A New Model for the Prediction of the Dog-bone Shape in Steel Mills

Transcription

1 ISIJ International, Vol. (), No. 6, pp. 9 7 New Model for te Prediction of te Dog-one Sape in Steel Mills Duckjoong YUN, ) Dongun L, ) Jaeoo KIM ) and Sangmoo HNG ) ) Department of Mecanical ngineering, Poang University of Science and Tecnology, San, Hyoja-dong, Nam-gu, Poang, Korea. -mail: smwang@postec.ac.kr ) Hyundai Heavy Industries Co., Jeona -dong, Dong-gu, Ulsan, Korea. (eceived on Novemer, ; accepted on January, ) Precision control of te widt of slas, plates and strips, is vital for product quality and production economy in steel mills. common practice in te production line is to perform vertical rolling and reduce te widt in te rouging mill. However, te formation of so called dog-one at te edge of te sla would affect te final widt after oriontal rolling tat follows. Terefore, it is essential to predict and control te dog-one sape. In tis paper we present a new model for te prediction of te dog-one sape during vertical rolling in rouging mills. Te model is developed on te asis of te minimum energy principle for a rigid-plastic material and a tree-dimensional admissile velocity field. Te predictions accuracy of te proposed model is examined via comparison wit predictions from finite element simulation and also wit experimental data. KY ODS: vertical rolling; finite element metod; dog-one sape; minimum energy principle; admissile velocity field.. Introduction ouging rolling, conducted in plate mills and ot strip mills, is to reduce te tickness of te slas prior to finis rolling. In addition to tat, simultaneously performed during rouging rolling, in general, is to reduce te widt of te slas so as to control te widt of te product as desired. ile te tickness reduction is acieved y oriontal rolling, te widt reduction is acieved y vertical rolling, or edge rolling, as illustrated in ig.. During vertical rolling, te sla is compressed in te widt direction, and as a result, partial tickening occurs near its edges, wic is known as dog-one sape, as illustrated in ig.. Te formation of te dog-one gravely affects te final widt after oriontal rolling tat follows, and terefore, its prediction is vital for sound process control. In te past, several models were proposed for te prediction of te dog-one sape. ) Taoe, ) et al. derived a matematical formula ased on te experimental data otained y Siaara, et al. ) Later, it was modified y Ginurg. ) Tose formulas were expressed in terms of te dimensions of te material to e rolled and roll geometry. Okado, et al. ) conducted a series of plasticine experiments in laoratoryscale mills, from wic tey came up wit a model in wic ig.. Vertical rolling (dge rolling) and oriontal rolling in a rouging mill. ig.. Te cross-sectional sape of te sla efore and after edge rolling. 9 ISIJ

