Analize Rolling Mills

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UDC 621.771.23.016.2

Analysis of Slab Temperature Change and Rolling Milldine Length in Quasi Continuous Hot Strip Mill Equippedwith Two Roughing Mills and Six Finishing Mills*By Natsuo HATTA,** un-ichi KOKADO,**Hiroshi NISHIMURA***and Keiji NISHIMURA***Synopsis

The length of the rolling mill line and the slab temperaturechange aretheoreticallycompared between he quasi continuous type equipped with 3rougher mills, of which the first one is reversible, and 6 finisher mills andthe quasi continuous4 passes type equipped with 2 rougher mills, of whichthe first is reversible, and 6 finisher mills for a mathematical model hat aslab is rolled rom the nitial thickness230 mm to the inal thickness 2.5 mm.

In the quasi continuous 4 passes type, the total length in the roughingprocesscan be shortened o about 60% comparedwith the case of the quasicontinuous ype. For a given reheating temperature, the slab temperatureat the entrance of the first finisher remains much higher in the quasi con-tinuous 4 passes type so that the reheating temperaturecan be reducedover100C. Finally, one conclusion s that the quasi continuous4 passes typeis muchprofitable not only or the shortness of the rolling mill line but alsofor the energy saving in the rolling process.I. Introduction

Hot strip mills have become large to receive theheavier slab weight and to meet the increased rollingvelocity, and the type of them was changed from thesemi continuous type to the full continuous type, sothat the total length of the rolling mill line becameremarkably long. Therefore, the hot strip mills builtrecently are the quasi continuous type, which areusually called the three quarters continuous type, inorder to shorten the rolling mill line. Generally,the quasi continuous type mills have 3 to 4 roughingmills, of which one is reversible, and 6 to 8 finishingmills in a rolling mill line.

The stand interval among these roughers is madeprogressively long in accordance with the decrease inthe thickness of the slab. Therefore, the slab tem-perature drop is more remarkable in the later stages.In a large hot strip mill, the length of the delay tablebetween the last rougher and the first finisher amountsto about 200 m, though it varies according to themaximum acceptable unit weight of the slab whichcan be handled in the mill. The slab temperaturedrop in such a long delay table is conspicuously large.

Usually, the slabs more than the initial thickness200 mm are rolled into 20 mm to 30 mm through 5to 6 passes in the roughing process, and then travelledto the finishing process. But, in the case of the hotstrip mill equipped with two roughers, of which thefirst one is reversible, four passes are performed inthe roughing process. In this case, the slab thicknesson the delay table must be large to a some extent, and

the slab temperature drop on the delay table can beconsequently reduced. Such a type hot strip mill isconsidered to be advantageous not only for the reduc-tion of the slab temperature drop but also the short-ness of the length of the rolling mill line.In this report, the temperature change of the rolledmaterial and the length of the rolling mill line areanalytically discussed for a mathematical model thatthe slab is reduced from 230 mm to 40 mm by fourpasses in the roughing process equipped with tworoughers and then rolled by the tandem type sixfinishers to the final thickness 2.5 mm.lI. Mill Arrangement in Quasi Continuous TypeHot Strip Mill

Figures 1 (a) and (b) show the arrangement of thetwo type mills : the quasi continuous type mill, whichis usually called the three quarters continuous mill,which is equipped with the three roughers (R1,2,3,R4,R5) and six finishers (F1, F2, F3, F4, F5, F6) here in(a), and the quasi continuous 4 passes type mill whichis equipped with two roughers (R1,2,3, R4) and sixfinishers (F1, F2, F3, F4, F5, F6) in (b).The size of the mill varies in accordance with themaximum acceptable slab unit weight. In both typemills, the first rougher R1,2,3 s the reverse mill, wherethree passes are performed.In the quasi continuous type (a), the table withthe length TL (6) is, so called, the delay table, andin the quasi continuous 4 passes type (b), the tablewith TL (5) plays a role of the delay table.TL (1) is the length between the exit of the reheat-ing furnace and the first rougher R1,2,3,and TL (4)is the length between R1,2,3 nd R4. Because the sameslab with the different length is twice travelled there,TL (1) and TL (4) are to set as follows :

TL(1) = ML(1)+ML(3)+2VOUT(3)+X(3) ...(1)TL(4) = ML(4)+2VOUT(4)+X(4) ...............(2)

where ML (1), ML (3), and ML (4) show the slablength before the first, the second, and the fourthpass, respectively; VOUT (3) and VOUT (4) indi-cate the exit velocity after the second and the thirdpass; 2 VOUT (3) is the distance, in which the slabtravels in 2 s with a velocity VOUT (3); X (3) is therequired distance, in which the slab is decelerated

* Originally published in Journal of The Japan Society or Technologyof Plasticity, 21 (1980), 230, in Japanese. English version receivedJune 11, 1980.** Department of Mineral Science and Technology , Faculty of Engineering, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto606.** * Graduate School of Department of Mineral Science and Technology , Faculty of Engineering, Kyoto University, Yoshida-honmachi,Sakyo-ku, Kyoto 606.

