Generalized Mathematical Model for Hot Rolling Process of Plate

J. Mater. Sci. Technol, Vol.19 Suppl.1, 200353Generalized Mathematical Model for Hot Rolling Process of PlateZhenshan CUI)t and Bingye XU3)1) Department of Plasticity Engineering, Shanghai Jiaotong University, Shanghai 200030, China2) Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China[ Manuscript received November 4, 2003)G3 AA generalized mathematical model is developed to predict the changes of temperature, rlling pressure, strain,strain rate, and austenite grain size for plate hot rlling and cooling processes. The model is established mainly byincorporating analytical and numerical method for differential equations under complicated boundary conditions. Anindustrial rlling and cooling process of plate is simulated by the model, in which the thickness of steel Q235B plateis rolled from initial 200 mm to final 12 mm by_ 13-passes in a two-high mill. The calculated results are in goodagreement with measured data. Different from FEM simulation, the model takes very short time in calculation andmakes the influence of rlling passes on precision to be very slight.KEY WORDS: Hot rlling, Microstructure, Thermomechanical coupling, Simulation1. Introductionwater. For the plate under rolling the upper and lower sur-faces are the main heat conduction surfaces that dominateTo meet the increasingly high requirements on plate prod-the temperature distribution and heat fAux, while the heatuct, many steel plate mills have been or will be equippedconduction along width and length are negligible. Assumingwith controlled-rolling and controlled-cooling techniques. Thesteady rolling state, the temperature governing equation canmain purpose of these techniques is to control the product mi-be simplified to be one-dimensional3) and yields the followingcrostructure and hence the mechanical properties. Recent de-solutions:velopments in the metallurgical theories for the recrystalliza-(a) for the time the plate not undergoing rollingtion, transformation and precipitation, and in the computersimulation technology for deformation, heat transfer makes itpossible to predict the final microstructure, and consequentlyT(y,t)= 2 Fex(act0co(.oN) +T。the properties of the hot- rolled products". In the past re-search works, many empirical and semi-theoretical models forthe metallic microstructure evolution during hot deformation(b) for the time the plate undergoing rllingand further the properties of the products were proposed bymetallurgical and material researchers. Among these mod-els the relationships of austenite grain size with temperature,Ti(,)= 2 Fxp(-alt0(nx)-strain and strain rate have reached a satisfactory stage andn=1can be used practically. The finite element simulation takes agreat role in investigation on the working parameters such astemperature, pressure, strain and strain rate during rling].kb,H，. b,H?By incorporating the processing parameters calculation and2a2y"+a2h+2a2+Iat(2)the metallurgical models many research works were proposed,wherea2 = k/pc, bi = +i/pc; i is the heat source generatedwhich have turned to be an important technology to increasethe product quality and to meet the requirement of the Cus-by plastic deformation and surface friction; y is coordinate intomers. However the FEM method has to take a long compu-thickness direction; H is the plate half thickness before en-tational time to get the simulated results while the precisiontering roll-gap and Hi is instantaneous half thickness duringis uncontrollable if many iterations are involved. Those makerolling; h denotes heat transfer coefficient for each heat trans-the engineers feel awkward to use the method in practicalfer state; T and Ti are ambient temperature; Fn aarea groupof corresponding Coffcients Bn are eigenvalues.production.In this research work, an integrated mathematical modelWhen the plate is undergoing rolling, the time intervalfor plate hot rolling was developed to predict the process-for an elementary volume to pass the roll-gap is divided intong parameters and product microstructures. The modelmultiple steps and in each step the thickness and ambientcomprises temperature model, deformation and rolling forcetemperature are assumed to be constants. Thus the tempera-model and microstructure model. These sub- models areture variation for any point within the plate can be evaluatedmainly based on the composition of analytical and numericalstep by step for the whole rolling process.methodserential equationsunder complicated bound-ary conditions. A 13-passes rolling of a plate in a two-high3. Deformation and Rolling Force Modelrily with measured data. Nevertheless, compared with finiteelement method, the model takes a very short time to accom-With the steady rolling state assumed, the distributionsplish the calculation and the round error between passes areof strain, strain rate and rolling force remain constant at anynegligible.spatial point during rolling. That makes the problem can besolved in Euler coordinate system.2. Temperature ModelDuring the rolling processing, the plate losses heat by3.