2 ISIJ International, Vol. (), No. 6 te dog-one sape is defined y four main parameters, eac of wic is given in terms of initial tickness of te sla and te amount of widt reduction. However, tese formulas are empirical, yet few attempts ave een made to predict te dog-one sape on te asis of te teory of plasticity. e feel tat in order to precisely predict te sla widt in a rouging mill, te model for te prediction of sla widt after oriontal rolling sould e capale of precisely taking into account te effect of edge rolling tat preceded oriontal rolling, or, more precisely, te effect of te detailed aspect of te dog-one profile. It is in tis regard tat we elieve it is essential to develop first a model wic can precisely predict te dog-one profile. Te finite element metod may e te est coice for tis purpose. However, it is not suitale for on-line prediction and control, due to te fact tat its application would require consumption of excessive computation time. or instance, finite element simulation of tree-dimensional deformation of a sla in rolling, using a steady-state formulation, may easily take several ours of computation time, and muc more if a non-steady formulation is employed. Instead, te approac adopted in tis paper is one ased on te minimum energy principle for a rigid-plastic material, comined wit an elegant tree dimensional admissile velocity field. vailale are te velocity field proposed y researcers 6 8) in association wit te application of te minimum energy principle to te analysis of plastic deformation of oriontal rolling and sape rolling. Unfortunately, tey are ardly applicale to vertical rolling, due to its extreme nature of deformation inomogeneity, and it is in tis regard tis paper presents peraps for te first time an admissile velocity field for te analysis of edge rolling. Te prediction accuracy of te proposed model is examined via comparison wit predictions from finite element simulation and also wit experimental data.. Minimum nergy Principle (Upper Bound Teorem) minimum energy principle as applied to a rigid-plastic material states tat for all admissile velocity fields u i, te actual field u i minimies n φ( ui ) σε dω+ λ ut utd dγc + k Δu dγl... () Ω Γ ls C Γl S s sown in ig., a sla is rolled troug a pair of vertical flat rolls of radius, and sla s widt is reduced from to Te axes x, y, and are cosen so as to represent te rolling direction, vertical direction, and te direction across te sla, respectively. Note tat, in te ite one, te tickness of te sla may e represented y (x,). Tus, te inlet tickness canges into (l,) after rolling, were l is te projected lengt of te roll-sla contact arc. Ten, te first term of q. (), wic is te energy dissipation due to plastic deformation, may e calculated from l ( x) x (, ) p σ ε, ε, T ε dy d dx... () S It is to e noted tat σ ε, ε, T, wic may e calculated from te flow stress curves, for example, otained y Sida, ) is not known since ε and ε, wic represent actual effective strain distriution and actual effective strain rate distriution in te ite one, are not known. ε may closely e approximated y te effective strain distriution predicted y -D elementary rolling teory wen calculating σ, or ε ( x) ln... () lso, assuming tat te admissile velocity field minimiing φ( u i ) under te condition ε ε may e sufficiently close to te actual velocity field, we approximate ε y ε ε... () Te second term, wic is te energy dissipation due to friction, may e calculated from l xx (, ) f μσ n ut utd dydx... () It is assumed tat te frictional stress is constant over te roll-sla contact surface, or, σ n /l, were denotes te roll force per unit sla tickness. Te magnitude of velocity of te sla relative to te roll at te roll-sla contact surface is given y were ig.. Definition sketc of te ite one in edge rolling. ut utd ( ux utdcosθ) + ( uy) + ( u utd cos θ )... (6) xl cosθ... (7) Te tird term, wic is te energy dissipation due to velocity discontinuity at te roll entry and at te roll exit, ecomes d y k u u dyd + x (, ) k u u dyd l + ( y) + ( ) x l... (8) were k σ / In order to derive an admissile field we consider te lateral strain defined y ISIJ

3 ISIJ International, Vol. (), No. 6 dw d da d d... (9) were da d denotes te cange in te lengt of an infinitesimal segment, and ww(x,) denotes te lateral displacement, as sown in ig.. rom te incompressiility condition, V d ux ( x, )... () ( x, ) da were V is inlet velocity of te sla. Sustituting q. (9) into q. () yields V dw ux ( x, ) x, d e also note tat u( x, ) ux( x, ) rom wic we otain u ( V x, ) x (, ) dw( x, ) dx dw d... ()... () dw dx... () Sustituting qs. () and () into te incompressiility condition u y u... () x x u (, ) ( x, ) y x and using te oundary condition u y at y, we otain w w w w x (, ) x ( x, ) x y x... (). Matematical epresentation of te Dog-one Sape Te admissile velocity field derived aove as two unknown functions. Tat is, (x,), wic may e called te dog-one sape function, and te lateral displacement field w(x,). In order to descrie (x,) we may divide te ite one into two ones, as sown in ig.. one I (<<(x) ) : te stem part of te dog one u ( x, y, ) y w V x ( x, ) ( x, ) + w x one II (x) <<(x)) : te end part of te dog-one, wic is round. ere is a constant and (x) is te sla widt in te ite one at x, wic may e given y (see ig. ) ( x) + xl... (6) Noting tat te dog-one would increase ot along x and direction, we may propose te dog-one function for one I as follows: were K, K and are constants, and (x,)k +K q(x)... (7) q(x) (x)... (8) lso, we may propose te dog-one function for te end part as follows: x, K q x K x x + + K x x (9) were constants K and K, as well as functions K (x), K (x) are to e determined. Note tat at te end part te sape is symmetric wit respect to te line ( x). Process simulation wit a tree dimensional rigid-viscoplastic finite element metod 9) reveals tat tis line is a close approximation of te line along wic te peak in te dog-one profile appears in te ite one, as sown in ig.. It is clear tat we ave in total eigt constants and functions (,K,) tat ave to e determined. On te oter and, te conditions to e satisfied are, from ig., K (,)... () (l, ) r... () (,)... () l,... () ig.. Definition of one I and one II of te dog-one profile, at roll exit. ig.. Te line along wic te peak in te dog-one profile appears in te ite one. Predictions from M and te present model. ISIJ