( 270 ) Technical Report


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at 0.4 m/s2 from the exit velocity after the second passto zero and X (4) is the distance, in which the slab isdecelerated at 0.4 m/s2 from the exit velocity at thethird pass to the entrance velocity at the fourth pass.Therefore, the length TL (1) and TL (4) respectivelyare not always the same between the quasi continuoustype and the quasi continuous 4 passes type, as thereduction schedule is different in two types.In the quasi continuous type, the table length TL(5) is set as shown in Eq. (2). Namely, the sum ofthe slab length ML (5) before the fifth pass, the re-quired distance X (5) for the deceleration, and thedistance 2 V0 UT (5), in which the slab travels in2 s with a velocity VOUT (5).

The length TL (6) of the delay table for the quasicontinuous type is the value calculated in a manneras for the determination of TL (4) or TL (5) plus10 m, which is the distance between the crop shearand the first finisher.In the quasi continuous 4 passes type, the lengthTL (5) of the delay table is the distance calculatedwith the similar method as for the determination ofTL (4) plus 10 m, which is the distance between thecrop shear and the first finisher.Table 1 indicates the specifications in the roughingprocess for the quasi continuous type and the quasicontinous 4 passes type hot strip mill, where the maxi-mum acceptable slab unit weight is 30 t/m, 20 t/m,

Fig. 1. Mill arrangement in quasi continuous typemill line (a) and in quasi continuous 4passes type (b).

Table 1. Reduction schedule, table length and material length of each standmills.

interval in roughing process of both type


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and 12 t/m. As shown in Table 1, the slab is reducedfrom the initial thickness 230 mm to the bar thickness25 mm through 5 passes in the roughing processin the quasi continuous type and to 40 mm through4 passes in the quasi continuous 4 passes type.The length from the reheating furnace to the firstfinisher F1 differs remarkably in two type mills.Namely, the length in the quasi continuous 4 passestype can be geometrically shortened to about 60%of the case of the quasi continuous type mill.It is concluded that the quasi continuous 4 passestype mill is considered to be very profitable forshortening the total length of the rolling mill line. Asit is assumed that the final thickness is set at 2.5 mm,and that the exit velocity at the final finisher F6 is12 m/s, the entrance velocity at the first finisher F1should be 1.2 m/s in the case of the quasi continuoustype and 0.75 m/s in the case of the quasi con-tinuous 4 passes type in consideration of the reduc-tion schedule as shown in Table 1.In both type mills, the slabs travelled through thedelay table to the finishing process are rolled by thetandem type six finishers (F1, F2, F3, F4, F5, F6) in-stalled at an interval of 5.5 m, respectively. Thereduction schedules in the finishing process will bementioned in the next section.III. Slab Temperature Change in Quasi Con-tinuous Type and Quasi Continuous 4 PassesType1. Slab TemperatureChange in the Roughing ProcessIn a hot strip mill, the security of the final rollingtemperature of the sheet steel, which is determinedaccording to the metallurgical requirements of thesteel, is one of the most important factors to guaranteethe stable mechanical quality of the products. Con-sequently, it is necessary to know the state of theslab temperature change in the quasi continuous 4passes type mill. It was concluded in the previoussection that this mill has a great advantage for shorten-ing the length of the rolling mill line. However,there is no more ground to discuss about this millwithout examining whether the slab can be rolled inthe allowable temperature range.The temperature of the slab travelled on each standinterval drops by the radiation and convection to theair and by the heat exchange with high pressurewater jets of the descalers at each roughing pass.The calculation method of the slab temperaturechange has been already reported1'2? : it can be cal-culated as an unsteady problem by rewriting the onedimensional differential equation for transient conduc-tion in the finite difference form assuming the heattransmission only in the slab thickness direction.

