchanical propertics of stccl

TCll1peraturc ko13) s Specific heat13) Elastic modulus") Yicld Slrcss") Poisson ’ s''')

('C) (W/mK) (J/kgK) (GPa) (MPa) rau。

20 206.5 303.0 0.29 100 57.8 490.5 203.0 275.0 0.3 1 200 53.2 527.5 200.0 240.0 0.33 300 49.4 563 .5 200.0 205.0 0.35 400 45.6 611 .8 165.0 17 1.0 0.37 500 41.0 685.0 100.0 136.0 0.39 600 36.8 782.3 60.0 10 1.0 0.41 700 33 ‘ l 986.0 40.0 65.0 0.43

<&;) 1998 ISIJ 1266

ISIJ International , Vo l. 38 (1998) , No. 11

Sr

1 뉴-+ -2. 00E+Ol 2 뉴-뉴 O.OOE+OO 3 뉴-l- +2.00E+Ol 4 뉴-+- +4.00E+Ol 5 f--+- +6.00E+Ol

Units : MPa

VALUE Experiment -- -- Isotropic cond uctivity -.-.-. Orthotropic stress independent conductivity --。미hotropic stress dependent conductivity 800

때

Q。

-

@」gm」φQE@·F

Distribution of radial thcrmal strcss in hot rollcd coil after I h of cooling

Fig. 12. 14

Time, hr Tεmperature variation at surface of hot rolled coil as a function of cooling time under air cooling condition

12 10 8 6 4 2 o

o

Fig. 10.

:,. .. -- .. ‘ ... 잉~-- ..

"‘~、、

... ... ’ .. - ‘、 r

·’ 、 ‘ .' 、 ‘、/ 、 - 、、 ‘ , , 、 (

‘./ I - Center Region 、、\

;' 1 -- --Intermed때:e Region 1 ‘、

rf I … .. - Su이a잉 Region \~ . I Orthotropic stress independent ν

Orthotropic stress dependent

40

30

20

10

iE~르 ·

〉)-〉-응그걷。。

-mEφ솥 은@-m〉-응버

_._._.- Surface Region

----- In!ermedia!e Region

o-L I --Ce띠er Region I •. ~ ‘i‘ 、 ‘ -:T~ 、\ '., 、“‘ ’/’ / 、、 “ , /. 1 ‘ ..... -._.-.-._._._._.’ ‘ι .. _._.ι

10

-10 mα---

φω@」im

」

14

Time, hr Variation of cquivalent thermal conductivity in hot rolled coil as a function of cooli ng time.

12 10 8 6 4 2 O

O

14 12 10 8 6 4 2

-20

-30 O

Fig. 13. Time, hr

Variation of radial thennal stress i n hot rolled coil as a function of cooling time

cooling. 1t can be seen that the radial stress is compressive in most region of hot ro l1ed coil except the region near to surface. The compressive radial thcrmal stress occurs owing to the larger thermal contraction of cold surface region than that of hot center region. Since the com­pressive stress increases the radial equivalent thermal conductivity, heat f1.ows more easily from hot core to cold surface than it did at the start of cooling. Therefore the cooling curve of orthotropic stress dependent con­ductivity becomes c10se to that of isotropic conductivi ty, as can be seen in Fig. 10.

Figure 13 shows the equivalent thermal conductivities of orthotropic stress dependent case at center, inter­mediate and surface region are calculated from Eq. (1 2) and shown in Fig. 13 as a function of cooling timc along with that of orthotropic stress independεnt case. At the start of cooling, the orthotropic stress dependent equivalent thermal conductivity is smaller than the radial thermal conductivity of orthotropic stress independent conductivity. As the cooling proceeds, the orthotropic stress dependent thermal conductivity increase owing to the increase of radial thermal stress and reach to their maximum in the l1l iddle of cooling process, then decrease at the end of cooling. The variation of orthotropic stress dependent ther l1lal conductivity shows the similar be­havior with radial thermal stress, because thermal con­ductivity is caJculated fl'om Eq. (12).

Figures 14(a)- 14(c) show the temperature distributions

(þ 1998 ISIJ 1267

Fig. 11.