4 ISIJ International, Vol. (), No. 6 (x,(x) ) (x,(x) )... () ( x ), x, x x... () were is te inlet tickness, is te maximum eigt of te dog-one, and r is te eigt of te dog-one at te edge of te sla. Note tat ot and r denote te values at te roll exit. rom qs. (), (), (), and (), we otain K... (6) r K... (7) ql ()( ) lso, from qs. () and (), we otain ( x ) K ( x) ( r ) ql () ql (). Determination of te Lateral Displacement ield... ()... () or te calculation for te lateral displacement, we may assume tat u x(x,) appearing in q. () may e replaced y V. inite element simulation reveals tat te approximation is acceptale, wit te maximum error eing % for te case investigated, as sown in ig. 6. Ten, from q. () we otain x, w( x, ) d... () K... (8) K... (9) ql () ( ) ( x) r ()( ) ql K ( x) r ()( ) ( x) ql () ql ( ) ( x) + r ()( ) ql It follows tat, for one I r w ( x, ) + nd for one II K + q( x) K w ( x, ) qxk ( x) ( x) ( x) + ( x ) () q x ql +... () qxk ( x) x + ( x ) + C ( x)... () rom te oundary condition w (x,(x) )w (x,(x) ) it may e sown tat r C ( x) + () qx ql... () lso, from te condition w (l, ) it may e sown tat I6( I I)+ I7( I I)+ I I I I I... (6) + + were + ( x ) ( ( x) ) + K + q( x) K ( ( x) ) qxk ( x) + { } x ( x) ( x ) + ( x) ( x) + ( x) { ( x) } { ( x) } { ( x) } qxk ( x) + ( x) x { } + ( x) { ( x) } { } ( x) { } ig. 6. Te distriution of u x(x,) in te ite one, predicted y M. Inlet velocity of te sla V. I (7) ISIJ

5 ISIJ International, Vol. (), No. 6 I +... (8) I ( ) ( ) + ( ) + ( )... (9). Determination of te Parameter... ()... ()... ()... () ssuming tat te material is plain caron steel, a matematical expression descriing te effect of te rolling conditions on te parameter may e given y f(μ,,,,ω,t,c,)... () were and ω denote te roll radius and te roll angular velocity, respectively, and T and C denote te sla temperature and te caron contents, respectively. ssuming tat te coefficient of friction is given, q. () may e written as, in a dimensionless form,... () f r T c c C ω,,,,, were c and c are proper units introduced to render T and ω dimensionless, and r... (6) Using te tree dimensional finite element code developed for te prediction of plastic deformation occurring during rolling, 9) a series of simulation is performed to investigate te effect of eac of te variales in q. (). Te rolling conditions selected for tis investigation are summaried in Tale. detailed aspect of te formation of te dog-one due to edge rolling, as predicted y simulation, is illustrated in ig. 7. It is revealed y te finite element simulation tat increases wit te increase of r and, wile it is I ( ) ( ) + I r + I I 7 6 ( )+ r r + r r + Tale. σ r Process conditions for simulation of deformation in edge rolling. kn/mm Process conditions flow stress for.% ~.% caron steel, see eference [] decreased wit te increase of, as sown in igs. 8. lso revealed is tat is little affected y te cange of T, ω and C, implying tat te effect of te flow stress c c caracteristics of te material on te formation of te dogone may e negligile. matematical formula may e derived from te regression of te data otained from a series of simulation. Te result is, for μ. 6 μ. mm mm 6 mm 6 T C 8 ω rad/s 6 Total numer of sets : ig. 7. T c ω c ormation of te dog-one due to edge rolling, as predicted y M r. 6 r r + 8. r 6. 6r 9. 76r 9. 86r r... (7) ISIJ