The calculation method of the slab temperaturechange during contacting with rolls is as follows : inthe case of the thick slab as in the roughing process,the calculation of the slab temperature change isperformed by taking only the heat generated by theplastic deformation into consideration in disregard ofthe heat loss by the contact with the rolls, and of theTechnical Report

friction heat generated by the surface slip betweenthe slab and the rolls, because the region of the influ-ence of the contact heat loss, and of the friction heatis considered to be limited only in the slab surfacenearby.The pass schedule used for the calculation is asshown in Table 1.The other assumptions and conditions for thecalculation will be indicated as follows the rolledmaterial is the killed steel containing 0.08% carbon,and its specific weight r=7 800 kgf/m3. As the heatconductivity A kcal/mhC and the specific heat c kcal/kgC differ in the kind of the steel and the tempera-ture, their values indicated in the special reportpublished by ISIJ were used in the calculation.3~The heat transfer coefficient a for the water jet bydescaler is 1 000 kcal/m2hC empirically, and the de-scaling time at each pass is 4 s. The temperature ofthe water and the air is 20C. Only the naturalconvection is taken into account, and the heat transfercoefficient an=7.2 kcal/m2hC was used.The value of the emissivity e is given by the formulathat has been already reported4~

= H(I)[0.58(H(1)/H(I)-1)+0.8]/H(1) ......(3)where, H (1) : the initial thickness of the slabH (I) : the thickness at a given point.For the calculation of the heat generated by theplastic deformation at each pass, it is necessary to givethe flow stress k (kgf/m2) of the rolled material. Theflow stress kf to be a function of the strain eN and themean temperature BmC s given as Eq. (4)5~

kf =1.15.1.7 1.15.1.7.e2.xp B 2 850,,+273 (4)The slab of the initial thickness 230 mm is reducedto the bar thickness 25 mm through 5 passes rough-ing in the quasi continuous type, whereas through 4

passes roughing to 40 mm in the quasi continuous 4passes type. Because the pass schedule is different inboth types, the slab temperature change in theroughing process in both type mills cannot be strictlycompared each other. However, it will be valid tograsp the amount of the temperature drop in theroughing process, i.e., the relation between thereheating temperature 6o and the temperature 8Fbefore the descaler installed at the first finisher F1 inboth types.Figure 2 shows the relation between 0o and eFfor the various maximum acceptable slab unit weightin the quasi continuous type mill. As shown in Fig.2, OF increases almost lineally with O. For a givenreheating temperature O, the temperature eF justbefore the descaler of the first finisher F1 is higher asthe maximum acceptable slab unit weight decreases,and the increasing rate of eF for 0 is apt to decreaseas the maximum slab unit weight increases.Figure 3 indicates the relation between Bo and eFin the quasi continuous 4 passes type mill. From thisfigure, OF ncreases almost lineally with 6o in this case,too. The increasing rate of OF for Bo s considered to


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be invariable, though the maximum acceptable slabunit weight changes.Z. Slab TemperatureChange n the Finishing Process

The final rolling temperature 0F6must be controlledto be above the A3 critical point, which is determinedaccording to the metallurgical requirements of thesteel, in order to guarantee the good mechanicalquality of the products. Consequently, the heat con-trol in the finishing process is the most importantfactor for the mechanical quality of the rolled prod-ucts.Here, first, the amount of the temperature dropdue to change in the descaling time before F1 must beexamined, because the thickness HFo mm at the en-trance of the first finisher is so small that the tem-perature difference between the temperature eF be-fore descaling at F1 and the temperature 0F1 afterdescaling changes by the descaling time.Figure 4 shows the relationship of the amount ofthe temperature drop due to change in the descalingtime at the entrance of the finishing train for the twokinds of the reheating temperature Bo=1 280C and1 120C. This calculation was performed on the as-sumption that the heat transfer coefficient a of thewater jets by the descaler is 1 000 kcal/m2hC. Theslab thickness HFo at the entrance of the finishingtrain is changed between two type rolling mills asmentioned in Table 1: HF0= 40 mm in the quasicontinuous 4 passes type, and HFo=25 mm in thequasi continuous type. For a given descaling time,the mean temperature in the case of the bar thick-ness HFo=25 mm drops at the rate of one and ahalf for the case of HFo=40 mm, and almost no signifi-cant difference in the temperature drop may be ob-

served, even though the reheating temperature Bochanges.Figure 5 shows the state of the temperature dropfrom eF before descaling to F1 after descaling due tochange in descaling time at the first finisher in thecase that the maximum acceptable slab unit weightis 30 t/m, eo=1200C, and HFo=40 mm in the quasicontinuous 4 passes type. The upper line in thisfigure indicates the state of the temperature change ofthe slab top end till the slab top end travels to thefirst finisher F1 from a position before 6 s. In thiscase, the calculation of this temperature change ofthe slab top end is performed taking only the convec-tion and the radiation into consideration. While thelower line shows the state of the temperature change ofthe slab top end on the assumption that the descalingis performed in the time range of 6 s. When nodescaling is performed at the entrance of the first

Fig. 2. Relation between reheating temperature t9o andtemperature OF before descaler of the first finisherdue to change in maximum acceptable slab unitweight in quasi continuous type mill. (Reductionschedule is shown in Table 1.)