Analysis of Cooling Proccdurcs

Figurc 10 shows the calculated cooling curves from three kinds of the radial thermal conductivity, comparing with the experimental data 1) measured at a surface of hot rolled coil denoted as coil 1 under air cooling

_I condition. The cool~~g. curve calculated from isotr_opic conductivity shows higher temperature than that from orthotropic conductivity and the experimental data. The cooling curve from orthotropic stress independent conductivity shows the lowest temperature. 1n case of orthotropic stress dependent conductivity, initially the calculated cooling curve is c10se to the curve of orthotropic stress independent conductivity. As cooling proceeds, the temperature becomes close to the tem­perature calculated llsing isotropic condllctivity. The change in the cooling curve of orthotropic stress dependent condllctivity is callsed by the variation of radial equivalent thermal conductivity due to the radial therl1lal stress. Figure II shows the calclllated variation ofradial therl1lal stress at center, interl1lediate and surface region in hot rol1ed coil as a function of cooling time in case of orthotropic stress dependent conductivity. The radial therl1lal stresses at center and interl1lediate region are compressive. Figure 12 shows the distriblltion of radial thermal stress in hot rolled coil calculated using orthotropic stress dependent conductivity after 1 h of

5.

ISIJ International. Vo l. 38 (1998) , No. 11

T

11-• - +4.00E+02 2 1-1-+4.60E+02 3 1-1- +5.20E+02 4 뉴-j- +5.60E+02 5 뉴-j- +6.40E+02 6 뉴-j- +7.00E+02

VALUE

Units :.c

(a) (b) (c)

Tcmpcralure disll’ibul ions in hOl rolled coil undcr “ir cooling condition aflcr 1 h of cooling calc띠alcd lIsing (a) isolropic conduclivily, (b) orthotropic slress indepcndcnt conductivily and (c) orthotropic slress dependcnl condllclivily.

Fig. 14.

Experiment -- -- Isotropic conductivity -.-.-. Orthotropic stress independent conductivit --Orthotropic stress dependent conductivity

• Experiment -- -- Isotropic conductivity -.-.-. Orthotropic stress independent conductivity - - Orthotropic stress dependent conductivity

•

600 鷹觸Measured Position

800

600

400

200

Q。-@』gm」φaE@LF

댄EfEJ l灣tion

;ik/밥

800

째

Q。

-

띤 공m」φg E

@L「

200

30

Time, hr Fig. 16. Temperatll rc va rialion al bottom SlIl 써cc of hol rolled

coil lIndcr water spray cooling al top surface as a fllnct ion of cooling time

25 20 15 10 5 O

O 3.0

Time, hr Fig. 15. Tcmperalllrc variation al slIrfacc of hot rollcd coil as

a fllnclion of cooling timc under slIcccssivc 3ir cooling HO min , watcl‘ c()oling 10min. air cooling 10min , waler cooling 2 min and air cooling 10 Ihc cnd con­ditions.

2.5 2.0 1.5 1.0 0.5 0

0.0

rapidly. When water spraying stops, surface temperatllrc increases due to the radia l heat Oow from hot core to cold sllrface region. The cooling curve calculatcd using isotropic conductivity shows highcr tempcra ture th“n cxperimental da ta du ring the whole proccss‘ becausc of I the large rad ia l thermal conductivity. On the other hand, orthotropic stress independent conductivity gives lower temperature due to the s111all radia l thcrmal conductivity The small radial thcrmal conductivity also Icads to thc slow incrcasc of tcmperature after the water spraying stops. In case of orthotropic stress dependent conduc-tivity, the ca lcula ted temperature shows rapid increase aftcr stop of water spraying, because thc large compres­sive radia l thermal stress occurs during water spraying.

Figure 16 shows the calcu!ated cooling curves, comparing with thc experimcn tal data1) measured a t bottom surface of hot rolled coil denoted as coil 3. The hot rolled coil stood vertica lly and water was spraycd to the upper surface. The cooling cu rve calculated from isotropic conductivity shows the lowest temperature and th at from o rthotropic strcss independent conductivity shows the highest tcmperature. Thc cooling curve from orthotropic stress dependcnt conductivity shows in tcr­mediate temperature and good agreement wi th the ex­perimenta l da ta. Figurcs 17(a)- 17(c) show thc tempcra-

in hot rolled coil undcr air coo1ing condi tion after 1 h of coo1ing which a re calculatcd using isotropic, orthotropic stress indcpcndent and orthotropic stress dependent conductivity, respectively. The isotherms in Fig. 14(a) show small temperature gradient in radial direction. T he small radia l therma l conductivity in orthotropic stress independent conductivity makes la rge temperaturc gradient in radial di rcction as can be seen in Fig. 14(b). In case of orthotropic stress dependent conductivity, radia l thermal conductivity becomes la rge in the region whcre the radia l thermal strcss is compressive and conscq ucntl y leads to small temperaturc gradient in radial directio n in outer region o f hot rolled coil as shown in Fig. 14(c).