6 ISIJ International, Vol. (), No. 6 ig. 8. Te effect of r on. ig.. Te values of, comparison etween teory and M. ig. 9. Te effect of on. plain caron steel may e represented in a form similar to q. (), or π(μ,c,t,ω,,,, )... (8) Suppose tat, in actual rolling, te widt of a sla is reduced from to. Now, for te same rolling geometry and te same sla consider a ypotetical mode of rolling, as sown in ig.. urter, assume tat te temperatures remain constant during rolling. Ten, e may define roll force, for ideal rolling in wic eac segment of te sla is uniformly compressed under te plane strain condition wile passing troug te roll ite, and no friction is present at te roll-sla interface. Te ideal roll force per unit tickness of te sla, may e given y φ σ ε, ε, T cosφdφ... (9) l φ sin... () ε... () ln φ ε tan ω ( φ)... () ( φ) ( cosφ)+... () ig.. Te effect of on. Te prediction error of te aove equation, wic is valid witin te range of rolling conditions given in Tale, is found to e less tan percent, as sown in ig.. 6. Model for te Prediction of oll orce in dge olling Te actual roll force per unit tickness of te sla for were φ is ite angle. rom qs. (8) and (9), te actual roll force may e expressed in a dimensionless form were C T srt... () ω π μ,,,,,, c c ld s + r t... ()... (6)... (7) ISIJ

7 ISIJ International, Vol. (), No esult and Discussion r Te values of as well as predicted y te model for various cases agree well wit predictions from simulation, wit te differences eing witin percent, as illustrated in ig.. Te predicted dog-one sapes are in good agreement wit te predictions from simulation, as sown in igs. and. Te predicted dog-one sape also agrees favoraly wit te Siaara s experiment, ) as sown in ig., indicating te validity of eiter te present model or M. rom tese figures, it is seen tat te dogl d... (8) were l d is te ite one lengt. Note tat c and c are introduced to render T and ω dimensionless, respectively. In tis paper, c C and c rad/sec are selected. rom a series of simulation, it was revealed tat is gravely affected y te dimensionless parameters descriing te rolling geometry and friction, ut very little y oter parameters. It follows tat, assuming tat for μ., wic is a close approximation of te actual friction coefficient for ot rolling, srt... (9) π (,, ) matematical formula may e derived from te regression of te data otained from a series of simulation. Te result is, t r s t rt r +. 77st sr s. 78rt. 7rt. 76st 86. 7srt 7. sr 6. 99st 87. sr+. 6r t +. 6srt +. 8sr t +. s t s rt sr 6. sr t. 98s rt 8. 6s r t s r t... (6) Te prediction error of te aove equation, wic is valid witin te range of rolling conditions descried y s.., r..9, and t, is found to e less tan percent. are sufficiently close to eac oter, stop. Te solution is d d. If not, go to Step-. or te calculation of f( d) φ( u i ), all te integrals appearing in q. () are calculated numerically, using ten points Gauss Quadrature formula. it r tus found, may e calculated from q. (6), using q. (7) for te estimation of te parameter. Ten, te dog one sape may e otained from qs. (7) and (9). 7. Numerical Procedure it all te development so far, te functional φ( u i ) to e minimied (see q. ()) as only two unknown variales, tat is, and r, since is given y q. (7), and is given y q. (6), as a function of r. it 8 eing selected for te present investigation, our model is reduced to e a prolem of finding te optimal value of r wic minimies φ( u i ).Te optimiation prolem may e solved y a tecnique known as quadratic curve fitting, ) as follows. Step-. Let te function to e minimied e f (d). Set a sufficiently large range for d, tat is, assume d L and d U, d L < d < d U. Step-. Let d i e an intermediate point in te interval d L < d < d U. Calculate f(d i), f(d L), and f(d U). Step-. Compute te coefficients a, a, a of te quadratic function q(d) a + a d + a d wic passes troug te tree points found in Step-. Compute d a / a at wic q(d) is minimied. lso, compute f( d). Step-. Let us consider te case in wic d < i d and f( d (ll te oter cases are not considered ere, i ) < f( d) since tey are treated essentially in te same manner). Ten, oviously, d, te point at wic te minimum value of f (d) is acieved is d < d <. It is in tis regard tat an L d improved guess for d L, d U, d i may e made. Tat is, d L d L, d, and d i d U d i. Step-. If two successive estimates of f( d) ig.. Te values of, comparison etween teory and M. ig.. Te dog-one sapes, teory and M. ntry sla widt 7 mm, exit sla widt 7 mm, sla tickness mm, roll radius mm, sla temperature C, roll velocity.9 rad/sec,.% plain caron steel. ISIJ