Fig. 3. Relation between reheating temperature Bo andtemperature OF before descaler of the first finisherdue to change in maximum acceptable slab unitweight in quasi continuous 4 passes type mill.(Reduction schedule is shown in Table 1.)

Fig. 4. Amount of temperature drop due to changedescaling time at the first finisher for two kindsreheating temperature (Bo=1 280 and 1 120C)both type mills.

inofin


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finisher F1, then OF BF1=1062C because the de-scaling time is zero. Or, when 4 s descaling is done ,then OF=1 067C, and consequently 0F1 can be easilyfound : F1=1027C, i.e., the amount of the tempera-ture drop is 40C. Therefore, the determination ofthe descaling time is important in operation for theenergy saving.The calculation of the slab temperature in thefinishing process can be made by means of the similarmethod as for the case of the roughing process. How-ever, in the case that a slab is in contact with rolls inthe finishing process, the heat generated by theplastic deformation, the heat loss by the contact withrolls, and the friction heat generated by the slip be-tween the slab and rolls should be taken into account,because the slab thickness in the finishing process isthin compared with the case of the roughing process.The temperature change of the slab during contactingwith rolls can be calculated not as an unsteady but asa quasi steady problem : these three factors may betreated respectively. The details about the calcula-tion method may be found in Ref. 6).The various heat constants and the flow stress neces-sary for this calculation were used as mentioned inthe case of the roughing process.The descaling time at the entrance of the firstfinisher is set at 2.5 s in the quasi continuous typemill, and at 4.0 s in the quasi continuous 4 passestype mill to make the mean temperature drop of the

slab by the descaler in both type mills almost equal .The roll diameter from F1 to F6 is 680 mm , re-spectively. The temperature of the rolls before thecontact with the material is 50C. The heat conduc-tivity and the specific heat of the rolls are : ~r=21.6 kcal/mhC, cr=0.128 kcal/kgC. The frictioncoefficient between the rolls and the material is:u=0.3.In the finishing process, the pass schedule differentby each type mill is applied : in the quasi continuous4 passes type mill the slab is reduced from the barthickness HF6=40 mm to the final thickness 2.5 mm,and in the quasi continuous type, from HFo=25 mmto 2.5 mm.Table 2 shows two kinds of pass schedules in thefinishing process applied to each type mill. The passschedule in (a) is the case that the reduction ratio isinvariable each for the first five finishing mills F1to F5, and 20% for the last finishing mill F6. Whilethe other pass schedule shown in (b) is the case thatthe reduction ratio decreases lineally with the laterpass and is finally 20% for the last finishing mill F6.Hereafter, the pass schedule indicated in (a) and (b)will be called (a) type and (b) type pass schedule,respectively.Figure 6 shows the relationship between the en-trance temperature OF and the final rolling tempera-ture 0F6 of the slab top end in the case that the slabis rolled from the bar thickness HFo=40 mm to thefinal thickness HF6=2.5 mm. Consequently, this caseis for the quasi continuous 4 passes type mill. Thisfigure shows that the final rolling temperature BF6increases almost lineally with OF in both (a) and (b)type pass schedules. The increasing rate of BF6 toOF is considered to be almost equal for the both passschedules : (a) and (b) type. The rolling in (a)type pass schedule is more profitable for the savingenergy than that in (b) type pass schedule. Forexample, the entrance temperature of the finishingprocess necessary to gain BF6 830C at the slab topend must be as follows: eF=910C in (a) type passschedule and OF=930C in (b) type.Figure 7 indicates the relationship between BF6and OF when the slab is reduced from HFo=25 mm toHF6= 2.5 mm in the quasi continuous type mill. This

Fig. 5. Temperature drop from BF before descalinafter descaling due to change in descalingthe first finisher.

g to BF1time at

Table 2. Pass schedules in f inishing process for both type mills.