中国煤化工; state should obey theradiation to the surrounding atmosphere, by conduction tothe work rollers, and by convection to the sprays and coolingfolloC N M H Gtal passing through anycross section In umt time remauns constant; and (2) Ex +ey=0t Prof, Ph.D, to whom correspondence should be ddresedfor plain strain problem. With these rules being satisfied, theE-mail: cuizs@vip.sina.comfollowing velocity functions for metal flow are established:54J. Mater. Sci. Technol, Vol.19 Suppl.1, 2003(1) for backward sliding area:(0y)=(l()"+(v2-告)+)1.2001/8 thicknessavengecalculatedcenlersurf. measured(r,)= {()".he + y }rvgBa_ }(3w 10000，900800(2) for forward sliding area:700(l()=(=()"(管一明)+登30;020 1580Time Isu(x,)=李(e()"-*'()y(x2 -据)+(4) Fig.1 Comparison of computed temperature with mcasuredduring plate rollingwhere C, (x) and C (x) are polynomial expressions character-2-measuredizing the metal flow patterns in each area; n. and n。arestants; 12o, hi and hz are respectively the thickness of theplate at entrance point, exit point and any point at x positionwithin the rlling arc; Px is the bite angle at x position; to isthe entering velocity.Then the strain rate and effective strain rate can be eval-uated according to the velocity feld. For a mass point, thestrain rate is the material time derivative of the strain. Bymapping the deformation zone onto a rectangular zone, the81Istrain rate components are related with the strain componentsPass numberby differential expression,θ∈x + ho 0εx0Ey。hg8eFig.2 Comparison of rolling force computed with measured比=8x+hy"Ev= 8x+ hey",phenomenon simulation such a8 dynamic and static recrys-tallization and grain growth during hot rolling processes and8γzy↓ho 8n;8E， ho 8Eixy=8x十h oy"'=8+h”(5outputs the grain size and recrystallization fraction. Whereasthe equations for recrystallization fraction and grain growthwhere y' is the corresponding y coordinate in the mappedare highly nonlinear, an effective temperature-compensatedrectangle. These equations are solved by numerical diferencetime is defined to deal with the time additivity for tempera-ture varying conditions,method.tee= 2 s,exp(1/Tr/e/ -1/T)(Q/R)]8)3.2 Rolling pressure3y using Orowan equilibrium equation for rolling, therolling pressures are established as5. ApplicationPa=p%f,"(2Kx岩- mKVh,-n h)he+2w(K=-K)The above models are integrated into a generalized math-(6)ematical model that is evaluated iteratively to account for thethermomechanical coupled problem in rolling process.The model is used to simulate the industrial hot rollingof plate in Qinhuangdao Plate Company. In this case, thepa=p=-f," (xK.,-+mKV,nN hx-hi后dhz- -2w(K1-Kx) Q235B steel plate was hot-olled from initial thickness of200 mm to final product thickness of 12 mm by 13-passes in(7a two-high mill. Figure 1 compares the model predicted tem-where Ko K1 and K。are shear strength, at entrance, exitperature with on site measuredduring the process froand any point at x position respectively; R is roller radiusthe stock dragged out of the furnace to the finished stage, inw=1+0.024 m-0.1995 m2 and m is shear friction factor, pRwhich the first temperature dropping is due to descale spraysand pH are rolling pressure, respectively at the entrance andand the last is due to on-line cooling. The steep thermalexit point that can be determined by boundary condition.gradients in the plate due to chilling of the rollers disappearCompound Simpson quadrature method with the addition ofrapidly in the inter-stand due to conduction from the interior.Romberg method at point ofh: = h is used to calculate theFigure 2 shows the computed rolling forces per unit width andpressure.measured data for some passes (the measured rolling force forothe I中国煤化工- figure due to the largemealit is evident that the co-4. Microstructure Modelincired data is satisfactory.FinYada model4 is used to calculate the microstructure evo-along contact arcs in pases 9~13. Figure 4 is a column chartlution for Q235B steel. The model includes the metallurgicalof calculated effective strain at center, surface and 1/4 thick-J. Mater. Sci. Technol, Vol.19 SuppL.1, 200355300160-enter2501/1/4 thickness10里120surface, 200150。10040 t0卜13-0.03820 160x /mTime IsFig.5 Austenite grain size evolution during rollingFig.3 Computed rlling pressure along contact are for pass9~136. Conclusioncenter0.5A generalized mathematical model for simulation of plateIsurfacerolling processes is proposed and used to simulate the changes邑04of temperature, strain, strain rate, rling force, and austenitegrain sizes during industrial plate rolling and cooling processy 03in a 13-pass schedule. The calculated temperature and rlling喜02force agrees with measured data satisfactorily that verifies thevalidity of the procedure. The method is much simpler andtakes quite shorter time in calculation than FEM especially出。1in multi-stands rolling and nevertheless can get reasonableprecision, it should be quite easy adopted by engineers.2Pass numberREFERENCESFig.4 Computed efective strain] O.Kwon: ISIJ International, 1992, 32(2), 350.2) L.M.Galantuc and L.J.Ticaic. Mater. Proc. Technol.ness. Figure 5 shows the calculated changes of austenite grain1999, 92, 494.size at the corresponding points during rolling and cooling.[3] C.Devadas and L.V.Samarasekera: lronmaking and Steelmak-An initial grain size of 160 pm is assumed which was deter-ing, 1986, 13, 311.mined from measurements on same temperature conditions.[4 ] T Senuma, H. Yada, Y.Matsumura and T.Futamura: Tetsu-to-The final grain size after rolling is predicted as about 30 um.Hagane, 1984, 70, 2112.中国煤化工MHCNMH G