Figurc J5 shows the calcllla ted cooling curves and comparing with thc expcrimental data 1) meas lIred a t sllrface of hot rolled coil denotcd as coil 2 under sllccessivc a ir cooli ng 80 min, water cooling 10 min, air cooling 10 min , water cooling 2 min and a ir cooling to thc end conditions. For the fi rst period of air cooling, thc cooling curves of calculatcd and measured data are similar to those as shown in Fig. 10. At the start o f water spraying starts, the surfacc temperature dccreascs

1268 @ 1998 ISIJ

ISIJ International, Vol. 38 (1998) , No. 11

T VALUE

1 f-I- +3.50E+02 2 1--1- +4.00E+ 02 3 1--1- +4.50E+02 4 1--1- +5.00E+02 5 뉴-j- +5.50E+02

Unils : oc

(a) (b) (c)

Fig. 17. Tcmpcralurc dis‘ribulions in hol rollcd coil undcr walcr spray cooling condition artcr 4 h of cooling calculatcd using (a) isolropic conductivily, (b) ortholropic Slrcss indcpcndcnl conduclivily and (c) orlholropic 씨rcss dcpendcnl conductivity.

ture distributions in hot rollcd coil after 4 h of walcr spray cooling which arc ωIculated using isolropic, o 1'thotropic stress independcnl and ortholropic slress dependent cond l1ctivily, respectively. 1t can be seen in Fig. 17(a) that the temperature gradient in axial direction

_. is the smallest in isolropic conductivily casc. T hc temperat l1re gradienls are intcrmcdiatc and the largest in orlholropic slrcss dependent and independent con­d l1ctivity cases, respectively as shown in Figs. 17(b) 17(c). Since thc thcrmal ∞nductivity in axial direction is samc for all cascs, lhc hcat flux in axia l dircction is the sma llest in isolropic conductivity case. Conseq uently thc tcm­perat l1re at lhe bollom s l1rfacc in isolropic condllctivity case is the lowest as shown in Fig. 16 to make ba lance the heat fl l1x from the boltom surface to ambient ai r with the heat flu x conducted to the bollom surface. On lhc other hand, the intermcd iate and the la rgest hcat flux result on the inlermcdiatc and lhc highcst temper“ture in orthotropic stress dependent and independent con­ductivity cases, respeclively.

6. Conclusion

A new cxprcssioll for thc cq ui va lent thermal conduc­tivity of the layered steel strips has been proposcd f1'om

J a unit layer model‘ Thc cquivalent thermal condllctivity can be given as a function of material properties, str ip thickness, surface characteristics of strip, normal com­prcssive stress and tempcralllrc. Thc modeled equivalent thcrmal conducti vity has been com pared with thc cx­perimental dala. Finite elcmcnt analyses for the cooling of hot rolled coil have been carried out under various cooling conditions using the cq llivalent thermal con­ductivity as thc thcrmal conductivity in radial dircction. A spccial calculation proccdurc llsing A BAQUS has bccn developcd wherc lhe radial thermal slrcss is takcn as lhe

1269

normal compressive stress. Thc calculated 1'es비ts have becn compared with data ca lculated using isotropic thermal conduclivity and ortholropic strcss indepenclcnt condllctivity and with ex perimental da la. The cooling curves calcu lated from lhc cqllivalcn l therma l co n­ductivity as the radial thermal conductivity show re“­sonablc cffccts of radia l therma l Slrcss and are in good agreemcnt with cxperimcnta l data.

Acknowledgments

T his work has been supported by Pohang Iron & Steel Co. LLd . lh rough Rcsearch fnslitulc fo 1' Aclvanccd Mate1'ia ls, Seolll National Univcrsily.

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<Q 1998 ISIJ