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figure shows that 0F6 ncreases lineally with OF n both(a) type and (b) type pass schedule. The degree ofinfluence of eF on F6 due to the difference of the passschedule is considered to be a little smaller for thecase of the quasi continuous 4 passes type mill.Figure 8 shows the reheating temperature Borequired to gain the final rolling temperature F6=830C of the slab top end for the various maximumacceptable slab unit weight G : it is assumed in thiscase that the A3 critical temperature of the rolledmaterial discussed here is 830C. These curves arederived from Figs. 3 and 6 in the quasi continuous4 passes type mill and from Figs. 2 and 7 in the quasicontinuous type mill. This figure shows that thereheating temperature can be lowered more over100C for the various slab unit weight in the quasicontinuous 4 passes type than in the quasi continuoustype. The reheating temperature in the case ofrolling slabs by the (a) type pass schedule in the finish-ing process can be lowered by about 30C in the quasicontinuous 4 passes type and by about 15C in thequasi continuous type than in the rolling by the (b)type pass schedule. Therefore, the rolling by the(a) type can be regarded to be more profitable thanby the (b) type pass schedule for the energy saving.

In future, hot strip mills in Japan will be introducedto the direction of the energy saving in rolling processand the shortening of the total length of the rollingmill line. In the semi continuous type mill, therequired total mill line length can be shortened verymuch and the mill arrangement is very simple.However, it is insufficient in the two points of therolling capacity and the energy saving in comparisonwith the case of the full or the quasi continuous typemills. In fact, the semi continuous type mills havenot been constructed lately. It is suggested on thebasis of the mill layout that the reforming from thesemi continuous type mills built already into the quasicontinuous 4 passes type mills discussed in this reportis not considered to be so difficult.IV. Conclusion

The length of the rolling mill line and the slabtemperature change are theoretically compared be-tween the quasi continuous type mill equipped with3 rougher mills, of which the first one is reversible,and 6 finisher mills and the quasi continuous 4 passestype equipped with 2 rougher mills, of which thefirst one is reversible, and 6 finisher mills for a mathe-matical model that a slab is rolled from the initialthickness 230 mm to the final thickness 2.5 mm. Asa result, the length of the rolling mill line includingthe delay table in the quasi continuous 4 passes typecan be shortened to about 60% of the case of thequasi continuous type mill, and for a given slab unitweight, the reheating temperature can be loweredover 100C in the quasi continuous 4 passes type millthan in the quasi continuous type mill. Other mainresults clarified by this study are as follows :

(1) Indeed, the increase of the reheating tempera-ture 8~ results in that of the entrance temperature BFat the first finisher F1, but in the quasi continuoustype mill, the effect lessens as the slab unit weight

Fig. 6. Relation between entrance temperature OF andfinal rolling temperature 0F 6 in finishing processof quasi continuous 4 passes type mill.

Fig. 7. Relation between entrancefinal rolling temperature 0F6quasi continuous type mill.

temperature 9 F andin finishing process of

Fig. 8. Reheating temperature 6o requiredrolling temperature &F6 830C formum acceptable slab unit weight Gmills.

to gain finalvarious maxi-in both type


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increases, while in the quasi continuous 4 passes typemill, eF increases with 80 at a nearly constant rate forthe various slab unit weight.(2) It results in the definite disadvantage for theenergy saving to make the descaling time needlesslylong, because the thickness of the slab is small at the

entrance of the first finisher. Therefore, the determi-nation of the descaling time must be considered to bevery important in operation.(3) The final rolling temperature F6 increasesalmost lineally with the temperature eF at the entranceof the finishing train in both type mills.(4) The relation between OF and 0F, changes bythe pass schedule : the (a) type and the (b) type passschedule.(5) For a given entrance temperature of at thefirst finisher, the final rolling temperature BFsremainshigher in the case of rolling the slab from the barthickness HFa=40 mm to the final thickness 2.5 mmthan in rolling from 25 mm to 2.5 mm.(6) It is suggested on the basis of the mill layoutthat the semi continuous type mills built already mayreformed into the quasi continuous 4 passes typemills discussed in this paper.

REFERENCES1) J. Kokado: Proc. ICSTIS, II, Suppl. to Trans. ISIJ, 11

(1971), 750.2) F. Hollander: "A Model to Calculate the Complete Tem-perature Distribution in Steel during Hot Rolling ", Int'l.Conf, on " Mathematical Models in Metallurgical ProcessDevelopment ", 151, London (1969), 46.3) Experiment and Calculation of Conduction Heat in Con-tinuous Slab Heating Furnace, ed. by ISIJ, (1971).