8 ISIJ International, Vol. (), No. 6 ig.. Te dog-one sapes, teory and M. ntry sla widt 7 mm, exit sla widt mm, sla tickness9 mm, roll radius mm, sla temperature C, roll velocity.9 rad/sec,.% plain caron steel. ig. 6. Te effect of r on. ig. 7. Te effect of on. ig.. Te dog-one sapes, comparison etween teory and experiment of Siaara et al. Te rolling conditions for te experiment wic are taken from Capter of reference. Tey are, entry sla widt mm, exit sla widt 8 mm, sla tickness98 mm, and roll radius mm. or oter conditions wic are not sown in te reference, ut required y te present model, it is assumed tat, sla temperature C, roll velocity. rad/sec,.% plain caron steel, and μ.. one eigt and te dog-one profile can e predicted wit muc iger level of accuracy tan tose predicted y Okado. ) It may e seen from ig. 6 tat increases linearly wit te increase in reduction, as may e expected. It is found tat also increases linearly wit, as sown in ig. 7, indicating tat te decrease in te dog-one eigt may e acieved y increasing te roll diameter. egarding te effect of, decreases wit te increase in, indicating tat wen r and are constant, te dog-one eigt increases wit te increase in te sla widt. ig. 8. Te effect of on. 9. Concluding emarks Precision control of te widt of slas, plates and strips, is vital for product quality and production economy in steel mills. common practice in te production line is to reduce te widt in te rouging mill y performing vertical rolling ISIJ 6

9 ISIJ International, Vol. (), No. 6 as well as oriontal rolling. However, te formation of so called dog-one at te edge of te sla would affect te final widt after oriontal rolling tat follows, and terefore, it is vital to predict te dog-one sape precisely for sound widt control. Te model presented troug tis investigation is developed rigorously on te asis of te scientific principle, and differs clearly from te existing models wic are mostly empirical. Comparison is made among Siaara et al s experiment, M, Okado et al s model, and te present model. Te results demonstrate te present model s remarkaly enanced level of accuracy wit regard to predicting te dog-one sape, compared to te conventional models. It is in tis regard tat te model proposed in tis paper will play an important role as an effective tool in acieving sound process control in steel mills. Yet, a need may e raised for improving te prediction accuracy of te present model, examining te end part of te dog-one represented y function (x,). It is expected tat te difference etween te prediction and te actual dogone sape (or M) may e reduced y selecting a iger order, asymmetric function for (x,), wic is to e left as a future work. Nomenclature roll force per unit sla tickness alf tickness of te sla at roll entrance (x,) alf tickness of te sla at point (x,) in te ite one k sear yield stress l projected lengt of roll/sla contact arc roll radius r widt reduction ratio T sla temperature V Inlet velocity of te sla tangential velocity of te sla at te roll-sla ut interface utd roll velocity ux, uy, u components of te admissile velocity vector u x,u y,u Components of te actual velocity vector alf widt of te sla at roll entrance alf widt of te sla at roll exit ε effective strain, calculated from te actual velocity field ε effective strain rate, calculated from te actual velocity field ε effective strain rate, calculated from te admissile velocity field μ coefficient of Coulom friction Ω volume representing te ite one Γ c roll-sla interface Γ ls interface were velocity discontinuity occurs in te admissile velocity field Δu σ n σ ω λ φ magnitude of te velocity discontinuity at Γ ls normal stress flow stress angular velocity of te roll frictional stress ite angle NCS ) N. Taoe, H. Honjyo, M. Takeuci and T. Ono: IS Spring Conf., IS, arrendale, P, (98). ) V. B. Ginurg: Hig-quality Steel olling Teory and Practice, Marcel Dekker, New York, (99), 7. ) T. Siaara, Y. Misaka, T. Kono, M. Koriki and H. Takemoto: J. Iron Steel Inst. Jpn., 67 (98), 9. ) M. Okado, T. riiumi, Y. Noma, K. Yauugi and Y. Yamaaki: J. Iron Steel Inst. Jpn., 67 (98), 6. ) S. Xiong, X. Liu, G. ang and. Zang, SM Int., 6 (997), 77. 6) S. I. O and S. Koayasi: Int. J. Mec. Sci., 7 (97), 9. 7) V. Nagpal: Trans. SM. J. ng. Ind., 99 (977), 6. 8) K. Komori: Sosei-to-Kako, (99), No. 9, 6 (in Japanese). 9) T. H. Kim,. H. Lee and S. M. Hwang: ISIJ Int., (), 97. ) J. S. rora: Introduction to Optimal Design, McGraw Hill, New York, (989). ) S. Sida: J. Jpn. Soc. Tecnol. Plast., (969), 6. 7 ISIJ