4) J. Kokado and N. Hatta: Trans. ISIJ, 19 (1979), 744.5) J. Kokado, N. Hatta and S. Yoshino: Proc. Jap. Spring

Conf, for Tech, of Plasticity, (1978), 114.6) N. Hatta and J. Kokado: J. Japan Soc,for Tech, of Plas-

ticity, 21 (1980), 59.AppendixCalculationMethod of TemperatureChangeof Rolled Mate-rial in the Finishing Process1. Temperature Drop by Heat Conduction through Con-tact between Rolled Material and RollsAssuming that no temperature gradient is in thethickness direction of a rolled material and in theradius direction of rolls before contact, the tempera-ture 00C of surface during contact is given by

Alai i2eol+A2a21i28o200= A1al /2+A2a21/2 ............( -1)where, e01: Temperature of rolled material beforecontact (C)802: Temperature of roll before contact (C)

Al Heat conductivity of rolled material(kcal JmhC)A2: Heat conductivity of roll (kcal/mhC)al: Thermal diffusivity of rolled material(m2/h)a2: Thermal diffusivity of roll (m2/h).The temperature of the rolled material oC during

contact is given by

= eo+(eol-O) erf x- t............(A-2)halwhere, x : Depth from surface of rolled material inthickness direction (m)

: Contact time (h).Therefore, the amount of the heat 4QQ~ cal/m2removed from unit surface area of the rolled materialto a roll in the contact time is as follows :2 ~aQ~ - Alao dt = tA1(ool-eo) ..(A-3)ax x-o

Consequently, the amount of the temperature dropd o~C is calculated by the following formula :4o~ = 24Q~s

where, S: Contact surface area (m2)V: Volume of the roll bite (m3)r: Specific weight of rolled material (kgf/m3): Specific heat of rolled material

kcal/kgC).2. Temperature Rise by Friction Heat between Rolls and

Rolled MaterialThe friction work per unit volume W kg-m/m3 ofthe rolled material is given as follows :W f = W F dt .....................(A-5)W = pp(O).yr ........................(A-6)

hcos~dt = d~ ..................(A-7)hnwcos ~L5nwhere, i : Frictional coefficient

p(c15) Roll pressure at a given roll bite angleVr: Relative velocity (m/s)h : Thickness of rolled material at a givenroll bite angle 0w Angular velocity

n : Neutral anglehn : Thickness of rolled material at neutralpoint.F is the contact area per unit volume of the rolledmaterial in roll bite and a function of h and 0, i.e.,

F = 2/(h cos 0) ...................(A-8)Putting Eqs. (A-6), (A-7) and (A-8) into Eq. (A-5)gives

Wf = 2jp(q5).vr d . ............(A-9)qahnwcos 0nThe relative velocity or m/s is

v Rw 1 hn cos 0=-................(Al0)i cos 0In hot rolling, the temperature of the rolls is by

far lower than that of the rolled material. Therefore,the value of heat conductivity of the rolls is considered


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to be about twice as large as that of the rolled material.Consequently, the distribution ratio of the frictionheat is considered to be 6O'70% for the rolls and3O40% for the rolled material. Finally, the tem-perature rise 40fC of the rolled material by thefriction heat is approximately given by

4e f = W A/(3rc) ..................(A-11)where, A indicates the heat equivalent of mechanicalwork.3. Temperature Rise by Plastic DeformationThe amount of work Wp kg-m/m3 by the plasticdeformation per unit volume of the rolled material isgiven by

E

where, kf : the flow stress (kgf/m2).The logarithmic strain s ish2+2R(1- cos ~b)= In =1n- - -- - - . ...(A-13)2Therefore,

dr _ 2R sin ~b A14)dO h2+2R(1 cos ~b) . ............(-Hence,

Wp 2R sin ~k2+2R(1- cos O) d ....'.(A-15)where, h2: the thickness of the rolled material afterrolling

R : the roll radius.The temperature rise 4BpC of the rolled material bythe plastic deformation is

4ep = WpA/(1. c) ................(A-16)From the above, the amount of the temperature

change 48C of the rolled material during contactwith rolls is as follows :

40 = 4ep+48 f-40C . ............(A-17)The temperature 82 of the rolled material at the exitof the each finishing pass can be calculated by adding4e to the temperature 81 at the entrance, i.e.,

02 = 01+4B ................... (A-18)

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Analize Rolling Mills